Kerala Plus One Maths Board Model Paper 2022 with Answers

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

Kerala Plus One Maths Board Model Paper 2022 with Answers

Time: 2 Hours
Total Scores: 60

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

Question 1.
(a) If a set A has 2 elements, then the number of subsets of A is ___________________ (1)
(i) 2
(ii) 4
(iii) 6
(iv) 8
(b) Write all subsets of {1, 2}. (1)
(c) Write the interval (6, 12] in set-builder form. (1)
Answer:
(a) (ii) 4
(b) {1}, {2}, {1, 2},
(c) {x : x ∈ R,6 < x ≤ 12}

Question 2.
(a) \(\frac{\pi}{4}\) radian = _____________ degree. (1)
(b) If sin x = \(\frac{3}{5}\), x lies in the second quadrant, find the values of cos x and tan x. (2)
Answer:
(a) 45

Question 3.
(a) Write the first four terms of the sequence whose nth term is a_n = 5n + 1. (1)
(b) Find the sum of the first n terms of the above sequence. (2)
Answer:
(a) a_n = 5n + 1
a₁ = -5 + 1 = 6
a₂ = 10 + 1 = 11
a₃ = 15 + 1 = 16
a₄ = 20 + 1 = 21
∴ First four terms are 6, 11, 16, 21
(b) a = 6, d = 5

Question 4.
(a) Find the slope of the line passing through the points (2, 1) and (4, 5). (1)
(b) Find the value of x for which the points (x, -1), (2, 1) and (4, 5) are collinear. (2)
Answer:
(a) Slope m = \(\frac{5-1}{4-2}=\frac{4}{2}\) = 2
(b) Given A(x, -1), B(2, 1), C(4, 5)
Slope of AB = Slope of BC

Question 5.
Find the equation of the circles with radius 5 whose centres lie on the x-axis and passing through the point (2, 3). (3)
Answer:

Let C(h, 0) be the centre.
∴ CP = 5
⇒ \(\sqrt{(h-2)^2+(3-0)^2}\) = 5
⇒ \(\sqrt{(h-2)^2+9}\) = 5
⇒ (h – 2)² + 9 = 25
⇒ (h – 2)² = 16
⇒ h – 2 = ±4
⇒ h – 2 = 4 or h – 2 = -4
⇒ h = 6 or h = -2
If the centre is (6, 0), the equation of a circle is (x – 6)² + y² = 25
If the centre is (-2, 0), the equation of a circle is (x + 2)² + y² = 25

Question 6.
(a) Coordinate planes divide the space into octants. (1)
(b) Find the distance between the points (-1, 3, -4) and (1, -3, 4). (2)
Answer:
(a) 8
(b) Given A(-1, 3, -4), B(1, -3, 4)
AB = \(\sqrt{(1+1)^2+(-3-3)^2+(4+4)^2}\)
= \(\sqrt{4+36+64}\)
= \(\sqrt{104}\)

Question 7.
Evaluate:
(a) \({Lt}_{x \rightarrow 4} \frac{4 x+3}{x-2}\) (1)
(b) \({Lt}_{x \rightarrow 1} \frac{x^3-1}{x^2-1}\) (2)
Answer:

Question 8.
(a) Write the negation of the statement ‘√7 is rational’. (1)
(b) Write the contrapositive and converse of the statement ‘If a number n is even, then n² is even’. (2)
Answer:
(a) It is not the case that √7 is rational.
(b) Contra positive: ‘If n² is not even then n is not even’.
Converse: ‘If n² is even then n is even’.

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

Question 9.
(a) If A and B are two sets such that A ⊂ B, then A ∪ B = _____________ (1)
(b) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}, find:
(i) A’ and B’. (1)
(ii) A ∪ B (1)
(iii) Verify that (A ∪ B)’ = A’ ∩ B’ (1)
Answer:
(a) B
(b) (i) A’ = {1, 3, 5, 7, 9}
B’ = {1, 4, 6, 8, 9}
(ii) A ∪ B = {2, 3, 4, 5, 6, 7, 8}
(iii) (A ∪ B)’ = {1, 9}
A’ ∩ B’ = {1, 9}
∴ (A ∪ B)’ = A’ ∩ B’

Question 10.
(a) Let A = {1, 2, 3, 4,…, 14}, define a relation R from A to A by R = {(x, y) : y = 3x, wherex, y ∈ A}. Write R in roster form. Write down the domain and range of R. (3)
(b) A function f is defined by f(x) = 2x – 5. Find the value of f(0). (1)
Answer:
(a) R = {(1, 3), (2, 6), (3, 9), (4, 12)}
Domain = {1, 2, 3, 4}
Range = {3, 6, 9, 12}
(b) f(x) = 2x – 5
f(0) = 2(0) – 5 = 0 – 5 = -5

Question 11.
Consider the statement
P(n): 1 + 3 + 3² + …… + 3^(n-1) = \(\frac{3^n-1}{2}\)
(a) Show that P(1) is true. (1)
(b) Prove by the principle of Mathematical Induction that P(n) is true for all n ∈ N. (3)
Answer:
(a) P(1): 1 = \(\frac{3^{\prime}-1}{2}=\frac{3-1}{2}=\frac{2}{2}\) = 1
∴ P(1) is true.
(b) Let us assume thatp(n) is true for n = k
(i.e) P(k): 1 + 3 + 3² + ……. + 3^k-1 = \(\frac{3^k-1}{2}\)
Now we have to prove that P(n) is true for n = k + 1

P(k + 1) is true.
∴ By the principle of Mathematical induction, P(n) is true for all n ∈ N.

Question 12.
(a) Evaluate \(\frac{7!}{5!}\). (1)
(b) How many 4 digit numbers can be formed using the digits 1 to 9 if repetition of digits is not allowed? (2)
(c) ¹⁷C₁₇ = __________ (1)
Answer:
(a) \(\frac{7!}{5!}=\frac{7 \times 6 \times 5!}{5!}\) = 42
(b) There are 9 ways to select units, 8 ways to select 10’s place, 7 ways to select 100’s place, and 6 ways to select 1000’s place.
∴ By fundamental principle of counting, required number of ways = 9 × 8 × 7 × 6 = 3024
(c) ¹⁷C₁₇ = 1

Question 13.
(a) The number of terms in the expansion of (a + b)⁴ is ______________ (1)
(b) Expand \(\left(x^2+\frac{3}{x}\right)^4\), x ≠ 0. (3)
Answer:
(a) 5

Question 14.
The sum of the first three terms of a Geometric Progression is \(\frac{39}{10}\) and their product is 1. Find the common ratio and the terms of the Geometric Progression. (4)
Answer:
Let the terms be \(\frac{a}{r}\), a, ar
∴ \(\frac{a}{r}\) + a + ar = \(\frac{39}{10}\) ……….(1)
\(\frac{a}{r}\) . a . ar = 1 …….(2)
⇒ a³ = 1
⇒ a = 1
Substituting this in equation (1),
\(\frac{1}{r}\) + 1 + r = \(\frac{39}{10}\)
Multiplying throughout by ’10r’
⇒ 10 + 10r + 10r² = 39r
⇒ 10r² – 29r + 10 = 0
⇒ 10r² – 25r – 4r + 10 = 0
⇒ 5r(2r – 5) – 2(2r – 5) = 0
⇒ (2r – 5)(5r – 2) = 0
⇒ r = \(\frac{5}{2}\) or r = \(\frac{2}{5}\)
If a = 1, r = \(\frac{5}{2}\) terms are \(\frac{2}{5}\), 1, \(\frac{5}{2}\)
If a = 1, r = \(\frac{2}{5}\) terms are \(\frac{5}{2}\), 1, \(\frac{2}{5}\)

Question 15.
(a) Write the equation of the x-axis. (1)
(b) Equation of a line is 3x + 2y – 12 = 0. Find its
(i) Slope (1)
(ii) x and y intercepts. (2)
Answer:
(a) y = 0
(b) (i) Slope = \(\frac{-A}{B}=\frac{-3}{2}\)
(ii) x intercept = \(\frac{-C}{A}=\frac{12}{3}\) = 4
y intercept = \(\frac{-C}{B}=\frac{12}{2}\) = 6

Question 16.
Find the coordinates of the foci. The eccentricity and the length of the latus rectum of the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}\) = 1. (4)
Answer:

Question 17.
One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be
(i) a diamond (1)
(ii) not an ace (2)
(iii) a black card (1)
Answer:

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

Question 18.
(a) Prove that \(\frac{\tan \left(\frac{\pi}{4}+x\right)}{\tan \left(\frac{\pi}{4}-x\right)}=\left(\frac{1+\tan x}{1-\tan x}\right)^2\). (3)
(b) Find the general solution for the equation cos 3x + cos x – cos 2x = 0. (3)
Answer:

(b) cos 3x + cos x – cos 2x = 0
⇒ 2 cos 2x . cos x – cos 2x = 0
⇒ cos 2x (2 cos x – 1) = 0
⇒ cos2x = 0 or 2 cos x – 1 = 0
Now, cos 2x = 0
⇒ 2x = (2n + 1)\(\frac{\pi}{2}\)
⇒ x = (2n + 1)\(\frac{\pi}{4}\); n ∈ Z
and, 2 cos x – 1 = 0
2 cos x = 1
cos x = \(\frac{1}{2}\)
cos x = cos\(\frac{\pi}{3}\)
x = 2nπ ± \(\frac{\pi}{3}\); n ∈ Z

Question 19.
(a) The value of i⁴ is _____________ (1)
(b) Find the multiplicative inverse of 1 – i in a + ib form. (2)
(c) Find the polar form of 1 – i. (3)
Answer:
(a) 1
(b) Let z = 1 – i

Question 20.
Solve the system of inequalities graphically. (6)
x + 2y ≤ 8
2x + y ≤ 8
x ≥ 0, y ≥ 0
Answer:

Question 21.
(a) Find the derivative of sin x from first principle. (3)
(b) Find the derivative of 5 sin x – 6 cos x + 7. (3)
Answer:

(b) f(x) = 5 sin x – 6 cos x + 7
f'(x) = 5 cos x + 6 sin x

Question 22.
Consider the following data:

Classes	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
Frequency	5	8	15	16	6

Find:
(i) Mean (2)
(ii) Variance and standard deviation. (4)
Answer: