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CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi

CBSE Previous Year Question Papers Class 12 Maths 2011 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.

**Answer is not given due to the change in present syllabus

CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi Set I

Section – A

Question 1.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not. [1]
Solution:
Given,
A = {1, 2, 3},
B = {4, 5, 6, 7}
f = {(1, 4), (2, 5), (3, 6)}
f : A → B is defined as
∴ f(1) = 4, f(2)= 5, f(3) = 6.
Different points of the domain have different f-image in the range.
∴ f is one-one.

Question 2.
What is the principal value of
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 1[1]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 2

Question 3.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 3[1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 4

Question 4.
If A = \left[\begin{array}{cc}{2} & {3} \\ {5} & {-2}\end{array}\right]write A-1 in terms of A. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 5

Question 5.
If a matrix has 5 elements, write all possible orders it can have. [1]
Solution:
Since a matrix of order m × n has mn elements therefore, to find all possible orders of a matrix with 5 elements, we have to fill all possible ordered pairs (m, n) of positive integers whose product is 5. Hence possible orders are 1 × 5 and 5 × 1.

Question 6.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 6[1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 7

Question 7.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 8[1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 9
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 10

Question 8.
Write the direction-cosines of the line joining the points (1, 0, 0) and (0, 1, 1). [1]
Solution:
The d.r’s of line joining points (1, 0, 0) and (0, 1, 1) are 0 -1, 1 – 0, 1 – 0 i.e. -1, 1, 1
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 11

Question 9.
Write the projection of the vector \hat{i}-\hat{j}on the vector \hat{i}+\hat{j}. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 12

Question 10.
Write the vector equation of a line given by
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 13[1]
Solution:
The given line is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 14

Section – B

Question 11.
Let f : R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that gof=fog = IR. [4]
Solution:
It is given that f : R → R is defined as
f(x) = 10x + 7
One-one
Let f(x) = f(y), where x, y ϵ R
⇒ 10x + 7 = 10y + 7
⇒ x = y
∴ f is a one-one function.
Onto:
For y ϵ R, let y = 10x + 7
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 15
OR
A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 16
Show that zero is the identity for this operation and each element ‘a’ of the set is invertible with 6 – a, being the inverse of ‘a’.**

Question 12.
Prove that:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 17[4]
Solution:
L.H.S
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 18
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 19
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 20

Question 13.
Using properties of determinants, solve the following for x:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 21[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 22
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 23

Question 14.
Find the relationship between ‘a’ and ‘b’ so that the function ‘f’ defined by:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 24
is continuous at x = 3. [4]
Solution:
∵ f(x) is continuous at x = 3,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 25
OR
If xy = ex-y, show that \frac{d y}{d x}=\frac{\log x}{(\log (x e))^{2}}.
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 26
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 27

Question 15.
Prove that y = \frac{4 \sin \theta}{(2+\cos \theta)}-\thetais an increasing function on \left[0, \frac{\pi}{2}\right]. [4]
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 28
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 29
OR
If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.
Solution:
Let r be the radius of sphere and ∆r be the error in measuring the radius.
Then r = 9 cm, ∆ r = 0.03 cm.
Now surface area S of the sphere is
S = 4πr2
Differentiating w.r. t r, we get
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 30
This is the approximate error in calculating surface area.

Question 16.
If x = \tan \left(\frac{1}{a} \log y\right), show that
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 31[4]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 32

Question 17.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 33[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 34
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 35

Question 18.
Solve the following differential equation :
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 36[4]
Solution:
The given differential equation is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 37
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 38
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 39

Question 19.
Solve the following differential equation:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 40[4]
Solution:
The given differential equation is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 41
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 42

Question 20.
Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). [4]
Solution:
The vertices of triangle ABC are given as A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)
Let O be the origin of triangle
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 43
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 44
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 45

Question 21.
Find the shortest distance between the following lines whose vector equations are:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 46[4]
Solution:
Given equation are
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 47
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 48
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 49

Question 22.
A random variable X has the following probability distribution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 50
Determine: [4]
(i) K
(ii) P(X < 3)
(iii) P(X > 6)
(iv) P(0 < X < 3).
Solution:
It is known that the sum of probability distribution of variable is one.
(i) ∴ Σ P(X) =1 .
Therefore,
P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) = 1
0 + K + 2K + 2K + 3K + K2 + 2K2 + 7K2 + K = 1
⇒ 10K2 + 9K – 1 = 0
⇒ 10K2 + 10K – K – 1 = 0
⇒ 10K (K +1) – 1(K + 1) = 0
⇒ (K +1) (10 K – 1) = 0
⇒ K + 1 = 0
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 51
OR
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:
Here, n = 6
p = p (getting 6)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 52
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 53

Section – C

Question 23.
Using matrices, solve the following system of equations:
4x + 3y + 2z = 60
x + 2y + 3z = 45
6x + 2y + 3z = 70. [6]
Solution:
4x + 3y + 2z =60,
x + 2y + 3z = 45,
6x + 2y + 3z = 70.
The equation of system can be written in matrix form
AX = B ….(i)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 54
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 55
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 56

Question 24.
Show that the right circular cone of least curved surface and given volume has an altitude equal to  \sqrt{{2}} times the radius of the base.
Solution:
Let radius of cone = r
Height of cone = h
and slant height of cone = l
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 57
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 58
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 59
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 60
Hence, altitude is equal to  \sqrt{{2}} times the radius of base. Hence Proved.
OR
A window has the shape of a rectangle surmounted by an equilateral mangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.
Solution:
Let ABCD be a rectangle and let the side of an equilateral triangle be AB = b (length of the rectangle) and BC = a(width of the rectangle)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 61
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 62
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 63

Question 25.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 64[6]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 65
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 66
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 67
OR
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 68
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 69
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 70
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 71

Question 26.
Sketch the graph of y = | x + 3| and evaluate the area under the curve y= | x + 3| above x-axis and between x = – 6 to x = 0. [6]
Solution:
For drawing a sketch of the graph of y = | x + 3|, we construct the following table of values
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 72
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 73
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 74

Question 27.
Find the distance of the point (-1, -5, -10), from the point of intersection of the line
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 75
Solution:
Equation of the line is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 76
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 77

Question 28.
Given three identical boxes I, II and III each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold ? [6]
Solution:
Let E1i be box I is chosen, E2 be box II is chosen and E3 be box HI be chosen and A be the coin drawn is of gold.
We have,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 78
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 79

Question 29.
A merchant plans to sell two types of personal computer- a desktop model and a portable model that will cost ₹ 25,000 and ₹ 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than ₹ 70 lakhs and his profit on the desktop model is ₹ 4,500 and on the portable model is ₹ 5,000. Make an L.P.P. and solve it graphically. [6]
Solution:
Let the merchant stock x desktop computers and y portable computers. We construct the following table :
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 80
∴ The LPP is
Maximize Z = 4,500 x + 5,000 y
Subject to constraints:
x + y ≤ 250
25,000 x + 40,000 y ≤ 70,00,000
⇒ 5x + 8y ≤ 1,400
and x ≥ 0 , y ≥ 0
First we draw the lines AB and CD whose equations are
x + y = 250 …(i)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 81
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 82
The feasible region is OBPCO which is shaded in the figure.
The vertices of feasible region are O(0, 0), B(250, 0), P(200, 50) and C(0, 175).
P is the point of intersection of the lines.
x + y = 250
and 5x + 8y = 1400
Solving these equation we get point P(200, 50).
∴ The value of objective function
Z = 4500 x + 5000y
At these vertices are as follows :
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 83
Hence, the profit is maximum at ₹ 11,50,000 when 200 desktop computers and 50 portable computers are stocked.

CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi Set II

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

Section – A

Question 9.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 84[1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 85
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 86

Question 10.
Write a unit vector in the direction of the vector
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 87[1]
Solution:
The given vector is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 88

Section – B

Question 19.
Prove the following:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 89[4]
Solution:
L.H.S
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 90

Question 20.
Using properties of determinants, solve the following for x:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 91[4]
Solution:
The determinant is
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 92

Question 21.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 93[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 94
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 95

Question 22.
Solve the following differential equation:
xdy – (y + 2x2)dx = 0 [4]
Solution:
Given,
xdy – (y + 2x2)dx = 0
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 96

Section – C

Question 28.
Using matrices, solve the following system of equations: [6]
x + 2y + z = 7
x + 3z =11
2x – 3y = 1
Solution:
The given equations are x + 2 y + z = 7
x + 3z = 11
2x – 3y = 1
The given system of equations can be written in matrix form
AX = B …(i)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 97
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 98

Question 29.
Find the equation of the plane passing througjh the line of intersection of the planes \vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1 \text { and } \vec{r} \cdot(2 \hat{i}+3 \hat{j}-\hat{k})+4=0and parallel to x-axis. [6]
Solution:
The given equation of planes are
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 99
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 100

CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi Set III

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

Section – A

Question 1.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 101[1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 102

Question 2.
Write the angle between two vectors \vec{a}and \vec{b}magnitudes with \sqrt{3}and 2 respectively having \vec{a} \cdot \vec{b}=\sqrt{6}. [1]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 103
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 104

Section – B

Question 11.
Prove that: \tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{8}\right)=\frac{\pi}{4}. [4]
Solution:
L.H.S
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 105
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 106

Question 12.
Using properties of determinants, solve the following for x:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 107[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 108

Question 13.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 109[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 110
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 111

Question 14.
Solve the following differential equation:
xdy + (y – x3)dx = 0 [4]
Solution:
We have,
xdy + (y – x3)dx = 0
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 112

Section – C

Question 23.
Using matrices, solve the following system of equations: [6]
x + 2y – 3z = -4
2x + 3y + 2z = 2
3x – 3y – 4z = 11
Solution:
The given system of equations can be written in matrix form as
AX = B …(i)
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 113
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 114
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 115

Question 24.
Find the equation of the plane passing through the line of intersection of the planes 2x + y – z = 3 and 5x – 3y + 4z + 9 = 0 and parallel to the line
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 116[6]
Solution:
Given planes are
2x + y – z – 3 = 0    …..(i)
5x – 3y + 4z + 9 = 0    …..(ii)
Any plane passing through the line of intersection of (i) and (ii) can be taken as
2x + y – z – 3 + λ(5x – 3y + 4z + 9) = 0
(2 + 5λ)x + (1 – 3λ)y + (-1 + 4λ)z – 3 + 9λ = 0 …(iii)
The plane is parallel to the line
CBSE Previous Year Question Papers Class 12 Maths 2011 Outside Delhi 117

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