Kerala Plus Two Maths Board Model Paper 2022 with Answers

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

Kerala Plus Two Maths Board Model Paper 2022 with Answers

Time: 2 Hours
Total Score: 60 Marks

Part – I

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

Question 1.
The function f : R → R defined by f(x) = x.
Find for (x).
Answer:
Given f(x) = x
(fof)(x) = f(f(x))
= f(x) = x

Question 2.
Find the value of cos(sec^-1 x + cosec^-1 x), |x| ≥ 1.
Answer:
cos(sec^-1 x + cosec^-1 x)
= cos \(\frac{\pi}{2}\) = 0

Question 3.
Let Abe a square matrix of order 3 × 3, then which among the following is the value of |kA|?
(a) k²|A|
(b) k|A|
(c) k³|A|
(d) 3k|A|
Answer:
(c) k³|A|
(Hint : |kA| = kⁿ . |A|)

Question 4.
The rate of change of area of a circle with respect to its radius, when radius 6 cm is
(a) 10π
(b) 12π
(c) 8π
(d) 11π
Answer:
(b) 12π
[Hint : A = πr²
\(\frac{d A}{d r}\) = 2πr
\(\left.\frac{d A}{d r}\right]_{r=6}\) = 12π

Question 5.
Write the value of the definite integral \(\int_{-\pi / 2}^{\pi / 2} \sin x \mathrm{~d} x\).
Answer:
\(\int_{-\pi / 2}^{\pi / 2} \sin x \mathrm{~d} x\) = 0
(since sin x is an odd function)

Question 6.
Write degree of the differential equation \(\frac{d^2 y}{d x^2}\) + y = 0
Answer:
Degree = 1

Question 7.
If \(\vec{a}\) = 2î + ĵ + 3k̂ and \(\vec{b}\) = 8î + 4ĵ + 12k̂, then \(\vec{a}\) × \(\vec{b}\) is ___________ .
Answer:
\(\vec{a}\) × \(\vec{b}\) = \(\vec{0}\)
(since \(\vec{a}\) is parallel to \(\vec{a}\))

Question 8.
The Cartesian equation of a line is \(\frac{x}{2}\) = \(\frac{y}{4}\) = \(\frac{z}{2}\). Write the corresponding vector equation.
Answer:
\(\vec{a}\) =(0î + 0ĵ + 0k̂) + λ(2î + 4ĵ + 2k̂)

Question 9.
If A⊂B, then the value of P(B/A) is __________ .
Answer:
Since A⊂B, A∩B = A
∴ P(B/A) = \(\frac{P(A \cap B)}{P(A)}[latex] = [latex]\frac{P(A)}{P(A)}[latex] = 1

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

Question 10.
Write the principal value of sin^-1([latex]\frac{x}{2}\)).
Answer:
sin^-1(\(\frac{1}{2}\) = \(\frac{\pi}{6}\)

Question 11.
If A is a singular matrix, then the value of |A| is __________ .
Answer:
|A| = 0

Question 12.
If y = e^{log x}, find \(\frac{d y}{d x}\).
Answer:
e^{log x} = x
∴ \(\frac{d y}{d x}\) = 1

Question 13.
If l, m, n are direction cosines of a line in space, then l² + m² + n² _____________ .
Answer:
l² + m² + n² = 1

Part – II

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

Question 14.
Construct a 2 × 2 matrix A = [a_ij] whose elements are given by a_ij – 2i – j.
Answer:
A = \(\left[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right]\)
a_ij = 2i – j
a₁₁ = 1 a₁₂ = 0 a₂₁ = 3 a₂₂ = 2
∴ A = \(\left[\begin{array}{ll}
1 & 0 \\
3 & 2
\end{array}\right]\)

Question 15.
Find the intervals in which the function f given by f(x) = x² – 4x + 6 is strictly increasing.
Answer:
f(x) = x² – 4x + 6
f'(x) = 2x – 4
f'(x) = 0 ⇒ 2x – 4 = 0
2x = 4
x = 2

Intervals are (-∞, 2), (2, ∞)
In (-∞, 2), f'(x) < 0 In (2, ∞), f'(x) > 0
∴ f(x) is increasing in (2, ∞)

Question 16.
Find the equation of normal to the curve y = x³ at (1, 1)
Answer:
y = x³
\(\frac{d y}{d x}\) = 3x²
\(\left.\frac{d y}{d x}\right]_{(1,1)}\) = 3(1)² = 3
Eqn. of normal is
y – y₀ = \(\frac{-1}{f^{\prime}\left(x_0\right)}\) (x – x₀
y – 1 = \(\frac{-1}{3}\)(x – 1)
3y – 3 = – x + 1
x + 3 y – 4 = 0

Question 17.
From the differential equation corresponding to the family of straight lines y = mx where m is an arbitrary constant.
Answer:
Given y = mx ________ (1)
Diff: w.r.t x,
\(\frac{d y}{d x}\) = m
Sub. in (1)
y = \(\frac{d y}{d x}\) . x

B. Answer any two questions from ¡8 to 20. Each carries two scores. (2 × 2 = 4)

Question 18.
Find \(\frac{d y}{d x}\), if x – y = π.
Answer:
x – y = π
Diff: w.r.t x,
1 – \(\frac{d y}{d x}\) = 0
\(\frac{d y}{d x}\) = 1

Question 19.
Solve the differential equation \(\frac{d y}{d x}\) = \(\frac{x + y}{x}\)
Answer:
\(\frac{d y}{d x}\) = \(\frac{x + y}{x}\)
\(\frac{d y}{d x}\) = 1 + \(\frac{y}{x}\) ________ (1)
Which is homogeneous diff. eqn.
Put \(\frac{y}{x}\) = v
or y = vx
\(\frac{d y}{d x}\) = v + x\(\frac{d v}{d x}\)
Sub. these in (1)
v + x\(\frac{d v}{d x}\) = 1 + v
x\(\frac{d v}{d x}\) = 1
dv = \(\frac{d x}{x}\)
Which is variable separable
∴ Solution is ∫dv = ∫\(\frac{d x}{x}\)
v = log x + C
\(\frac{y}{x}\) = log x + C
or y = x(log x + C)

Question 20.
Show that the vectors \(\vec{a}\) = î – 2ĵ + 3k̂, \(\vec{b}\) = 2î + 3ĵ – 4k̂ and \(\vec{c}\) = î – 3ĵ + 5k̂ are coplanar.
Answer:
Given \(\vec{a}\) = î – 2ĵ + 3k̂,
\(\vec{b}\) = 2î + 3ĵ – 4k̂
\(\vec{c}\) = î – 3ĵ + 5k̂
[\(\vec{a}\)\(\vec{b}\)\(\vec{c}\)] = \(\left|\begin{array}{ccc}
1 & -2 & 3 \\
-2 & 3 & -4 \\
1, & -3 & 5
\end{array}\right|\)
= 1(3) + 2(-6) + 3(3)
= 3 – 12 + 9 = 0
∴ \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar.

Part – III

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

Question 21.
Show that the relation R in the set {1, 2, 3} given by R= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.
Answer:
Let A = {1, 2, 3}
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
(a, a) ∈ R for all a ∈ A
∴ R is reflexive
(1, 2) ∈ R, but (2, 1) ∉ R
∴ R is not symmetric
(1, 2) ∈ R, (2, 3) ∈ R but (1, 3) ∉ R
∴ R is not transitive

Question 22.
If A = \(\left[\begin{array}{ll}
3 & -2 \\
4 & -2
\end{array}\right]\) and I = \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\). Find K so that A² = KA – 2I.
Answer:

Question 23.
Find the, area of a parallelogram whose adjacent sides are given by the vectors:
\(\vec{a}\) = 3î + ĵ + 4k̂ and \(\vec{b}\) = î – ĵ + k̂
Answer:
\(\vec{a}\) = 3î + ĵ + 4k̂
\(\vec{b}\) = î – ĵ + k̂
\(\vec{a}\) × \(\vec{a}\) \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
3 & 1 & 4 \\
1 & -1 & 1
\end{array}\right|\)
= î(5) – ĵ(-1) + k̂(-4)
= 5î + ĵ – 4k̂
Area of parallelogram = |\(\vec{a}\) × \(\vec{a}\)|
= \(\sqrt{25 + 1 + 16}\)
= \(\sqrt{42}\) sq. units

Question 24.
Probability of solving specific problem independently by A and B are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively. 1f both try to solve the problem independently. Find the prob
ability that exactly one of them solves the problem.
Answer:
Let P(A) = \(\frac{1}{2}\), P(B) = \(\frac{1}{3}\)
∴ P(A’) = \(\frac{1}{2}\), P'(B) = \(\frac{2}{3}\)
P(exactly one solves)
= P(A∩B’∪A’∩B)
= P(A∩B’) + P(A’∩B)
= P(A) . P(B’) + P(A’) . P(B)
= \(\frac{1}{2}\) . \(\frac{2}{3}\) + \(\frac{1}{2}\) . \(\frac{1}{3}\)
= \(\frac{2}{6}\) + \(\frac{1}{6}\) = \(\frac{3}{6}\) = \(\frac{1}{2}\)

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

Question 25.
Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧.
Answer:

Question 26.
By using elementary operations, find the inverse of the matrix A = \(\left[\begin{array}{cc}
1 & 2 \\
2 & -1
\end{array}\right]\)
Answer:

Question 27.
Find \(\int_0^2 x^2 d x\) as 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 + 2h) = (0 + 2h)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 three questions from 28 to 31. Each carries four scores. (3 × 4 = 12)

Question 28.
Show that:
(i) tan^-1(\(\frac{1}{2}\)) + tan^-1(\(\frac{2}{11}\)) =
tan^-1(\(\frac{3}{4}\))
(ii) cos^-1(4x³ – 3x) = 3cos^-1 x, x ∈ [\(\frac{1}{2}\), 1]
Answer:

(ii) Put x = cos θ ⇒ θ = cos^-1 x
LHS = cos^-1 (4 cos³ θ – 3 cos θ)
= cos^-1 (cos 3θ)
= 3θ = 3 cos^-1 x = RHS

Question 29.
Verify Mean Value Theorem for the function f(x) = x² in the interval [2, 4]
Answer:
Given f(x) = x²
Since f(x) is a polynomial function, it is continuous in [2, 4]
f'(x) = 2x
It is differentiable in (2, 4)
f(a) = f(2) = 4
f(b) = f(4) = 16
f'(c) = \(\frac{f(b)-f(a)}{b-a}\) = \(\frac{16 – 4}{2}\) = \(\frac{12}{2}\) = 6
2c = 6
c = 3 ∈ (2, 4)
Hence mean value theorem verified.

Question 30.
Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis in the first quadrant.
Answer:
Given y² = x, x = 1, x = 4
Required area

Question 31.
Find the shortest distance between the lines whose vector equations are
\(\vec{r}\) = î + ĵ + λ̂,(2î – ĵ + k̂) and
\(\vec{r}\) = 2î + ĵ – k̂ + µ(3î – 5ĵ + 2k̂)
Answer:
Given \(\vec{r}\) = î + ĵ + λ̂,(2î – ĵ + k̂) and
\(\vec{r}\) = 2î + ĵ – k̂ + µ(3î – 5ĵ + 2k̂)
∴ \(\vec{a}\)₁ = î + ĵ,
\(\vec{b}\)₁ = 2î – ĵ + k̂
\(\vec{a}\)₂ = 2î + ĵ – k̂
\(\vec{b}\)₂ = 3î – 5ĵ + 2k̂
\(\vec{a}\)₂ – \(\vec{a}\)₁ = î – k̂
\(\vec{b}\)₁ × \(\vec{b}\)₂ = \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & -1 & 1 \\
3 & -5 & 2
\end{array}\right|\)
= î(3) – ĵ(1) + k̂(-7)
= 3î – ĵ – 7k̂
\(\vec{b}\)₁ × \(\vec{b}\)₂ = \(\sqrt{9 + 1 + 49}\)
= \(\sqrt{59}\)
(\(\vec{a}\)₂ – \(\vec{a}\)₁) . (\(\vec{b}\)₁ × \(\vec{b}\)₂) = 3 + 0 + 7 = 10
Shortest Distance = \(\left|\frac{\left(\bar{a}_2-\bar{a}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)}{\left|\bar{b}_1 \times \bar{b}_2\right|}\right|\)
= \(\frac{10}{\sqrt{59}}\) units

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

Question 32.
Random variable Xhas following probability distribution.

(a) Find the value of k. (2)
(b) Find P (X < 3). (2)
Answer:
(a) 0.1 + k + 2k + 2k + k = 1
6k + 0.1 = 1
6k = 1 – 0.1 = 0.9
k = \(\frac{0.9}{6}\) = \(\frac{9}{60}\) = \(\frac{3}{20}\)

(b) p(x < 3) = p(0) + p(1) + p(2)
= 0.1 + k + 2k
= 0.1 + 3k
= \(\frac{1}{10}\) + \(\frac{9}{20}\)
= \(\frac{2}{20}\) + \(\frac{9}{20}\)
= \(\frac{11}{20}\)

Question 33.
Find the Cartesian and vector equation of the plane with intercepts 2, 3, 4 on the x, y, z axis respectively.
Answer:
a = 2, b = 3, c = 4
Eqn is
\(\frac{x}{a}\) + \(\frac{y}{b}\) + \(\frac{z}{c}\) = 1
\(\frac{x}{2}\) + \(\frac{y}{3}\) + \(\frac{z}{4}\) = 1
Multiplying throughout by 12
6x + 4y + 3z = 12
\(\vec{r}\).(6î + 4ĵ + 3k̂) = 12

Part – V

Answer any two questions from 34 to 36. Carries six scores. (2 × 6 = 12)

Question 34.
Solve the following system of equations by matrix method:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
Answer:
Let A = \(\left[\begin{array}{ccc}
3 & -2 & 3 \\
2 & 1 & -1 \\
4 & -3 & 2
\end{array}\right]\), x = \(\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\), B = \(\left[\begin{array}{l}
8 \\
1 \\
4
\end{array}\right]\)
∴ System of equations can be written as AX = B
|A| = 3(-1) + 2(8) + 3(-10)
= – 3 + 16 – 30
= -17 ≠ 0
∴ Unique solution is given by X = A^-1B
A₁₁ = -1 A₁₂ = -8 A₁₃ = -10
A₂₁ = -5 A₁₂ = -6 A₂₃ = 1
A₃₁ = -1 A₃₂ = 9 A₃₃ = 7

Question 35.
Solve the following linear programming problem graphically:
Maximize Z = 4x + y subject to the constraints x + y ≤ 50, 3x + y ≤ 90, x ≥ 0 , y ≥ 0
Answer:
x + y = 50

x	0	50
y	50	0

3x + y = 90

x	0	30
y	90	0

∴ z has maximum when x = 30, y = 0
Maxmimum value = 120

Question 36.
Find
(i) \(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^2} d x\) (2)
(ii) \(\int_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\) (2)
(iii) \(\int \frac{d x}{x^2-16}\) (2)
Answer: