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CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi

CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi

Time allowed: 3 hours
Maximum marks : 100

General Instructions:

  • All questions are compulsory.
  • The question paper consists of 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
  • All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
  • There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.
  • Use of calculators is not permitted. You may ask for logarithmic tables, if required.

CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi Set I

Section – A

Question 1.
Write the principal value of tan-1( \sqrt{{3}} ) – cot-1( \sqrt{{3}} ). [1]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 1

Question 2.
Write the value of tan-1[2 sin(2cos-1\frac{\sqrt{3}}{2})]. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 2

Question 3.
For what value of x, is the matrix A = \left[\begin{array}{ccc}{0} & {1} & {-2} \\ {-1} & {0} & {3} \\ {x} & {-3} & {0}\end{array}\right]a skew symmetric matrix ? [1]
Solution:
We see that,
a31 = x
Given that the matrix ‘A’ is knew symmetric
∴ aij = -aji
⇒ a31 = – a13
∴ x = (-2) = 2.

Question 4.
If matrix A = \left[\begin{array}{cc}{1} & {-1} \\ {-1} & {1}\end{array}\right]and A2 = KA, then write the value of K. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 3

Question 5.
Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant. [1]
Solution:
We have, y = mx
On differentiating, we get
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 4

Question 6.
If Aij is the cofactor of the element aij of the determinant \left|\begin{array}{ccc}{2} & {-3} & {5} \\ {6} & {0} & {4} \\ {1} & {5} & {-7}\end{array}\right|then write the value of a32, A32. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 5

Question 7.
P and Q are two points with position vectors 3 \vec{a}-2 \vec{b} \text { and } \vec{a}+\vec{b}respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2 : 1 externally. [1]
Solution:
Position vector of point
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 6

Question 8.
Find |\vec{x}|if for a unit vector \vec{a},(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})= 15. [1]
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 7

Question 9.
Find the length of the perpendicular drawn from the origin to the plane 2x – 3y + 6z + 21 = 0. [1]
Solution:
Length of perpendicular
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 8

Question 10.
The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupee) received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5 and write which value does the question indicates. [1]
Solution:
Total revenue, R(x) = 3x2 + 36x + 5
Marginal revenue,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 9
Money for welfare of employees is a nice step, there should be a growth in raising funds for the welfare of the employees.

Section – B

Question 11.
Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f-1 of f given by f-1 (y) = \sqrt{y-4}where R+ is the set of all non-negative real numbers. [4]
Solution:
Given, f(x) = x2 + 4
f(x1) = f(x2)
x21 + 4 = x22 + 4
x1 = x2
Thus, f(x) is one-one.
Since, x2 + 4 is a real number. Thus, for every y in the co-domain of f, there exists a number x in R+ such that
f(x) = y = x2 + 4
Thus, we can say that f(x) is onto.
Now, f(x) is one-one and onto. Hence, f(x) is invertible.
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 10

Question 12.
Show that: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 11
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 12
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 13
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 14
OR
Solve the following equation:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 15
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 16
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 17

Question 13.
Using properties of determinants prove the following: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 18
Solution:
L. H. S
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 19
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 20

Question 14.
If yx = ey-x, prove that \frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y}. [4]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 21
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 22

Question 15.
Differentiate the following with respect to x: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 23
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 24
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 25

Question 16.
Find the value of k, for which
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 26
continuous at x = 0. [4]
Solution:
At x = 0,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 27
OR
If x = a cos3θ and y = a sin3θ, then find the value of \frac{d^{2} y}{d x^{2}} \text { at } \theta=\frac{\pi}{6}.
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 28
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 29

Question 17.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 30
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 31
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 32
OR
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 33
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 34
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 35
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 36

Question 18.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 37
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 38
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 39
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 40

Question 19.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 41
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 42

Question 20.
If \vec{a}=\hat{i}-\hat{j}+7 \hat{k} \text { and } \vec{b}=5 \hat{i}-\hat{j}+\lambda \hat{k}then find the value of λ, so that \vec{a}+\vec{b} \text { and } \vec{a}-\vec{b}are perpendicular vectors. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 43

Question 21.
Show that the lines
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 44
are intersecting. Hence find their point of intersection. [4]
Solution:
Consider,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 45
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 46
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 47
OR
Find the vector equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x – 2y + 4z = 10.
Solution:
Equation of plane passing through (2, 1, -1) is
a(x – 2) + b(y – 1) + c(z + 1) = 0 …(i)
This plane passes through (-1, 3, 4)
Thus, a(-1 – 2) + b (3 – 1) + c(4 + 1) = 0
⇒ -3a + 2b + 5c = 0 ……..(ii)
Also, the above plane is perpendicular to the plane
x – 2y + 4z = 10
∴ a(1) + b(-2) + c(4) = 0
⇒ a – 2b + 4c = 0 …(iii)
Now, we have
-3a + 2b + 5c = 0
and a – 2b + 4c = 0
Solving the above equation by cross multiplication, we get
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 48

Question 22.
The probabilities of two students A and B coming to the school in time are \frac{3}{7} \text { and } \frac{5}{7}respectively.
Assuming that the events, A coming in time and B coming in time are independent. Find the probability of only one of them coming to the school in time. Write at least one advantages of coming to school in time. [4]
Solution:
Let probability that A comes on time = P(A)
Let probability that B comes on time = P(B)
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 49

Section – C

Question 23.
Find the area of the greatest rectangle that can be inscribed in an ellipse \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1. [6]
Solution:
Equation of ellipse is
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 50
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 51
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 52
OR
Find the equations of tangents to the curve 3x2 – y2 = 8 which pass through the point \left(\frac{4}{3}, 0\right).
Solution:
The equation of the curve is 3x2 – y2 = 8.
On differentiating, we get
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 53
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 54
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 55

Question 24.
Find the area of the region bounded by the parabola y = x2 and y = |x|. [6]
Solution:
Given parabola y = x2 which is symmetrical about Y-axis and passes through O(0,0) and the curve y = | x |. The point of intersection of parabola, y = x2 and line, y = x in the first quadrant is P(1, 1). The given area is symmetrical about Y-axis.
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 56
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 57
Question 25.
Find the particular solution of the differential equation (tan-1 y – x)dy = (1 + y2)dx, given that when x = 0, y = 0. [6]
Solution:
The given differential equation
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 58
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 59
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 60

Question 26.
Find the equation of the plane passing through the line of intersection of the planes \vec{r} \cdot(\hat{i}+3 \hat{j})-6=0 \text { and } \vec{r} \cdot(3 \hat{i}-\hat{j}-4 \hat{k})=0whose perpendicular distance from origin is unity. [6]
Solution:
The equation of the plane passes through the intersection of given planes is
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 61
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 62
OR
Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes \vec{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=5 \text { and } \vec{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=6.
Solution:
The equation of line passing through the point (1, 2, 3) is
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 63
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 64

Question 27.
In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not. [6]
Solution:
Probability of getting a six by the captains of both the teams A and B is
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 65
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 66
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 67
The decision of the referee was not fair as whosoever starts throwing the die gets an upper hand.

Question 28.
A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively; which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹ 100 and ₹ 120 per unit respectively, how should he use his resources to maximize the total revenue ? Form the above as an LPP and solve graphically.
Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate? [6]
Solution:
Let x units of the goods A and y units of goods B be produced to maximize the total revenue. Then the total revenue is Z = 100x + 120y. This is a linear function which is to be maximized. Hence it is the objective function. The constraints are as per the following table :
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 68
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 69
2x + 3y = 30
and 3x + y = 17
Solving these equations, we get point P(3, 8).
The vertices of the feasible region are O(0, 0), D(17/3), P (3,8) and A(o,
10).
The value of objective function Z = 100x + 120y at these vertices are as follows :
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 70
∴ The maximum revenue ₹ 1260 at the point P(3, 8) i.e., when 3 units of goods A and 8 units of goods B are produced.
Yes, I agree with the view of the manufacturer.

Question 29.
The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others using matrix method, find the number of awardees of each category. Apart from these values namely honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards. [6]
Solution:
From equation,
x + y + z = 12 …(i)
2x + 3 (y + z) = 33 ⇒ 2x + 3y + 3z = 33 …(ii)
x + z = 2y ⇒ x – 2y + z = 0 …(iii)
∴ The given equations can be written in matrix form
AX = B …(iv)
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 71
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 72
The colony management must includes cleanliness for awards.

CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi Set II

Note : Except for the following questions, all the remaining questions have been asked in previous set

Section – A

Question 9.
If matrix A = \left[\begin{array}{cc}{2} & {-2} \\ {-2} & {2}\end{array}\right]and A2 = pA, then write the value of p. [1]
Solution:
Given that
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 73

Question 10.
A and B are two points with position vectors 2 \vec{a}-3 \vec{b} \text { and } 6 \vec{b}-\vec{a}respectively. Write the position vector of a point P which divides the line segment AB internally in the ratio 1 : 2. [1]
Solution:
Position vector of point
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 74

Section – B

Question 19.
If xy= ex-y, prove that \frac{d y}{d x}=\frac{\log x}{(1+\log x)^{2}}. [4]
Solution:
Given, xy= ex-y
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 75
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 76

Question 20.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 77
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 78

Question 21.
Evaluate : [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 79
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 80
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 81

Section – C

Question 28.
Find the area of the region {(x, y) :y2 ≤ 6ax and x2 + y2 ≤ 16 a2} using method of integration. [6]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 82
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 83

Question 29.
Show that the differential equation \left[x \sin ^{2}\left(\frac{y}{x}\right)-y\right]dx + xdy = 0 is homogeneous.
Find the particular solution of this differential equation, given that y = \frac{\pi}{4}when x = 1. [6]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 84
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 85

CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi Set III

Note : Except for the following questions, all the remaining questions have been asked in previous sets.

Section – A

Question 9.
If matrix A = \left[\begin{array}{cc}{3} & {-3} \\ {-3} & {3}\end{array}\right]and A2 = λ A, then write the value of X. [1]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 86
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 87

Question 10.
L and M are two points with position vectors 2 \vec{a}-\vec{b} \text { and } \vec{a}+2 \vec{b}respectively. Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally. [1]
Solution:
The position vectors of the points are \mathrm{L}(2 \vec{a}-\vec{b}) \text { and } \mathrm{M}(\vec{a}+2 \vec{b}).
We have to divide segment LM at N externally in the ratio 2 : 1.
Position of vector of point N
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 88

Section – B

Question 19.
Using vectors, find the area of the triangle ABC, whose vertices are A(1, 2, 3), B(2, -1, 4) and C(4, 5, -1). [4]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 89
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 90

Question 20.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 91
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 92
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 93

Question 21.
If x sin (a + y) + sin a cos (a + y) = 0, prove that \frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}. [4]
Solution:
Here, x sin (a + y) + sin a cos (a + y)
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 94

Question 22.
Using properties of determinants prove the following: [4]
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 95
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 96

Question 28.
Find the area of the region, {(x, y): y2 ≤ 4x, 4x2 + 4y2 ≤ 9} using method of integration. [6]
Solution:
R = {(x, y): y2 ≤ 4x, 4x2 + 4y2 ≤ 9}
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 97
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 98
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 99
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 100

Question 29.
Find the particular solution of the differential equation \frac{d x}{d y}+ x cot y = 2y + y,sup>2 cot y, (y ≠ 0), given that x = 0 when y = \frac{\pi}{2}. [6]
Solution:
The given differential equation is
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 101
CBSE Previous Year Question Papers Class 12 Maths 2013 Outside Delhi 102

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