Plus One Maths Improvement Question Paper 2018

Kerala Plus One Maths Improvement Question Paper 2018

Time Allowed: 2½ hours
Cool off time: 15 Minutes
Maximum Marks: 80

General Instructions to Candidates:

• There is a ‘cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘cool off time’ to familiarize yourself with the questions and plan your answers.
• Read instructions carefully.
• Read questions carefully before you answer.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices, except non-programmable calculators, are not allowed in the Examination Hall.

Answer any 6 questions from question numbers 1 to 7. Each carries 3 scores

Question 1.
a. If A= {2, 3, 4, 5} and B = {4, 5, 6, 7}, then write :
i. A ∪ B
ii. A ∩ B
b. Which one of the following is equal to {x: x∈ R, ∂< x ≤ 4 }?
i. {2,3,4}
ii. {3,4}
iii. [2, 4]
iv.(2, 4]

Question 2.
Consider the set
A = {x : x is an integer, 0 ≤ x < 4}
a. Write A in Roser form
b. If B = {5, 6}, then write A x B
c. Write the number of possible relations from A to B

Question 3.
(a) Prove that \(\frac{\cos 9 x-\cos 5 x}{\sin 17 x-\sin 3 x}=\frac{-\sin 2 x}{\cos 10 x}\).
(b) Evaluate \(\lim _{x \rightarrow 0} \frac{\cos 9 x-\cos 5 x}{\sin 17 x-\sin 3 x}\)

Question 4.
Solve the inequality \(\frac{x}{2} \geq \frac{5 x-2}{3}-\frac{7 x-3}{5}\)

Question 5.
Find the polar form of the complex number \(\frac{1+i}{1-i}\).

Question 6.
How many terms of the GP, 3, 3/2, 3/4,…………………..are needed to give the sum \(\frac { 3069 }{ 512 } \)?

Question 7.
Consider the real-valued function f(x) = \(\frac{x-3}{x^2-x-6}\)
(a) Find the domain of f(x).
(b) Evaluate \(\lim _{x \rightarrow 3} f(x)\)

Answer any 8 from questions 8 to 17. Each question carries 4 scores.

Question 8.
a. If U = {1, 2, 3,4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
B = {2, 3, 5, 7}
Verify (A ∪B) = A’ ∩ B’
b. If A and B are two disjoint sets with n(A) = 4 and n(B) = 2, then n (A-B) = …………

Question 9.
Consider the statement P(n) : 1-3 + 32 +……….. \({ 3 }^{ n-1 }=\frac { { 3 }^{ n-1 } }{ 2 } \)
a. Show that P(1) is TRUE
b. Prove by principle of Mathematical induction, that P(n) is TRUE for all n ∈ N

Question 10.
Solve the following inequalities graphically
2x + y ≥ 4
x + y ≤ 3 and
2x – 3y ≤ 6

Question 11.
Find the square roots of the complex number 3 + 4i.

Question 12.
a. Insert five numbers between 8 and 26 such that the resulting sequence is an AP.
b. Find the sum to n terms of the series 1 x 2 + 2 x 3 + 3 x 4+…………..

Question 13.
a. Find the equation of the perpendicular bisector of the line joining the points (0, 0) and (-3, 4).
b. Find the coordinates of the point on the line y = 3x-2 that is equidistant from (0,0) and (-3, -4)

Question 14.
a. Reduce the equation x – y = 4 into normal form.
b. Write the distance of this line from the origin

Question 15.
a. Find the derivative of f(x) = x Sin x concerning x.
b. Find the derivative of the function y = √x concerning x by using first principles.

Question 16.
Consider the points A (3, 8, 10) and B (6, 10, -8).
a. Find the ratio in which the line segment joining A and B is divided by the YZ coordinate plane.
b. Find the coordinates of the point of division.
c. Which coordinate plane divides the line segment AB internally? Justify your answer.

Question 17.
a. Write the contrapositive of the statement: “If the integer n is odd, then n2 is odd”,
b. Prove by the method of contradiction ‘ √7 is irrational’

Answer any 5 questions from question numbers 18 to 24. Each carries 6 scores

Question 18.
(a) If \(\left(\frac{x+3}{2}, \frac{y-1}{3}\right)\) = (4, 2), find the value of x and y.
b. Consider the function f (x) = |x| = 3, draw the graph of f(x)
c. Write the domain and range of f(x)

Question 19.
a. Find the value of Sin (75°)

b. In the given figure. ∠AOB = 30° and the radius of the circle is d units. Find the length of area APB.
c. Find the length of chord AB

Question 20.
a. Find the number of words with or without meaning, which can be made by using all the letters of the word GANGA.
b. If these words are written as in a dictionary, what will be the 26th word?
c. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team should have at least 3 girls?

Question 21.
a. Write the expansion of (a + b)ⁿ.
b. Find the coefficient of x⁵ y⁷in the expansion of (x-2y)¹²
c. Show that 9^n-1 – 8n – 9 is divisible by 64.

Question 22.
Focii of the ellipse in the given figure are (± √12,0) and vertices are (±4, 0).

a. Find the equation of the ellipse.
b. Write the equation of a circle with centre (0, k) and radius r.
c. The circle in the figure passes through the points A, B, and C on the ellipse. Find the equation of a circle.

Question 23.
Consider the following table

a. Find the arithmetic mean of marks given in the above data
b. Find the standard deviation of marks in the above data.
c. Find the coefficient of variation.

Question 24.
a. Consider the experiment in which a coin is tossed repeatedly until a head comes up. Write the sample space.
b. If A and B are two events of a sample space with P (A) = 0.54, P(B) = 0.69 and P (A ∩ B) = 0.35. Find P (A’ ∩ B’).
c. 3 cards are drawn from a well shuffled pack of 52 cards. Find the prohibility that
i. all the 3 cards are diamond.
ii. at least one of the cards is non diamond
iii. one card is king and two are jacks.

Answers

Answer 1.
a. A ∪ B = {2, 3, 4, 5, 6, 7}
A∩B = {4, 5}
b. iv) (2, 4]

Answer 2.
a. A = {0, 1,2,3}
b. A x B = {(0, 5),(0, 6),(1, 5),(1, 6),(2, 5), (2, 6),(3, 5),(3, 6)}
c. No. of possible relations from A to B
= 2^mn = 2^{4 x 2} = 256

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.
a. A ∪ B = {2, 3, 4, 5, 6, 7, 8}
(A ∩ B)’ = {1, 9}………….(1)
A’= {1, 3, 5, 7, 9}, B’={1, 4, 6, 8, 9} A’∩B’= {1, 9}…………….(2)
From (1) and (2), we have (A + B)’ = A’ ∩ B’
b. n (A – B) = n (A) = 4

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.
(a) Let the ratio be k : 1
Point of division is \(\left(\frac{6 k+3}{k+1}, \frac{10 k+8}{k+1}, \frac{-8 k+10}{k+1}\right)\)
Since the line joining A and B divides the YZ plane, the point of intersection has x coordinate equal to zero.
\(\frac{6 k+3}{k+1}\) = 0
⇒ 6k + 3 = 0
⇒ k = \(-\frac{1}{2}\)
Hence YZ plane divides externally in the ratio 1 : 2.
(b) Substituting the value of k, we get the point.


(c) The point A(3, 8, 10) and B(6, 10, 4) have z coordinates opposite in sign. So the point A will lie above and point B will lie below the XY plane. Hence the XY plane divides the line segment joining A and B internally.

Answer 17.
a. If n2 is not even, then the integer n is not even.
b. Let us assume that √7 is a rational number.
√7 =a/b, where a and b are co-prime, i.e., a and b have no common factors. Squaring we have,
7b² = a² ⇒ 7 divides a.
∴ Let a = lk
∴a²= 49k²  ⇒ 7b²= 49k² ⇒ b² = 7k² ⇒7  divides b.
i.e., 7 divides both a and b, which is a contradiction to our assumption that a and b have no common factor. our supposition is wrong.
∴ √7 is an irrational number.

Answer 18.
(a) \(\frac{x+3}{2}\) = 4, \(\frac{y-1}{3}\) = 2
⇒ x + 3 = 8; y – 1 = 6
⇒ x = 5; y = 7

(c) From the graph of f(x) is the domain and range is clear.
Domain = R
Range = [-3, ∞)

Answer 19.

Answer 20.
a. Number of words = \(\frac { 5! }{ 2!2! } \) = 30
b. If A is fixed, the remaining 4 letters can 4!
be permuted in \(\frac { 4! }{ 2! } \) = 12 ways.
If G is fixed, the ramaining 4 letters
can be permuted in \(\frac { 4! }{ 2! } \) = 12 ways.
∴ the 25^th word is NAAGG.
∴ the 26^th word is NAGAG
c. No. of selections = ⁴C₃ x ⁷C₂ + ⁴C₄ + ⁷C₄ = 4 x 21 + 1 x 7 = 84 + 7 = 91

Answer 21.

Answer 22.
a. (± c ,0) = (±√2 , 0) ; (± a, 0) = (± 4, 0)
c² = a² – b² ⇒ b² = a²– c² = 16 – 12 = 4
Equation of the ellipse is \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 4 } \) = 1
b.. (x – 0)²+(y – k)²= r² ⇒ x² + y² – 2ky + k² – r² = 0
c. The co-ordinates of the points are :
A(-4,0), B(4,0) and C(0, -2).
Equation of the circle is
x² + y² + 2gx + 2fy + c = 0 ………….. (1)
(1) passes through A(-4, 0)
: 16 + 0 – 8g + c = 0 …………… (2)
(1) passes through B (4, 0)
: 16 + 0 + 8g + c = 0…………. (3)
passes through C (0, -2)
: 4 + 0-4f + c = 0 ……………… (4)
(2) + (3) ⇒ 32+2c=0
⇒ 2c = -32
⇒ c = -16 when c= -16
⇒ 16 + 0-8g – 16 = 0
⇒- 8g=0 ⇒ g=0
when c=-16 ⇒ 4 + 0-4f – 16 = 0
⇒- 4f = 12 ⇒ f = -3
Equation of the circle is
x² + y² + 2 (0) x + 2(-3) y+ – 16 = 0
⇒ x² + y²-6y – 16 = 0

Answer 23.

Answer 24.
a. Sample space = {H, TH, TTH, TTTH, …………………… }
b. P(A’∩B’)= 1-P(A ∪ B)=1 -[P(A) + P(B)-P(A ∩ B)]
=1 – [0.54 + 0.69-0.35] =1-0.88=0.12
c. i. P(all cards are diamond)

Plus One Maths Previous Year Question Papers and Answers