Plus One Maths Model Question Papers Paper 4 are part of Plus One Maths Previous Year Question Papers and Answers. Here we have given Plus One Maths Model Question Papers Paper 4.

Board |
SCERT |

Class |
Plus One |

Subject |
Maths |

Category |
Plus One Previous Year Question Papers |

## Plus One Maths Model Question Papers 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.

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