Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 16 Probability.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability

Plus One Maths Probability 3 Marks Important Questions

Question 1.
(i) If \(\frac{2}{11} \) is the probability of an event A, then what is the probability of the event ‘not A’? (MARCH-2011)
(ii) If P(A) = \(\frac{3}{5} \) and P(B) = \(\frac{1}{5} \) , then find P(A∪B), if A and B are mutually exclusive events.
(iii) A coin is tossed twice. What is the probability that atleast one tail occurs?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 1

Question 2.
A bag contains 9 balls of which 4 are red, 3 are blue and 2 are yellow. The balls are similar in shape and size. A ball is drawn at random from the bag. Calculate the probability that the ball drawn will be (MARCH-2013)
(i) Red.
(ii) Not yellow.
(iii) Either red or yellow.
Answer:
(i) P (Red) = \(\frac{4}{9} \)
(ii) P(No yellow) \(\frac{7}{9} \)
(iii) P(Either red or yellow) = \(\frac{4+2}{9}=\frac{6}{9}=\frac{2}{3}\)

Question 3.
A and B are two events in a random experiment such that \(P(A)=\frac{1}{3} ; P(B)=\frac{1}{5} ; P(A \cup B)=\frac{7}{15}\) (IMP-2014)
(i) Find P(A∩B)
(ii) Find P(A’)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 2

Question 4.
(i) The probability of a sure event is ……… (IMP-2014)
(ii) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is 8?
Answer:
(i) 1
(ii) A = {(2,6),(3,5),(4,4),(5,3),(6,2)}
\(P(A)=\frac{5}{36}\)

Question 5.
If A and B are two events such that P(A) = 0.42, P(B) = 0.48 andP(yinS) = 0.16 then, find: (IMP-2014)
(i) P(not A)
(ii) P(not B)
(iii) P(A∪B)
Answer:
(i) P(not A) = P(A’) = 1-P(A) = 1-0.42=0.58
(ii) P(not B) = P(B’) -1 -P(B) = 1 -0.48=0.52
(iii) P(A∪B) = P(A) + P(B)-P(A∩B)
= 0.42 + 0.48-0.16 = 0.74

Plus One Maths Probability 4 Marks Important Questions

Question 1.
Two students A and B appeared in an examination. The probability that A passes the examination is 0.25 and that B passes is 0.45. Also the probability that both will pass is 0.1. Find the probability that: (MARCH-2010)
i) Both will not pass.
ii) Only one of them will pass.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 3

Question 2.
If M and N are events such that: (IMP-2010)
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 4
Find
i) P(M or N)
ii) P(not M and not N)
(Imp (Science) – 2010)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 5

Question 3.
A and B are two events associated with (MARCH-2013)
i) a random experiment such that P(A) = 0.3, P(B) = 0.4 and P(A∪B) = 0.5
a) Find P(A∩B)
b) Find P(A’∪B’)
ii) A coin is tossed twice. What is the probability that at least one tail occurs?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 6

Question 4.
If A and B are two events in a random (IMP-2013)
experiment, then P(A) + P(B)-P(A∩B) = ……….
ii) Given P(A) = 0.5, P(B) = 0.6 and P(A∩B)=03. Find P(A∪B) and P(A’)
iii) Two dice are thrown simultaneously.
Find the probability of getting a doublet.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 7
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 8

Question 5.
The probability that Ramu pass the examination in both Mathematics and Physics is 0.5, the probability of passing neither Mathematics nor Physics is 0.1, the probability of passing Mathematics is 0.75 (MARCH-2014)
i) What is the probability of passing Mathematics or Physics?
ii) What is probability of passing Physics?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 9

Question 6.
If A and B are two events such that (MARCH-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 10
then find;
4 2 8
i) P(A’)
ii) P(A∪B)
iii) P(A∩B)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 11

Question 7.
The number of outcomes in the sample space of the random experiment of throwing two dice is… (MARCH-2015)
a) 6³
b)6
c) 6²
d)12
ii) Two students, Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05
Answer:
i) 6²
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 12

Question 8.
i) If A and B are mutually exclusive and exhaustive events then
P(A) + P(B) = ……….. (IMP-2015)
a) 0
b) 1
c) 1/2
d) 2
ii) Two students A and B appeared in an examination. The probability that A will qualify the examination is 0.25 and B will qualify is 0.45 and both will qualify the examination is 0.1. Find the probability that: (IMP-2015)
a) Both A and B will not qualify the examination.
b) One of them will qualify the examination.
Answer:
i) b) 1
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 3

Question 9.
i) In a random experiment, 6 coins are tossed simultaneously. Write the number of sample points in the sample space. (MARCH-2016)
(a) 2
(b) 24
(c) 26
(d) 25
ii) Given that P(A) = 0.5, P(B) = 0.6,
P(A∩B) = 0.3
Find P(A’),P(A∪B),P(A’∩B’) and P(A’∪B’)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 27

Question 10.
i) If P(A’) = 0.8 .write the value of P(A). (SAY-2017)
ii) In a class of 60 students; 30 selected for NCC, 32 selected for NSS and 24 selected for both NCC and NSS. If one of these students is selected at random, find the probability that:
a) the students selected for NCC or NSS.
b) the students has selected neither NCC nor NSS.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 13

Plus One Maths Probability 6 Marks Important Questions

Question 1.
Two dice are thrown. Let A be an event to get an even number on first die and B be an event to get sum of the numbers obtained on two dice is 8. (IMP-2011)
i) Write the sample space.
ii) Write the outcomes favorable to the event A, the event B.
iii) Find P(A or B).
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 14

Question 2.
A box contains 6 red, 5 blue and 4 green balls. 3 balls are drawn from the box. Find the probability that (IMP-2011)
i) All are blue.
ii) All balls are either red or blue.
iii) Atleast one green ball.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 15

Question 3.
i) A coin is drawn repeatedly until a tail comes up. What is the sample space (IMP-2012)
a) no head
b) exactly one head.
c) atleast one head.
d) atleast two heads.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 16

Question 4.
If E and F are two events such that (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 17
find
i) P(E);P(F)
ii) P(E or F)
iii) P(not E and not F)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 18
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 19

Question 5.
John and Mary appeared in an examination. The probability that John will qualify the examination is 0.05 and that Mary will qualify the examination is 0.10. The probability that both will qualify is 0.02. Find the probability that (MARCH – 2012)
i) John or Mary qualifies the examination.
ii) Both John and Mary will not qualify the examination.
iii) Atleast one of them will not qualify the examination.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 20

Question 6.
In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find: (MARCH – 2012)
i) The probability that the student opted for NCC or NSS.
ii) The probability that the student has opted for exactly one of NCC or NSS.
Answer:
i) Let the events be defined as A – NCC and B – NSS
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 21.
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 22

Question 7.
i) A coin is tossed repeatedly until a head comes up. Write the sample space,
ii) If A and B are two events in a random experiment, then (MARCH – 2014)
P(A∪B) = P(A) + P{B) – …………
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 23
Find P(not A and not B).
iv) A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. A disc is drawn at random from the bag. Calculate the probability that it will be
a) Red.
b) Not yellow.

Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 24.

Question 8.
Match the following: (MARCH-2017)
i)
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 25
ii) Two dice are thrown at random. Find the probability of
a) Getting a doublet.
b) Getting sum of the numbers on the
dice 8.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 16 Probability 26

Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 15 Statistics.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics

Plus One Maths Statistics 3 Marks Important Questions

Question 1.
Consider the numbers 4,7,8,9,10,12,13,17 (MARCH-2010)
i) Find the mean of the numbers.
ii) Find the mean deviation about the mean.
iii) Find the standard deviation.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 1
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 2

Question 2.
Consider the following data; 35,49,30,32,50,41,34,45,36 (MARCH-2013)
i) Find its median.
ii) Find its mean deviation about median
Answer:
i) 30, 32, 34, 35, 36, 41, 45, 49, 50
Median is the 5th observation when the data is arranged in ascending order. Hence median = 36
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 3

Question 3.
The mean and standard deviation of marks obtained by 50 students of 50 students in a class in two subjects mathematics are given below:  (IMP-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 4
Which one of the subject shows highest variability in marks? Why?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 5
Thus Accountancy with highest CV shows highest variability and Mathematics with lowest CV shows lowest variability.

Plus One Maths Statistics 4 Marks Important Questions

Question 1.
A public Opinion polling agency surveyed 200 government employees. The following table shows the ages of the employees interviewed: (MARCH-2011)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 6
i) Calculate the mean age of the employees interview.
ii) Compute the mean deviation of the ages about the mean age.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 7
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 8

Question 2.
Consider the following frequency table. (IMP-2011)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 46
i) Find the mean.
ii) Find the mean deviation about mean.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 9

Question 3.
Consider the following data in respect of marks of 50 students in Mathematics and Physics. (IMP-2011)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 10
i) Find coefficient of variation of Mathematics and Physics.
ii) Which subject shows more variability?
iii) Which subject shows more consistent?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 11
ii) Greater CV more variability; therefore Mathematics is more variable than Physics.
iii) Less CV more consistent, therefore Physics is more consistent.

Question 4.
Find the Standard deviation for the following data: (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 12
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 13

Question 5.
Consider the frequency distribution. (MARCH-2013)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 14
i) Find the mean.
ii) Calculate the variance and the standard deviation.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 15
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 16
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 17

Question 6.
Consider the frequency distribution. (MARCH-2013)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 18
i) Find the mean.
ii) Calculate the variance and standard deviation.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 19
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 20

Question 7.
Consider the following frequency table (MARCH-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 21
i) Find the mean.
ii) Find the mean deviation about the mean.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 22

Question 8.
Find the standard deviation of the data: (IMP-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 23
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 24

Plus One Maths Statistics 6 Marks Important Questions

Question 1.
The scores of two batsmen A and B in 5 innings during a certain match are as follows: (IMP-2010)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 25
Find:
i) Mean score of each batsman.
ii) Standard deviation of the scores of each batsman.
iii) Which of the batsman is more consistent?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 26
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 27

Question 2.
Calculate mean, variance and standard deviation for the following distribution. (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 28
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 29
Standard Deviation = √20100 = 141.8

Question 3.
Calculate the median and Mean deviation about median for the following data. (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 30
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 31
Median class is the class in which the \(\left(\frac{50}{2}=25\right)^{t h}\) observation lies. Therefore median class is 20 – 30.
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 32

Question 4.
Consider the following distribution; (MARCH-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 33
i) Calculate the mean of the distribution.
ii) Calculate the standard deviation of the distribution.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 34
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 35

Question 5.
Consider the following distribution. (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 36
i) Find the mean.
ii) Find the standard deviation.
iii) Find the coefficient of variation of marks.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 16
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 15
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 17
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 37

Question 6.
Consider the frequency distribution (MARCH-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 38
i) Find the mean.
ii) Calculate the variance and the standard deviation.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 39
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 40

Question 7.
i) If \(overline { x }\) is the mean and a is the standard deviation of a distribution, then the coefficient of variation is ……… (MARCH-2015)
a) \(\frac{\bar{x}}{\sigma} \times 100\)
b) \(\frac{\sigma}{\bar{x}}\)
c) \(\frac{\sigma}{\bar{x}} \times 100\)
d) \(\frac{\bar{x}}{\sigma} \times 50\)
ii) Find the standard deviation for the following data:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 41
Answer:
i)
\(\frac{\bar{x}}{\sigma} \times 100\)
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 13

Question 8.
i) The sum of all the deviations of the observations of a data from its A.M. is …………. (IMP-2013)
a) Zero
b) Maximum
c) Minimum
d) Negative number
ii) Calculate the Mean. Variance and Standard deviations of the following frequency distribution.
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 42
Answer:
i) a) zero
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 15
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 17
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 37

Question 9.
i) Suppose the mean of certain number of observation is 50 and the sum of all the observations is 450. Write down the number of observations (MARCH-2016)
ii) Find the mean deviation about mean for the following data:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 43
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 47

Question 10.
i) If the variance of a certain distribution is 8, write its standard deviation. (MAY-2017)
ii) Find the mean, standard deviation and coefficient of variation for the following frequency distribution.
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 48
Answer:
i) √8
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 15
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 17
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 37

Question 11.
i) Find the variance for the observations 2,4,6,8 and 10. (MARCH-2017)
ii) Consider the frequency distribution
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 44
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 45
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 15 Statistics 49

Plus One Maths Chapter Wise Previous Questions Chapter 14 Mathematical Reasoning

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 14 Mathematical Reasoning.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 14 Mathematical Reasoning

Plus One Maths Mathematical Reasoning 3 Marks Important Questions

Question 1.
i) Write the converse of the statement. (IMP-2010)
p: If a divides b then b is a multiple of a.
ii) Consider the compound statement,
p: 2 + 2 is equal to 4 or 6
1) Write the component statements.
2) Is the compound statement true? Why?
Answer:
i) Converse statement is “If a is a multiple of b then a divides b.”
ii) 1) q: 2+2 is equal to 4
r. 2+2 is equal to 6.
2) q is true and r is false, so p is true.

Question 2.
Verify by method of contradiction p. √2 is irrational. (IMP-2012)
Answer:
Assume that √2 is rational. Then can be written in the form √2 = \(\frac { p }{ q }\) , where p and q are integers without common factors.
Squaring; 2 = \(\frac { p² }{ q² }\)
=> 2q² = p²
=> 2 divides p² => 2 divides p
Therefore, p = 2k for some integer k.
=> p² = 4k²
=> 2q² = 4k²
=> q² = 2k²
=> 2 divides q² => 2 divides q
Hence p and g have common factor 2, which
contradicts our assumption. Therefore, √2 is irrational.

Question 3.
i) Write the negation of the following . statement, ‘Every natural number is an integer’. (MARCH-2013)
ii) Write the contrapositive and converse of the following statement, ‘If x is a prime number, then x is odd ’.
Answer:
Negation of the statement is ‘Every natural ’ number is not an integer’,
ii) The contrapositive statement, ‘If x is not odd, then x is not a prime number.’
The converse of the statement, ‘If x is odd, then x is a prime number’.

Question 4.
i) Write the component statement of the  following statement: “All rational
numbers are real and all real numbers are complex. (IMP-2014)
ii) Write the contrapositive and converse
of the following statement: ‘If a number is divisible by 9, then it is divisible by 3.’
(Imp (Commerce) – 2014)
Answer:
i) p: All rational numbers are real,
q: All real numbers are complex,
ii) Contrapositive:
If a number is not divisible by 3, it is not divisible by 9.
Converse:
If a number is divisible by 3 then it is divisible by 9.

Question 5.
i) Write the negation of the statement: the sum of 3 and 4 is 9. (IMP-2014)
ii) Write the component statements of ‘Chandigarh is the capital of Haryana and Uttar Pradesh.’
iii) Write the converse of the statement: ‘If a number n is even, then n² is even.’
Answer:
i) Negation: ‘The sum of 3 and 4 is not equal to 9.’
ii) p: Chandigarh is the capital of Haryana.
q: Chandigarh is the capital of Uttar Pradesh.
iii) Converse: If a number n² is even then n is even.

Plus One Maths Mathematical Reasoning 4 Marks Important Questions

Question 1.
i) Write the negation of the statement.“Both the diagonals of a rectangle have the same length.” (MARCH-2013)
ii) Prove the statement, “Product of two odd integers is odd,” by proving its contrapositive.
Answer:
i) “Both the diagonals of a rectangle do not have the same length.”
ii) Let us name the statements as below p: ab is odd. q: a, b is odd.
We have to check p => q is true or not, that is by checking its contrapositive statement
~ q =>~ p
~ q: ab is even.
Let a and b be two even numbers. Then, a = 2n and b = 2m, where m and n are any integer.
a x b = 2n(2m) = 4 nm
Then product of a and b is even. That is ~ p is true. Hence by the contrapositive principle we say that “Product of two odd integers is odd,”

Question 2.
Consider the compound statement “ √5 is a rational number or irrational number”. (IMP-2011)
i) Write the component statements of
above and check whether these component statements are true or false.
ii) Check whether the compound statement is true or false.
(Imp (Commerce) – 2011)
Answer:
i) The component statements are
p: √5 is a rational number
q: √5 irrational number.
Here p is false and q is true,
ii) In this compound statement “or” is exclusive, p is false and q is true and therefore compound statement is true.

Question 3.
i) Write the converse of the statement: “If a number n is even, then n² is even” (MARCH-2011)
ii) Verify by method of contradiction: “ √2 is irrational”.
Answer:
i) “If n² is even, then n is even”
ii) Assume that √2 is rational. Then √2 can
be written in the form √2=\(\frac { p }{ q }\), where p and q are integers without common factors.
Squaring; 2 = \(\frac { p² }{ q² }\)
=> 2q²= p²
=> 2 divides p² => 2 divides p
Therefore, p = 2k tot some integer k.
=> p² = 4k²
=> 2q² = 4k²
=> q² = 2k²
=> 2 divides
q² => 2 divides q
Hence p and g have common factor 2, which contradicts our assumption.
Therefore, √2 is irrational.

Question 4.
Which of the following sentences are statements? Give reason for your answer. (IMP-2012)
a) The cube of a natural number is an odd number.
b) The product of (- 4) and (- 5) is 20.
Write the negation of the following statements and check whether the resulting statements are true.
a) √2 is rational.
b) Every natural number is greater than zero.
Answer:
a) This sentence is a statement. Since for a particular natural number it is true
and for other it is false. (1)³ = 1 and 2³ = 8
b) This sentence is a statement. Since the product is always 20 and true.
a) √2 is not rational. The negation statement is true.
b) Every natural number is not greater than zero. The negation statement is false.

Question 5.
Consider the statement, “If x is an integer and x² is even, then x is also even.” (MARCH-2012)
i) Write the converse of the statement.
ii) Prove the statement by the contra-positive method.
Answer:
i) Converse of the statement is “If x is an even number, then x is an integer and x² is even.”
ii) The contrapositive of a statement p => q is the statement ~ q => ~ p .
The contrapositive statement is “If x is an odd integer, then x² is odd.”
Let x is an odd number.
Then x= 2n+1
x² =(2n+1)² =4n² +4n+1
= 4(n² +n)+1
Which is odd.
ie; if q is not true then p is not true.

Question 6.
i) Write the negation of the following statement: “All triangles are not equilateral triangles”. (MARCH-2013)
ii) Verify by the method of contradiction. p:√7 is irrational.
Answer:
i) All triangles are equilateral triangles
ii) Assume that is rational. Then √7 can be written in the form√7 =\(\frac { p }{ q }\), where p and q are integers without common factors.
Squaring; 7 = \(\frac { p² }{ q² }\)
=> 7q² = p²
=> 7 divides p² => 7 divides p
Therefore, p = 7k tot some integer k.
=>p² = 49k²
7q² = 49k²
=>q² = 7k²
=>7 divides q² =>7 dividesq
Hence p and q have common factor 7, which contradicts our assumption.
Therefore, √7 is irrational.

Question 7.
i) Write the contrapositive of the statement. “If x is a prime number, then x is odd.” (IMP-2013)
ii) Verify by the method of contradiction p : √5 is irrational.
Answer:
i) Contrapositive statement is “If x is not odd, then x is not prime number.”
ii) Assume that √5 is rational. Then √5 can be written in the form √5 = \(\frac { p }{ q }\), where p and q are integers without common factors.
=> 5q² = p²
=> 5 divides p²=> 5 divides p
Therefore, p = 5k for some integer k.
=>p² = 25k²
=> 5q² = 25k²
=> q² = 5k²
=> 5 divide q² => 5 divides q
Hence p and g have common factor 5, which contradicts our assumption.
Therefore, √5 is irrational.

Question 8.
i) Write the negation of the following statement : “ √5 is not a complex number.” (MARCH-2014)
ii) Verify using the method of contradiction:
“p: √2 is irrational number.”
Answer:
i) Negation statement:” √5 is a complex number.”
ii) Assume that √2 is rational. Then √2 can
be written in the form √2=\(\frac { p }{ q }\), where p and q are integers without common factors.
Squaring; 2 = \(\frac { p² }{ q² }\)
=> 2q²= p²
=> 2 divides p² => 2 divides p
Therefore, p = 2k tot some integer k.
=> p² = 4k²
=> 2q² = 4k²
=> q² = 2k²
=> 2 divides
q² => 2 divides q
Hence p and g have common factor 2, which contradicts our assumption.
Therefore, √2 is irrational.

Question 9.
i) Write the negation of the statement: “√7 is rational.”  (MARCH-2015)
ii) Prove that“√7 is rational.” by the method of contradiction.
(March – 2015)
Answer:
i) Negation is : “ √7 is not rational.”
ii) Assume that is rational. Then √7 can be written in the form√7 =\(\frac { p }{ q }\), where p and q are integers without common factors.
Squaring; 7 = \(\frac { p² }{ q² }\)
=> 7q² = p²
=> 7 divides p² => 7 divides p
Therefore, p = 7k tot some integer k.
=>p² = 49k²
7q² = 49k²
=>q² = 7k²
=>7 divides q² =>7 dividesq
Hence p and q have common factor 7, which contradicts our assumption.
Therefore, √7 is irrational.

Question 10.
i) Which of the following is the contrapositive of the statement (IMP-2015)
a) q => p
b) ~ p =>~ q
c) ~ q =>~ p
d) p =>~ q
ii) Prove by contrapositive method. “If x is an integer and x² is even then x is also even.”
(Imp-2015)
Answer:
i) c) ~ q =>~ p
ii) The contrapositive of a statement p => q is the statement ~ q => ~ p .
The contrapositive statement is “If x is an odd integer, then x² is odd.”
Let x is an odd number.
Then x= 2n+1
x² =(2n+1)² =4n² +4n+1
= 4(n² +n)+1
Which is odd.
ie; if q is not true then p is not true.

Question 11.
i) Write the negation of the statement: “Every natural number is greater than zero.” (MARCH-2016)
ii) Verify by the method of contradiction: “P: √13 is irrational.”
Answer:
i) Negation of the statement: “It is false that every natural number is greater than zero.”
ii) Assume that √13 is rational. Then √13 can be written in the form √13 = \(\frac { p }{ q }\) ,
where p and q are integers without common factors.
Squaring; 13 = \(\frac { p² }{ q² }\)
=>13q² = p²
=>13 divides p² => 13 divides p
Therefore, p = 13k for some integer k.
=> p² = 169k²
=> 13q² = 169k²
q²= 13k²
=>13 divides q²
=> 13 divides q
Hence p and q have common factor 13, which contradicts our assumption.
Therefore, √13 is irrational.

Question 12.
i) Write the negation of the statement
“ √2 is not a complex number.”
ii) Prove by the method of contradiction,
P: √11 is irrational.”
Answer:
i) “√2 is a complex number.”
ii) Assume that √11 is rational. Then √11 can be written in the form√11 = \(\frac { p }{ q }\),
where p and q are integers without common factors.
Squaring; 11 = \(\frac { p² }{ q² }\)
=> 11q² =p²
=> 11 divides p² => 11 divides p
Therefore, p =11k for some integer k.
=>p² =121 k²
=>11q² = 121k²
=> q² = 11k²
=> 11 divides q² => 11 divides q
Hence p and q have common factor 11, which contradicts our assumption. Therefore,
√11 is irrational.

Question 13.
i) Write the contrapositive of the statement “If a number is divisible by 9, then it is divisible by 3.”
ii) Prove by the method of contradiction.
“P: √5 is irrational.”
Answer:
i) “If a number is not divisible by 9, then it is not divisible by 3.”
ii) Assume that √5 is rational. Then √5 can be written in the form √5 = \(\frac { p }{ q }\), where p and q are integers without common factors.
=> 5q² = p²
=> 5 divides p²
=> 5 divides p
Therefore, p = 5k for some integer k.
=>p² = 25k²
=> 5q² = 25k²
=> q² = 5k²
=> 5 divide q² => 5 divides q
Hence p and g have common factor 5, which contradicts our assumption.
Therefore, √5 is irrational.

Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 13 Limits and Derivatives.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives

Plus One Maths Limits and Derivatives 3 Marks Important Questions

Question 1.
Find the derivative of y = tan x from first principles. (MARCH-2010)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 1

Question 2.
Choose the most appropriate answer from those given in the bracket (IMP-2010)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 2
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 3

Question 3.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 4
(IMP-2010)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 5

Question 4.
Using the first principle of derivatives, find the derivatives of \(\frac { 1 }{ x }\) (MARCH-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 6

Question 5.
Using the quotient rule find the derivative mof f(x) = cot x (MARCH-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 7

Question 6.
Find the derivatives of the following: (MARCH-2011)
https://i2.wp.com/www.aplustopper.com/wp-content/uploads/2019/06/Plus-One-Maths-Chapter-Wise-Previous-Questions-Chapter-13-Limits-and-Derivatives-8.png?resize=172%2C73&ssl=1
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 9

Question 7.
Prove that (MARCH-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 10
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 11

Question 8.
Find the derivative of y = cotx from first principles. (MARCH-2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 12

Question 9.
i) The value of \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) (MARCH-2013)
ii) Evaluate \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{3 x}\)
Answer:
i) 1
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 13

Question 10.
i) The value of \(\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}\) (MARCH-2013)
ii) Evaluate \lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 14

Question 11.
Find the derivative of f(x) = sin x from the first principle. (MARCH-2013)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 15

Question 12.
Find the derivative of \(\frac{x+\cos x}{\tan x}\) (MARCH-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 16

Question 13.
Find the derivatives of f(x) = sinx using the first principle. (MARCH-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 17

Question 14.
Find the derivative of \(\frac{x^{5}-\cos x}{\sin x}\) using the quotient rule. (MARCH-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 18

Question 15.
Using the first principle, find the derivative of cosx . (IMP-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 19

Question 16.
Find the derivative of \(\frac{\cos x}{2 x+3}\) (IMP-2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 20

Plus One Maths Limits and Derivatives 4 Marks Important Questions

Question 1.
Evaluate (MARCH-2010)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 21
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 22

Question 2.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 23(MARCH-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 24

Question 3.
Compute the derivative of sec x with respect to x from first principle. (IMP-2010)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 25

Question 4.
Find \(\lim _{x \rightarrow 2} \frac{x^{4}-4 x^{2}}{x^{2}-4}\) (IMP-2011)
ii) If y = sin 2x .Prove that \(\frac{d y}{d x}=\) = 2cos2x
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 26

Question 5.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 27
(IMP-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 28

Question 6.
Find the derivative of y = cosec x from first principle. (IMP-2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 29

Question 7.
Find the derivative of y = cosec x from first principle. (IMP-2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 29

Question 8.
Find the derivative of \(\frac{x+1}{x-1}\) from first principle (IMP-2013)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 30
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 31

Question 9.
i) The value of \(\lim _{x \rightarrow 0} \frac{\sin 5 x}{5 x}\) (MARCH-2014)
ii) Evaluate \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, a, b \neq 0\)
Answer:
i) 1
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 32

Plus One Maths Limits and Derivatives 6 Marks Important Questions

Question 1.
Find the derivative of \(\frac{1}{x}\) from first principle. (IMP-2010)
Find the derivative of
(ax + b)n (ax + c)m
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 33

Question 2.
i) Find \(\lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x^{2}+3 x+2}\) (IMP-2011)
ii) Find f ‘(x) given f(x) = \(\frac{x^{2}+5 x+6}{x^{2}+3 x+2}\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 34

Question 3.
i) Evaluate \(\lim _{x \rightarrow 3}\left(\frac{x^{3}-27}{x^{2}-9}\right)\) (MARCH-2012)
ii) Evaluate \(\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{\sin ^{3} x}\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 35

Question 4.
i) Evaluate \(\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 3 x}\) (MARCH-2013)
ii) Find the derivate of y = cosx from the first principle.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 36
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 19

Question 5.
i) Find the derivative of \(\frac{\sin x}{x+\cos x}\) (MARCH-2014)
ii) Match the following:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 37
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 38
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 39

Question 6.
i) \(\frac{d}{d x}(\tan x)\) = ……… (IMP-2014)
ii) Find the derivative of 3 tan x + 5 sec x
iii) Find the derivative of /(x) = (x² + 1)sinx
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 40
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 41

Question 7.
i) Match the following (MARCH-2015)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 42
ii) Find the derivative of tanx using first principle.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 43
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 44

Question 8.
i) Match the following: (MARCH-2015)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 45
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 46
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 47

Question 9.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 48
iii) Using first principles, find the derivative of cos x. (IMP-2015)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 49
iii)

Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 19

Question 10.
i) Derivative of x² – 2 at x = 10 is (IMP-2016)
a) 10
b) 20
c) -10
d) -20
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 50
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 51

Question 11.
i) \(\frac{d}{d x}\left(\frac{x^{n}}{n}\right)\) = ………… (MARCH-2016)
ii) Differentiate \(y=\frac{\sin x}{x+1}\) with respect to x
iii) Use first principles, find the derivative of cosx.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 52
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 19

Question 12.
i) \(\frac{d}{d x}(-\sin x)\) = ………….. (MARCH-2016)
ii) Find\(\frac{d y}{d x}\) if \(y=\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x\) where a, b are constants.
iii) Using first principles, find the derivative of sinx.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 53
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 17

Question 14.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 54
iii) Using the first principle, find the derivative of cosx (MAY-2017)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 55
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 19

Question 15.
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 56
(MARCH-2017)
Answer:
i) cos x
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 60
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 57

Question 16.
i) \(\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}=\) …….(MARCH-2017)
a) 0
b) 1
c) 2
d) 3
ii) Find
\(\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}\)
iii) Find the derivative of f(x) = sin x by using first principal.
Answer:
i) b) 1
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 58
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 13 Limits and Derivatives 17

Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry

Plus One Maths Three Dimensional Geometry 3 Marks Important Questions

Question 1.
Consider the triangle with vertices (0,7,- 10), (1,6,- 6) and (4,9,- 6) (MARCH-2010)
i) Find the sides AB, BC and CA.
ii) Prove that the triangle is right triangle.
iii) Find the centroid of the triangle.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 1

Question 2.
i) Find the co-ordinates of the points which trisect the line segment joining the points P(4,0,1) and Q(2,4,0). (IMP-2010)
ii) Find the locus Of the set of points P such that the distance from A(2,3,4) is equal to twice the distance from B(-2,1,2).
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 2
Let R and S be two points which trisect the line join of PQ. Therefore PR = RS = SQ
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 3

Question 3.
i) Write the coordinate of the centroid of the triangle whose vertices are
(x1, y1, z1) ; (x1, y1, z1)and(x1, y1, z1(IMP-2011)
ii) If the centroid of the triangle ABC is (1,1,1) and A and B are (3,-5,7), (1,1,2) then find the coordinate of C.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 4

Question 4.
Given three points A(- 4,6,10), B(2,4,6) and C(14,0,- 2) (IMP-2012)
i) Find AB.
ii) Prove that the points A, B and C are collinear.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 5
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 6

Question 5.
Name the octants in which the points A(1,6,- 6) and B(- 1,- 6,- 6). Find the distance between A and B. (IMP-2012)
Answer:
ASP A(1,6,- 6) and B(-1 ,-6,-6) in the octants XOYZ’ and X’OY’Z’ respectively.
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 7

Question 6.
i) If P is a point in YZ-plane, then its x coordinate is ………….. (IMP-2013)
ii) Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8).
Answer:
i) Zero
ii) Let the ratio be kA. Since the point lies on the YZ plane, its x-coordinate. will be zero. Hence
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 8
Therefore the ratio is 2:3.

Question 7.
i) Find the distance between the points (2- 1,3) and (- 2,1,3) (MARCH-2013)
ii) Find the coordinate of the point which divides the line segment joining the points (- 2,3,5) and (1,- 4,6) internally in the ratio of 2:3.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 9

Question 8.
i) Name the octant in which the points (3,- 2,1) and (- 5,- 6,1) lie. (MARCH-2014)
ii) Find the distance between the points P(1,- 3,4) and Q(- 4,1,2).
(March(Commerce) – 2014)
Answer:
i) (3,-2,1) lie on octant XOYZ and (-5,-6,1) lie on octant X’OY’Z.
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 10

Question 9.
Find the centroid of the triangle with vertices (3,- 5,7), (- 1,7,- 6) and (1,1,2). (IMP-2010)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 11
= (1,1,1)

Question 10.
Show that the points (-2,3,5), (1,2,3) and (7,0,-1) are collinear. (IMP-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 12

Question 11.
Find the coordinate of the points which divides the line segment joining the points (- 2,3,5) and (1,- 4,6) in the ration 2 : 3 internally. (IMP-2014)
Answer:
coordinate of the point is:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 13

Question 12.
i) State whether the following is TRUE or FALSE. (MAY-2017)
“The point (4,-2,-5) lies in the eight octant.”
ii) Find the equation of the set of points such that its distances from the points A (3,4,- 5) and B (- 2,1,4) are equal.
Answer:
i) True
ii) PA = PB
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 14

Question 13.
i) The distance between the point (1,-2,3) and (4,1,2) is …………. (MARCH-2017)
(a) √2
(b) √19
(c) √11
(d) √15
ii) the centroid of the triangle ABC is at the point (1,2,3). If the coordinates of A and B are (3,- 5,7) and (- 1,7,- 6) respectively. Find the coordinates of the point C.
Answer:
i) (b) √19
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 15

Plus One Maths Three Dimensional Geometry 4 Marks Important Questions

Question 1.
Consider the points A(- 2,3,5), B(1,2,3) and C(7,0,- 1) (MARCH-2011)
i) Using the distance formula, show that the points A, B and C are collinear.
ii) Find the ratio in which B divides the line segment AC.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 16

Question 2.
i) The x – coordinate of the point in the YZ plane is ………….. (MARCH-2013)
ii) Find the ratio in which the YZ plane divides the line segment joining the points (- 2,4,7) and (3,- 5,8).
Answer:
i) zero
ii) Let the ratio be k:1. Since the point lies on the YZ plane, its Xrcoordinate will be zero. Hence
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 17

Question 3.
i) Find the distance between the points (2,3,5) and (4,3,1). (MARCH-2014)
ii) Find the ratio in which the line segment joining the points A(4,8,10) and B (6,10,-8) is divided by the XY plane. (March (Science) – 2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 18

Question 4.
A point in the XZ plane is (MARCH-2015)
a) (1,1,1)
b) (2,0,3)
c) (2,3,0)
d) (-1,2,3)
ii) Show that the points A(1,2,3), B(- 1,- 2,- 1), C(2,3,2) and D(4,7,6) are the vertices of a parallelogram.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 19
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 20
Here; AB = CD and BC = DA, therefore ABCD is a parallelogram

Question 5.
i) Which of the following lies in the sixth octant? (MARCH-2016)
a) (- 3,- 2,- 2)
b) (- 3,1,- 2)
c) (3,- 1,2)
d) (3,- 1,-2)
ii) Find the ratio in which the YZ plane divides the line joining the points (- 2, 4, 7) and (3,- 5,8)
Answer:
i) b) (- 3,1,- 2)
ii) Let the ratio be k:1. Since the point lies on the YZ plane, its Xrcoordinate will be zero. Hence
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 17

Question 6.
i) Which one of the following points lies in the sixth octant? (IMP-2015)
a) (-4,2,-5)
b) (-4,-2,-5)
c) (4,-2,-5)
d) (4,2,5)
ii) Find the ratio in which the YZ plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8).
Answer:
i) a) (-4,2,-5)
ii) Let the line joining the points A(-2,4,7) and B(3,-5, 8) is divided by the yz-plane in the ratio k: 1.
Then the coordinate
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 21

Plus One Maths Three Dimensional Geometry 6 Marks Important Questions

Question 1.
i) If \(\left(\frac{5}{3}, \frac{22}{3}, \frac{-22}{3}\right)\) is the centriod of is the centroid of ∆PQR with vertices P(a,7,-10), Q(1,2b,-6) and R(4,9,3c), Find the value of a, b, c. (MARCH-2012)
ii) Prove that ∆PQR is isosceles.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 12 Introduction to Three Dimensional Geometry 23

Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 11 Conic Sections.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections

Plus One Maths Conic Sections 3 Marks Important Questions

Question 1.
1. Find the equation of the Hyperbola where foci (0,±8)are and the length of the latus rectum is 24.(IMP-2012)
Answer:
Since foci (0,±8)
=> ae = 8
Latus rectum = 24= \(\frac {2b² }{ a }\)
=> 12a = b²
b² =a²(e² -1)
=> b² – a²e² -a²
=>12a = 64 – a²
=>a²+12a-64 = 0
=> a = – 16,4
acceptable value is => a = 4
=> 48 = b²
Hence equation is
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 1

Question 2.
Find the equation of the circle with centre (- a,- b)and radius \(\sqrt{a^{2}+b^{2}}\) . (IMP-2012)
Answer:
We have the equation of a circle as;
(x-h)² + (y-k)² – r²
=> (x + a)² +(y + b)² = a² + b²
=> x² +2 ax + a² + y² +2 by + b² =a² +b²
=> x² +2ax + y² +2by = 0

Question 3.
Find the coordinate of the foci, the length of the major axis, minor axis, latus rectum and eccentricity of the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) . (MARCH-2013)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 2

Question 4.
Consider the parabola y² =12x. (MARCH-2015)
i) Find the coordinate of the focus.
ii) Find the length of the latus rectum.
Answer:
i) Given; y² =12x comparing with y² = 4ax We have 4a = 12 => a = 3 Then; Focus is (3,0)
ii) Length of latus rectum = 4a = 12

Question 5.
Find the foci, vertices, the eccentricity and the length of the latus rectum of the hyperbola 16x² – 9y² =144. (SAY-2017) 
Answer:
The equation of the hyperbola is of the form
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 3
=>a² =9,b² =16
=>c² = a² +b² =9 + 16 = 25
=>c = 5
Coordinate of foci are (±5,0)
Coordinate of vertices are (±a,0) => (±3,0)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 4

Question 6.
Directrix of the parabola x² = – 4ay is ……….. (MARCH-2014)
a) x + a = 0
b) x – a = 0
c) y – a = 0
d) y + a = 0
Find the equation of the ellipse whose length of the major axis is 20 and foci are (0 ±5)
(March-2015)
Answer:
i) y-a = 0
ii) The equation of the ellipse is of the form;
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 5

Question 7.
Find the coordinates of the focii, vertices, eccentricity and the length of the Latus Rectum of the ellipse 100x² + 25y² = 2500 (IMP-2015)
Answer:
Given: 100x² +25y² = 2500
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 6

Question 8.
Find the foci, vertices, length of the major axis and eccentricity of the ellipse: \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) (MARCH-2016)
Answer:
Since 25 > 9 the standard equation of the ellipse is \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) => a² =25;b² =9
c² =a² – b² =25 – 9 = 16
=>c = 4
Coordinate of foci are (±4,0)
Coordinate of vertex are (±5,0)
Length of major axis = 2a = 2 x 5 = 10
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 7

Plus One Maths Conic Sections 4 Marks Important Questions

Question 1.
An ellipse whose major axis as x-axis and the centre (0,0) passes through (4,3) and (- 1,4). (MARCH-2010)
i) Find the equation of the ellipse.
ii) Find is eccentricity.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 8
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 9

Question 2.
Consider the conic find 9y² -4x² = 36 (IMP-2010)
i) The foci.
ii) Eccentricity.
iii) Length of latus rectum.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 10

Question 3.
Find the equation of the circle with center (2,2) and passing through the point (4,5). (MARCH-2011)
Find the eccentricity and the length of latus rectum of the ellipse 4x² + 9y² =36
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 11

Question 4.
For the hyperbola 9x² – 16y² =144 (IMP-2011)
i) find eccentricity.
ii) find the latus rectum.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 12
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 13

Question 5.
A hyperbola whose transverse axis is x-axis, centre (0,0) and foci (±√10,0) passes through the point (3,2) (MARCH-2012)
i) Find the equation of the hyperbola.
ii) Find the eccentricity.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 14
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 15

Question 6.
Find the centre and radius of the circle. (IMP-2013)
x² +y² – 8x + 10y – 12 = 0.
ii) Determine the eccentricity and length of latus rectum of the hyperbola —–
Answer:
i) Comparing with the general equation we have
g = – 4; f = 5; c = – 12
Centre – (- g,- f) => (4,- 5)
\(\sqrt{g^{2}+f^{2}-c} \)= \(\sqrt{16+25+12}=\sqrt{53}\)
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 16

Question 7.
Consider the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\). Find the coordinate of the foci, the length of the major axis, the length of the minor axis, latus rectum and eccentricity. (MARCH-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 17

Question 8.
Which one of the following equations (IMP-2014)
represents a parabola which is symmetrical about the positive Y-axis?
a) y² = 4x
b) y² = – 8x
c) x² + 4y = 0
d) x² – 4y = 0
ii) Find the equation of the ellipse vertices are (±13,0) and foci are (±5,0)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 18

Question 9.
Match the following. (IMP-2014)
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 19
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 20

Question 10.
i) Find the equation of the parabola with focus (6,0) and equation of the directrix is x = – 6. (MARCH-2017)
ii) Find the coordinate of the foci, vertices, the length of transverse axis, conjugate axis and eccentricity of the hyperbola \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\)
(MARCH -2017)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 11 Conic Sections 21

Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 10 Straight Lines.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines

Plus One Maths Straight Lines 3 Marks Important Questions

Question 1.
i) Find the slope of the line joining (- 2,6) and (4,8). (MARCH-2010)
ii) Find the value of x if the above line is perpendicular to the line joining (8,12) and (x,24).
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 1
ii) Slope of line through (8,12)and (x,24)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 2
Since both are perpendicular to each other
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 3

Question 2.
i) Write the equation of y-axis. (IMP-2010)
ii) Find the distance between the lines
8x + 15y – 5 = 0 and 8x + 15y + 12 = 0
(Imp (Science) – 2010)
Answer:
i) Equation of y-axis is x = 0
ii) Both are parallel lines, we have;
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 4

Question 3.
The vertices of ∆ ABCare A(2,1), B(- 3,5) and C(4,5). (IMP-2012)
i) Write the coordinates of the midpoint of AC.
ii) Find the equation of the median through the vertex B.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 5
ii) Equation of the median through the vertex B is the equation of a line passing through the midpoint of AC and the vertex B. ie; through (3,3)and (-3,5)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 6

Question 4.
Find the slope of the straight lines √3x + y = 1, x + √3y = 1 (IMP-2012)
Also find the angles between them.
(Imp (Science) – 2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 7
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 8

Question 5.
The vertices of ∆ABC are A(-2,3), B(2,-3) and C(4,5). (MARCH-2012)
i) Find the slope of BC.
ii) Find the equation of the altitude of ∆ABC passing through A.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 9

Question 6.
i) Find the slope of the line joining the points (2,2) and (5,3). (MARCH-2013)
ii) Find the equation of the line joining the points (2,2) and (5,3).
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 10

Question 7.
i) If two lines are perpendicular, then the product of their slopes is ______. (MARCH-2013)
ii) Find the equation of a line perpendicular to the line x – 2y + 3 = 0 and passing through the points (1 ,- 2).
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 11

Question 8.
Consider the line joining the points P(- 4,1) and Q(0,5) (IMP-2013)
i) Write the coordinate of the line passing through the midpoint of PQ .
ii) Find the equation of the line passing through the midpoint of PQ and parallel to the line 3x – 4y + 2 = 0
Answer:
i) Midpoint of PQ = Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 12
ii) The equation of the parallel line is of the form 3x – 4y + k = 0. Since it passes through (- 2,3) we have;
3(- 2) – 4 + k = 0
=> – 6 – 12 + A: = 0
=>k = 18
Hence the equation is 3x – 4y + 18 = 0

Question 9.
i) Find the slope of the line y = 2x – 3. (MARCH-2013)
ii) Find the equation of the line which makes intercepts – 3 and 2 on the X and Y axes respectively. Find its slope.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 13

Question 10.
Consider the lines 2x – 3y + 9 = 0 and 2x – 3y + 7 = 0
i) Find the distance from the origin to these two lines.
ii) Find the distance between these two lines.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 14

Plus One Maths Straight Lines 4 Marks Important Questions

Question 1.
i) Find the slope of the line \(\frac{x}{a}+\frac{y}{b}=1\) (IMP-2010)
ii) If the lines joining the points and are perpendicular (0,0), (1,1) and (2,2), (4,y) find y.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 15

Question 2.
Find slope of the line through the points (5,-1) and (6,4). (IMP-2011)
ii) Find the equation of the line through (5,-1) and (6,4).
iii) Find x intercept and y intercept of this line.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 16

Question 3.
i) Find the slope of the line joining the points (3,- 1) and (4,- 2). (IMP-2012)
ii) Find the angle between the positive x-axis and the line joining the points (3,- 1) and (4,- 2).
iii) Find the equation of the line joining the points (3,- 1) and (4,- 2).
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 17
ii) Slope=- 1
=>tanθ = – 1
=>θ = 135°
iii) Equation of the line joining the points (3,1) and (4,- 2) is y +1 = – 1(x – 3)
=>y + 1 = – x + 3
=> x + y = 2

Question 4.
i) Find the point of intersection of the lines 2x + y – 3 = 0,3x – y – 2 = 0.(MARCH-2012)
ii) Find the equation of the line passing through the above point of intersection and parallel to the linex + y + 1 = 0
Answer:
i)
2x + y = 3 ……..(1)
3x-y = 2 …………(2)
(1) + (2)
=> 5x = 5
=>x = 1
(1)=> y = 3 – 2 =1
Intersection point is (1,1)
ii) Equation of the parallel line x + y + k = 0 Since it passes through (1,1) we have;
1 + 1 + A = 0
=>k = -2
Equation is x + y- 2 = 0

Question 5.
Consider the line x + 3y – 7 = 0 (IMP-2013)
i) The slope of the line is ……….
ii) Find the image of the point (3,8) with respect to the given line.
Answer:
i) We have; x + 3>’-7 = 0
\(y=-\frac{1}{3} x+\frac{7}{3}\)
Hence the slope = – \(\frac { 1 }{ 3 }\)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 18
ii) The equation of the perpendicular to the given and passing through (3,8) is
(y – 8) = 3(x – 3)
=>y – 8 = 3x – 9
=> 3x – y = 1
Solving
=> 3x – y = 1 and x + 3y = 7
we get the coordinate of D, which is the midpoint of the points (3,8) and (x,y).
3x – y = 1
3x + 9y = 21
– 10y = – 20
=> y – 2;
x + 3y = 7
=> x = – 3 + 7 = 1
Hence midpoint is (1,2)
Therefore the coordinate of the image is
\(\left(\frac{x+3}{2}, \frac{y+8}{2}\right)=(1,2)\)
=>x + 3 = 2
=>x = – 1
=>y + 8 = 4
=>y = – 4
Hence (- 1,- 4)

Question 6.
Find the slope of the line 3x – 4y + 10 = 0 (MARCH-2014)
ii) Find the equation of the line passing through the points (1,3) and (5,6).
iii) Find the equation of the line parallel to x – 2y + 3 = 0 and passing through the point (1,- 2).
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 19

Question 7.
i) Find the slope of the line passing through the points (3,- 2) and (- 1,4). (MARCH-2014)
ii) Find the distance of the point (3,- 5) from the line 3x – 4y – 26 = 0
iii) Consider the equation of the line 3x – 4y + 10 = 0
Find its
a) Slope.
b) x and y intercepts.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 20
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 21

Question 8.
i) Find the equation of the line passing through (4,2) with a slope 2. (IMP-2014)
ii) Convert the above equation into intercept form. Find x and y intercepts.
Answer:
i) The equation of the line is
y – 2 = 2(x – 4)
=> y – 2 = 2x-8
=> 2x – y – 6 = 0
ii) Given
=> 2x – y – 6 = 0
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 22
x – intercept = 3;
y – intercept = – 6

Question 9.
i) Find the equation of the line passing through the two points (1,-1) and (3,5). (IMP-2014)
ii) Find the angles between the lines
y – √3x – 5 = 0 and √3y – x + 6 = 0
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 23

Question 10.
Slope of the line L : 2x + 3y + 5 = 0 is. (IMP-2015)
a) \(\frac { 2 }{ 3 }\)
b) – \(\frac { 2 }{ 3 }\)
c) – \(\frac { 3 }{ 2 }\)
d) \(\frac { 3 }{ 2 }\)
ii) Find the equation of the line L’ parallel to L and passing through (2, 2). Find the distance of the lines L and L’
from the origin. Also find the distance between the lines L and L’.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 24

Question 11.
The slope of the line passing through the points (3,-2) and (7,-2) is (MARCH-2017)
(a) – 1
(b) 2
(c) 0
(d) 1
ii) Reduce the equation 6x + 3y – 5 = 0 into slope intercept form and hence find it slope and y-intercept.
iii) Find the point on the x-axis which equidistant from the points (7,6) and (3,4).
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 25

Plus One Maths Straight Lines 6 Marks Important Questions

Question 1.
Reduce the equation3x + 4y – 12 = 0 into intercept form. (MARCH-2010)
Find the distance of it from the origin. Find the distance of the above line from the line 6x + 8y – 18 = 0
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 26

Question 2.
Consider the straight line 3x + 4y + 8 = 0 (MARCH-2011)
i) What is the slope of the line which is perpendicular to the given line?
ii) If the perpendicular line passes through (2,3) from its equation.
iii) Find the foot of the perpendicular drawn from (2,3) to the given line.
Answer:

Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 27

Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 28

Question 3.
i) Find the equation of the line passing through the points (3,- 2) and (- 1,4). (MARCH-2015)
ii) Reduce the equation √3x + y – 8 = 0 into normal form.
iii) If the angle between two lines is \(\frac { π }{ 4 }\) and slope of one of the lines is \(\frac { 1 }{ 2 }\), find the slope of the other line.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 29
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 30
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 31

Question 4.
i) The Slope of a line ‘ L1‘ making an angle 135° with direction of the positive direction of x-axis is (IMP-2015)
(a)1
(b)- 1
(c)√3
(d)-√3
ii) Find the equation of the line L2 perpendicular to L1 and passing through the point (- 2, 3).
iii) Find the equation of a line passing
through the intersection of x + 2y – 3 = 0 and 4x – y + 7 = 0 and which is parallel to 5x + 4y – 20 = 0 .
Answer:
i) tan(135°) = tan(90 + 45) = -1
ii) Slope of line L2= 1
Equation of line L2 passing through (- 2,3)
is y – 3 = 1(x + 2)
=>y – 3 = x + 2
=>x – y + 5 = 0
iii) let the equation line passing through the intersecting point is
x + 2y – 3 + k( 4x – y + 7) = 0
=> (1 + 4k)x + (2 – k)y – 3 + 7k = 0
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 32

Question 5.
Which one of the following pair of straight lines are parallel? (MARCH-2016)
a) x – 2y – 4 = 0;2x – 3y – 4 = 0
b) x – 2y – 4 = 0;x – 2y – 5 = 0
c) 2x – 3y – 8 = 0,3x – 3y – 8 = 0
d) 2x – 3y – 8 = 0;3x – 2y – 8 = 0
ii) Equation of a straight line is 3x – 4y + 10=0. Convert it into the intercept form and write the x-intercept and write the x-intercept and y- intercept.
iii) Find the equation of the line perpendicular to the line x – 7y + 5 = 0 and having x-intercept 3.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 33
iii) The equation of the perpendicular line will be 7x + y + k = 0.
Since x – intercept is 3, the line passes through the point (3,0). So we have;
7(3) + 0+k = 0
=>21 + 0 + k = 0
=>k = – 21
Therefore the equation is 7x + y – 21 = 0.

Question 6.
Which is the slope of the line perpendicular to the line with slope –\(\frac { 3 }{ 2 }\)?(MAY-2016)
(a) –\(\frac { 3 }{ 2 }\)
(b) –\(\frac { 2 }{ 3 }\)
(c) \(\frac { 3 }{ 2 }\)
(d) \(\frac { 2 }{ 3 }\)
ii) Find the equation of the line intersecting the x-axis at a distance of 3 units to the left of origin with slope – 2.
iii) Assume that straight tines work as the plane mirror for a point, find the image of the point (1,2) in the line x – 3y + 4 = 0
Answer:
i) (d) \(\frac { 2 }{ 3 }\)
ii) Slope is m=- 2 and point is (- 3,0)
Equation is y – yx = mix – xx)
=> y – 0 = – 2(x + 3)
=> y = – 2x – 6
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 10 Straight Lines 34

Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 9 Sequences and Series.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series

Plus One Maths Sequences and Series 3 Marks Important Questions

Question 1.
Consider the GP 3,32,33, _______. (IMP-2014)
i) Find the sum to n terms of this GP.
ii) Find the value of n so that the sum to n terms of this GP is 120.
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 1
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 2

Plus One Maths Sequences and Series 4 Marks Important Questions

Question 1.
Given sum of three consecutive terms in an AP is 21 and their product is 280  (IMP-2011)
i) Find the middle term of the above terms.
ii) Find the remaining two terms of the above AP.
Answer:
i) Let the three consecutive terms be
a-d, a, a + d
a-d + a + a + d = 21
=>3a = 21
=>a = 7
ii) Then the AP becomes 7 – d,7, 7 + d
Given product is 280;
(7 – d)(7)(7 + d) = 280
=> (7 – d)(7 + d) = 40
==> 49 – d² = 40
=> <d² = 9 => d= 3,- 3
Therefore the AP is 4,7,10 or 10,7,4.

Question 2.
Consider the GP 3,6,12  (IMP-2011)
i) Which term of this GP is 96?
ii) Find the value of n so that sum to n terms of this GP is 381.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 3

Question 3.
i) What is the sum of the first ‘n’ natural numbers?  (IMP-2012)
ii) Find the sum to ‘n’ terms of the series
3 x 8 + 6 x 11 + 9 x 14 + ______.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 4
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 5

Question 4.
If the sum of the first n terms of an Arithmetic progression is ——,where X and Y are constants, find  (IMP-2012)
i) S1 and S2
ii) The first term and common difference.
iii) The nth term.
Answer:
i) S1 = X
S2 =2X + 1/2(2 – 1)Y=2X + Y
ii) First term = a, = Sx = X
S2 =2 X + Y
=> a1 +a2 =2 X + Y
=> a2 =2X + Y
=>a2 = X+ Y
Common difference =
a2 – a1 =X + Y – X = Y
iii) nthterm = an = a + (n-1)d = X + (n – 1)Y

Question 5.
Find the sum to n terms of the series;  (IMP-2012)
2² + 5² + 8² +_______
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 7

Question 6.
i) Write the first four terms of the sequence whose nth term \(a_{n}=\frac{n}{n+1}\) (MARCH-2013)
ii) The sum of the first three terms of a GP is \(\frac {12}{13}\) and their product is -1. Find the common ratio and the terms.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 8
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 9

Question 7.
If the numbers \(\frac { 5 }{ 2 }\) x \(\frac { 5 }{ 8 }\) are three consecutive terms of a GP, then find x. (MARCH-2014)
Find the sum of the first n-terms of the series. 2 +22+222 + _____
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 10
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 11

Question 8.
i) Find the 5th term of the sequence whose nth term is \(a_{n}=\frac{n(n-2)}{(n+3)}\) (MARCH-2014)
ii) Write the sum of first n natural numbers.
iii) The 5th, 8th and 11th terms of a GP are p, q and s respectively. Prove that q2 – ps
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 12

Question 9.
i) A man starts repaying a loan as a first instalment of Rs. 1,000. If he increases the instalment by Rs. 150 every month, what amount will he pay in the 30th instalment?  (IMP-2014)
ii) Find the sum to n terms of the sequence:
7,77,777,7777 ______.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 13

Question 10.
i) Consider the AP 4,10,16,22…….. Find its common difference and the 7th terms.  (IMP-2014)
ii) If the mth term of an AP is \(\frac { 1 }{ n }\) and the nth term is \(\frac { 1 }{ m }\) , prove that the sum of the first ‘mn’ terms is \(\frac { 1 }{ 2 }\)(mn +1)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 14

Question 11.
The 6th term of the sequence whose nth term is \(t_{n}=\frac{2 n-3}{6}\) is _____. (MARCH-2015)
a) 3
b) \(\frac { 1 }{ 2 }\)
c) \(\frac { 3 }{ 2 }\)
d) \(\frac { 1 }{ 3 }\)
ii) Find the sum to infinity of the sequence 1,\(\frac { 1 }{ 3 }\) ,\(\frac { 1 }{ 9 }\), ………
iii) If a, b, c are in AP and \(a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}\), prove that x, y, z are in AP.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 15

Plus One Maths Sequences and Series 6 Marks Important Questions

Question 1.
i) In an AP, the first term is 2 and the sum of the first five terms is one fourth the sum of the next five terms. (MARCH-2010)
a) Find the common difference.
b) Find the 20th term.
ii) If AM and GM of two numbers are 10 and 8 respectively, find the numbers.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 16

Question 2.
i) In an AP if mth term is ‘n’ and nth term is ‘m’ .find the (m + n)th term.  (IMP-2010)
ii) If 3rd, 8th and 13th terms of a GP are x,y,z respectively, prove that x,y,z are in GP.
iii) Prove that x,y,z in the above satisfies the equation \(\frac{y^{10}}{(x z)^{5}}=1\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 17
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 18

Question 3.
Which of the following is the nth term of an AP? (MARCH-2011)
a) 3 – 2n
b)n² – 3
c) 3n – 2
d) 2 – 3n²
ii) Find the 10th term of the sequence
– 6,- \(\frac { 11 }{ 2 }\), – 5,….
iii) The sum of the first three terms of a GP is \(\frac { 39 }{ 10 }\) and their product is 1. Find the common ratio and the terms.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 19
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 20

Question 4.
Find the 10th term of an AP whose nth \(\frac{2 n-3}{6}\) term is (MARCH-2012)
ii) Find the sum of the first 10 terms of the above AP.
iii) Find the sum of the first 10 terms of a GP, whose 3rd term is 12 and 8th term is 384.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 21

Question 5.
i) Find the 5th term of the sequence whose nth term, \(a_{n}=\frac{n^{2}-5}{4}\) (MARCH-2013)
ii) Find 7 + 77 + 777 +……. to n terms.
iii) Find the sum to n terms of the series.
1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + ………
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 22
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 23

Question 6.
i) Find the sum of multiple of 7 between 200 and 400.  (IMP-2013)
ii) The sum of first 3 terms of a GP is \(\frac { 39 }{ 10 }\) and their product is 1. Find the terms.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 24
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 25

Question 7.
If ‘a’ is the first term and ‘cf is the common difference of an AP, then the nth term of the AP, an = ……. (MARCH-2014)
ii) In an AP, if the mth‘ term is ‘n’ and the nth term is ‘m’, where , prove that its pth term is n + m – p.
iii) Find the sum to ‘n’ terms of the series:
1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + _______.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 26
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 27

Question 8.
i) If the sum of certain number of terms of the AP 25,22,19 is 116, then find the last term.  (IMP-2014)
ii) Find the sum to n terms of the series
1 x 2 x 3 + 2 x 3 x 4 + 3 x 4 x 5 + ………
(Imp (Science) – 2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 28
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 29

Question 9.
i) The 3rd term of the sequence whose nth term is (MARCH-2015)
ii) Insert three numbers between 1 and 256 so that the resulting sequence is a GP.
iii) If pth term of an AP is q and qth term is ‘p’, where p ≠ qfind rth term.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 30

Question 10.
i) Geometric mean of 16 and 4 is ______.  (IMP-2015)
(a) 20
(b) 4
(c) 10
(d) 8
ii) Find the sum to n terms of the series: 5 + 55 + 555 + ________.
iii) Find the sum to n terms of the AP,
whose Kth term is ak = 5K +1
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 31
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 32

Question 11.
i) If the first three terms of an AP is x – 1,x + 1, 2x + 3, then x is  (IMP-2015)
(a)- 2
(b) 2
(c) 0
(d) 4
ii) Find the sum to n terms of the sequence.
1 x 2 + 2 x 3 + 3 x 4 + _______
iii) The nth term of the GP 5,- \(\frac { 5 }{ 2 }\),\(\frac { 5 }{ 4 }\),\(\frac { 5 }{ 8 }\),….. is \(\frac { 5 }{ 1024 }\) find ‘n’.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 33

Question 12.
The nth term of the GP 5,25,125 (MARCH-2016)
is
(a) n5
(b) 5n
(c) (2n)5
(d) (5)2n
ii) Find the sum of .all natural numbers between 200 and 1000 which are multiples of 10.
iii) Calculate the sum of n-terms of the series whose n81 term is an = n(n + 3)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 34

Question 13.
i) Which among the following represents the sequence whose nth terms is \(\frac { n}{ n+1 }\) ? (MAY-2017)
a) 1,2,3,4,5,6
b) 2,3,4,5,6
c) 2,\(\frac { 3 }{ 2 }\),\(\frac { 4 }{ 3 }\),\(\frac { 5 }{ 4 }\),\(\frac { 6 }{ 5 }\)
d) \(\frac { 1 }{ 2 }\),\(\frac { 2 }{ 3 }\),\(\frac { 3 }{ 4 }\),\(\frac { 4 }{ 5 }\),\(\frac { 5 }{ 6 }\)
ii) Using progression, find the sum of first five terms of the series 1 + \(\frac { 2 }{ 3 }\) + \(\frac { 4 }{ 9 }\) + …..
iii) Calculate: 0.6 + 0.66 + 0.666 + ………. n terms.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 35
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 36

Question 14.
The sum of the infinite series is 1, \(\frac { 1 }{ 3 }\),\(\frac { 1 }{ 9 }\) ………is ________. (MARCH-2017)
(a) \(\frac { 3 }{ 2 }\)
(b) \(\frac { 5 }{ 2 }\)
(c) \(\frac { 2 }{ 3 }\)
(d) \(\frac { 7 }{ 2 }\)
ii) Find the sum of all natural numbers between 100 and 1000 which is a multiple of 5.
iii) Find the sum to n terms of the series 8,88,888 ………
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 37

Question 15.
The 6th term of the GP \(\frac { 1 }{ 2 }\),\(\frac { 1 }{ 4 }\),\(\frac { 1 }{ 8 }\), ………. (MARCH-2017)
a) \(\frac { 1 }{ 32 }\)
b) \(\frac { 1 }{ 64 }\)
c) \(\frac { 1 }{ 16 }\)
d) \(\frac { 1 }{ 128 }\)
ii) The sum of 1st 3 terms of a G.P is \(\frac { 13 }{ 12 }\) and their product is – 1. Find the common ratio and terms.
iii) Find the sum to n terms of the series \(3 \times 1^{2}+5 \times 2^{2}+7 \times 3^{2}\) + ………
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 38
Plus One Maths Chapter Wise Previous Questions Chapter 9 Sequences and Series 39

Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 8 Binomial Theorem.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem

Plus One Maths Binomial Theorem 3 Marks Important Questions

Question 1.
i) The number of terms in the expansion of \(\left(\frac{x}{3}+9 y\right)^{10}\) is _____.(IMP-2013)
ii) Find the middle term in the above expansion.
Answer:
i) 11
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 1

Question 2.
i) Find the general term in the expansion of (x + y)n
ii) Find the middle term in the expansion of \(\left(2 x+\frac{1}{3 y}\right)^{18}\) (MARCH-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 2

Question 3.
i) Write the general term in the expansion of (a + b)
ii) Find the 9th term in the expansion of \(\left(\frac{x}{2}+\frac{6}{x^{2}}\right)^{12}\) (IMP-2014)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 3

Plus One Maths Binomial Theorem 4 Marks Important Questions

Question 1.
i) Find the general term in the expansion of \(\left(3 x^{2}-\frac{1}{3 x}\right)^{9}\) (MARCH-2010)
ii) Find the term independent of x in the above expansion.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 4
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 5

Question 2.
Consider the expansion of \(\left(x^{2}-\frac{1}{3 x}\right)^{9}\) (IMP-2010)
i) Find the coefficient of x9
ii) Find the term which is independent of x.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 6

Question 3.
Consider the expansion of \(\left(\frac{x}{9}+9 y\right)^{2 n}\) (MARCH-2011)
i) The number of terms in the expansion is _____
(a) 2n
(b) n+1
(c) 2n+1
(d) 2/7-1
ii) What is its (n+1)th term?
iii) If n = 5, find its middle term.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 7

Question 4.
i) Write the general term in the expansion (1 + x)44   
Write 21st and ,22nd terms in the expansion of (1 + x)44
iii) If 21st and 22nd terms in the expansion of (1 + x)44 are equal then find the value of x. (IMP-2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 8

Question 5.
8. Find(x + y)4 – (x – y)4(IMP-2012)
Hence evaluate: (√5 + √6)4 – (√5 – √6)4
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 9

Question 6.
i) How many terms are there in the expansion of (1 + x)2n (n is a positive integer)? (IMP-2012)
ii) Show that the middle term in the (1 + x)2n
expansion of is \(\frac{1.3 .5 \ldots(2 n-1)}{n !} 2^{n} x^{n}\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 10
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 11

Question 7.
i) Find the general term in the expansion of \(\left(\frac{x}{2}-\frac{2}{x}\right)^{10}\) (MARCH-2012)
ii) Find the terms independent of x in the above expansion.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 12

Question 8.
i) Find the number of terms in the expansion of \(\left(x-\frac{1}{x}\right)^{14}\) (MARCH-2013)
ii) Find the general term in the expansion of \(\left(x-\frac{1}{x}\right)^{14}\)
iii) Find the term independent of x in the above expansion.
Answer:
i) 15
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 13

Question 9.
i)Write the number of terms in the expansion of (a -b)2n
ii) Find the general term in the expansion of \(\left(x^{2}-y x\right)^{12}, x \neq 0\) (MARCH-2014)
iii) Find the coefficient of x6y3 in the expansion of (x + 2y)9
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 26

Question 10.
i) Write the expansion of (a + n)n, where n is any positive integer. (IMP-2014)
ii) Find the value of ‘a’ if the 17th term and 18th term in the expansion of (2 +a)50 are equal.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 15

Question 11.
i) The number of term in the expansion of \(\left(x-\frac{1}{x}\right)^{2 n}\) is ______. (MARCH-2015)
(a) n+1
(b) n
(c) 2n+1
(d) 2n+2
ii) Find a, if the 17th term and 18th term of the expansion of (2 +a)50 are equal.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 16

Question 12.
i) Number of terms in the expansion of \(\left(x+\frac{1}{x}\right)^{20}\) (IMP-2016)
(a) 19
(b) 20
(c) 21
(d) 22
Consider the expansion of \(\left(3 x^{2}-\frac{1}{3 x}\right)^{9}\)
find the coefficient of x6 and the term independent of x.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 25

Question 13.
The 8th term in the expression (MARCH-2016)
of(√2 + √3)7 is
a) 27√2
b) 27√3
c) 72√2
d) 72√3
ii) Find the term independent of x in the expansion of \(\left(x+\frac{1}{2 x}\right)^{18} ; x>0\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 18

Question 14.
Write the expansion of (a + b)4  (MAY-2017)
Evaluate: (√5 + √6)4+ (√5 – √6)4
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 19
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 20

Question 15.
Consider the expansion of \(\left(x+\frac{1}{x}\right)^{10}\) (MARCH-2017)
i) The number of terms in the expansion is _____.
(a) 10
(b) 9
(c)11
(d) 12
ii) Find the term which is independent of x in the above expansion.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 21

Plus One Maths Binomial Theorem 6 Marks Important Questions

Question 1.
i) Write the number of terms in the expansion of (a + b)n
ii) Expand \(\left(\frac{x}{3}+\frac{1}{x}\right)^{5}\) (MARCH-2013)
iii) Find the general term in the expansion of \(\left(x^{2}-y\right)^{6}\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 22
Plus One Maths Chapter Wise Previous Questions Chapter 8 Binomial Theorem 23

Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations

Kerala State Board New Syllabus Plus One Maths Chapter Wise Previous Questions and Answers Chapter 7 Permutation and Combinations.

Kerala Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations

Plus One Maths Permutation and Combinations 3 Marks Important Questions

Question 1.
i) if \({ }^{n} C_{9}={ }^{n} C_{8}\) find ‘n’ and \({ }^{n} C_{17}\) (IMP-2014)
ii) How many chords can be drawn, through 23 points on a circle?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 1

Plus One Maths Permutation and Combinations 4 Marks Important Questions

Question 1.
i) Simplify (MARCH-2011)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 2
ii) In how many different ways can the letters of the word HEXAGON be permuted?
iii) In how many different ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
(March (Science) – 2011)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 3
Total number of ways = 10 x 4 = 40

Question 2.
i) If \(\frac{1}{8 !}+\frac{1}{9 !}=\frac{x}{10 !}\) (MARCH-2011)
then find x.
ii) How many 4 digit numbers are there with no digit repeated?
iii) If nC8 = nC2, then find nC3 ?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 4

Question 3.
Consider all the letters of the word ‘FALIURE’.  (IMP-2011)
i) How many words can be formed using these letters?
ii) How many words can be formed so that the vowels being together?
iii) How many words begin with A and end with E?
(Imp (Commerce) – 2011)
Answer:
i) ‘FALIURE’ word has 7 letters in it, can be arranged in 7! Ways = 7.6.5.4.3.2.1 = 5040
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 5
A,E,I,U are vowels in the word, should be kept together, so should be treated as on block. Hence there are 4 such blocks can be arranged in 4! ways. These 4 vowels can be arranged in 4! Ways.
Hence the total words = 4! x 4! = 24 x 24 = 576
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 6
The only possible arrangement is for 5 blocks; hence total number of ways is 5! = 120.

Question 4.
i) Find the value of n if (IMP-2012)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 7
ii) How many words, with or without meaning,
can be formed using all the letters of the word CHEMISTRY, using each letter exactly once? How many of them start with C and end with Y?
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 8
n —10,- 3
n = – 3 is not possible since negative son = 10
ii) Total number of words = 9!
Total number of words starting by C and ending by Y= 7!

Question 5.
i) If 2nC3 : nC3 = 12 : 1 find n. (IMP-2012)
ii) What is the total number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards of the same suit?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 9

Question 6.
i) If nC9 = nC8, find nC17 
ii) A committee of 5 person is to be selected from a group of 4 men and 5 women. In how many ways can this be done? How many of these committees would consist of 2 men and 3 women? (IMP-2012)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 10
Number of different selection = 6 x 10 = 60

Question 7.
i) If nC9 = nC8, find nC17 
ii) How many three digit number can be formed using the digits 1,2,3,4,5 if repetition is not allowed? (MARCH-2013)
iii) In How many ways can a team of 4 boys and 3 girls be selected from 6 boys and 4 girls?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 11

Question 8.
i) If nC5 = nC4, find nC8
ii) How many chords can be drawn through 20 points on a circle? (MARCH-2014)
iii) A bag contains 6 red and 5 blue balls. In how many ways can one choose 3 red and 2 blue balls from this bag?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 12

Question 9.
i) Find the number of permutation of the letters of the word ALLAHABAD.
ii) Find r,if \({ }^{5} P_{r}=2 \times{ }^{6} P_{r-1}\) (IMP-2014)
Answer:
i) Total number of letters is 9.
A: 4 times; L: 2 times.
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 13
Therefore r = 3.

Plus One Maths Permutation and Combinations 6 Marks Important Questions

Question 1.
i) Find the value of n such that (MARCH-2010)
nP5 = 42 x nP3 for n > 4
ii) A committee of 3 persons is to be constituted from a group of 2 men and 3 women.
a) In how many ways can this be done?
b) How many of these committees would consist of 1 man and 2 women? (March (Science) – 2010)
Answer:
i) nP5 = 42 x nP2
=> n(n – 1)(n – 2 )(n – 3)(n – 4) = 42 x n(n -1)(n- 2)
=> (n – 3)(n – 4) = 42
=>n2 – 7n + 12 = 42
=> n2 – 7n – 30 = 0
=> (n -10)(n + 3) = 0
=> n = 10; n = – 3
n can’t be negative, so the acceptable value is n = 10
ii) a) 3 person can be selected from 5 in 5C3 = 10
(b) 1 man can be selected from 2 in 2C1 = 2 ways.
2 women can be selected from 3 in 3 C2 = 3 ways.
Total ways = 2×3 = 6

Question 2.
i) lf \({ }^{n} C_{2}:{ }^{2 n} C_{1}=3: 2\), find n. (MARCH-2010)
ii) a) Find the number of words that can be
formed from the letters of the word MALAYALAM.
b) How many of these arrangements start with Y?
(March (Science) – 2010)
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 14
ii) a) MALAYALAM, this word has 9 letters
M- repeated 2 times.
A- repeated 4 times.
L- repeated 2 times.
Number of words formed by these 9 letters
= \(\frac{9 !}{2 ! \times 4 ! \times 2 !}\)
b) If the word starts with Y, then total number of letters that can be arranged become 8. Number of words formed which begin with Y
= \(\frac{8 !}{2 ! \times 4 ! \times 2 !}\)

Question 3.
i) if \(5 \times 4 P_{r}=6 \times 5 P_{r-1}\) find ‘r’, (IMP-2014)
ii) How many 3 digit number can be formed with the digits 0,1,2,3 and 4?
iii) In a Panchayath there are 10 Panchayath members. Ladies contested only in the 50 % reserved constituency. If the post president and vice president are reserved for ladies, in how many ways both the president and vice president can be selected?
(Imp (Science) – 2010)
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 15
ii)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 16
Total 3 digit numbers = 4x5x5 = 100 (IMP-2010)
iii) The 10 member Panchayath has 5 men and 5 ladies. The president and vice president are to be selected from these ladies in 5C12 = 10 ways.

Question 4.
i) Prove that nCr = nCn-r
ii) Twenty eight matches were played in a volley ball tournament. Each team playing one against each of others. How many teams were there? (IMP-2010)
iii) If the letters of the word ‘TUTOR’ be . permuted among themselves and arranged as in a dictionary, then find the position of the word ‘TUTOR’.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 17
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 18

Question 5.
A student is instructed to answer any 8 out of 12 questions. (IMP-2011)
i) How many different ways he can choose the questions?
ii) How many different ways he can choose the questions so that question no.1 will be included?
iii) How many different ways, he can choose the questions so that question no.1 will be included and question no.10 will be excluded?
Answer:
i) 8 out of 12 questions can be selected
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 19
ii) Since question no.1 is included, the possible is selection is from 11 questions and the number of questions to be selected becomes 7.
Hence the total selection
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 20
iii) Question no.10 will be excluded so total questions become 11. Question no.1 is included again total questions reduced to 10. Now we have to select 7 questions out of 10, can be done in 10C7 = 10C3 = \(\frac{10 \times 9 \times 8}{1 \times 2 \times 3}\) = 120.

Question 6.
Solve for the natural n; (MARCH-2012)
12.(n-1)P10C3 =5.(n + 1)P10C3
In how many ways seven althlets can be chosen out of 12?
iii) The English alphabets has 5 vowels and 21 consonants. How many words with two different vowels and two different consonants can be formed without repetition of letters?
Answer:
i)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 21
ii) 7 athletes be chosen out of 12 in 12C7 =12C5 ways
iii) Two different vowels can be selected in 5C2. Two different consonants can be selected in 21C2.
Therefore total numbers of words
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 22

Question 7.
i) Find r if 5Pr = 6Pr-1.
ii) If there are 12 persons in a party and each of them shake hands with all others, what is the total number of handshakes? (MARCH-2012)
iii) In How many ways can a committee of 3men and 2 women be selected out of 7 men and 5 women?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 23
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 24

Question 8.
i) Find the value of n such that (MARCH-2013)
3.nP4 = 5.(n -1 )P4,n > 4
ii) In how many ways can 5 students be seated on a bench?
iii) Find the number of different 8-letter arrangements that can be made from the letters of the word, ‘DAUGHTER’ so that:
a) All vowels are occur together.
b) All vowels do not occur together.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 25

Question 9.
i) Determine n if 2nC3 = 11.nC
ii) In how many ways can a cricket team of 11 of players be selected from 15 players? (MARCH-2013)
iii) A bag contains 5 white, 6 red and 4 blue balls. Determine the number of ways in which 2 white, 3red and 2 blue balls can be selected.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 26
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 27

Question 10.
i) The number of 3 digit numbers can be formed from the digits 1,2,3,4,5 assuming that repetition of the digits is not allowed is _______.
ii) If \(\frac{1}{6 !}+\frac{1}{7 !}=\frac{x}{8 !}\), find x. (IMP-2013)
iii) How many words, with or without meaning, can be formed using all the letters of the word ‘FRIDAY’, using each letter exactly once? How many of them have first letter is a vowel?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 28

Question 11.
i) nC7=nC5,n =
ii) A bag contains 5 blue and 6 white balls. Determine the number of ways in which 3 blue and 4 white balls can be selected. (IMP-2013)
iii) What is number of choosing 3 cards from a pack of 52 playing cards? In how many of these 3 cards of the same colour?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 29
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 30

Question 12.
i) if \(\frac{1}{8 !}+\frac{1}{9 !}=\frac{x}{10 !}\) find x? (MARCH-2014)
ii) How many four digit numbers can be formed using the digits 4,5,6,7,8 if repetition of digits is not allowed?
iii) Find the number of arrangements that can be made from the letters of the word ‘MOTHER’ so that all vowels occur together.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 31

Question 13.
i) In how many ways can the letters of the word PERMUTATIONS be arranged if; (MARCH-2014)
a) the word starts with P and ends with S?
b) there are always 4 letters between P and S?
ii) In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?
iii) How many chords can be drawn through 21 points?
Answer:
i) a) In the word T is repeated twice
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 32
b) P should move from 1 to 7th position and S should move from 6th to 12th position. Hence the arrangements
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 33
ii) First arrange 5 girls in 5P5 ways. Arrange 3 boys in 6P3.
Hence the total arrangements
= 5P5 x 6P3 = 14400
iii) Chord is the join of two points. Hence selection 2 points from 21, which can be done in 21C2 =210

Question 14.
i) What is the minimum number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these
a) are 4 cards of the same suit? (MARCH-2014)
b) do 4 cards belong to 4 different suits?
ii) Find the number of permutation of the letters of the word, ALLAHABAD.
iii) How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 34

Question 15.
i) Find ‘n’ if \(9 \times^{(n-1)} P_{3}={ }^{n} P_{4}\) (IMP-2014)
ii) Find the number of words that can be formed from the letters of the word, COMMERCE’.
iii) In how many ways can a group of 12 students be selected from 15 students? How many of these groups would include one particular student?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 35

Question 16.
i) \(\frac{0 !}{1 !}\) = _____ (MARCH-2015)
a) 0
b) 1
c) 2
d) 3
ii) Find r, if \(5 \times^{4} P_{r}=6 \times^{5} P_{r-1}\)
iii) Find the number of 8-letter arrangements that can be made from the letters of the word DAUGHTER so that all vowels do not occur together.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 36
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 37

Question 17.
nCn – 1 = _____ (MARCH-2015)
(a) n-1
(b) n
(c) 0
(d) 1
If nC9=nC8,find nC2
How many ways can a team of 5 persons be selected out of a group of 4 men and 7 women, if the team has at least one man and one women?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 38

Question 18.

i) \({ }^{7} P_{7}\) = _____ (IMP-2014)
a) 7
b) 7!
c) 1
d) 77
ii) Find the number of words that can be formed from the letters of the word “MALAYALAM”. How many of them start with Y?
iii) \({ }^{2 n} C_{3}=11 \times{ }^{n} C_{3}\) Find ’n’.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 39
iii)
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 49

Question 19.
i) \({ }^{29} C_{29}\) = ______ (IMP-2015)
(a) 0
(b) 1
(c) 2
(d) 3
ii) Prove that \({ }^{61} C_{57}-{ }^{60} C_{56}={ }^{60} C_{3}\)
iii) In how many ways can the letters of the word ‘ARRANGE’ be arranged such that two A’s do not occur together?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 40

Question 20.
Write the value of \({ }^{7} C_{5}\) (MARCH-2016)
Find the vale of n, if \(3 \times^{n} P_{4}=5 \times^{n-1} P_{4}\)
What is the number of ways of choosing 4 cards from a pack of 52 cards, provided all 4 cards belong to 4 different suits? (March (Science) – 2016)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 41

Question 21.
i) \({ }^{29} C_{29}\) = ______ (MARCH-2016)
a) 0
b) 1
c )2
d )3
ii) Find the value of n,
if \(12 \times^{n-1} P_{3}=5 \times^{n+1} P_{3}\)
iii) A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has at least one boy and one girl?
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 42

Question 22.
i) How many 4 digit numbers can be formed using the digits 9, 8, 7, 6, 5, 4, if no digits are repeated? (MAY-2017)
(a) 630
(b) 603
(c) 306
(d) 360
ii) In how many ways a committee of 3 persons can be formed from a group of 2 men and 3 women?
iii) Find the value of n,
if \({ }^{2 n} C_{3}=11 \times{ }^{n} C_{3}\)
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 43

Question 23.
i) \({ }^{569} \mathrm{C}_{569}\) = ______. (MAY-2017)
ii) \({ }^{2 n} C_{3}:{ }^{n} C_{3}=12: 1\) Find n.
iii) If the letters of the word EQUATION are arranged, find the number of arrangements in which no two consonants occur together?
Answer:
i) 1
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 44

Question 24.
i) if \(\frac{1}{6 !}+\frac{1}{7 !}=\frac{x}{8 !}\) , then x is ______.
(a) 32
(b) 16
(c) 64
(d) 8
ii) Given 5 flags of different colour, how many different signals can be generated if each other.
iii) Find r, if \({ }^{5} P_{r}=2 \times{ }^{6} P_{r-1}\)
Answer:
i) c) 64
ii) Number of ways = 5 x 4 = 20
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 45
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 46

Question 25.
i) lf nC9=nC8, then n = ______.
a) 9
b) 8
c) 17
d) 1
ii) How many chords can be drawn through 12 point on a circle?
iii) What is the number of way of choosing 4 cards from a pack 52 playing cards? In how many of these:
a) Four cards are of the same suit.
b) Cards are of the same colour.
Answer:
Plus One Maths Chapter Wise Previous Questions Chapter 7 Permutation and Combinations 47