## Kerala Plus One Maths Improvement Question Paper 2018

Time Allowed: 2 1/2 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 get familiar with the questions and to plan your answers.
- Read instructions carefully .
- Read questions carefully before you answering.
- 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.

Prove that

Question 4.

Question 5.

Find the polar from of the complex number

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

a. Find the domain of f(x).

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 origin

Question 15.

a. Find the derivative of f(x) = x Sin x with respect to x.

b. Find the derivative of the function y = √x with respect to 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.

find the valueof x and y.

b. Consider the function f (x) = |x| = 3, draw the graph of f (x)

c. Write the domain and range of (x)

Question 19.

a. Find the value of Sin (75°)

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

c. Find the length of chord AB

Question 20.

a. Find the number 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 x^{5 }y^{7}in 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 ellipse. Find the equation of a circle.

Question 23.

Consider the following table

a. Find the arithmatic 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.

∴ n = 10

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.

a. Let the required line l is perpendicular bisector of the line joining A(0,0) and B (-3, 4).

Since l is perpendicular bisector of AB, it passes through the midpoint of AB.

Answer 14.

Answer 15.

Answer 16.

a. Since the line joining A and Bis divided by the YZ plane, x = 0

∴ the points divides the YZ plane in the ratio 1 : 2 externally.

b. R divides AB in the ratio 1:2 externally,

Since m:n = – z_{1} : z_{2} = -10 : -8 = 5:4

XY plane divides the line segment internally.

Answer 17.

a. If n2 is not even, then the integer n is not even.

b. Let us assu,e 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,

7b^{2} = a^{2} ⇒ 7 divides a.

∴ Let a = lk

∴a^{2}= 49k^{2}* ⇒ *7b* ^{2}= 49*k

^{2}⇒ b

^{2 }= 7k

^{2 }⇒7 divides b.

i.e., 7 divides both a and b, which is contradition to our assumption that a and b have no common factor. our supposition is wrong.

∴ √7 is an irrational number.

Answer 18.

Answer 19.

a. sin 75 = sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30

=1.05 units

The radius of the circle is not given

properly. Type mismatching happened,

c. OA = OB = 2 units.

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

= ^{4}C_{3 }x ^{7}C_{2 }+ ^{4}C_{4 }+ ^{7}C_{4} = 4 x 21 + 1 x 7 = 84 + 7 = 91

Answer 21.

Answer 22.

a. (± c ,0) = (±√2 , 0) ; (± a, 0) = (± 4, 0)

c^{2 }= a^{2 }– b^{2} ⇒ b^{2} = a^{2}– c^{2 }= 16 – 12 = 4

Equation of the ellipse is \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 4 } \) = 1

b.. (x – 0)^{2}+(y – k)^{2}= r^{2} ⇒ x^{2 }+ y^{2 }– 2ky + k^{2 }– r^{2 }= 0

c. The co-ordinates of the points are :

A(-4,0), B(4,0) and C(0, -2).

Equation of the circle is

x^{2 }+ y^{2 }+ 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

x^{2 }+ y^{2 }+ 2 (0) x + 2(-3) y+ – 16 = 0

⇒ x^{2} + y^{2}-6y – 16 = 0

Answer 23.

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)