Kerala Plus One Maths Board Model Paper 2018 with Answers

Reviewing Kerala Syllabus Plus One Maths Previous Year Question Papers and Answers Pdf Board Model Paper 2018 helps in understanding answer patterns.

Kerala Plus One Maths Board Model Paper 2018 with Answers

Time: 2 Hours
Total Scores: 60

Answer any six questions from 1 to 7. Each carries 3 scores.

Question 1.
Consider the Venn diagram given below:

(a) Write A’, B’, (A ∩ B)’ (2)
(b) Verify that (A ∩ B)’ = A’ ∪ B’ (1)
Answer:
(a) A’ = {5, 6, 7, 8, 9}, B = {1, 2, 7, 8, 9}
(A ∩ B)’ = {1, 2, 5, 6, 7, 8, 9}
(b) A’ ∪ B’ = {1, 2, 5, 6, 7, 8, 9}
Hence (A ∩ B)’ = A’ ∪ B’

Question 2.
For any triangle ABC, prove that \(\sin \left(\frac{B-C}{2}\right)=\frac{b-c}{a} \cos \left(\frac{A}{2}\right)\)
Answer:

Question 3.
Consider the complex number z = 3 + 4i
(a) Write the conjugate of z. (1)
(b) Verify that z\(\bar{z}\) = |z|² (2)
Answer:

Question 4.
(a) Solve the inequality -5 ≤ \(\left(\frac{5-3 x}{2}\right)\) ≤ 8 (2)
(b) Represent the solution on a number line. (1)
Answer:

Question 5.
4 cards are drawn from a well shuffled pack of 52 cards.
(a) In how many ways can this be done? (1)
(b) In how many ways can this be done if all 4 cards are of the same colour? (2)
Answer:
(a) No. of ways = ⁵²C₄
(b) No. of ways, all 4 are same colour = ²⁶C₄ + ²⁶C₄ = 2 × ²⁶C₄

Question 6.
Consider the equation of the ellipse 9x² + 4y² = 36. Find
(a) Focii (1)
(b) Eccentricity (1)
(c) Length of latus rectum (1)
Answer:

Question 7.
Evaluate \(\lim _{x \rightarrow-1} \frac{2 x+8}{x^2+x-12}\)
Answer:

Answer any eight questions 8 to 17. Each carries 4 scores each.

Question 8.
Consider A = {x : x is an integer, 0 < x ≤ 3}
(a) Write A in roster form. (1)
(b) Write the power set of A. (1)
(c) The number of proper subsets of A. (1)
(d) Write the number of possible relations from A to A. (1)
Answer:
(a) A = {1, 2, 3}
(b) P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
(c) Number of proper subset = 2³ – 1 = 7
(d) Number of relations = 2^3×3 = 512

Question 9.
Consider the statement P(n): \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots \ldots .+\frac{1}{n(n+1)}=\frac{n}{n+1}\)
(a) Show that P(1) is true. (1)
(b) Prove that P(n) is true for all n ∈ N using the principle of Mathematical induction. (3)
Answer:


Hence, by using the principle of mathematical induction true for all n ∈ N.

Question 10.
Consider the complex number z = \(\frac{1+3 i}{1-2 i}\)
(a) Write z in the form a + ib. (1)
(b) Write z in polar form. (3)
Answer:

Question 11.
Solve the following inequalities graphically:
x – 2y ≤ 3; 3x + 4y ≥ 12; x ≥ 0; y ≥ 1
Answer:

Question 12.
(a) How many 3-letter words with or without meaning can be formed using 26 letters in the English alphabet, if no letter is repeated? (1)
(b) Find the number of permutations of letters of the word MATHEMATICS. (2)
(c) How many of them begin with the letter C? (1)
Answer:
(a) No. of words = ²⁶P₃ = 15600
(b) In the word MATHEMATICS, the letters M-2, A-2, T-2, H-1, E-1, I-1, C-1, S-1 are repeated.
No. of permutation = \(\frac{11!}{2!\times 2!\times 2!}\)
(c) If the words begin with C = \(\frac{10!}{2!\times 2!\times 2!}\)

Question 13.
Consider the figure given below. A(3, 0) and B (0, 2) are two points on the axes. The line is perpendicular to AB.

(a) Find the slope of OP. (1)
(b) Find the coordinate of the point P. (3)
Answer:

Question 14.
Equation of the parabola given in the figure is y² = 8x.

(a) Find the focus and length of latus rectum of the parabola. (2)
(b) The latus rectum of the parabola is a chord to the circle centered at the origin as shown in the figure. Find the equation of the circle. (2)
Answer:
(a) The equation parabola is y² = 8x
⇒ a = 2
Length of latus rectum = 4a = 4 × 2 = 8
(b) A and B are extremities of the latus rectum. Hence the coordinate of A is (a, -2a) = (2, -4) and that of B is (a, 2a) = (2, 4)
Therefore, the radius of the circle is the distance from A or B
Radius = \(\sqrt{(2-0)^2+(4-0)^2}=\sqrt{4+16}=\sqrt{20}\)
Equation of the circle is x² + y² = 20

Question 15.
Let L be the line x – 2y + 3 = 0
(a) Find the equation of the line L₁ which is parallel to L and passing through (1, -2). (2)
(b) Find the distance between L and L₁. (1)
(c) Write the equation of another line L₂ which is parallel to L, such that the distance from the origin to L and L₂ are the same. (1)
Answer:
(a) Slope of the line L is \(\frac{1}{2}\)
The slope of parallel line L₁ is \(\frac{1}{2}\)
Therefore, the equation of line L₁
y – (-2) = \(\frac{1}{2}\) (x – 1)
⇒ 2(y + 2) = x – 1
⇒ 2y + 4 = x – 1
⇒ x – 2y – 5 = 0
(b) Distance between L and L₁ = \(\left|\frac{-5-3}{\sqrt{(1)^2+(-2)^2}}\right|=\frac{8}{\sqrt{5}}\)
(c) The equation of the line L₂ will be on the other side of the origin.
So the equation of L₂ can be obtained by changing the sign of the constant term in the equation of L = x – 2y – 3 = 0

Question 16.
Consider a point A(3, 2, -1) in space
(a) Write the octant in which A belongs to (1)
(b) If B (1, 2, 3) is another point in Space, find the distance between A and B. (1)
(c) Find the coordinate of the point R which divides AB in the ratio 1 : 2 internally. (2)
Answer:
(a) 5th octant.

Question 17.
a) Write the contrapositive of the statement. (1)
p: If a triangle is equilateral, then it is isosceles.
b) Verify by method of contradiction: ‘√3 is irrational’ (3)
Answer:
(a) If a triangle is not isosceles, then it is not equilateral.
(b) Assume that √3 is rational. Then √3 can be written in the form \(\sqrt{3}=\frac{p}{q}\), where p and q are integers without common factors.
Squaring: 3 = \(\frac{p^2}{q^2}\)
⇒ 3q² = p² …….(1)
⇒ 3 divides p²
⇒ 3 divides p
Therefore, p = 3k for some integer k.
⇒ p² = 9k²
(1) ⇒ 3q² = 9k²
⇒ q² = 3k²
⇒ 3 divides q²
⇒ 3 divides q
Hence p and q have a common factor 3, which contradicts our assumption. Therefore, √3 is irrational.

Answer any 5 questions from 18 to 24. Each carries 6 scores.

Question 18.
(a) If A = {a, b} write A × A × A. (2)
(b) If R = {(x, x³) : x is a prime number less than 10}. Write R in roster form. (2)
(c) Find the domain and range of the function f(x) = 2 + \(\sqrt{x-1}\) (2)
Answer:
(a) A × A × A = {(a, a, a), (a, a, b), (a, b, a), (a, b, b), (b, a, a),(b, a, b), (b, b, a), (b, b, b)}
(b) R = {(2, 8), (3, 27), (5, 125), (7, 343)}
(c) Domain = [1, ∞)
Range = [2, ∞)

Question 19.
(a) The minute hand of a watch is 3 cm long. How far does its tip move in 40 minutes? (2)
(Use π = 3.14)
(b) Solve the trigonometric equation sin 2x – sin 4x + sin 6x = 0. (4)
Answer:

Question 20.
(a) Find the sum of all 3-digit numbers which are multiples of 5. (2)
(b) How many terms of the GP: 3, 3², 3³, …… are needed to give the sum 120? (2)
(c) Find the sum of the first n terms of the series whose nth term is n(n + 3). (2)
Answer: