Kerala Plus One Maths Board Model Paper 2021 with Answers

Reviewing Kerala Syllabus Plus One Maths Previous Year Question Papers and Answers Pdf Board Model Paper 2021 helps in understanding answer patterns.

Kerala Plus One Maths Board Model Paper 2021 with Answers

Time: 2 Hours
Total Scores: 60

Answer any 6 questions from 1 to 12. Each carries 3 scores. (6 × 3 = 18)

Question 1.
Let A = {x : x is a prime number less than 10}
(i) Write A in the roster form. (2)
(ii) Number of subsets of A = ____________ (1)
(a) 16
(b) 8
(c) 32
(d) 4
Answer:
(i) A = {2, 3, 5, 7}
(ii) 24 = 16

Question 2.
If A and B are two sets such that n(A) = 100, n(B) = 150 and n(A ∩ B) = 50. Find the following values:
(i) n(A ∪ B) (2)
(ii) n(A – B) (1)
Answer:
(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 100 + 150 – 50
= 200
(ii) n(A – B) = n(A) – n(A ∩ B)
= 100 – 50
= 50

Question 3.
Consider the statement p(n): 7ⁿ – 3ⁿ is divisible by 4, n ∈ N
(i) Check whether P(1) is true. (1)
(ii) If P(k) is true. Prove that P(k + 1) is also true. (2)
Answer:
P(n): 7ⁿ – 3ⁿ is divisible by 4
(i) P(1): 7¹ – 3¹ = 7 – 3 = 4, divisible by 4
∴ P(1) is true.
(ii) P(k) = 7^k – 3^k is divisible by 4
ie. 7^k – 3^k = 4m ………(1)
P(k+1): 7^k+1 – 3^k+1
= 7^k . 7 – 3^k. 3
= 7^k . 7 – 7^k. 3 + 7^k . 3 – 3^k. 3
= 7^k (7 – 3) + 3(7^k – 3^k)
= 7^k . 4 + 3 × 4m
= 4(7^k + 3m), divisible by 4
∴ P(k+1) is true whenever P(k) is true.

Question 4.
Using Binomial Theorem expand \(\left(x^2+\frac{3}{x}\right)^5\), x ≠ 0. (3)
Answer:

Question 5.
Finds the middle term in the expansion of \(\left(\frac{x}{3}+9 y\right)^{10}\). (3)
Answer:

Question 6.
In an A.P. 8th term is 16 and 16th term is 48.
(i) Find the common difference of the A.P. (1)
(ii) Find the 25th term of the A.P. (2)
Answer:
(i) a + 7d = 16 …………(1)
a + 15d = 48 …………(2)
(2) – (1) ⇒ 8d = 32
⇒ d = 4
(ii) a + 28 = 16
⇒ a = -12
a₂₅ = a + 24d
= -12 + 24 × 4
= -12 + 96
= 84

Question 7.
(i) The slope of the line joining the points (-2, 6) and (4, 8) is ____________ (1)
(a) 3
(b) \(\frac{1}{3}\)
(c) -3
(d) \(-\frac{1}{3}\)
(ii) If the above line is perpendicular to the line-joining the points (8, 12) and (x, 24) then find the value of ‘x’. (2)
Answer:

Question 8.
(i) A point (x, y, z) is on the XY-plane, then which of the following is always true:
(a) x = 0
(b) y = 0
(c) z = 0
(d) z ≠ 0
(ii) Find the distance between the points A(-2, 3, 5) and B(1, 2, 3). (2)
Answer:
(i) (c) z = 0
(ii) A (-2, 3, 5), B(1, 2, 3)

Question 9.
Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the YZ plane. (3)
Answer:
Let the YZ plane divide in the ratio k : 1
x = 0

∴ YZ plane divides externally in the ratio 2 : 3.

Question 10.
Consider the parabola y² = 16x.
(i) Length of the jatus rectum of the parabola is ____________ (1)
(a) 4
(b) 16
(c) 8
(d) 32
(ii) Write the co-ordinates of the focus and equation of the directrix of the above parabola. (2)
Answer:
(i) (b) 16
(ii) focus (a, 0) = (4, 0)
Directrix: x = -a
⇒ x = -4
⇒ x + 4 = 0

Question 11.
Evaluate \(\lim _{x \rightarrow 2}\left(\frac{x^5-32}{x^3-8}\right)\). (3)
Answer:

Question 12.
Consider the statement ‘p: If a number is divisible by 10, then it is divisible by 5’. Write the
(i) Negation of the statement (1)
(ii) Converse of the statement (1)
(iii) Contrapositive of the statement (1)
Answer:
(i) Negation: “It is false that if a number is divisible by 10, then it is divisible by 5”.
(ii) Converse: “If a number is divisible by 5, then it is divisible by 10”.
(iii) Contra positive: “If a number is not divisible by 5, then it is not divisible by 10”.

Answer any 6 questions from 13 to 24. Each carries 4 scores. (6 × 4 = 24)

Question 13.
U = {1, 2, 3, 4, 5, 6} be the Universal set and A = {1, 2, 3, 4}, B = {3, 4, 5} are two subsets of U.
(i) Write the sets A’ and B’. (2)
(ii) Verify that A – B = A ∩ B’. (2)
Answer:
(i) A’ = {5, 6}
B’ = {1, 2, 6}
(ii) A – B = {1, 2}
A ∩ B’ = {1, 2, 3, 4} ∩ {1, 2, 6} = {1, 2}
∴ A – B = A ∩ B’

Question 14.
Let A = {-1, 0, 1, 2, 3}, B = {1, 2, 3, 4, 5} be two sets. R be a relation defined from A to B by R = {(x, y) : x + y = 3, x ∈ A, y ∈ B}
(i) Write R in the Roster form. (2)
(ii) Write the domain and range of R. (2)
Answer:
(i) R = {(-1, 4), (0, 3), (1, 2), (2, 1)}
(ii) Domain = {-1, 0, 1, 2}
Range = {4, 3, 2, 1}

Question 15.
(i) Choose one of the possible values of f: R → R, f(x) = sin x from the following values: (1)
(a) \(-\frac{1}{2}\)
(b) 3
(c) -3
(d) 2
(ii) If sin x = \(\frac{3}{5}\), x in the second quadrant. Find the values of cos x and tan x. (3)
Answer:
(i) (a)

(ii) cos x = \(\frac{-4}{5}\)
tan x = \(\frac{-3}{4}\)

Question 16.
Using the principle of Mathematical Induction, prove that for all n ∈ N. (4)
1 + 2 + 3 + …… + n = \(\frac{n(n+1)}{2}\)
Answer:
p(n): 1 + 2 + 3 + …….. + n = \(\frac{n(n+1)}{2}\)
p(1): 1 = \(\frac{1(1+1)}{2}\) = 1
∴ p(1) is true.
Let p(k) be true
(i.e) p(k): 1 + 2 + 3 +….+ k = \(\frac{k(k+1)}{2}\)
We have to prove that p(k + 1) is true
(i.e) p(k + 1): 1 + 2 + 3 +….. + (k + 1) = \(\frac{(k+1)(k+2)}{2}\)
LHS = 1 + 2 + 3 +…..+ k + (k + 1)
= \(\frac{k(k+1)}{2}\) + (k + 1)
= (k + 1) [\(\frac{k}{2}\) + 1]
= \(\frac{(k+1)(k+2)}{2}\)
= RHS
∴ p(k + 1) is true
Hence by P.M.I, p(n) is true for all n ∈ N.

Question 17.
(i) If i = \(\sqrt{-1}\), then i⁴ = ____________ (1)
(i) 1
(b) -1
(c) i
(d) -i
(ii) Write z = \(\frac{2+3 i}{1+2 i}\) in the form a + ib. (3)
Answer:
(i) (a)

Question 18.
(i) Let z = 1 + i, then |z| = ____________ (1)
(a) 1
(b) √2
(c) 2
(d) 0
(ii) Represent the above complex number z in the polar form. (3)
Answer:
(i) (b) √2
(ii) r = |z| = √2
⇒ r cos θ = 1, r sin θ = 1
⇒ cos θ = \(\frac{1}{\sqrt{2}}\), sin θ = \(\frac{1}{\sqrt{2}}\)
⇒ θ = \(\frac{\pi}{4}\)
∴ 1 + i = \(\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)

Question 19.
How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, and 5, assuming that
(i) Repetition of the digits is allowed. (2)
(ii) Repetition of the digits is not allowed. (2)
Answer:
(i) If digits can be repeated, there are 5 ways each for selecting 4 digits.
∴ By F.P.C. required no. of ways = 5 × 5 × 5 × 5 = 5⁴
(ii) If digits cannot be repeated, required no. of ways = 5 × 4 × 3 × 2 = 120

Question 20.
(i) If ²⁰C₁₂ = ²⁰C_r then one of the possible value of r = ____________ (1)
(a) 9
(b) 8
(c) 11
(d) 10
(ii) A bag contains 7 red balls and 5 white balls. Determine the number of ways in which 3 red balls and 2 white balls can be selected. (3)
Answer:
(i) (b) 8
(ii) Required no. of ways = ⁷C₃ × ⁵C₂
= 35 × 10
= 350

Question 21.
Using method of contradiction prove that ‘√2 is irrational’. (4)
Answer:
Assume √2 is rational
√2 =\(\frac{a}{b}\), where a and b are coprime.
a = √2b
a² = 2b² ………(1)
∴ 2 divides a²
Implies 2 divides a
∴ a = 2c
Substituting in (1)
(2c)² = 2b²
4c² = 2b²
2c² = b²
(i.e) 2 divides b²
Hence 2 divides b which is a contradiction to the fact that a and b are coprime.
Hence our assumption is wrong.
∴ √2 is irrational.

Question 22.
A line cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).
(i) Find the equation of the line. (3)
(ii) Find the slope of the line obtained in (i). (1)
Answer:
(i) Let the equation be \(\frac{x}{a}+\frac{y}{b}\) = 1
Since a = b, \(\frac{x}{a}+\frac{y}{b}\) = 1
(i.e) x + y = a ………..(1)
Since (1) passes through (2, 3),
2 + 3 = a
(i.e) a = 5
∴ Equation of line is x + y = 5
(ii) Slope of line = \(\frac{-1}{1}\) = -1

Question 23.
(i) Write the equation of the ellipse with vertices (±5, 0) and foci (±4, 0). (3)
(ii) Find the length of latus rectum of the above ellipse. (1)
Answer:

Question 24.
Find \(\frac{d y}{d x}\) if y = \(\frac{x^2+1}{x+1}\). (4)
Answer:

Answer any 3 questions from 25 to 30. Each carries 6 scores. (3 × 6 = 18)

Question 25.
Consider the function f: R → R, f(x) = |x – 2|.
(i) The value of f(0) = ____________ (1)
(a) 0
(b) 2
(c) -2
(d) 1
(ii) Draw the graph of the function f. (4)
(iii) Hence write the range of the function f. (1)
Answer:
(i) (b)

(iii) Range = [0, ∞)

Question 26.
(i) Find the value of sin 75°. (3)
(ii) Prove that \(\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}\) = tan 4x. (3)
Answer:

Question 27.
Solve the following system of inequalities graphically: (6)
x + 2y ≤ 8
2x + y ≥ 8
x > 0, y ≥ 0
Answer:

Question 28.
(i) If the numbers \(\frac{1}{4}\), x, 4 are three consecutive terms of a G.P. Find the value of x. (2)
(ii) Find the sum to n terms of the series 5 + 55 + 555+ ……. (4)
Answer:

Question 29.
Find the variance and standard deviation for the following data:

xi	3	8	13	18	23
fi	7	10	15	10	6

Answer:

Question 30.
In a class of 60 students, 30 opted for NCC, 32 opted for NSS, and 24 opted for both NCC and NSS. One student is selected at random from the class. Find the probability that a student opted for.
(i) NCC (1)
(ii) NCC or NSS (3)
(iii) Neither NCC nor NSS (2)
Answer:
A → Set of students who opted for NCC
B → Set of students who opted for NSS
n(A) = 20, n(B) = 32, n(A ∩ B) = 24