2nd PUC Maths Previous Year Question Paper June 2017

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Karnataka 2nd PUC Maths Previous Year Question Paper June 2017

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.
Find the identity element for the binary operation *, defined on the set Q of rational numbers by a * b = \frac{a b}{4}
Solution:
Let e be the identity element.
a * e = a
\frac{a e}{4} = a
⇒ e = 4

Question 2.
Write the values of x for which \tan ^{-1} \frac{1}{x}=\cot ^{-1} x, holds.
Solution:
x > 0

2nd PUC Maths Previous Year Question Paper June 2017

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 June 2017 Q3

Question 4.
Find the values of x for which \left|\begin{array}{ll}3 & x \\x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\4 & 1\end{array}\right|
Solution:
3 – x2 = 3 – 8
⇒ x2 = 8
⇒ x = ±√8 = ±2√2

Question 5.
Find \frac{d y}{d x}, if y = sin (x2)
Solution:
\frac{d y}{d x} = cos(x2) . 2x

2nd PUC Maths Previous Year Question Paper June 2017

Question 6.
Find ∫cos 3x dx
Solution:
\frac{\sin 3 x}{3}+c

Question 7.
Find unit vector in the direction of vector \hat{i}+\hat{j}+2 \hat{k}
Solution:
\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{\hat{i}+\hat{j}+2 \hat{k}}{\sqrt{6}}

Question 8.
Write the direction cosines of the y-axis.
Solution:
0, 1, 0

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.

2nd PUC Maths Previous Year Question Paper June 2017

Question 10.
If P(A) = 0.8 and P(B|A) = 0.4 then find P(A∩B).
Solution:
P(A∩B) = P(A) × P(B/A)= 0.8 × 0.4 = 0.32

Part – B

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

Question 11.
Show that if f : A → B and g : B → C are onto, then gof : A → C is also onto.
Solution:
Given an arbitrary element z – C, there exists a pre-image y of z under g such that g(y) = z i.e., g is onto. Further, for y ∈ B, there exists an element x in A with f(x) = y, since f is onto. Therefore, gof(x) = g(fx)) = g(y) = z, showing that gof is onto.

Question 12.
Show that 2 \tan ^{-1} x=\cos ^{-1} \frac{1-x^{2}}{1+x^{2}}, x \geq 0
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q12

Question 13.
Find the value of \sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q13

Question 14.
Using the determinant method, find the area of the triangle whose vertices are (1, 0), (6, 0) and (4, 3).
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q14

Question 15.
Differentiate (sin x)x with respect to x.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q15

2nd PUC Maths Previous Year Question Paper June 2017

Question 16.
Find \frac{d y}{d x}, if 2x + 3y = sin y.
Solution:
Differentiate both sides w.r.t x
2nd PUC Maths Previous Year Question Paper June 2017 Q16

Question 17.
Find the point on the curve \frac{x^{2}}{4}+\frac{y^{2}}{25}=1 at which the tangents are parallel to x-axis.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q17

Question 18.
Evaluate: \int \frac{\sqrt{\tan x}}{\sin x \cos x} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q18
2nd PUC Maths Previous Year Question Paper June 2017 Q18.1

Question 19.
Evaluate: \int \frac{x-3}{(x-1)^{3}} e^{x} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q19

Question 20.
Find the order and degree, if defined of the differential equation \frac{d^{4} y}{d x^{4}}+\sin \left(\frac{d^{3} y}{d x^{3}}\right)=0
Solution:
Order = 4
Degree = not defined.

2nd PUC Maths Previous Year Question Paper June 2017

Question 21.
If \vec{a} is a unit vector and (\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=8, then find \vec{x}|.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q21

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

Question 23.
Find the angle between the pair of lines given by \vec{r}=2 \hat{i}-5 \hat{j}+\hat{k}+\lambda(3 \hat{i}+2 \hat{j}+6 \hat{k}) and \vec{r}=7 \hat{i}-6 \hat{k}+\mu(\hat{i}+2 \hat{j}+2 \hat{k})

2nd PUC Maths Previous Year Question Paper June 2017

Question 24.
If A and B are two independent events, then prove that the probability of occurrence of at least one of A and B is given by 1 – P(A’) P(B’).
Solution:
P(Atleast one of A or B) = P(A ∪ B) [∵ A and B are independent events]
= P(A) + P(B) – P(A ∩ B)
= P(A) + P(B) – P(A) . P(B)
= P(A) + P(B) – (1 – P(A))
= 1 – P(A’) + P(B) P(A’)
= 1 – [P(A’) + P(B)P(A’)]
= 1 – P(A’) [1 – P(B)]
= 1 – P(A’) P(B’)
= RHS.

Part – C

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

Question 25.
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive or symmetric.
Solution:
R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
2 ∈ A, But (2, 2) ∉ R
∴ R is not reflexive
(2, 3) ∈ R, But (3, 2) ∉ R
∴ R is not Symmetric

2nd PUC Maths Previous Year Question Paper June 2017

Question 26.
Solve : tan-1 2x + tan-1 3x = \frac{\pi}{4}. Find the value of ‘x’.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q26

Since x = -1 does not satisfy the equation, x =\frac{1}{6} is the only solution of the given equation.

Question 27.
By using elementary transformations, find the inverse of A = \left[\begin{array}{rr}1 & 2 \\2 & -1\end{array}\right]
Solution:
A = I A
2nd PUC Maths Previous Year Question Paper June 2017 Q27
2nd PUC Maths Previous Year Question Paper June 2017 Q27.1

Question 28.
Find \frac{d y}{d x}, if x = a (cos t + log tan\frac{t}{2}], y = a sin t.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q28
2nd PUC Maths Previous Year Question Paper June 2017 Q28.1

2nd PUC Maths Previous Year Question Paper June 2017

Question 29.
Verify Mean Value Theorem for the function f(x) = x2 in the interval [2, 4].
Solution:
The function f(x) = x2 is continuous in {2, 4} and differentiable in {2, 4} as its derivative f(x) = 2x is defined in (2, 4).
Now f(2) = 4 and f(4) = 16. Hence
2nd PUC Maths Previous Year Question Paper June 2017 Q29
MVT states that there is a point c ∈ (2, 4) such that f'(c) = 6. But f'(x) = 2x which implies c = 3.
Thus at c = 3 ∈ (2, 4), we have f'(c) = 6.

Question 30.
Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

Question 31.
Evaluate: \int \frac{x}{(x+1)(x+2)} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q31

Question 32.
Evaluate: \int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q32
2nd PUC Maths Previous Year Question Paper June 2017 Q32.1

2nd PUC Maths Previous Year Question Paper June 2017

Question 33.
Find the area bounded by the curve y = cos x between x = 0 and x = 2π.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q33

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}}
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q34

Question 35.
Show that the position vector fo thepoint P, which divides the line 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}.

2nd PUC Maths Previous Year Question Paper June 2017

Question 36.
Find x such that the four point 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 June 2017 Q36
2nd PUC Maths Previous Year Question Paper June 2017 Q36.1

Question 37.
Find the vector and cartesian equations of the plane which passes through the points (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q37

Question 38.
A man is known to speak truth 3 out of 4 times. He throws a die 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 June 2017 Q38

Part - D

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

Question 39.
Prove that the function f : R → R defined by f(x) = 4x + 3 is invertible and find the inverse of f.

2nd PUC Maths Previous Year Question Paper June 2017

Question 40.
2nd PUC Maths Previous Year Question Paper June 2017 Q40
Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q40.1
2nd PUC Maths Previous Year Question Paper June 2017 Q40.2
2nd PUC Maths Previous Year Question Paper June 2017 Q40.3

Question 41.
Solve the following system of equations by matrix method.
x - y + z = 6
y + 3z = 11
x - 2y + z = 0
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q41

Question 42.
If y = 3 cos(log x) + 4 sin(log x), show that x2 y2 + x y1 + y = 0.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q42
2nd PUC Maths Previous Year Question Paper June 2017 Q42.1

2nd PUC Maths Previous Year Question Paper June 2017

Question 43.
Sand is pouring from a pipe at the rate of 12 cm3/s. 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 the sand cone at any time t.
2nd PUC Maths Previous Year Question Paper June 2017 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 \sqrt{a^{2}+x^{2}} with respect to x and hence evaluate \int \sqrt{1+x^{2}} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q44

Question 45.
Find the smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2.
Solution:
The smaller area enclosed by the circle, x2 + y2 = 4 and the line x + y = 2 is represented by the shaded area ACBA.
2nd PUC Maths Previous Year Question Paper June 2017 Q45
The intersection points of circle and line are A(2, 0) and B(0, 2).
Required area (shown ¡n shaded region)
= Area OACBO - Area (∆OAB)
2nd PUC Maths Previous Year Question Paper June 2017 Q45.1

2nd PUC Maths Previous Year Question Paper June 2017

Question 46.
Find the general solution of the differential equation y dx - (x + 2y2) dy = 0.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q46
2nd PUC Maths Previous Year Question Paper June 2017 Q46.1

Question 47.
Derive the equation of a line in space passing through two given points both in vector and cartesian form.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q47
2nd PUC Maths Previous Year Question Paper June 2017 Q47.1

Question 48.
If a fair coin is tossed 10 times, find the probability of
(i) exactly six heads and
(ii) at least six heads.
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q48

Part - E

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

Question 49(a).
Minimize z = -3x + 4y subject to the constraints
x + 2y ≤ 8
3x + 2y ≤ 12
x ≥ 0, y ≥ 0
by graphical method.
Solution:
Draw the graph of the line, x + 2y = 8
2nd PUC Maths Previous Year Question Paper June 2017 Q49(a)
Putting (0, 0) in the inequality x + 2y ≤ 8, we have
0 + 0 ≤ 8
0 ≤ 8 (which is true)
So, the half-plane is towards the origin,
Since, x, y ≥ 0
So, the feasible region lies in the first quadrant.
Draw the graph of the line, 3x + 2y = 12
2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).1
Putting (0, 0) in the inequality 3x + 2y ≤ 12, we have
3 x 0 + 2 x 0 ≤ 12 ⇒ 0 ≤ 12 (which is true)
So, the half-plane is towards the origin,
The feasible region is OABCO.
On solving equations x + 2y = 8 and x = 2 and y = 3
Intersection point B is (2, 3)
The corner points of the feasible region are O(0, 0), A (4, 0) and B(2, 3) and C(0, 4).
The values of Z at these points are as follows:
2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).2
Therefore, the maximum value of Z is -12 at point A(4, 0).
2nd PUC Maths Previous Year Question Paper June 2017 Q49(a).3

2nd PUC Maths Previous Year Question Paper June 2017

Question 49(b).
Prove that: \left|\begin{array}{ccc}1 & a & a^{2} \\1 & b & b^{2} \\1 & c & c^{2}\end{array}\right|=(a-b)(b-c)(c-a)
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q49(b)
2nd PUC Maths Previous Year Question Paper June 2017 Q49(b).1

Question 50(a).
Prove that:
2nd PUC Maths Previous Year Question Paper June 2017 Q50(a)
Solution:
2nd PUC Maths Previous Year Question Paper June 2017 Q50(a).1
f(x) = x3 + x cos x
f(-x) = (-x)3 - x cos (-x) = -x3 - x cos x = -[x3 + x cos x] = -f(x)
∴ f(x) is odd
\int_{-\pi / 2}^{\pi / 2}\left(x^{3}+x \cos x\right) d x=0

2nd PUC Maths Previous Year Question Paper June 2017

Question 50(b).
For what value of λ is the function defined by
2nd PUC Maths Previous Year Question Paper June 2017 Q50(b)
Solution:
When x < 0, f(x) = λ(x2 - 2x) being a polynomial is continous. When x > 0, f(x) = 4x + 1 being a polynomial is continous.
At x = 0
f(0) = λ(02 - 2 × 0) = 0
2nd PUC Maths Previous Year Question Paper June 2017 Q50(b).1
∴ LHL ≠ RHL
∴ lim f(x) does not exists for x = 0
∴ f is discontinous at x = 0 for any value of λ.

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