Kerala Plus One Maths Board Model Paper 2023 with Answers

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

Kerala Plus One Maths Board Model Paper 2023 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.
(i) The interval representing {x : x ∈ R, -2 < x ≤ 3} is (1)
(A) [-2, 3]
(B) (-2, 3]
(C) (-2, 3)
(D) [-2, 3]
(ii) Write all the subsets of {a, b, c}. (2)
Answer:
(i) B (-2, 3]
(ii) {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}, φ

Question 2.
Consider A = {1, 2, 3, 4, 6}. Let R be a relation defined by R = {(a, b): a, b ∈ A, b is exactly divisible by a}
(i) Write R in roster form. (2)
(ii) Is R a function? Justify. (1)
Answer:
(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}
(ii) R is not a function. Since images of different elements A are not unique.

Question 3.
(i) Solve the inequality 2(x – 1) < 3(x – 2). (2)
(ii) Show the solution of the inequality on a number line. (1)
Answer:
(i) 2x – 2 < 3x – 6
2x – 3x < -6 + 2
-x < -4 x > 4
(ii)

Question 4.
(i) If ⁿC₃ = ⁿC₇, find ⁿC₂. (2)
(ii) If there are 10 persons in a meeting and each of them shakes hands with all others, what is the total number of handshakes? (1)
Answer:
(i) ⁿC₃ = ⁿC₇
⇒ n = 3 + 7 = 10
∴ ⁿC₂ = ¹⁰C₂
= \(\frac{10 \times 9}{1 \times 2}\)
= 45
(ii) Number of shake hands = ¹⁰C₂ = 45

Question 5.
(i) The length of latus rectum of the parabola y² = 10x is _______________ (1)
(ii) Find the equation of the parabola with focus (0, -3) and directrix y = 3. (2)
Answer:
(i) 10
(ii)

Equation is x² = -4ay
Put a = 3
∴ Equation is x² = -12y

Question 6.
(i) Which of the following is a point on XZ-plane?
(A) (1, -2, 0)
(B) (1, 0, -2)
(C) (0, 1, 0)
(D) (0, 1, -2)
(ii) Verify (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an isosceles triangle. (2)
Answer:
(i) B(1, 0, -2)
(ii) Let A(0, 7, -10), B(1, 6, -6), C(4, 9, -6)

AB = BC
∴ ABC is an isosceles triangle.

Question 7.
(i) \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) = _______________ (1)
(ii) Evaluate \(\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\cos x}{\frac{\pi}{2}-x}\right)\) (2)
Answer:
(i) 1
(ii) \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\frac{\pi}{2}-x}=\lim _{\frac{\pi}{2}-x \rightarrow 0} \frac{\sin \left(\frac{\pi}{2}-x\right)}{\frac{\pi}{2}-x}\) = 1

Question 8.
If P(A) = \(\frac{1}{2}\), P(B) = \(\frac{1}{2}\) and P(A ∩ B) = \(\frac{1}{8}\), find
(i) P(not A) (1)
(ii) P(A or B) (1)
(iii) P(not A and not B) (1)
Answer:
(i) P(not A) = 1 – P(A)
= 1 – \(\frac{1}{4}\)
= \(\frac{3}{4}\)

(ii) P(A or B) = P(A ∪ B)
= P(A) + P(B) – P(A ∩ B)
= \(\frac{1}{4}+\frac{1}{2}-\frac{1}{8}\)
= \(\frac{5}{8}\)

(iii) P(not A and not B) = P(A’ ∩ B’)
= P[(A ∪ B)’]
= 1 – P(A ∪ B)
= 1 – \(\frac{5}{8}\)
= \(\frac{3}{8}\)

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

Question 9.
(i) If A ⊂ B, then A ∩ B = _______________ (1)
(ii) Consider the sets U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10}. Verify that (A ∪ B)’ = A’ ∩ B’. (3)
Answer:
(i) A
(ii) A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
(A ∪ B)’ = {7, 9}
A’ = {6, 7, 8, 9, 10}
B’ = {1, 3, 5, 7, 9}
A’ ∩ B’ = {7, 9}
∴ (A ∪ B)’ = A’ ∩ B’

Question 10.
(i) Draw the graph of the function f(x) =|x| – 2 (2)
(ii) Write the range of f. (1)
(iii) Write the domain of the function g(x) = \(\sqrt{9-x}\) (1)
Answer:
(i)

(ii) Range of f = [-2, ∞)
(iii) 9 – x ≥ 0
-x ≥ -9
x ≤ 9
Domain of g(x) = (-∞, 9]

Question 11.
(i) Express z = i⁹ + i¹⁸ into a + ib form. (1)
(ii) Write the conjugate of z = i⁹ + i¹⁸. (1)
(iii) Find the multiplicative inverse of 2 – 3i. (2)
Answer:
(i) z = i⁹ + i¹⁸
= i – 1
= -1 + i
a = -1, b = 1
(ii) Conjugate of z = \(\bar{z}\) = -1 – i
(iii) z = 2 – 3i
Multiplicative inverse of z

Question 12.
Find the number of different 8 letter arrangements that can be made from the letters of the work ‘DAUGHTER’ so that
(i) all vowels occur together. (2)
(ii) all vowels do not occur together. (2)
Answer:
Total number of 8 letter words = 8! = 40,320
(i) Take the three vowels (A, U, E) as a single object. Then there are 6 objects which can be arranged in 6! ways and the vowels can be arranged in 3! ways.
∴ Required no. of ways = 6! × 3! = 4,320
(ii) Number of words in which all vowels do not occur together = 40,320 – 4,320 = 36,000

Question 13.
Find (a + b)⁴ – (a – b)⁴. Hence evaluate (√3 + √2)⁴ – (√3 – √2)⁴. (3)
Answer:
(i) (a + b)⁴ – (a – b)⁴ = (a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴) – (a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴)
= 8a³b + 8ab³
= 8ab (a² + b²)
∴ (√3 + √2)⁴ – (√3 – √2)⁴ = 8 × √3 × √2(3 + 2) = 40√6

Question 14.
Find the sum to n terms of the sequence 3, 33, 333,….. (4)
Answer:
3 + 33 + 333 + ……….n terms
= 3(1 + 11 + 111 + ………n terms)

Question 15.
Find focii, vertices, eccentricity and length of latus rectum of the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\). (4)
Answer:
a = 5, b = 3, c = \(\sqrt{25-9}\) = 4
Foci = (±c, 0) = (±4, 0)
Vertices = (±a, 0) = (±5, 0)
Eccentricity = \(\frac{c}{a}=\frac{4}{5}\)
Length of Latus rectum = \(\frac{2 b^2}{a}=\frac{18}{5}\)

Question 16.
A fair coin is tossed 3 times. Write the sample space. Find the probability of getting (1)
(i) 3 heads (1)
(ii) exactly two heads (1)
(iii) one head and two tails. (1)
Answer:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
(i) P(3 heads) = \(\frac{1}{8}\)
(ii) P(exactly 2 heads) = \(\frac{3}{8}\)
(iii) P(1 head and 2 tails) = \(\frac{3}{8}\)

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

Question 17.
(i) Find the radian measure of 240°. (1)
(ii) If tan x = \(\frac{-5}{12}\) and x lies in second quadrant, find the values of sin x and cos x. (2)
(iii) Prove that \(\frac{\cos 7 x+\cos 5 x}{\sin 7 x-\sin 5 x}\) = cot x. (3)
Answer:

Question 18.
(i) Find the equation of the line passing through (-3, 5) and perpendicular to the line through the points (2, 5) and (-3, 6). (4)
(ii) Find the distance of the point (-3, 5) from the line 3x – 4y + 26 = 0. (2)
Answer:
(i) Let P(-3, 5), A(2, 5), B(-3, 6) be the given points
Slope of AB = \(\frac{6-5}{-3-2}=\frac{1}{-5}\)
∴ Slope of required line = 5
Equation is y – 5 = 5(x + 3)
⇒ y – 5 = 5x + 15
⇒ 5x – y + 20 = 0

Question 19.
(i) Find the derivative of cos x using first principle. (3)
(ii) Find the derivative of f(x) = \(\frac{1+\sin x}{\cos x}\) (3)
Answer:

Question 20.
Calculate Arithmetic Mean, Variance and Standard Deviation of the following data. (6)

Class	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70
Frequency	3	7	12	15	8	3	2

Answer: