Kerala Plus Two Maths Question Paper March 2022 with Answers

Reviewing Kerala Syllabus Plus Two Maths Previous Year Question Papers and Answers Pdf March 2022 helps in understanding answer patterns.

Kerala Plus Two Maths Previous Year Question Paper March 2022

Time: 2 Hours
Total Score: 60 Marks

Part – I

A. Answer any 5 questions from 1 to 9. Each carries 1 score. (5 × 1 = 5)

Question 1.
Which of the following relations on A = {1, 2, 3} is an equivalence relation?
(a) {(1, 1), (2, 2), (3, 3)}
(b) {(1, 1), (2, 2), (3, 3), (1, 2)}
(c) {(1, 1), (3, 3), (1, 3), (3, 1)}
(d) None of these
Answer:
(a) {(1, 1), (2, 2), (3, 3)}

Question 2.
The value of sin^-1(sin(\(\frac{1}{2}\))) = …………………..
(a) \(\frac{1}{2}\)
(b) π – \(\frac{1}{2}\)
(c) –\(\frac{1}{2}\)
(d) \(\frac{\pi}{6}\)
Answer:
(a) \(\frac{1}{2}\)

Question 3.
If A is a 3 × 3 matrix, then |adj(A)| = ……………………..
(a) |A|
(b) |A|²
(c) |A|³
(d) 3|A|
Answer:
(b) |A|²

Question 4.
A fair, die is rolled. If the events are E = {1, 3, 5}, F = {2, 3}, then P(E|F) = …………
Answer:
P(E|F) = \(\frac{P(E \cap F)}{P(F)}\) = \(\frac{\frac{1}{6}}{\frac{2}{6}}\) = \(\frac{1}{2}\)

Question 5.
The area bounded by the curve y = 2x between x = 0, x = 2 and x-axis is ………………
Answer:
Required area = \(\int_0^2 y d x\)
= \(\int_0^2 2 x d x\) = \(\left[x^2\right]_0^2\) = 4 sq. units

Question 6.
Slope of the tangent to the curve y = x² + 1 at the point (2, 5) is ……………….
Answer:
y = x² + 1
\(\frac{d y}{d x}\) = 2x
\(\left.\frac{d y}{d x}\right]_{(2,5)}\) = 4
∴ Slope of tangent = 4

Question 7.
Write the vector from the point A(1, 3, 5) to B (4, 3, 2)
Answer:
\(\overrightarrow{A B}\) = 3î – 3k̂

Question 8.
Which of the following is a point on the plane 3x + 2y + 4z = 0?
(a) (1, 2, 1)
(b) (2, 3, 2)
(c) (2, 1, -2)
(d) (2, 1, 2)
Answer:
(c) (2, 1, -2)

Question 9.
Write the degree of the differential equation
2\(\frac{d^2 y}{d x^2}\) + (\(\frac{d y}{d x}\)) = 0
Answer:
Degree = 1

B. Answer all questions from 10 to 13. Each carries 1 score. (4 × 1 = 4)

Question 10.
The value of sin^-1(\(\frac{1}{\sqrt{2}}\)) = ………………….
Answer:
sin^-1(\(\frac{1}{\sqrt{2}}\)) = \(\frac{\pi}{4}\)

Question 11.
The vertices of a triangle are (0, 2), (0, 3), (4, 6), then area of the triangles ____________
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
Area = \(\frac{1}{2}\)\(\left|\begin{array}{lll}
0 & 2 & 1 \\
0 & 3 & 1 \\
4 & 6 & 1
\end{array}\right|\) = |\(\frac{1}{2}\)[4 × -1]|
= |\(\frac{-4}{2}\)| = |-2|
= 2 sq. units

Question 12.
Find the direction cosines of the vector 3î – 2ĵ + 5k̂
Answer:
l = \(\frac{3}{\sqrt{38}}\), m= \(\frac{-2}{\sqrt{38}}\), n = \(\frac{5}{\sqrt{38}}\)

Question 13.
Derivative of log (x³) is ………………….
Answer:
y = log (x³)
\(\frac{d y}{d x}\) = \(\frac{1}{\dot{x}^3}\) × 3x² = \(\frac{3}{x}\)

Part – II

A. Answer any 2 questions from 14 to 17. Each carries 2 scores. (2 × 2 = 4)

Question 14.
\(\left[\begin{array}{cc}
x+y & 2 \\
5+x & 8
\end{array}\right]\) = \(\left[\begin{array}{ll}
5 & 2 \\
6 & 8
\end{array}\right]\), find x and y.
Answer:
5 + x = 6
∴ x = 1

x + y = 5
1 + y = 5
y = 4

Question 15.
The length x of a rectangle is increasing at the rate of 4 cm/s and the width y is decreasing at the rate of 5 cm/s. Find the rates of change of its area when x =10 cm and y = 5 cm.
Answer:
\(\frac{d x}{d t}\) = 4 cm/s. \(\frac{d y}{d t}\) = -5 cm/s
A = xy
\(\frac{d A}{d t}\) = x . \(\frac{d y}{d t}\) + y . \(\frac{d x}{d t}\)
= – 5x + 4y
\(\left.\frac{d A}{d t}\right]_{x=10, y=5}\) = -50 + 20 = -30
Area decreases at the rate of 30 cm2/s

Question 16.
Show that the function f(x) = x³ + 3x + 5 is strictly increasing on R.
Answer:
f(x) = x³ + 3x + 5
f'(x) = 3x² + 3
= 3(x² + 1) > 0 for all x ∈ R
∴ f(x) is increasing on R

Question 17.
Solve the differential equation \(\frac{d y}{d x}\) = \(\frac{2 x}{y^2}\)
Answer:
Given \(\frac{d y}{d x}\) = \(\frac{2 x}{y^2}\)
y² dy = 2x dx,
which is variable separable.
Solution is \(\int y^2 d y\) = \(\int 2 x d x\)
\(\frac{y^3}{3}\) = x² + C

B. Answer any 2 questions from 18 to 20. Each carries 2 scores. (2 × 2 = 4)

Question 18.
Find the value of λ if the vectors î – ĵ + k̂, 3î + ĵ + 2k̂ and î + λĵ – 3k̂ are coplanar.
Answer:
Let \(\bar{a}\) = î – ĵ + k̂, \(\bar{b}\) = 3î + ĵ + 2k̂, \(\bar{c}\) = î + λĵ – 3k̂
Since \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar,
[\(\bar{a}\), \(\bar{b}\), \(\bar{c}\)] = 0
\(\left|\begin{array}{ccc}
1 & -1 & 1 \\
3 & 1 & 2 \\
1 & \lambda & -3
\end{array}\right|\) = 0
1(- 3 – 2λ) + 1(-11) + 1(3λ – 1) = 0
– 3 – 2λ – 11 + 3λ – 1 = 0
– 15 + λ = 0
λ = 15

Question 19.
If y = x^{sin x} find \(\frac{d y}{d x}\)
Answer:
Let y = x^{sin x}
log y = sin x . log x
\(\frac{1}{y}\)\(\frac{d y}{d x}\) = sin x.\(\frac{1}{x}\) + log x cos x
\(\frac{d y}{d x}\) = y[\(\frac{\sin x}{x}\) + log x cos x]
= x^{sin x}[\(\frac{\sin x}{x}\) + log x cos x]

Question 20.
Find the integrating factor of the differential equation
x\(\frac{d y}{d x}\) – y = 2x²
Answer:
Given x \(\frac{d y}{d x}\) – y = 2x²

Part – III

A. Answer any 3 questions from 21 to 24. Each carries 3 scores. (3 × 3 = 9)

Question 21.
Express the matrix A = \(\left[\begin{array}{ccc}
3 & 3 & -1 \\
-2 & -2 & 1 \\
-4 & -5 & 2
\end{array}\right]\) as the sum of a symmetric matrix and a skew symmetric matrix.
Answer:

Question 22.
R = {(x, y): x, y ∈ Z, (x – y) is an integer}. Show that R is an equivalence relation.
Answer:
Given R = {(x, y): x, y ∈ Z, x – y is an integer} 0 is an integer
⇒ x – x is an integer for all x ∈ Z
⇒ (x, x) ∈ R for all x ∈ Z

∴ R is reflexive
Let (x, y) ∈ R ⇒ x – y is an integer
⇒ -(x – y) is an integer
⇒ y – x is an integer
⇒ (y, x) ∈ R for all x, y ∈ Z

∴ R is symmetric
Let (x, y) ∈ R, (y, z) ∈ R
⇒ x – y is an integer and y – z is an integer
⇒ x – y + y – z is an integer
⇒ x – z is an integer
⇒ (x, z) ∈ R for all x, y, z ∈ R
∴ R is transitive
Hence R is an equivalence relation

Question 23.
Bag 1 contains 5 red and 3 black balls while another Bag 2 contains 3 red and 7 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag 2.
Answer:

Let E₁ : ball is drawn from bag I
E₂ : ball is drawn from bag II
A : Getting a red ball.

Question 24.
Consider the vector \(\vec{a}\) = 2î + ĵ + k̂ and \(\vec{b}\) = î + ĵ + k̂
(a) Find \(\vec{a}\) × \(\vec{b}\) (2)
(b) Find a unit vector perpendicular to \(\vec{a}\) and \(\vec{b}\) (1)
Answer:
(a) \(\vec{a}\) × \(\vec{b}\) = \(\left|\begin{array}{lll}
\hat{i} & \hat{j} & \hat{k} \\
2 & 1 & 1 \\
1 & 1 & 1
\end{array}\right|\)
= î(0) – ĵ(1) + k̂(1)
= ĵ + k̂

(b) Unit vector perpendicular to \(\vec{a}\) and \(\vec{b}\)
= \(=\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}\) = \(\frac{-\hat{j}+\hat{k}}{\sqrt{2}}\)
= \(\frac{-1}{\sqrt{2}}\)ĵ + \(\frac{1}{\sqrt{2}}\)k̂

B. Answer any 2 questions from 25 to 27. Each carries 3 scores. (2 × 3 = 6)

Question 25.
Using elementary operations, find the inverse of the matrix A = \(\left[\begin{array}{ll}
1 & 3 \\
2 & 7
\end{array}\right]\)
Answer:

Question 26.
If * is a binary operation of R defined by a * b = \(\frac{a b}{3}\),
(a) Find the identity element of*. (1)
(b) Find the inverse of 3. (2)
Answer:
(a) Let ‘e’ be the identity element.
∴ a * e = a
\(\frac{a e}{3}\) = a
e = 3

(b) Let ‘b’ be the inverse of 3
∴ 3 * b = e
\(\frac{3 b}{3}\) = 3
b = 3

Question 27.
Evaluate \(\int_0^2 x^2 d x\) as the limit of a sum.
Answer:
a = 0, b = 2, nh = b – a = 2
f(x) = x²
f(a) = f(0) = 0
f(a + h) = f(0 + h) = (0 + h)² = h²
f(a + 2h) = f(0 + 2 h) = (0 + 2 h)² = 4h²
f(a + 3h) = f(0 + 3h) = (0 + 3h)² = 9h²
f(a + (n – 1 )h = f(0 + (n – 1)h = (n – 1)²h²

Part – IV

A. Answer any 3 questions from .28 to 31. Each carries 4 scores. (3 × 4 = 12)

Question 28.
Show that 2 tan^-1(\(\frac{1}{2}\)) + tan^-1(\(\frac{1}{7}\)) = tan^-1(\(\frac{31}{17}\)).
Answer:

Question 29.
Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
Answer:

Question 30.
(a) Discuss the continuity of the function (2)

(b) Verify Rolle’s theorem for the function
f(x) = 2x² – 12x + 1 in [2, 4] (2)
Answer:

LHL = 3(3) + 1 = 9 + 1 = 10
RHL = (3)2 +1 = 9 + 1 = 10
f(3) = 9 + 1 = 10
LHL = RHL = f(3)
∴ f(x) is continuous at x = 3
Hence f(x) is a continuous function.

(b) f(x) = 2x² – 12x + 1
Since f(x) is a polynomial function, it is continuous in [2, 4]
f'(x) = 4x – 12
∴ It is differentiable in (2, 4)
f(a) = f(2) = 8 – 24 + 1 = -15
f(b) = f(4) = 32 – 48 + 1 = -15
∴ f(a) = f(b)
∴ f'(c) = 0
⇒ 4c – 12 = 0
4c = 12
c = 3 ∈ (2, 4)
Hence Rolle’s theorem verified.

Question 31.
(a) Find the equation of the line passing through the pints (2, 1, 0) and (4, 4, 3)
(b) Find the equation of the plane which is perpendicular to the above line and passing through the point (1, 1, 2)
Answer:
(a) (x₁, y₁, z₁) = (2, 1, 0)
(x₂, y₂, z₂) = (4, 4, 3)
Equation of line is
\(\frac{x-x_1}{x_2-x_1}\) = \(\frac{y-y_1}{y_2-y_1}\) = \(\frac{z-z_1}{z_2-z_1}\) = λ
(ie) \(\frac{x-2}{2}\) = \(\frac{y-1}{3}\) \(\frac{z}{3}\) = λ

(b) < a, b, c > = < 2, 3, 3 >
(x₁, y₁, z₁,) = (1, 1, 2)
Equation of plan is
a(x – x₁) + b(y – y₁) + c(z – z₁) = 0
2(x – 1) + 3(y – 1) + 3(z – 2) = 0
2x – 2 + 3y – 3 + 3z – 6 = 0
2x + 3y + 3z – 11 = 0

B. Answer any 1 question from 32 to 33. Carries 4 scores. (1 × 4 = 4)

Question 32.
Find the mean of the number obtained on a throw of an unbiased die. (4)
Answer:
Let X be the number obtained.
∴ X = 1, 2, 3, 4, 5, 6

Mean = Σx_ip_i = \(\frac{21}{6}\) = \(\frac{7}{2}\) = 3.5

Question 33.
Consider the planes 3x – 2y + z + 6 = 0 and 2x + y + 2z – 6 =0;
(a) Find the angle between the planes. (2)
(b) Find the equation of the plane passing through the line of intersection of above planes and through the point (0, 0, 0). (2)
Answer:
p₁ : 3x – 2y + z + 6 = 0
p₂ : 2x + y + 2z – 6 = 0

(b) Eqn. of plane is p₁ + λp₂ = 0
(3x – 2y + z + 6) + λ(2x + y + 2z – 6) = 0 ………….. (1)
Since it passes through (0,0,0)
6 + λ(-6) = 0
-6λ = -6
λ = 1
Sub in (1),
3x – 2y + z + 6 + 1(2x + y + 2z – 6) = 0 5x – y + 3z = 0

Part – V

Answer any 2 questions from 34 to 36. Each carries 6 scored. (2 × 6 = 12)

Question 34.
Solve the following system of equations by matrix method.
x – y + 2z = 1
2y – 3z = 1
3x – 2y + 4z = 2
Answer:
A = \(\left[\begin{array}{ccc}
1 & -1 & 2 \\
0 & 2 & -3 \\
3 & -2 & 4
\end{array}\right]\), X = \(\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\), B = \(\left[\begin{array}{l}
1 \\
1 \\
2
\end{array}\right]\)

∴ System of equations can be written as AX = B.
|A| = 1(8 – 6) + 1(0 + 9) + 2(0 – 6)
= 2 + 9 – 12
= -1 ≠ 0
∴ System is consistent
Unique solution is X = A^-1B
A₁₁ = 2, A₁₂ = -9, A₁₃ = -6
A₂₁ = 0, A₂₂, = -2, A₂₃ = -1
A₃₁ = -1, A₃₂, = 3, A₃₃ = -2

Question 35.
Find the following integrals:
(a) \(\int \frac{x}{(x+1)(x+2)} d x\) (3)
(b) \(\int_0^{\frac{\pi}{2}} \frac{\sin ^4 x}{\sin ^4 x+\cos ^4 x} d x\) (3)
Answer:

Question 36.
Solve the following linear programming problem graphically
Maximise Z = 3x + 2y
Subject to
x + 2y ≤ 10
3x + y ≤ 15
x, y ≥ 0
Answer:
x + 2y = 10

x	0	10
y	5	0

3x + y = 15

x	0	5
y	15	0

∴ z has maximum, when x = 5, y = 0
Maximum value = 15