## Kerala Plus One Maths Model Question Paper 4

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.

Questions 1 to 7 carry 3 scor each. Answer any 6.

Question 1.

Consider the Venn-diagram given below.

Question 2.

For any triangle ABC, prove that

Question 3.

Consider the complex number z = 3 + 4i

a. Write the conjugate of z.

b. Verify that z z =|z|^{2}

Question 4.

a. Solve the inequality

b. Represent the solution in real line

Question 5.

4 cards are drawn from a wellshuffled pack of 52 cards.

a. In how many ways can this be done?

b. In how many ways can this be done if all 4 cards are of the same colour?

Question 6.

Consider the equation of the ellipse 9x^{2}+ 4y^{2} = 36. Find

a. Focii

b. Eccentricity

c. Length of latus rectum

Question 7.

Questions from 8 to 17 carry 4 scor each. Answer any 8.

Question 8.

Consider A = {x : x is an integer, 0 < x ≤ 3}

a. Write A in roster form.

b. Write the power set of A.

c. The number of proper subsets of A =

d. Write the number of possible relations from A to A.

Question 9.

Consider the statement

a. Show that P(1) is true.

b. Prove that P(n) is true for all n ∈ N using principle of mathematical induction.

Question 10.

Consider the complex number, z =

\(\frac { 1 + 3i }{ 1 + 2 i}\)

a. Write z in the form a + ib.

b. Write z in polar form

Question 11.

Solve the following inequalities graphically

Question 12.

a. How many 3 letter words with or without meaning can be formed using 26 letters in English alphabet, if no letter is repeated?

b. Find the number of permutations of letters of the word MATHEMATICS

c. How many of them begin with the letter C?

Question 13.

Consider the figure given below. A (3, 0) and B (0, 2) are two points on axes. The line OP is perpendicular to AB.

a. Find the slope of OP.

b. Find the co-ordinates of the point P.

Question 14.

Equation of the parabola given in the figures is y= 8x.

a. Find the focus and length of latus rectum of the parabola.

b. The latus rectum of the parabola is a chord to the circle centered at origin as shown in the figure. Find the equation of the circle.

Question 15.

Let L be the line x – 2y+3 = 0.

a. Find the equation of the line L_{1} which is parallel to L and passing through (1, -2).

b. Find the distance between L and L_{2}.

c. Write the equation of another line L_{2} which is parallel to L, such that the distance from origin to L and L_{2} are the same.

Question 16.

Consider the points A (3, 2, 01).

a. Write the octant in which A belongs to

b. If B (1, 2, 3) is another point in space, find distance between A and B.

c. Find the coordinates of the point R which divides AB in the ratio 1 : 2 internally.

Question 17.

a. Write the contrapositive of the statement:

P : If a triangle is equilateral, then it is isosceles.

b. Prove by the method of contradiction ‘ √3 is irrational’

Questions from 18 to 24 carry 6 score each. Answer any 5.

Question 18.

a. If A = {a, b} write A x A x A

b. If R = {(x,x^{3}) : x is a prime number, less than 10}. Write R in roster form.

c. Find the domain and range of the function f (x) = 2+ √x-1

Question 19.

a. The minute hand of a watch is 3 cm long. How far does its tip move in 40 minutes? (Use π = 3.14).

b. Solve the trigonometric equation sin 2x-sin 4x + sin 6x = 0.

Question 20.

a. Find the sum of all 3 digit numbers which are multiples of 5.

b. How many terms of the GP 3, 3^{2}, 3^{3}, …. are needed to give the sum 120?

c. Find the sum of first n terms of the series whose n* term is n(n + 3).

Question 21.

a. Expand using binomial theorem,

b. Find (a+b)4 – (a-b)^{4}

c. Hence find (√3 + √2)4 -(√3-√2)4

Question 22.

a. Find the derivative of the function y = 1/x from first principles.

b. Differentiate f(x) = ,\(\frac { cos x }{ 1 + sinx }\)

with respect to x.

Question 23.

Calculate the mean deviation about median for the following data.

Question 24.

Consider a bag containing 3 red balls R_{1}, R_{2}, R_{3} and 2 black balls B_{1}, B_{2}, which are identical. 2 balls are drawn simultaneously at random from the bag.

a. Write the sample space of the random experiment.

b. Write the event

A : Both balls are red B : One is red and one is black

c. Show that A and B are mutually exclusive

d. Find P(A) and P(B)

Answers

Answer 1.

a. A’= {5, 6, 7, 8, 9}

B’= {1, 2, 7, 8, 9}

(A ∩ B)’ = {l,2,5,6, 7, 8, 9}

b. LHS = (A ∩ B)’ = {1,2, 5, 6, 7, 8,9}

RHS=A’∪B’= {1,2, 5, 6, 7, 8, 9}

∴(A∩B)’= A’∪B’

Answer 2.

Answer 3.

a. conjugate of z = 3 + 4i = 3 – 4i

b. z *z̄* = (3+4i) (3-4i) = 9 – 16i^{2} = 9 – (-16) = 25

\({ \left| z \right| }^{ 2 }={ \left( \sqrt { { 3 }^{ 2 }+{ 4 }^{ 2 } } \right) }^{ 2 }={ \left( \sqrt { 25 } \right) }^{ 2 }=25\)

\(\Rightarrow z=\bar { z } ={ \left| z \right| }^{ 2 }\)

Answer 4.

Answer 5.

a. No. of ways = ^{52}C_{4} = 270725

b. No. of ways of all cards of the same colour = ^{26}C_{4}+^{26}C_{4} = 2 x ^{26}C_{4} = 29900

Answer 6.

The standard form of the ellipse is \({ x }^{ 2 }+{ y }^{ 2 }\) = 1, is an ellipse whose major axis is on the y-axis.

a = 3, b = 2

c = \(\sqrt { { a }^{ 2 }-{ b }^{ 2 } } \) = \(\sqrt { 9-4 } \) = √5

a. Focii = (0, ± c) = (0, +√5)

b. Eccentricity = c/a = √5/3

c. Length of latus rectum = \(\frac { { 2b }^{ 2 } }{ a } =\frac { { 2\times 2 }^{ 2 } }{ 3 } =\frac { 8 }{ 3 } \)

Answer 7.

Answer 8.

a. A = {1, 2, 3}

b. Power set of A = P(A)={ 1,2,3}, {1,2}, {1,3}, {2,3}, {1}, {2}, {3}, φ}

c. No. of proper subject of A = 2^{n}-1 = 2^{3}-1 = 7

d. No. of relations from A to A = 2^{mxm}=2^{3×3} = 512

Answer 9.

Answer 10.

Answer 11.

Answer 12.

a. No. of words = ^{26}P_{3} = 15600

b. The word MATHEMATICS has

c. If the letter C is fixed first, the remaining 10 letters can be permuted as

Answer 13.

a. Slope of OP

b. Equation of OP is

Answer 14.

Equation of parabola is y^{2} = 8x ⇒ a = 2

a. Focus = (a, 0) = (2, 0)

Length of latus rectum = 4a = 4 x 2 = 8

b. Coordinates of A is (a, -2a) = (2, -4) and that of B iss (1, 2a) = (2, 4)

∴ Radius of the circle

Answer 15.

a. Equation of L is x-2y + 3 = 0

its given that distance from origin to L and L_{2} are same. We have,

Answer 16.

Answer 17.

a. Contrapositive statement: If a triangle is not isosceles, then it is not equilateral.

b. Let us assume that √3 be rational ∴√3 = a/b, where a and b are co-prime, i.e., a and b have no common factors othe than 1.

3b^{2} = a^{2} ⇒ 3 divides a.

∴ there exists an integer ‘k’ such that a = 3k

∴ a^{2 }= 9k^{2} ⇒ 3b^{2 }= 9k^{2 }⇒b^{2 }= 3k^{2 }⇒3divides b.

i.e., 3 divides both a and b, which is contradiction to our assumption that a and b have no common factor.

∴ our supposition is wrong.

Answer 18.

a. Ax Ax A={(a, a, a), (a, a, b), (a, b, a),

(a, b, b), (b, a, a), (b, a, b), (b, b, a), (b, b, b)}

b. R = {(2, 8), (3, 27), (5, 125), (7, 343)}

c. Domain=[1, ∞ ),{x/x ≥ 1}

Range = [2,∞), {y/y ≥ 2}

Answer 19.

a. The required distance moved

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.