Kerala Plus One Maths Question Paper June 2022 with Answers

Reviewing Kerala Syllabus Plus One Maths Previous Year Question Papers and Answers Pdf June 2022 helps in understanding answer patterns.

Kerala Plus One Maths Previous Year Question Paper June 2022

Time: 2 Hours
Total Scores: 60

Answer any 6 questions from 1 to 8. Each carries 3 scores. (6 × 3 = 18)

Question 1.
(i) If A is any set, then A ∩ A’ = _______________________ (1)
(a) A
(b) φ
(c) A’
(d) U
(ii) A = {x: x is a natural number less than 3}
(a) Write A in roster form. (1)
(b) Write all the subsets of A. (1)
Answer:
(i) (b) φ
(ii) (a) A = {1, 2}
(b) {1}, {2}, {1, 2}, φ

Question 2.
(i) 25° = ______________ radian. (1)
(ii) If tan x = \(\frac{5}{12}\), x lies in 3rd quadrant, then find the value of sin x and cos x. (2)
Answer:
(i) 25° = 25 × \(\frac{\pi}{180}\) = \(\frac{5 \pi}{36}\) radian

Question 3.
(i) For what values of x, the numbers \(\frac{4}{3}\), x, \(\frac{3}{4}\) are in Geometric progression? (1)
(ii) Find the nth term of the Geometric Progression: (2)
√3, 3, 3√3,….
Answer:

Question 4.
Find the angle between the lines y – √3x – 5 = 0 and √3y – x + 6 = 0. (3)
Answer:

Question 5.
(i) Focus of the parabola y² = 8x is ______________ (1)
(a) (4, 0)
(b) (0, 2)
(c) (0, -4)
(d) (2, 0)
(ii) Find the centre and radius of the circle x² + y² + 6x – 4y – 3 = 0. (2)
Answer:
(i) (d) (2, 0)
(ii) Given circle is of the form x² + y² + 2gx + 2fy + c = 0
2g = 6, 2f = -4
⇒ g = 3, f = 2, c = -3
Centre = (-g, -f) = (-3, 2)
Radius = \(\sqrt{g^2+f^2-c}\)
= \(\sqrt{9+4+3}\)
= 4

Question 6.
Find the ratio in which the yz-plane divides the line segment formed by joining the points (-2, 4, 7) and (3, -5, 8). (3)
Answer:
Let the yz plane divides the line segment joining A(-2, 4, 7) and B(3, -5, 8) in the ratio k : 1

∴ yz plane divides AB internally in the ratio 2 : 3

Question 7.
Evaluate the following limits:
(i) \(\lim _{x \rightarrow 2} x^2-4\) (1)
(ii) \(\lim _{x \rightarrow 2} \frac{x^2-4}{x-2}\) (1)
(iii) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{x}\) (1)
Answer:

Question 8.
Prove by the method of contradiction that √3 is irrational. (3)
Answer:
Assume that √3 is rational
∴ √3 = \(\frac{a}{b}\) where a and b are coprime.
∴ a = √3b
a² = 3b² ________(1)
i.e 3 divides a²
∴ 3 divides a
∴ There exists an integer c such that a = 3c
∴ a² = 9c²
Substituting this in equation (1),
9c² = 3b²
b² = 3c²
∴ 3 divides b²
hence 3 divides b.
∴ 3 is a common factor of a and b.
This is a contradiction to the fact that a and b are coprime.
∴ Our assumption is wrong.
Hence √3 is irrational.

Answer any 6 questions from 9 to 17. Each carries 4 scores. (6 × 4 = 24)

Question 9.
(i) Which one of the following is equal to {x: x ∈ R, -4 < x ≤ 5}? (1)
(a) (-4, 5]
(b) (-4, 5)
(c) [-4, -5]
(d) [-4, 5)
(ii) If U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 3, 4, 6}, B = {3, 4, 5}, then verify that (A ∪ B)’ = A’ ∩ B’. (3)
Answer:
(i) (a) (-4, 5]
(ii) A ∪ B = {2, 3, 4, 5, 6}
(A ∪ B)’ = {1, 7}
A’ = {1, 5, 7}
B’ = {1, 2, 6, 7}
A’ ∩ B’ = {1, 7}
∴ (A ∪ B)’ = A’ ∩ B’

Question 10.
(i) Let A = {1, 2, 3, 4, 5, 6, 7, 8}. A relation R from A to A is defined by R = {(x, y): 2x – y = 0 where x, y ∈ A}. Write down its domain and range. (2)
(ii) Draw the graph of the function f: R → R defined by f(x) = |x| + 1. (2)
Answer:
(i) R = {(1, 2), (2, 4), (3, 6), (4, 8)}
Domain = {2, 3, 4}
Range = {2, 4, 6, 8}
(ii) f(x) = |x| + 1

Question 11.
Consider the statement:
P(n): \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{2^n}=1-\frac{1}{2^n}\)
(i) Show that P(1) is true. (1)
(ii) Prove by principle of Mathematical Induction that P(n) is true for all n ∈ N. (3)
Answer:


∴ P(k + 1) is true.
Hence by Principle of Mathematical Induction, P(n) is true for all n ∈ N.

Question 12.
(i) If ⁿC₉ = ⁿC₈, then n = ______________ (1)
(ii) ⁿP_r = ______________ (1)
(iii) Find the number of permutations using all the letters of the word “MATHEMATICS”. (2)
Answer:
(i) n = 9 + 8 = 17
(ii) ⁿP_r = \(\frac{n!}{(n-r)!}\)
(iii) In the given word MATHEMATICS
Total letters = 11
M → 2
A → 2
T → 2
H → 1
E → 1
I → 1
C → 1
S → 1
∴ Required number of words = \(\frac{11!}{2!2!2!}\)
= \(\frac{39916800}{8}\)
= 4989600

Question 13.
Consider the expansion of (x + 9y)¹⁰. Find its
(i) number of terms (1)
(ii) general term (2)
(iii) 5th term (1)
Answer:
(i) 11

Question 14.
Find the sum of the sequence 8, 88, 888,……… to n terms. (4)
Answer:

Question 15.
(i) Find the slope of the line x – 7y + 5 = 0. (1)
(ii) Find the equation of the line perpendicular to the above line having x-intercept 3. (3)
Answer:
(i) Slope = \(\frac{-A}{B}=\frac{-1}{-7}=\frac{1}{7}\)
(ii) Slope of required line = -7
(x₁, y₁) = (3, 0)
Equation is y – y₁ = m(x – x₁)
⇒ y – 0 = -7(x – 3)
⇒ y = -7x + 21
⇒ 7x + y – 21 = 0

Question 16.
Find the coordinates of the foci, vertices, the length of the major axis and the length of the latus rectum of the ellipse \(\frac{x^2}{36}+\frac{y^2}{16}\) = 1 (4)
Answer:
Given \(\frac{x^2}{36}+\frac{y^2}{16}\) = 1
a = 6, b = 4
c = \(\sqrt{a^2-b^2}=\sqrt{20}\)
Foci = (±c, 0) = (±√20, 0)
Vertices = (±a, 0) = (±6, 0)
Length of major axis = 2a = 12
Length of Latus Rectum = \(\frac{2 b^2}{a}=\frac{2 \times 16}{6}=\frac{16}{3}\)

Question 17.
If A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35, then find
(i) P(A or B) (2)
(ii) P(not A and not B) (2)
Answer:
(i) P(A or B) = P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
= 0.54 + 0.69 – 0.35
= 0.88
(ii) P(not A and not B) = P(A’ ∩ B’)
= P(A ∪ B)’
= 1 – P(A ∪ B)
= 1 – 0.88
= 0.12

Answer any 3 questions from 18 to 22. Each carries 6 scores. (3 × 6 = 18)

Question 18.
(i) Prove that \(\frac{\cos 9 x-\cos 5 x}{\sin 17 x-\sin 3 x}=\frac{-\sin 2 x}{\cos 10 x}\) (3)
(ii) Find the principal and general solution of the equation sin x = \(\frac{-\sqrt{3}}{2}\). (3)
Answer:

Question 19.
(i) Represent the complex number Z = -1 + i√3 in the polar form. (3)
(ii) Solve the equation √5x² + x + √5 = 0 (3)
Answer:

Question 20.
(i) Solve the inequality \(\frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}\). (2)
(ii) Solve the following inequalities graphically: (4)
x + 3y ≤ 9
2x + y ≤ 12
x ≥ 0; y ≥ 0
Answer:
(i) \(\frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}\)
9(x – 2) ≤ 25(2 – x)
9x – 18 ≤ 50 – 25x
9x + 25x ≤ 50 + 18
34x ≤ 68
x ≤ 2
∴ Solution is (-∞, 2]
(ii) x + 3y = 9, 2x + y = 12

Question 21.
(i) Find the derivative of cos x using first principle. (3)
(ii) Find the derivative of \(\frac{x^2}{3 x-1}\) (3)
Answer:
(i) f(x) = cos x
f(x + h) = cos(x + h)

Question 22.
Consider the following table:

Classes	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Frequency	6	8	14	16	4	2

(i) Find the mean. (2)
(ii) Find the variance. (3)
(iii) Find the standard deviation. (1)
Answer:


Variance = (Standard Deviation)² = 160