Kerala Plus One Maths Question Paper Sep 2021 with Answers

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

Kerala Plus One Maths Previous Year Question Paper Sep 2021

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 natural less than 6} and B = {1, 2}
(i) Write A in roster form. (1)
(ii) Find A ∩ B. (1)
(iii) Find A – B. (1)
Answer:
(i) A = {1, 2, 3, 4, 5}
(ii) A ∩ B = {1, 2}
(iii) A – B = {3, 4, 5}

Question 2.
Find the sum of all natural numbers between 100 and 1000 multiples of 5. (3)
Answer:

Question 3.
Using Binomial Theorem, expand the expression (2x + 3)⁵. (3)
Answer:

Question 4.
Find \(\lim _{x \rightarrow 0} f(x)\) where \(f(x)=\left\{\begin{array}{l}
2 x+3, x \leq 0 \\
3(x+1), x>0
\end{array}\right.\) (3)
Answer:

Question 5.
In a group of 400 students, 250 can speak Hindi and 200 can speak English. Also each can speak atleast one of these two languages. How many students can speak both Hindi and English? (3)
Answer:
H → Set of students who can speak Hindi
E → Set of students who can speak English
n(H) = 250, n(E) = 200, n(E ∪ H) = 400
n(E ∩ H) = n(E) + n(H) – n(E ∪ H)
= 250 + 200 – 400
= 50

Question 6.
Consider the line 2x + 3y – 6 = 0. Find its
(i) Slope (2)
(ii) y-intercept (1)
Answer:
(i) Slope = \(\frac{-2}{3}\)
(ii) y-intercept = \(\frac{-C}{B}=\frac{6}{3}\) = 2

Question 7.
Consider the equation y² = 12x. Find
(i) The coordinates of the focus. (1)
(ii) Equation of the directrix. (1)
(iii) Length of latus rectum. (1)
Answer:
(i) Focus = (a, 0) = (3, 0)
(ii) Directrix, x = -a
⇒ x = -3
⇒ x + 3 = 0
(iii) Length of Latus Rectum = 4a = 12

Question 8.
(i) The point (0, 2, 3) lies in
(a) XY-plane
(b) YZ-plane
(c) XZ-plane
(d) None of these (1)
(ii) Find the distance between the points P(-3, 7, 2) and Q(2, 4, -1). (2)
Answer:
(i) (b)

Question 9.
Using the principle of mathematical induction, prove that 7ⁿ – 3ⁿ is divisible by 4 for all n ∈ N. (3)
Answer:
p(n): 7ⁿ – 3ⁿ is divisible by 4
p(1): 7¹ – 3¹ = 7 – 3 = 4, divisible by 4
∴ p(1) is true.
Let p(k) be true
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 10.
Consider the expansion of (x – 2y)¹². Find its
(i) general term (2)
(ii) 4th term (1)
Answer:

Question 11.
Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the XY-plane. (3)
Answer:

Let the required ratio be k : 1
In the XY plane, Z = 0
(i.e) \(\frac{-8+10}{k+1}\) = 0
⇒ -8k + 10 = 0
⇒ -8k = -10
⇒ k = \(\frac{-10}{-8}=\frac{5}{4}\)
∴ XY plane divides in the ratio 5 : 4 internally.

Question 12.
(i) Write the negation of statement “Every natural number is greater than zero”. (1)
(ii) Write the converse and contrapositive of the statement “If a number n2 is even then n is even”. (2)
Answer:
(i) “It is false that every natural number is greater than zero”
(ii) Converse: “If n is even then n2 is even”
Contra positive: “If n is not even then n2 is not even”

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

Question 13.
Let A = {1, 2, 3}
(i) No. of subsets of A is ___________ (1)
(a) 3
(b) 6
(c) 8
(d) 9
(ii) Write all subsets of A having 2 elements. (2)
(iii) If the given set A is a subset of the universal set U = {1, 2, 3, 4, 5, 6}, then write A’. (1)
Answer:
(i) (c)
(ii) {1, 2}, {2, 3}, {1, 3}
(iii) A’ = {4, 5, 6}

Question 14.
(i) If (x + 1, y – 2) = (3, 1), find the values of x and y. (2)
(ii) Let A = {1, 2, 3} and B = {3, 4}. Find A × B. (2)
Answer:
(i) x + 1 = 3, y – 2 = 1
⇒ x = 2, y = 3
(ii) A × B = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}

Question 15.
(i) If cos x = \(\frac{-1}{2}\), x lies in 3rd quadrant, find the values of sin x and tan x. (2)
(ii) Prove that \(\sin ^2 \frac{\pi}{6}+\cos ^2 \frac{\pi}{3}=\frac{1}{2}\) (2)
Answer:

Question 16.
For all n ∈ N, Let P(n): 1 + 3 + 3² +……+ 3^n-1 = \(\frac{\left(3^n-1\right)}{2}\)
(i) Prove that P(1) is true. (1)
(ii) Prove that the statement P(n) is true for all natural numbers using the principle of mathematical induction. (3)
Answer:
(i) p(1): 1 = \(\frac{3^{\prime}-1}{2}=\frac{2-1}{2}\) = 1
∴ p(1) is true.
(ii) Let p(k) is true
(ie) p(k): 1 + 3 + 3²+… + 3^k-1 = \(\frac{3^k-1}{2}\)
We have to prove that p(k + 1) is true
p(k+1): 1 + 3 + 3²+…+ 3^k-1 + 3^k
= \(\frac{3^k-1}{2}+3^k=\frac{3^k-1+2 \times 3^k}{2}\)
= \(\frac{3 \cdot 3^k-1}{2}=\frac{3^{k+1}-1}{2}\)
∴ p(k + 1) is true.
Hence by P.M.I, p(n) is true for all n ∈ N

Question 17.
(i) Which of the following is the value of i⁹? (1)
(a) -i
(b) i
(c) -1
(d) 1
(ii) Express the complex number 3(7 + i7) + i(7 + i7) in a + ib form. (3)
Answer:
(i) (b)
(ii) 3(7 + i7) + i(7 + i7)
= 21 + 21i + 7i + 7i²
= 21 + 28i – 7
= 14 + 28i
∴ a = 14, b = 28

Question 18.
Represent the complex number z = 1 + i√3 in the polar form. (4)
Answer:

Question 19.
(i) If ⁿC₈ = ⁿC₂ then n is ______________ (1)
(a) 6
(b) 16
(c) 1
(d) 10
(ii) How many chords can be drawn through 21 points on a circle? (3)
Answer:
(i) n = 8 + 2 = 10
(ii) No. of chords = ²¹C₂
= \(\frac{21 \times 20}{1 \times 2}\)
= 210

Question 20.
(i) How many 3 digit numbers can be formed using the digits 1, 2, 3, 4 and 5 assuming that the repetition of the digits is not allowed? (2)
(ii) Find the number of permutations using all the letters of the word ALLAHABAD. (2)
Answer:
(i) Since repetition is not allowed,
No.of 3 digit numbers = 5 × 4 × 3 = 60
(ii) No. of permutations = \(\frac{9!}{4!2!}\) = 7560

Question 21.
Find the equation of the line perpendicular to the line x – 7y + 5 = 0 and passing through (2, -3). (4)
Answer:
Slope of given line = \(\frac{-1}{-7}=\frac{1}{7}\)
Slope of required line = -7
(x₁, y₁) = (2, -3)
Equation is y – y₁ = m(x – x₁)
⇒ y + 3 = -7(x – 2)
⇒ y + 3 = -7x + 14
⇒ 7x + y – 11 = 0

Question 22.
Consider an ellipse whose vertices are (±5, 0) and foci (±4, 0).
(i) Write the equation of the ellipse. (3)
(ii) Find the eccentricity of the ellipse. (1)
Answer:
(i) a = 5, c = 4
b² = a² – c²
= 25 – 16
= 9
Equation is \(\frac{x^2}{a^2}+\frac{y^2}{b^2}\) = 1
\(\frac{x^2}{25}+\frac{y^2}{9}\) = 1
(ii) e = \(\frac{c}{a}=\frac{4}{5}\)

Question 23.
Find the derivatives of
(i) x(x² + 2x + 1) (2)
(ii) \(\frac{x+1}{x}\) (2)
Answer:

Question 24.
Prove by the method of contradiction √5 is irrational. (4)
Answer:
Assume √5 is rational
∴ √5 = \(\frac{a}{b}\), where a and b are coprime
⇒ a = √5b
⇒ a² = 5b² ………(1)
∴ 5 divides a²
Implies 5 divides a
∴ a = 5c
Substituting in (1)
25c² = 5b²
5c² = b²
(i.e) 5 divides b²
hence 5 divides b
Which is a contradiction to the fact that a and b are coprime.
Hence our assumption is wrong.
∴ √5 is irrational.

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

Question 25.
(i) Draw the graph of the function f: R → R defined by f(x) = |x|. (3)
(ii) Let A = {1, 2, 3, 4, 5, 6} and R is a relation defined from A to A by R = {(x, y): y = x + 1}
(a) Depict this relation using an arrow diagram. (2)
(b) Write the domain of R. (1)
Answer:
(i) f(x) = |x|

(b) Domain = {1, 2, 3, 4, 5}

Question 26.
(i) Evaluate 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)
2x + y ≥ 6
3x + 4y ≤ 12
x ≥ 0, y ≥ 0
Answer:

Question 28.
(i) Find the 12th term of the geometric progression 5, 25, 125,…. (2)
(ii) Find the sum to n terms of the sequence 8, 88, 888,…… (4)
Answer:

Question 29.
Consider the following table:

Class	30 – 40	40 – 50	50 – 60	60 – 70	70 – 80	80 – 90	90 – 100
Frequency	3	7	12	15	8	3	2

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

Question 30.
(i) A coin is tossed twice. What is the probability that at least one tail occurs? (2)
(ii) If E and F are two events such that P(E) = \(\frac{1}{4}\), P(F) = \(\frac{1}{2}\) and P(E ∩ F) = \(\frac{1}{8}\). Find
(a) P(E or F) (2)
(b) P(not E and not F) (2)
Answer:
(i) S = {HH, HT, TH, TT}
E = atleast one tail = {HT, TH, TT}
P(E) = \(\frac{n(E)}{n(S)}=\frac{3}{4}\)

(ii) P(E) = \(\frac{1}{4}\), P(F) = \(\frac{1}{2}\), P(E ∩ F) = \(\frac{1}{8}\)
(a) P(E or F) = P(E) + P(F) – P(E ∩ F)
= \(\frac{1}{4}+\frac{1}{2}-\frac{1}{8}=\frac{2+4-1}{8}=\frac{5}{8}\)
(b) P(not E and not F) = 1 – P(E or F)
= 1 – \(\frac{5}{8}\)
= \(\frac{3}{8}\)