2nd PUC Maths Previous Year Question Paper March 2018

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Karnataka 2nd PUC Maths Previous Year Question Paper March 2018

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.
Define the bijective function.
Solution:
A function f: A → B is said to be bijective if it is both one-one and onto.

Question 2.
Write the principal value brach of cos-1 x.
Solution:
(o, π)

2nd PUC Maths Previous Year Question Paper March 2018

Question 3.
Construct a 2 × 2 matrix, A = [aij], whose elements are given by, aij = \frac{i}{j}
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q3

Question 4.
If A is an invertible matrix of order 2 then find |A-1|.
Solution:
|A-1| = \frac{1}{|A|}

Question 5.
If y = e^{x^{3}}, find \frac{d y}{d x}
Solution:
\frac{d y}{d x}=e^{x^{2}} \cdot 3 x^{2}

Question 6.
Find: \int \frac{x^{3}-1}{x^{2}} d x
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q6

Question 7.
Find the unit vector in the direction of the vector \vec{a}=\hat{i}+\hat{j}+2 \hat{k}
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q7

Question 8.
If a line makes angle 90°, 60° and 30°with the positive direction of x, y and z-axis respectively, find its direction cosines.
Solution:
cos 90° = 0, cos 60° = \frac {1}{2}, cos 30° = \frac { \sqrt { 3 } }{ 2 }
dc’s are 0, \frac {1}{2}, \frac { \sqrt { 3 } }{ 2 }

2nd PUC Maths Previous Year Question Paper March 2018

Question 9.
Define the optimal solution in a 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{7}{13}, P(B) = \frac{9}{13} and P(A ∩ B) = \frac{4}{13}, find P(A/B)
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q10

Part – B

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

Question 11.
Let * be a binary operation on Q, defined by a * b = \frac{a b}{2}, ∀ a, b ∈ Q

Question 12.
sin(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x) = 1 then find the value of x.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q9

Question 13.
Write the simplest from of \tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), 0 < x < \frac{\pi}{2}
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q13

Question 14.
Find the area of the triangle whose vertices are (-2, -3), (3, 2) and (-1, -8) by using the determinant method.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q14

Question 15.
Differentiate xsin x, x > 0 with respect to x.
Solution:
Let y = xsin x
Taking logarithm on both sides, we have
log y = sin x log x
2nd PUC Maths Previous Year Question Paper March 2018 Q15
2nd PUC Maths Previous Year Question Paper March 2018 Q15.1

2nd PUC Maths Previous Year Question Paper March 2018

Question 16.
Find \frac{d y}{d x}, if x2 + xy + y2 = 100
Solution:
x2 + xy + y2 = 100
Differentiating with respect to x.
2x + x \frac{d y}{d x} + y(1) + 2y \frac{d y}{d x} = 0
⇒ (x + 2y) \frac{d y}{d x} = -(2x + y)
\frac{d y}{d x}=-\frac{(2 x+3)}{x+2 y}

Question 17.
Find the slope of the tangent to the curve y = x3 – x at x = 2.
Solution:
\frac{d y}{d x} = 3x2 – 1
At x = 2, \frac{d y}{d x} = 3(4) – 1 = 11 = slope of the tangent.

Question 18.
Integrate \frac{e^{{tan}^{-1}} x}{1+x^{2}} with respect to x.
Solution:
\frac{e^{{tan}^{-1}} x}{1+x^{2}}
2nd PUC Maths Previous Year Question Paper March 2018 Q18

Question 19.
Evaluate \int_{2}^{3} \frac{x d x}{x^{2}+1}
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q19

Question 20.
Find the order and degree of the differential equation:
2nd PUC Maths Previous Year Question Paper March 2018 Q20
Solution:
Order = 3, degree = 2.

Question 21.
Find the projection of Jhe vector \hat{i}+3 \hat{j}-7 \hat{k} on the vector 7 \hat{i}+\hat{j}+8 \hat{k}
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q21

Question 22.
Find the area of the parallelogram whose adjacent sides are determined by the vectors \vec{a}=3 \hat{i}+\hat{y}+4 \hat{k} and \vec{b}=\hat{i}-\hat{j}+\hat{k}

2nd PUC Maths Previous Year Question Paper March 2018

Question 23.
Find the angle between the planes whose vector equations are \vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 and \vec{r} \cdot(3 \hat{i}+3 \hat{j}-5 \hat{k})=3
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q23

Question 24.
A random variable X has the following probability distribution:
Determine:
(i) k
(ii) P(X ≥ 2)
Solution:
The probability distribution of X is
2nd PUC Maths Previous Year Question Paper March 2018 Q24
(i) We know that \sum_{i=1}^{n} p i=1
Therefore 0.1 + k + 2k + 2k + k = 1
⇒ k = 0.15

(ii) P(you study at least two hours) = P(X ≥ 2)
= P(X = 2) + P(X = 3) + P(X = 4)
= 2k + 2k + k
= 5k
= 5 × 0.15
= 0.75
P(you study exactuly two hours) = P(X = 2)
= 2k
= 2 × 0.15
= 0.3
P(you study at most two hours) = P(X ≤ 2)
= P(X = 0) + P(X = 1) + P(X = 2)
= 0.1 + k + 2k
= 0.1 + 3k
= 0.1 +3 × 0.15
= 0.55

Part – C

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

Question 25.
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation.
Solution:
|a – a| = |o| = 0 is even
(a, a) ∈ R.
∴ R is reflexive.
Let (a, b) ∈ R ⇒ |a – b| is even
⇒ |-(a – b)| is even
⇒ |(b – a)| is even
⇒ (b, a) ∈ R
∴ R is symmetric.
Let (a, b) ∈ R and (b, c) ∈ R
⇒ |a – b| is even and |b – c| is even
⇒ a – b is even
⇒ b – c is even
⇒ a – b + b – c is even
⇒ a – c is even
⇒ |a – c| is even
∴ (a, c) ∈ R.
∴ R is transitive
∴ R is an equivalence relation.

2nd PUC Maths Previous Year Question Paper March 2018

Question 26.
Prove that 2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17} in the simplest form.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q26
2nd PUC Maths Previous Year Question Paper March 2018 Q26.1

Question 27.
By using elementary transformations, find the inverse of the matrix A = \left[\begin{array}{ll}1 & 3 \\2 & 7\end{array}\right]
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q27
2nd PUC Maths Previous Year Question Paper March 2018 Q27.1

Question 28.
If x = sin t, y = cos 2t then prove that \frac{d y}{d x} = -4 sin t

Question 29.
Verify Rolle’s theorem for the function f(x) = x2 + 2, x ∈ [-2, 2]
Solution:
The function y = x2 + 2 is Continuous in {-2, 2} and differentiable in (-2, 2). Also f(-2) = f(2) = 6 and hence the value of f(x) at -2 and 2 coincide.
Rolle’s theorem states that there is a point c ∈ (-2, 2) where f'(c) = 0. Since f'(x) = 2x, we get c = 0. Thus at c = 0, we have f'(c) = 0 and c = 0 ∈ (-2, 2).

Question 30.
Find two numbers whose sum is 24 and whose product is as large as possible.
Solution:
Let one number be x. Then, the other number is (24 – x).
(∵ Sum of the two numbers is 24)
Let y denotes the product of the two numbers. Thus, we have
y = x(24 – x) = 24x – x2
On differentiating twice w.r.t. x, we get
\frac{d y}{d x} = 24 – 2x and \frac{d^{2} y}{d x^{2}} = -2
Now, put \frac{d y}{d x} = 0
⇒ 24 – 2x = 0
⇒ x = 12
At x = 12, \frac{d^{2} y}{d x^{2}} = -2 < 0
∴ By second derivative test, x = 12 is the point of local mxima of y. Thus, the product of the number is maximum when the numbers are 12 and 24 – 12 = 12.
Hence, the numbers are 12 and 12.

2nd PUC Maths Previous Year Question Paper March 2018

Question 31.
Find: \int \frac{x d x}{(x+1)(x+2)}

Question 32.
Find: ∫ex sin x dx
Solution:
Take ex as the first function and sin x as second function. Then integrating by parts, we have
I = ∫ex sin x dx = ex (-cos x) + ∫ex cos x dx = -ex cos x + I1 (say) …… (1)
Taking ex and cos x as the first and second functions, respectively, in I1, we get
I = ex sin x – ∫ex sin x dx
Substituting the value of I1 in (i), we get
I = -ex cos x + ex sin x – 1 or 2I = ex (sin x – cos x)
Hence I = ∫ex sin x dx = \frac{e^{x}}{2} (sin x – cos x) + C

Question 33.
Find the area of the region bounded by the curve y = x2 and the line y = 4.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q33
2nd PUC Maths Previous Year Question Paper March 2018 Q33.1

Question 34.
Form the differential equation representing the family of curves y = a sin(x + b), where a, b are arbitrary constants.

Question 35.
Show that the position vector of the point P, which divides the fine joining the points A and B having position vectors \vec{a} and \vec{b} internally in the ratio m : n is \frac{m \vec{b}+n \vec{a}}{m+n}

Question 36.
Find x such that the fojir 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 2018 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).

2nd PUC Maths Previous Year Question Paper March 2018

Question 38.
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Part – D

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

Question 39.
Lef R+ be the set of all non-negative real numbers. Show that the function f : R+ → [4, ∞) given by f(x) = x2 + 4 is invertible and write the inverse of f.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q39
2nd PUC Maths Previous Year Question Paper March 2018 Q39.1

Question 40.
If A = \left[\begin{array}{rrr}0 & 6 & 7 \\-6 & 0 & 8 \\7 & -8 & 0\end{array}\right], B = \left[\begin{array}{lll}0 & 1 & 1 \\1 & 0 & 2 \\1 & 2 & 0\end{array}\right], C = \left[\begin{array}{c}2 \\-2 \\3\end{array}\right], calculate AC, BC and (A + B)C. Also verify that (A + B)C = AC + BC.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q40
2nd PUC Maths Previous Year Question Paper March 2018 Q40.1
2nd PUC Maths Previous Year Question Paper March 2018 Q40.2

Question 41.
Solve the following system of linear equations by matrix method.
x – y + 2z = 7, 3x + 4y – 5z = -5 and 2x – y + 3z = 12.

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 2018 Q42

Question 43.
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground is 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 the sand cone at any time t.
2nd PUC Maths Previous Year Question Paper March 2018 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.

2nd PUC Maths Previous Year Question Paper March 2018

Question 44.
Find the integral of \frac{1}{x^{2}+a^{2}} with respect to x and hence find \int \frac{1}{x^{2}-6 x+13} d x
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q44

Question 45.
Using integration find the area of the region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1).
Solution:
Area = Area of ∆ABD + Area of trapezium BDEC – Area of ∆AEC
Now equation of the sides AB, BD and CA are given by
y = 2(x – 1), y = 4 – x, y = \frac{1}{2}(x – 1) respectively
Hence, area of ∆
2nd PUC Maths Previous Year Question Paper March 2018 Q45
2nd PUC Maths Previous Year Question Paper March 2018 Q45.1

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 2018 Q46
2nd PUC Maths Previous Year Question Paper March 2018 Q46.1

Question 47.
Derive the equation of the line in space passing through two given points, both in vector and Cartesian form.
Solution:
Vector Form
Let \overrightarrow { a } and \overrightarrow { b } be the position vectors of two points A (x1, y1, z1) and B(x2, y2, z2) respectively that are lying on a line.
Let \overrightarrow { r } be the position vector of an arbitrary point P(x, y, z), then P is a point on the line if and only if \overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}} and \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}} are collinear vectors. Therefore, P is on the line if and only
2nd PUC Maths Previous Year Question Paper March 2018 Q47
2nd PUC Maths Previous Year Question Paper March 2018 Q47.1
2nd PUC Maths Previous Year Question Paper March 2018 Q47.2

Question 48.
If a fair coin is tossed 10 times, find the probability of
(i) exactly six heads
(ii) at least six heads.
Solution:
Let X denote the number of heads in an experiment of 10 trials.
2nd PUC Maths Previous Year Question Paper March 2018 Q48

Part – E

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

Question 49(a).
Prove that \int_{0}^{a} f(x) d x=\int_{0}^{1} f(a-x) d x and hence evaluate \int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q49(a)
2nd PUC Maths Previous Year Question Paper March 2018 Q49(a).1

Question 49(b).
2nd PUC Maths Previous Year Question Paper March 2018 Q49(b)
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q49(b).1

Question 50(a).
Solve the following problem graphically:
Minimise and Maximise: z = 3x + 9y
Subject to the constraints:
x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x ≥ 0, y ≥ 0.
Solution:
Let us graph the feasible region of the system of linear inequalities (2) to (5).
The coordinates of the corner point A, B, C and D are (0, 10), (5, 5), (15, 15) and (0, 20) respectively.
2nd PUC Maths Previous Year Question Paper March 2018 Q50(a)
2nd PUC Maths Previous Year Question Paper March 2018 Q50(a).1
We now find the minimum and maximum value of Z. From the table, we find that the minimum value of Z is 60 at point B (5, 5) of the feasible region.
The maximum value of Z on the feasible region occurs at the two comer points C(15, 15) and D(0, 20) and it is 180 in each case.

2nd PUC Maths Previous Year Question Paper March 2018

Question 50(b).
Find the relationship between a and b so that function f defined by 2nd PUC Maths Previous Year Question Paper March 2018 Q50(b) is continuous at x = 3.
Solution:
2nd PUC Maths Previous Year Question Paper March 2018 Q50(b).1
3a + 1 = 3b + 3 = 3a + 1
⇒ 3a + 1 = 3b + 3
⇒ 3a = 3b + 3 – 1
⇒ 3a = 3b + 2
⇒ a = b + \frac{2}{3}

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