2nd PUC Maths Previous Year Question Paper March 2020

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Karnataka 2nd PUC Maths Previous Year Question Paper March 2020

Time: 3 Hrs 15 Min
Max. Marks: 100

Instructions:

  • The question paper has five parts namely A, B, C, D, and E. Answer all the parts.
  • Use the graph sheet for the question on Linear programming in Part E

Part – A

Answer ALL the following questions: (10 × 1 = 10)

Question 1.
Let * be the binary operation on N given by a * b = LCM of a and b. Find 5 * 7.
Solution:
5 * 7 = LCM of 5 and 7 = 35

Question 2.
Write the range of the function y = sec-1 x.
Solution:
[0, π] – {\(\frac{\pi}{2}\)}

Question 3.
If a matrix has 5 elements, what are the possible orders it can have?
Solution:
1 × 5 and 5 × 1

2nd PUC Maths Previous Year Question Paper March 2020

Question 4.
Find the values of x for which \(\left|\begin{array}{cc}x & 2 \\18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\18 & 6\end{array}\right|\)
Solution:
x2 – 36 = 36 – 36
⇒ x2 = 36
⇒ x = ±6

Question 5.
If y = tan(√x), find \(\frac{d y}{d x}\).
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q5

Question 6.
Find ∫(2x2 + ex) dx.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q6

Question 7.
Define the Negative of a vector.
Solution:
A vector whose magnitude is the same as that of a given vector but direction is opposite to that of it is called negative of a vector.

2nd PUC Maths Previous Year Question Paper March 2020

Question 8.
If a line makes angles 90°, 135° and 45° with the X, Y and Z-axes respectively, find its direction cosines.
Solution:
Let direction cosines of the line be l, m and n with the X, Y, and Z-axes respectively, and given that
α = 90°, β = 135° and γ = 45°
Then, l = cos α = cos 90° = 0
m = cos β = cos 135° = \(-\frac{1}{\sqrt{2}}\)
and n = cos γ = cos 45° = \(\frac{1}{\sqrt{2}}\)
Therefore, the direction cosines of the line are 0, \(-\frac{1}{\sqrt{2}}\) and \(\frac{1}{\sqrt{2}}\).

Question 9.
Define optimal solution in Linear Programming Problem.
Solution:
Any feasible solution of LPP which maximizes or minimizes the objective function is called an optimal solution.

Question 10.
If P(A) = \(\frac{3}{5}\) and P(B) = \(\frac{1}{5}\). find P(A ∩ B) if A and B are independent events.
Solution:
P(A ∩ B) = P(A) × P(B) = \(\frac{3}{5} \times \frac{1}{5}=\frac{3}{25}\)

Part – B

Answer any TEN questions: (10 × 2 = 20)

Question 11.
If f : R → R and g : R → R are given by f(x) = cos x and g(x) = 3x2. Find gof and fog.
Solution:
gof(x) = g[f(x)] = g[cosx] = 3cos2 x
fog(x) = f[g(x)] = f[3x2] = cos 3x2

2nd PUC Maths Previous Year Question Paper March 2020

Question 12.
Prove that cot-1(-x) = π – cot-1 x, ∀ x ∈ R.
Solution:
Let cot-1(-x) = α
⇒ -cot α = -x
⇒ -cot α = x
⇒ cot(π – α) = x
⇒ π – α = cot-1 x
⇒ α = π – cot-1 x
⇒ cot-1 (-x) = π – cot-1 x

Question 13.
Find the value of sin-1(sin \(\frac{3 \pi}{5}\))
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q13

Question 14.
Find the area of the triangle whose vertices are (-2, -3), (3, 2) and (-1, -8) using the determinant method.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q14

Question 15.
Find \(\frac{d y}{d x}\), if sin2 x + cos2 y = 1.
Solution:
2 sin x cos x + 2 cos y(-sin y) \(\frac{d y}{d x}\) = 0
⇒ \(\frac{d y}{d x}=\frac{2 \sin x \cos x}{2 \sin y \cos y}=\frac{\sin 2 x}{\sin 2 y}\)

Question 16.
If y = xx, find \(\frac{d y}{d x}\).
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q16

Question 17.
Find the interval in which the function f given by f(x) = x2 – 4x + 6 is strictly decreasing.
Solution:
We have f(x) = x2 – 4x + 6
⇒ f'(x) = 2x – 4
Therefore, f'(x) = 0 gives x = 2.
Now the point x = 2 divides the real line into two disjoint intervals namely (-∞, 2) and (2, ∞).
In the interval (-∞, 2), f'(x) = 2x – 4 < 0. Therefore, f is strictly decreasing in this interval. Also, in the interval (2, ∞) f'(x) > 0 and so the function f is strictly increasing in this interval.

2nd PUC Maths Previous Year Question Paper March 2020

Question 18.
Find ∫cot x log(sin x) dx.
Solution:
Put log sin x = t
2nd PUC Maths Previous Year Question Paper March 2020 Q18

Question 19.
Find ∫x sec2 x dx.
Solution:
x tan x – log |sec x| + c

Question 20.
Find the order and degree (if defined) of the differential equation:
2nd PUC Maths Previous Year Question Paper March 2020 Q20
Solution:
Order = 2, Degree is not defined.

Question 21.
Find the projection of the vector \(\vec{a}=\hat{i}+3 \hat{j}+7 \hat{k}\) on the vector \(\vec{b}=7 \hat{i}-\hat{j}+8 \hat{k}\).
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q21

Question 22.
Find the area of the parallelogram whose adjacent sides are determined by the vectors \(\vec{a}=\hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}\)
Solution:
Area = \(|\vec{a} \times \vec{b}|\) = 15√2 sq.units

Question 23.
Find the equation of the plane with intercepts 2, 3 and 4 on the X, Y and Z-axes respectively.
Solution:
\(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1\)

2nd PUC Maths Previous Year Question Paper March 2020

Question 24.
A random variable X has the following probability distribution:
2nd PUC Maths Previous Year Question Paper March 2020 Q24
Find the value of K.
Solution:
ΣP(x) = 1
⇒ 0.1 + K + 2K + 2K + K = 1
⇒ 0.1 + 6K = 1
⇒ 6K = 1 – 0.1 = 0.9
⇒ K = 0.9/6 = 0.15

Part – C

Answer any TEN questions: (10 × 3 = 30)

Question 25.
Show that the relation R defined in the set A of all triangles as R = {(T1, T2); T1 is similar to T2} is an equivalence relation.
Solution:
T1 is similar to T2 ⇒ R is reflexive.
Let (T1, T2) ∈ R ⇒ T1 is 111 to T2
⇒ T2 111 T1
⇒ (T2, T1) ∈ R
∴ R is symmetric
Let (T1, T2) ∈ R; (T2, T3) ∈ R
⇒ T1 is 111 to T2 and T2 is 111 to T3
⇒ T1 is 111 to T3
⇒ (T1, T2) ∈ R
∴ R is transitive
∴ R is an equivalence relation.

2nd PUC Maths Previous Year Question Paper March 2020

Question 26.
Prove that \(2 \tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{7}\right)=\tan ^{-1}\left(\frac{31}{17}\right)\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q26
2nd PUC Maths Previous Year Question Paper March 2020 Q26.1

Question 27.
If F(x) = \(\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\\sin x & \cos x & 0 \\0 & 0 & 1\end{array}\right]\) then show that F(x) F(y) = F(x + y).
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q27

Question 28.
If x = 2at2, y = at4 then find \(\frac{d y}{d x}\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q28

Question 29.
Verify Mean Value Theorem for the function f(x) = x2 – 4x – 3, x ∈ [1, 4].
Solution:
f(x) is a polynomial in x.
It is continuous in {1, 4} and differentiable in {1, 4) and f'(x) = 2x – 4.
There exists at least one value c ∈ (1, 4) such that f'(c) = \(\frac{f(b)-f(a)}{b-a}\)
a = 1, f(a) = f(1) = 12 – 4(1) – 3 = 1 – 4 – 3 = 1 – 7 = -6
b = 4, f(b) = f(4) = 42 – 4(4) – 3 = 16 – 16 – 3 = -3
f'(c) = 2c – 4
2nd PUC Maths Previous Year Question Paper March 2020 Q29
Mean Value theorem is verified.

Question 30.
Use differential to approximate √36.6.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q30

Question 31.
Find \(\int \frac{(x-3)^{e^{x}}}{(x-1)^{3}} d x\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q31
2nd PUC Maths Previous Year Question Paper March 2020 Q31.1

Question 32.
Evaluate: \(\int_{0}^{\pi / 2} \cos ^{2} x d x\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q32

Question 33.
Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axes in the first quadrant.
Solution:
The given curve x2 = 4y is a parabola which is symmetrical about Y-axis (∵ it contains even power of x) only and passes through the origin.
The area of the region bounded by the curve x2 = 4y, y = 2 and y = 4 and the Y-axis are shown in the figure.
2nd PUC Maths Previous Year Question Paper March 2020 Q33
Required area (shaded region) = \(\int_{y=a}^{y=b}|x| d y\)
(Here, |x| = √4y and a = 2, b = 4)
2nd PUC Maths Previous Year Question Paper March 2020 Q33.1

Question 34.
Find the equation of a curve passing through the point (-2, 3), given that the slope of the tangent to the curve at any point (x, y) is \(\frac{2 x}{y^{2}}\).

2nd PUC Maths Previous Year Question Paper March 2020

Question 35.
Find a unit vector perpendicular to each of the vector \((\vec{a}+\vec{b})\) and \((\vec{a}+\vec{b})\) where \(\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}\).
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q35
2nd PUC Maths Previous Year Question Paper March 2020 Q35.1

Question 36.
Find x such that the four points A(3, 2, 1), B(4, x, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q36

Question 37.
Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Question 38.
A man is known to speak truth 3 out of 4 times. He throws a dice and reports that it is a six. Find the probability that it is actually a six.
Solution:
P(S1) = Probability that six occurs = \(\frac{1}{6}\)
P(S2) = Probability that six does not occur = \(\frac{5}{6}\)
P(E | S1) = Probability that man reports that six occurs when six has actually on the die = Probability that the truth occurs = \(\frac{3}{4}\)
2nd PUC Maths Previous Year Question Paper March 2020 Q38

Part – D

Answer any SIX questions: (6 × 5 = 30)

Question 39.
Show that the function f : R → R given by f(x) = 4x + 3 is invertible. Find the inverse of f.
Solution:
Let f(x1) = f(x2) = 4x1 + 3 = 4x2 + 3
⇒ 4x1 = 4x2
⇒ x1 = x2
∴ f is one-one.
Let y ∈ Y ∃ x ∈ N such that y = 4x + 3
∴ f is onto.
∴ f is bijective.
Consider an arbitrary element y of Y. By the definition of Y, y = 4x + 3, for some x in the domain N.
x = \(\frac{y-3}{4}\)
Define a function g : Y → N by g(y) = \(\frac{y-3}{4}\)
2nd PUC Maths Previous Year Question Paper March 2020 Q39
⇒ gof = IN and fog = IY
⇒ f is invertible and g is the inverse of f.
∴ f-1 = g.

2nd PUC Maths Previous Year Question Paper March 2020

Question 40.
If A = \(\left[\begin{array}{ccc}1 & 2 & -3 \\5 & 0 & 2 \\1 & -1 & 1\end{array}\right]\), B =\(\left[\begin{array}{ccc}3 & -1 & 2 \\4 & 2 & 5 \\2 & 0 & 0\end{array}\right]\) and C =\(\left[\begin{array}{ccc}4 & 1 & 2 \\0 & 3 & 2 \\1 & -2 & 3\end{array}\right]\) then compute (A + B) and (B – C). Also verify that A + (B – C) = (A + B) – C. Also verify (A + B) C = AC + BC.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q40
2nd PUC Maths Previous Year Question Paper March 2020 Q40.1

Question 41.
Solve the following system of linear equations by matrix method:
2x + 3y + 3z = 5
x – 2y + z = -4
3x – y – 2z = 3
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q41
2nd PUC Maths Previous Year Question Paper March 2020 Q41.1
x = 1, y = 2, z = -1

Question 42.
If y = (tan-1 x)2, show that (x2 + 1)2 y2 + 2x(x2 + 1) y1 = 2
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q42

Question 43.
Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Solution:
Let r be the radius, h be the height and V be the volume of sand cóne at any time t.
2nd PUC Maths Previous Year Question Paper March 2020 Q43
Hence, when the height of the sand cone is 4 cm, its height is increasing at the rate of \(\frac{1}{48 \pi}\) cm/s

Question 44.
Find the integral of \(\frac{1}{x^{2}+a^{2}}\) w.r.t x and hence evaluate \(\int \frac{1}{x^{2}+2 x+2} d x\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q44
2nd PUC Maths Previous Year Question Paper March 2020 Q44.1

Question 45.
Using the method of integration, find the area of the smaller region bounded by the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and the line \(\frac{x}{3}+\frac{y}{2}=1\)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q45

Question 46.
Find the general solution of the differential equation x\(\frac{d y}{d x}\) + 2y = x2 log x.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q46
2nd PUC Maths Previous Year Question Paper March 2020 Q46.1

2nd PUC Maths Previous Year Question Paper March 2020

Question 47.
Derive the equation of a line in space passing through a given point and parallel to a given Vector and Cartesian form.
Solution:
Vector Form
Let \(\vec{a}\) be the position vector of the given point A with respect to the origin O of the rectangular
coordinate system. Let l be the line which passes through point A and is parallel to a given vector \(\vec{b}\). Let \(\vec{r}\) be the position vector of an arbitrary point P on the line Fig.
Then \(\vec{AP}\) is parallel to the vector \(\vec{b}\), i.e., \(\overrightarrow{\mathrm{AP}}=\lambda \vec{b}\), where λ is some real number.
But \(\overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{OP}}-\overrightarrow{\mathrm{AO}}\)
i.e., \(\lambda \vec{b}=\vec{r}-\vec{a}\)
Conversely, for each value of the parameter λ, this equation gives the position vector of a point P on the line. Hence, the vector equation of the line is given by
\(\vec{r}=\vec{a}+\lambda \vec{b}\) ……. (1)
2nd PUC Maths Previous Year Question Paper March 2020 Q47
Cartesian Form
Let the coordinates of the given point A be (x1, y1, z1) and the direction ratios of the line be a, b, c.
Consider the coordinates of any point P be (x, y, z). Then \(\vec{r}=x \hat{i}+y \hat{j}+z \hat{k} ; \vec{a}=x_{1} \hat{i}+y_{1} \hat{j}+z_{1} \hat{k}\) and \(\vec{b}=a \hat{i}+b \hat{j}+c \hat{k}\)
Substituting these values in (1) and equating the coefficients of \(\hat{i}, \hat{j} \text { and } \hat{k}\), we get
x = x1 + λa ; y = y1 + λb ; z = z1 + λc …….. (2)
These are parametric equations of the line. Eliminating the parameter λ from (2), we get
\(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\)
This is tle Cartesian equation of the line.

Question 48.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \(\frac{1}{100}\). What is the probability that he will win a prize (a) exactly once (b) at least once?
Solution:
Let X denotes the number of wins
2nd PUC Maths Previous Year Question Paper March 2020 Q48
2nd PUC Maths Previous Year Question Paper March 2020 Q48.1
2nd PUC Maths Previous Year Question Paper March 2020 Q48.2

Part – E

Answer any ONE question: (1 × 10 = 10)

Question 49(a).
2nd PUC Maths Previous Year Question Paper March 2020 Q49(a)
2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).1
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).2
2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).3
2nd PUC Maths Previous Year Question Paper March 2020 Q49(a).4

Question 49(b).
2nd PUC Maths Previous Year Question Paper March 2020 Q49(b)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q49(b).1
2nd PUC Maths Previous Year Question Paper March 2020 Q49(b).2

2nd PUC Maths Previous Year Question Paper March 2020

Question 50(a).
Maximise z = 4x + y Subject to constraints: x + y ≤ 50, 3x = y ≤ 90, x ≥ 0, y ≥ 0 by graphical method.
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q50(a)
2nd PUC Maths Previous Year Question Paper March 2020 Q50(a).1
Optimal Solution:
Z is maximum at D(30, 0) & Zmax = 120

Question 50(b).
Find the value of K,
2nd PUC Maths Previous Year Question Paper March 2020 Q50(b)
Solution:
2nd PUC Maths Previous Year Question Paper March 2020 Q50(b).1