2nd PUC Maths Previous Year Question Paper March 2017

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Karnataka 2nd PUC Maths Previous Year Question Paper March 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.
Let * be the binary operation on N, given by a * b = LCM of a and b. Find 20 * 16.
Solution:
20 * 16 = LCM of 20 and 16 = 80

Question 2.
Find the principal value of cosec-1 (-√2)
Solution:
cosec-1 (-√2) = -cosec √2 = -\pi / 4

Question 3.
Construct a 2 × 2 matrix, A = [aij], where elements are given by, aij = \frac{i}{j}
Solution:
a11 = \frac{1}{1} = 1
a12 = \frac{1}{2}
a21 = \frac{2}{1} = 2
a22 = \frac{2}{2} = 1
∴ A = \left[\begin{array}{ll}1 & 2 \\1 / 2 & 1\end{array}\right]

2nd PUC Maths Previous Year Question Paper March 2017

Question 4.
If A is a square matrix with |A| = 8 then find the value of |AA’|.
Solution:
|AA’| = |A| |A’| = (8) (8) = 64

Question 5.
If y = cos√x, find \frac{d y}{d x}
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q5

Question 6.
Find \int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q6

Question 7.
Define collinear vectors.
Solution:
Collinear vectors Two or moe non-zero vectors are said to be collinear if they are parallel to the same line. Two vectors a and b are collinear if a = λb for some scalar λ.

2nd PUC Maths Previous Year Question Paper March 2017

Question 8.
Find the direction cosines of a line which makes equal angles with the positive co-ordinates axes.
Solution:
α = β = r
2nd PUC Maths Previous Year Question Paper March 2017 Q8

Question 9.
Define a feasible region in a linear programming problem.
Solution:
The region containing the set of points satisfying all the constraints of an LPP is called the feasible region.

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

Part – B

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

Question 11.
If f : R → R defined by f(x) = 1 + x2, then show that f is neither 1-1 nor onto.
Solution:
f(1) = 1 + 12 = 2
f(-1) = 1 + (-1)2 = 2
∴ f(1) = f(-1) But 1 ≠ -1
∴ f is not one-one
Let y ∈ R ∃ x ∈ R such that
f(x) = y
⇒ 1 + x2 = y
⇒ x2 = y – 1
⇒ x = \sqrt{y-1}
if y = 0 then x = \sqrt{-1} \notin \mathrm{R}
∴ 0 has no pre-image
∴ f is not onto.

2nd PUC Maths Previous Year Question Paper March 2017

Question 12.
2nd PUC Maths Previous Year Question Paper March 2017 Q12.1
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q12

Question 13.
Solve the equaton \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x,(x>0)
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q13

Question 14.
Find the values of k, if the area of the triangle is 4 sq. units and vertices are (k, 0), (4, 0) and (0, 2) using determinant.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q14
⇒ -2k + 8 = 8
⇒ 2k =
⇒ k = 0
on taking -ve sign we get -2k + 8 = -8
⇒ 2k = 16
⇒ k = 8
∴ k = 0, 8.

Question 15.
If ax + by2 = cos y, find \frac{d y}{d x}
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q15

Question 16.
Verify Rolle’s theorem for the function f(x) = x2 + 2x – 8, x ∈ [-4, 2]
Solution:
f(x) is a polynomial in x. Flence it is continuous over {-4, 2} and differentiable over (-4, 2).
f(-4) = (-4)2 + 2(-4) – 8 = 16 – 8 – 8 = 0
f(2) = 4 + 4 – 8 = 0
∴ f(-4) = f(2)
∴ All the conditions of the Rolle’s theorem are satisfied.
∴ there exists a c ∈ [-4, 2] such that
f'(c) = 0
f'(x) = 2x + 2
f'(c) = 2c + 2
f'(c) = 0
⇒ 2c + 2 = 0
⇒2c = -2
⇒ c = -1 ∈ [-4, 2]
∴ Rolle’s theorem is verified.

2nd PUC Maths Previous Year Question Paper March 2017

Question 17.
Find the approximate change in the volume of a cube of side x metres caused by increasing the side by 3%.
Solution:
v = x3
∆x = 3% of x = (0.03)x
\frac{d v}{d x} = 3x2
∆v = \frac{d v}{d x} . ∆x
= (3x2) (0.03) x
= 0.09 x3 m3

Question 18.
Integrate \frac{\tan ^{4} \sqrt{x} \sec ^{2} \sqrt{x}}{\sqrt{x}} with respect to x.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q18

Question 19.
Evaluate \int_{0}^{2 / 3} \frac{d x}{4+9 x^{2}}
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q19

Question 20.
Find the order and degree of the differential equation \left( \frac { dy }{ dx } \right) ^{ 2 }+\frac { dy }{ dx } { sin }^{ 2 }y=0
Solution:
Order = 1, Degree = 2.

2nd PUC Maths Previous Year Question Paper March 2017

Question 21.
Find the position vector of a point R which divides the lipe joining two points P and Q whose position vectors are \hat{i}+2 \hat{j}-\hat{k} and -\hat{i}+\hat{j}+\hat{k} respectively in theratio 2 : 1
(i) Internally.
(ii) Externally.
Solution:
The position vector of a point R divided the line segment joining two points P and Q in the ratio m : n is given by
2nd PUC Maths Previous Year Question Paper March 2017 Q21
2nd PUC Maths Previous Year Question Paper March 2017 Q21.1

Question 22.
Find the area of the parallelogram whose adjacent sides are determined by the vector \vec{a}=\hat{i}-\hat{j}+3 \hat{k} and \vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q22
Area of the parallelogram = |\vec{a} \times \vec{b}| = √450 sq units.

Question 23.
Find the vector and Cartesian equation of the line that passes through the points (3, -2, 5) and (3, -2, 6).
Solution:
Let a and b be the position vectors of points (3, -2, -5), and (3, -2, 6) respectively.
\vec{a}=3 \hat{i}-2 \hat{j}-5 \hat{k} and b=3 \hat{i}-2 \hat{j}+6 \hat{k}
We know that the vector equation of a line passing through the points having position vectors a and b is r=\vec{a}+\lambda(\vec{b}-\vec{a})
2nd PUC Maths Previous Year Question Paper March 2017 Q23

2nd PUC Maths Previous Year Question Paper March 2017

Question 24.
Find the probability distribution of the number of heads in two tosses of a coin.
Solution:
When one coin is tossed twice, the sample space is
S = {HH, HT, TH, TT}.
Let X denotes, the number of heads in any outcome in S,
X(HH) = 2, X(HT) = 1, X(TH) = 1 and X(TT) = 0
Therefore, X can take the value of 0, 1 or 2. It is known that
P(HH) = P(HT) = P(TH) = P(TT) = \frac{1}{4}
P(X = 0) = P (tail occurs on both tosses) = P({TT}) = \frac{1}{4}
P(X = 1) = P(one head and one tail occurs) = P({TH, HT}) = \frac{2}{4} = \frac{1}{2}
and P(X = 2) = P (head occurs on both tosses) = P({HH}) = \frac{1}{4}
Thus, the required probability distribution is as follows.
2nd PUC Maths Previous Year Question Paper March 2017 Q24

Part – C

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

Question 25.
Show that the relation R in R (set of real numbers) is defined as R = {(a, b) : a ≤ b} is reflexive and transitive but not symmetric.
Solution:
a ≤ a is always true
∴ (a, a) R a R
∴ R is reflexive
Let (a, b) ∈ R ⇒ a ≤ b which does not imply b ≤ a
∴ (b, a) ∉ R
∴ R is not symmetric
Let (a, b) ∈ R and (b, c) ∈ R
⇒ a ≤ b and b ≤ c
⇒ a ≤ c
⇒ (a, c) ∈ R
∴ R is transitive

2nd PUC Maths Previous Year Question Paper March 2017

Question 26.
Write \tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right), x ≠ 0 in the simplest form.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q26

Question 27.
If A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if AB = BA.
Solution:
Let A and B are symmetric
∴ A’ = A and B’ = B
Let AB is symmetric
∴ (AB)’ = AB, B’A’ = AB, BA = AB
Conversely, Let AB = BA
(AB)’ = B’A’ = BA = AB
∴ AB is symmetric

Question 28.
Differentiate (log x) cos x with respect to x.
Solution:
Let cos x
y = (log x) cos x
log y = log (log x)
log y = cos x (log x)
2nd PUC Maths Previous Year Question Paper March 2017 Q28

Question 29.
Differentiate sin2 x with respect to ecos x.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q29

Question 30.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Solution:
Let the two numbers be x, y and P = xy3
Given x + y = 60 ⇒ x = 60 – y
On putting this value in P = xy3, we get
P = (60 – y) y3 ⇒ P = 60y3 – y4
On differentiating twice w.r.t. y, we get
\frac{d P}{d y} = 180y2 – 4y3
and \frac{d^{2} P}{d y^{2}} = 360y – 12y2
For maxima, we must have \frac{d P}{d y} = 0
⇒ 180y2 – 4y3 = 0
⇒ 4y2(45 – y) = 0
⇒ y = 0, 45
But y ≠ 0, so y = 45
At y = 45, \left(\frac{d^{2} P}{d y^{2}}\right)_{y=45} = 360 × 45 – 12 × (45)2
=16200 – 24300
= -8100 < 0
⇒ P has afocal maxima at y = 45
∴ By second derivative test, x = 45 is a point of local maxima of P. Thus, the function xy3 is maximum when y = 45 and x = 60 – 45 = 15.
Hence, the required numbers are 15 and 45.

2nd PUC Maths Previous Year Question Paper March 2017

Question 31.
Evaluate \int \frac{2 x}{x^{2}+3 x+2} d x
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q31

Question 32.
Evaluate ∫ex sin x dx.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q32

Question 33.
Find the area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q33
2nd PUC Maths Previous Year Question Paper March 2017 Q33.1

Question 34.
Form the differential equation of the family of circles having a centre on y-axis and radius 3 units.
Solution:
The equation of the family circles having centre on y-axis and radius 3 unit is
x2 + (y – b)2 = 9 …….(i)
On differentiating Eq. (i) w.r.t x, we get
2x + 2(y – b) y’ = 0
⇒ y – b = -\frac{x}{y^{\prime}} ……. (ii)
On substituting this value of (y – b) in Eq. (i) we get
x^{2}+\left(-\frac{x}{y^{\prime}}\right)^{2}=9
⇒ x2[(y’)2 + 1] = 9(y’)2
⇒ (x2 – 9)(y’)2 + x2 = 0
which is the required differential equation.

2nd PUC Maths Previous Year Question Paper March 2017

Question 35.
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 2017 Q35
2nd PUC Maths Previous Year Question Paper March 2017 Q35.1

Question 36.
2nd PUC Maths Previous Year Question Paper March 2017 Q36
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q36.1
2nd PUC Maths Previous Year Question Paper March 2017 Q36.2

Question 37.
Find the shortest distance between the lines \vec{r}=\hat{i}+2 \hat{j}+\hat{k}+\lambda(\hat{i}-\hat{j}+\hat{k}) and \vec{r}=2 \hat{i}-\hat{j}-\hat{k}+\mu(2 \hat{i}+\hat{j}+2 \hat{k})
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q37
2nd PUC Maths Previous Year Question Paper March 2017 Q37.1

Question 38.
Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
Solution:
When dice is thrown, number of observations in the sample space S = 6 × 6 = 36 (equally likely sample events)
i.e., n(S) = 36
Let E : set of numbers in which numbers appearing on the two dice are different
Then, E = {(1, 3), (2, 2), (3, 1)}
⇒ n(E) = 3
2nd PUC Maths Previous Year Question Paper March 2017 Q38
n(F) = 30
Here, F contains all points of S except {(1, 1), (2, 2), (3, 3),(4, 4), (5, 5), (6, 6)}
E ∩ F = {(1, 3), (3, 1)}
2nd PUC Maths Previous Year Question Paper March 2017 Q38.1

Part – D

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

Question 39.
Let f : N → R be a function defined as f(x) = 4x2 + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Find the inverse of f.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q39
2nd PUC Maths Previous Year Question Paper March 2017 Q39.1
2nd PUC Maths Previous Year Question Paper March 2017 Q39.2

Question 40.
If A = \left[\begin{array}{lll}1 & 0 & 2 \\0 & 2 & 1 \\2 & 0 & 3\end{array}\right] prove that A3 – 6A2 + 7A + 2I = 0.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q40

2nd PUC Maths Previous Year Question Paper March 2017

Question 41.
Solve the following system of linear equations by matrix method.
x – y + 2z = 1, 2y – 3z = 1 and 3x – 2y + 4z = 2.
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q41
2nd PUC Maths Previous Year Question Paper March 2017 Q41.1

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 2017 Q42

Question 43.
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute when x = 8 cm and y = 6 cm, find the rate of change of
(i) The perimeter and
(ii) The are of the rectangle.
Solution:
At any instant of time t, let length, breadth, perimeter and area of the rectangle are x, y, P and A respectively, then
P = 2(x + y) and A = xy …… (i)
It is given that \frac{d x}{d t} = -5 cm/mm and \frac{d y}{d t} = -5 = 4 cm/min
(-ve sign shows that the length is decreasing)
(i) Now, P = 2(x + y).
On differentiating w.r.t. t, we get Rate of change of pert meter
\frac{d P}{d t}=2\left(\frac{d x}{d t}+\frac{d y}{d t}\right)
= 2(-5 + 4) cm/min
= -2 cm/min [∴ \frac{d x}{d t} = -5 and \frac{d y}{d t} = 4]
Hence, perimeter of the rectangle is decreasing (-ve sign) at the rate of 2 cm/min.
(ii) Here, area of rectangle A = xy.
On differentiating w.r.t. t, we get
Rate of change \frac{d A}{d t}=x \frac{d y}{d t}+y \frac{d x}{d t}
= 8 × 4 + 6 × (5) [∴ \frac{d x}{d t} = -5 and \frac{d y}{d t} = 4]
= 32 – 30
= 2 cm2/min
Hence, area of the rectangle is increasing at the rate of 2 cm2/min.
Note: It rate of change is increasing, we take positive sign and if the rate of change is decreasing, then we take a negative sign.

2nd PUC Maths Previous Year Question Paper March 2017

Question 44.
Find the integral of \sqrt{x^{2}-a^{2}} with respect to x and hence evaluate \int \sqrt{x^{2}-8 x+7} dx
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q44
2nd PUC Maths Previous Year Question Paper March 2017 Q44.1
2nd PUC Maths Previous Year Question Paper March 2017 Q44.2
2nd PUC Maths Previous Year Question Paper March 2017 Q44.3

Question 45.
Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.
Solution:
Given equation of sides of the triangle are y = 2x + 1, y = 3x + 1 and x = 4.
On solving these equations, we obtain the vertices of the triangle as A(0, 1), B(4, 13) and C(4, 9).
Required area (shown in the shaded region)
= Area (OLBAO) – Area (OLCAO)
2nd PUC Maths Previous Year Question Paper March 2017 Q45
= 8 sq.units
2nd PUC Maths Previous Year Question Paper March 2017 Q45.1

2nd PUC Maths Previous Year Question Paper March 2017

Question 46.
Solve the differential equation \cos ^{2} x \frac{d y}{d x}+y=\tan x\left(0 \leq x<\frac{\pi}{2}\right)
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q46

Question 47.
Derive the equation of a plane perpendicular to a given vector and passing through a given point both in vector and Cartesian form.
Solution:
Vector Form
Let a plane pass through a point A with position vector \overrightarrow{\mathrm{a}} and perpendicular to the vector \overrightarrow{\mathrm{N}}
Let \overrightarrow{\mathrm{r}} be the position vector of any point P(x, y, z) in the plane.
Then the point P lies in the plane if and only if \overline{\mathrm{AP}} is perpendicular to \overrightarrow{\mathrm{N}}
i.e. \overline{\mathrm{AP}} \cdot \overline{\mathrm{N}}=0
But \overline{\mathrm{AP}}=\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}
Therefore (\overrightarrow{\mathbf{r}}-\overrightarrow{\mathrm{a}}) \cdot \overrightarrow{\mathrm{N}}=0 …….. (1)
This is the vector equation of the plane.
2nd PUC Maths Previous Year Question Paper March 2017 Q47
Cartesian form
Let the given point A be (x1, y1, z1), P be (x, y, z) and direction ratios of
\overline{\mathrm{N}} are A, B and C. Then
\overrightarrow{\mathrm{a}}=x_{1} \hat{\mathrm{i}}+\mathrm{y}_{1} \hat{\mathrm{j}}+\mathrm{z}_{1} \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{r}}=x \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}} and \overline{\mathrm{N}}=\mathrm{A} \hat{\mathrm{i}}+\mathrm{B} \hat{\mathrm{j}}+\mathrm{C} \hat{\mathrm{k}}
2nd PUC Maths Previous Year Question Paper March 2017 Q47.1

Question 48.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one will fuse after 150 days of use.
Solution:
Let X represents the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials. The trials are Bernoulli trials, n = 5p = p (success) = 0.05, q = 1 – p = 1 – 0.05 = 0.95
2nd PUC Maths Previous Year Question Paper March 2017 Q48

Part – E

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

Question 49(a).
2nd PUC Maths Previous Year Question Paper March 2017 Q49(a)
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q49(a).1

2nd PUC Maths Previous Year Question Paper March 2017

Question 49(b).
Show that \left|\begin{array}{ccc}x & x^{2} & y z \\y & y^{2} & z x \\z & z^{2} & x y\end{array}\right| = (x – y) (y – z) (z – x) (xy + yz + zx)
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q49(b)
2nd PUC Maths Previous Year Question Paper March 2017 Q49(b).1
= (y + x)(y – x)(z – x)(z3 + x2 + xz) – (z + x)(z – x)(y – x)(y2 + x2 + xy)
= (y – x)(z – x)[(y – x)(z3 + x2 + xz) – (z + x)(y2 + x2 + xy)]
= (y – x)(z – x) [(yz2 + yx2 + xyz + xz2 + x3 + x2z – zy2 – zx2 – xyz – xy2 – x3 – x2y]
= (y – x)(z – x)[yz2 + zy2 + xz2 – xy2]
= (y – x) (z – x) [yz (z – y) + x (z – y)(z + y)]
= (y – x) (z – x) [(z – y) (xy + yz + zx)]
= (x – y) (y – z) (z – x) (xy + yz + zx)
= RHS
Hence proved.

Question 50(a).
Minimize and maximize z = 600x + 400y
Subject to the constraints:
x + 2y ≤ 12
2x + y ≤ 12
4x + 5y ≥ 20 and x ≥ 0, y ≥ 0 by graphical method.
Solution:
x + 2y = 12
y = 0 ⇒ x = 12 ∴ P (12, 0)
x = 0 ⇒ y = 6 ∴ D (0, 6)
2x + 2y = 12
y = 0 ⇒ x = 6 ∴ B (6, 0)
x = 0 ⇒ y = 12 ∴ F (0, 12)
4x + 5y = 20
y = 0 ⇒ x = 5 ∴ A (5, 0)
x = 0 ⇒ y = 4 ∴ E (0, 4)
2nd PUC Maths Previous Year Question Paper March 2017 Q50(a)
2nd PUC Maths Previous Year Question Paper March 2017 Q50(a).1

2nd PUC Maths Previous Year Question Paper March 2017

Question 50(b).
Find the value of k, if
2nd PUC Maths Previous Year Question Paper March 2017 Q50(b)
Solution:
2nd PUC Maths Previous Year Question Paper March 2017 Q50(b).1

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