Kerala Plus One Maths Question Paper March 2023 with Answers

Reviewing Kerala Syllabus Plus One Maths Previous Year Question Papers and Answers Pdf March 2023 helps in understanding answer patterns.

Kerala Plus One Maths Previous Year Question Paper March 2023

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 and B are two sets such that A ⊂ B, then A ∪ B = _________________ (1)
(ii) Write the set {x: x is a positive integer and x2 < 40} in the Roster form. (1)
(iii) Write all the subsets of {2}. (1)
Answer:
(i) B
(ii) {1, 2, 3, 4, 5, 6}
(iii) {2}, φ

Question 2.
Solve: 3{1 – x} < 2(x + 4), Also represent the solutions on a number line. (3)
Answer:
3(1 – x) < 2(x + 4)
3 – 3x < 2x + 8
-3x – 2x < 8 – 3
-5x < 5x > -1

Question 3.
(i) If (x + 1, y – 4} = (3, 7), then find the values of x and y. (1)
(ii) The Cartesian product A × A has 9 elements among which 2 elements are (-a, 0) and (0, a). Write A. Also find A × A. (2)
Answer:
(i) x + 1 = 3, y – 4 = 7
x = 2, y = 11
(ii) A = {-a, 0, a}
A × A = {(-a, – a), (-a, 0), (-a, a) (0, -a), (0, 0), (0, a), (a, -a), (a, 0), (a, a)}

Question 4.
Find the number of arrangements of the letters of the word ‘INSTITUTE ’. How many of them begin with N? (3)
Answer:
Total no.of arrangements = \(\frac{9!}{2!3!}\) = 30,240
No. of arrangements begins with N = \(\frac{8!}{2!3!}\) = 3360

Question 5.
If f: R → R defined by \(f(x)=\left\{\begin{array}{l}
2 x+3 \text { if } x \leq 0 \\
3(x+1) \text { if } x>0
\end{array}\right.\). Evaluate \(\lim _{x \rightarrow 0} f(x)\). (3)
Answer:
LHL = \(\lim _{x \rightarrow 0^{-}} f(x)\) = 2 × 0 + 3 = 3
RHL = \(\lim _{x \rightarrow 0^{+}} f(x)\) = 3(0 + 1) = 3
LHL = RHL
∴ \(\lim _{x \rightarrow 0} f(x)\) = 3

Question 6.
(i) The point (0, 5, 7) lies in (1)
(a) XY-Plane
(b) YZ-Plane
(c) XY-Plane
(d) X-axis
(ii) Find the distance between (2, -3, -1) and (-2, 4, 3). (2)
Answer:
(i) (b) YZ plane
(ii) \(\sqrt{(-4)^2+(7)^2+(4)^2}=\sqrt{16+49+16}\)
= \(\sqrt{81}\)
= 9 units

Question 7.
If P(A) = 0.35, P(A ∩ B) = 0.25, P(A ∪ B) = 0.6, then find P(B) and P(not-B). (2)
Answer:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
∴ P(B) = P(A ∪ B) + P(A ∩ B) – P(A)
= 0.6 + 0.25 – 0.35
= 05
P(not B) = P(B’)
= 1 – P(B)
= 1 – 0.5
= 0.5

Question 8.
Find the centre and radius of the circle x² + y² + 8x + 10y – 8 = 0. (3)
Answer:
2g = 8, 2f = 10, c = -8
⇒ g = 4, f = 5
Centre = (-g, -f) = (-4, -5)
r = \(\sqrt{g^2+f^2-c}\)
= \(\sqrt{16+25+8}\)
= \(\sqrt{49}\)
= 7

Answer any 6 questions from 9 to 16. Each carries 4 scores. (6 × 4 = 24)

Question 9.
Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3}, B = {3, 4, 5}
(i) Find A ∪ B. (1)
(ii) Find A’ and B’. (1)
(iii) Verify (A ∪ B)’ = A’ ∩ B’. (2)
Answer:
(i) A ∪ B = {2, 3, 4, 5}
(ii) A’ = {1, 4, 5, 6}, B’ = {1, 2, 6}
(iii) (A ∪ B)’ = {1, 6}
A’ ∩ B’ = {1, 6}
∴ (A ∪ B)’ = A’ ∩ B’

Question 10.
(i) Let f: R → R, g: R → R defined by f(x) = x + 1, g(x) = 2x – 3. Find (f+g) (x) and (fg)(x). (1)
(ii) The function h: R → R defined by h(x) = |x|. Draw the graph of h(x). Also write its domain and range. (3)
Answer:
(i) (f + g)(x) = f(x) + g(x)
= x + 1 + 2x – 3
= 3x – 2
(f.g)(x) = (x + 1)(2x – 3)
= 2x² – 3x + 2x – 3
= 2x² – x – 3
(ii)

Domain = R
Range = [0, ∞)

Question 11.
(i) i^-35 = _________________ (1)
(ii) Find the multiplicative inverse and conjugate of \(\frac{1+i}{1-i}\). (3)
Answer:

Question 12.
4 cards are drawn from a pack of 52 playing cards.
(i) In how many ways can it be done? (1)
(ii) In how many ways can these 4 cards contain 2 red and 2 black? (3)
Answer:
(i) ⁵²C₄
(ii) ²⁶C₂ × ²⁶C₂

Question 13.
(i) Number of terms in the expansion of \(\left(x-\frac{1}{x}\right)^4\). (1)
(ii) Write the expansion of \(\left(x-\frac{1}{x}\right)^4\). (3)
Answer:
(i) 5

Question 14.
Insert 3 numbers between 1 and 256 so that the resulting sequence is a G.P. (4)
Answer:
Let x1, x2, x3 be the required numbers, Such that 1, x1, x2, x3, 256 is in a G.P.
a = 1, a5 = 256
ar⁴ = 256
⇒ r4 = 256
⇒ r = 4
∴ x₁ = 4, x₂ = 16, x₃ = 64
∴ Required numbers are 4, 16, 64

Question 15.
Find the co-ordinates of foci, vertices, eccentricity and length of latus rectum of the hyperbola \(\frac{x^2}{9}-\frac{y^2}{16}=1\). (4)
Answer:
a = 3, b = 4
c = \(\sqrt{a^2+b^2}\)
= \(\sqrt{9+16}\)
= 5
Foci = (±c, 0) = (±5, 0)
Vertices = (±a, 0) = (±3, 0)
Eccentricity = \(\frac{c}{a}=\frac{5}{3}\)
Length of Latus rectum = \(\frac{2 b^2}{a}=\frac{2 \times 16}{3}=\frac{32}{3}\)

Question 16.
A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random from the bag. Calculate the probability that it will be
(i) red
(ii) yellow
(iii) blue
(iv) not blue
Answer:
(i) P(red) = \(\frac{4}{9}\)
(ii) P(yellow) = \(\frac{2}{9}\)
(iii) P(blue) = \(\frac{3}{9}\) = \(\frac{1}{3}\)
(iv) P(not blue) = 1 – \(\frac{1}{3}\) = \(\frac{2}{3}\)

Answer any 3 questions from 17 to 20. Each carries 6 scores. (3 × 6 = 18)

Question 17.
(i) 25° = _________________ radian. (1)
(ii) Find the value of sin 15°. (2)
(iii) Prove that \(\frac{\sin 3 x+\sin x}{\cos 3 x+\cos x}\) = tan 2x. (3)
Answer:
(i) 25° = 25 × \(\frac{\pi}{180}\) = \(\frac{5 \pi}{36}\) radian
(ii) sin(15°) = sin(45° – 30°)
= sin 45° cos 30° – cos 45° sin 30°
= \(\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \times \frac{1}{2}\)
= \(\frac{\sqrt{3}-1}{2 \sqrt{2}}\)
(iii) LHS = \(\frac{\sin 3 x+\sin x}{\cos 3 x+\cos x}\)
= \(\frac{2 \sin 2 x \cos x}{2 \cos 2 x \cdot \cos x}\)
= \(\frac{\sin 2 x}{\cos 2 x}\)
= tan 2x
= RHS

Question 18.
(i) Find the equation of a line passing through the point (-4, 3) with slope \(\frac{1}{2}\). (2)
(ii) Write the equation of the line passing through the points (1, -1) and (3, 5). (2)
(iii) Find the angle between the lines obtained in (i) and (ii). (2)
Answer:
(i) (x₁, y₁) = (-4, 3), m = \(\frac{1}{2}\)
Equation is y – y₁ = m(x – x₂₎
⇒ y – 3 = \(\frac{1}{2}\)(x + 4)
⇒ 2y – 6 = x + 4
⇒ x – 2y + 10 = 0
(ii) (x₁, y₁) = (1, -1), (x₂, y2) = (3, 5)
Equation is \(y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\)
⇒ y + 1 = \(\frac{5+1}{3-1}\) (x – 1)
⇒ y + 1 = 3(x – 1)
⇒ y + 1 = 3x – 3
⇒ 3x – y – 4 = 0
(iii) m1 = \(\frac{1}{2}\), m2 = 3
tan θ = \(\left|\frac{m_1-m_2}{1+m_1, m_2}\right|=\left|\frac{\frac{1}{2}-3}{1+\frac{1}{2} \times 3}\right|\)
⇒ θ = 45°

Question 19.
(i) Find the derivative of tan x using 1st principles. (4)
(ii) If y = x’ sin x, find \(\frac{d y}{d x}\). (2)
Answer:
(i) f(x) = tan x
f(x + h) = tan(x + h)

(ii) y = x.sin x
\(\frac{d y}{d x}=x \cdot \frac{d}{d x}(\sin x)+\sin x \cdot \frac{d}{d x}(x)\)
= x cos x + sin x × 1
= x cos x + sin x

Question 20.
Consider the following table:

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

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

= 861 – 729
= 133
(iii) Standard deviation = σ = \(\sqrt{133}\) = 11.53