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

Question 3.
Construct a 2 × 2 matrix, A = [a_ij], whose elements are given by, a_ij = \(\frac{i}{j}\)
Solution:

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

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.

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:

Question 13.
Find the value of \(\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)\)
Solution:

Question 14.
Using the determinant method, find the area of the triangle whose vertices are (1, 0), (6, 0) and (4, 3).
Solution:

Question 15.
Differentiate (sin x)^x with respect to x.
Solution:

Question 16.
Find \(\frac{d y}{d x}\), if 2x + 3y = sin y.
Solution:
Differentiate both sides w.r.t x

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:

Question 18.
Evaluate: \(\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x\)
Solution:

Question 19.
Evaluate: \(\int \frac{x-3}{(x-1)^{3}} e^{x} d x\)
Solution:

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.

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:

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 \(\vec{r}=7 \hat{i}-6 \hat{k}+\mu(\hat{i}+2 \hat{j}+2 \hat{k})\)

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

Question 26.
Solve : tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\). Find the value of ‘x’.
Solution:

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

Question 28.
Find \(\frac{d y}{d x}\), if x = a (cos t + log tan\(\frac{t}{2}\)], y = a sin t.
Solution:

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

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:

Question 32.
Evaluate: \(\int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x\)
Solution:

Question 33.
Find the area bounded by the curve y = cos x between x = 0 and x = 2π.
Solution:

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:

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 \(\frac{m \vec{b}+n \vec{a}}{m+n}\).

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:

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:

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 = \(\frac{3}{4}\)

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.

Question 40.

Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.
Solution:

Question 41.
Solve the following system of equations by matrix method.
x – y + z = 6
y + 3z = 11
x – 2y + z = 0
Solution:

Question 42.
If y = 3 cos(log x) + 4 sin(log x), show that x² y₂ + x y₁ + y = 0.
Solution:

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.

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:

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.

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)

Question 46.
Find the general solution of the differential equation y dx – (x + 2y²) dy = 0.
Solution:

Question 47.
Derive the equation of a line in space passing through two given points both in vector and cartesian form.
Solution:

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:

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

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

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:

Therefore, the maximum value of Z is -12 at point A(4, 0).

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:

Question 50(a).
Prove that:

Solution:

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
∴ \(\int_{-\pi / 2}^{\pi / 2}\left(x^{3}+x \cos x\right) d x=0\)

Question 50(b).
For what value of λ is the function defined by

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

∴ LHL ≠ RHL
∴ lim f(x) does not exists for x = 0
∴ f is discontinous at x = 0 for any value of λ.