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CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi

CBSE Previous Year Question Papers Class 12 Maths 2015 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 2015 Outside Delhi Set I

Section – A

Question 1.
Write the value of [1]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 1
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 2

Question 2.
Write the sum of the order and degree of the following differential equation: [1]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 3
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 4
Since the order and degree of the differential equation is 2 and 1 respectively.
So, the sum of the order and degree is 3.

Question 3.
Write the integrating factor of the following [1]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 5
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 6
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 7

Question 4.
If \hat{a}, \hat{b} \text { and } \hat{c}are mutually perpendicular unit vectors, then find the value of |2 \hat{a}+\hat{b}+\hat{c}|. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 8

Question 5.
Write a unit vector perpendicular to both the vectors \vec{a}=\hat{i}+\hat{j}+\hat{k} \text { and } \vec{b}=\hat{i}+\hat{j}. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 9
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 10

Question 6.
The equations of a line are 5x – 3 = 15y + 7 = 3 – 10z. Write the direction cosines of the line. [1]
Solution:
Given line is 5x – 3 = 15y + 7 = 3 – 10z
Rewritting the eq. in standard form :
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 11

Section – B

Question 7.
To promote the making of toilets for women, an organisation tried to generate awareness through
(i) house calls
(ii) letters, and
(iii) announcements. The cost for each mode per attempt is given below:
(i) ₹ 50
(ii) ₹ 20
(ii) ₹ 40
The number of attempts made in three villages X, Y and Z are given below:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 12
Find the total cost incurred by the organisation for the three villages separately, using matrices. Write one value generated by the organisation in the society. [4]
Solution:
The number of attempts made in three villages X, Y and Z can be represented by the 3 × 3 matrix.
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 13
Hence the total cost incurred by the organisation for the three villages separately are ₹ 30,000, ₹ 23,000 and ₹ 39,000.
The organisation in the society generated the value of cleanliness for the women welfare.

Question 8.
Solve for x:
tan-1(x + 1) + tan-1 (x -1) = tan-1\frac{8}{31}[4]
Solution:
Given, tan-1(x + 1) + tan-1 (x – 1)
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 14
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 15
OR
Prove the following:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 16
Solution:
L . H. S
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 17

Question 9.
Using properties of determinants, prove the following: [4]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 18
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 19
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 20

Question 10.
Find the adjoint of the matrix:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 21
and hence show that A. (adj A) = | A | I3. [4]
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 22
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 23
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 24

Question 11.
Show that the function f(x) = |x – 1| + |x + 1|, for all x ϵ R, is not differentiable at the points x = -1 and x = 1. [4]
Solution:
Given,
f(x) = | x – 1 | + | x + 1 |
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 25
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 26
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 27

Question 12.
If y = em sin-1x, then show that [4]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 28
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 29
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 30

Question 13.
If f(x) = \sqrt{x^{2}+1}; g(x) = \frac{x+1}{x^{2}+1}and h(x) = 2x – 3, then find f’ [h’ {g'(x)}]. [4]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 31

Question 14.
Evaluate: [4]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 32
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 33
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 34
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 35
OR
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 36
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 37
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 38

Question 15.
Find [4]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 39
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 40
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 41
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 42

Question 16.
Find [4]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 43
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 44
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 45

Question 17.
If \vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}+\hat{j} \text { and } \vec{c}=3 \hat{i}-4 \hat{j}-5 \hat{k}then find a unit vector perpendicular to both of the vectors (\vec{a}-\vec{b}) \text { and }(\vec{c}-\vec{b}). [4]
Solution:
Here,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 46

Question 18.
Find the equation of a line passing through the point (1, 2, – 4) and perpendicular to two lines
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 47
Solution:
Let the direction ratios of required line be a, b, c, since, the line is perpendicular to
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 48
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 49
OR
Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line \frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}.
Solution:
The equation of a plane passing through (-1, 2, 0) is
a (x + 1) + b (y – 2) + c (z – 0) = 0 …(i)
It passes through (2, 2, -1)
∴ a(2 + 1) + b(2 – 2) + c(-1 – 0) = 0
3a + 0b – c = 0 …(ii)
The given line is
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 50

Question 19.
Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence find the mean of the distribution. [4]
Solution:
Let X denote the number of spades when three cards are drawn, then, X is a random variable that can take values 0, 1, 2, 3.
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 51
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 52
OR
For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9P(X = 4) = P(X = 2). Find the probability of success.
Solution:
Let p denote the probability of getting success and q be the probability of failure.
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 53
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 54

Section – C

Question 20.
Consider f : R+ → [- 9, ∞) given by f(x) = 5x2 + 6x – 9. Prove that f is invertible with [6]
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 55
Solution:
To prove f is invertible we have to prove that f is one-one and onto.
For one-one
Let x1, x2 ϵ R+, then
f(x1) = f(x2)
⇒ 5x12 + 6x1 – 9 = 5x22 + 6x2 – 9
⇒ 5(x12 – x22) + 6(x1 – x2) = 0
⇒ (x1 – x2) (5x1 + 5x2 + 6) = 0
⇒ x1 – x2 = 0 as 5x1 + 5x2 + 6 ≠ 0
⇒ x1 = x2
i.e., f is one-one function.
For onto
Let f(x) = y
∴ y = 5x2 + 6x – 9
∴ 5x2 + 6x – (9 + y) = 0
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 56
OR
A binary operation * is defined on the set x = R – {-1} by x * y = x + y + xy, ∀ x, y ϵ X. Check whether * is commutative and associative. Find its identity element and also find the inverse of each element of X.**

Question 21.
Find the value of p for which the curves x2 = 9p(9 – y) and x2 = p(y + 1) cut each other at right angles. [6]
Solution:
Given,
x2 = 9p(9 – y) …(i)
and x2 = p (y +1) …(ii)
From (i) and (ii), we get
9p (9 – y) = p(y + 1)
⇒ 81p – 9py = py + p
⇒ 10py =80 p
⇒ y = 8
x2 = 9p
Now, differentiating (i) and (ii) w.r.t. x, we get
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 57

Question 22.
Using integration, prove that the curves y2 = Ax and x2 = 4y divide the area of the square bounded by x = 0, x = 4, y = 0 and y = 4 into three equal parts. [6]
Solution:
Given,
y2 = Ax …(i)
and x2 = 4y ……(ii)
Solving (i) and (ii), we get the point of intersection (0, 0) and (4, 4).
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 58
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 59
From (iii), (iv) and (v), it can be concluded that the given curves divide the area of the square bounded by x = 0, x = 4, y = 0 into three equal parts. Hence Proved.

Question 23.
Show that the differential equation \frac{d y}{d x}=\frac{y^{2}}{x y-x^{2}}is homogeneous and also solve it. [6]
Solution:
We have
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 60
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 61
OR
Find the particular solution of the differential equation (tan-1y – x) dy = (1 + y2) dx, given that x = 1, when y = 0.
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 62
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 63

Question 24.
Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, -4, -5) and B(2, -3, 1) intersects the plane 2x + y + z = 7. [6]
Solution:
Equation of the line joining the points A(3, -4, -5) and B(2, – 3, 1) is
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 64

Question 25.
A company manufactures three kinds of calculators : A, B and C in its two factories I and II. The company has got an order for manufacturing at least 6400 calculators of kind A, 4000 of kind B and 4800 of kind C. The daily output of factory I is of 50 calculators of kind A, 50 calculators of kind B, and 30 calculators of kind C. The daily output of factory II is of 40 calculators of kind A, 20 of kind B and 40 of kind C. The cost perdaytorunfactorylis ₹ 12,000 and of factory II is ₹ 15,000. How many days do the two factories have to be in operation to produce the order with the minimum cost ? Formulate this problem as an LPP and solve it graphically. [6]
Solution:
Let the factories I and II work for x and y number of days respectively.
Thus, the given linear programming problem is Minimize Z = ₹ (12000x + 15000y)
Subject to the constraints
50x + 40y ≥ 6400
50x + 20y ≥ 4000
30x + 40y ≥ 4800
x ≥ 0
and y ≥ 0 .
i.e. 5x + 4y ≥ 640
5x + 2y ≥ 400
3x + 4y ≥ 480
x ≥ 0, y ≥ 0.
To solve this L. P. P,
Let us consider the equations
L1 : 5x + 4y = 640 …(i)
L2 : 5x + 2y = 400 …(ii)
L3 : 3x + 4y = 480 …(iii)
The point of intersection of L1 and L2 is D(32, 120) and that of L1 and L3 is C (80,60)
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 65
The shaded region is the solution region of the given L. P. P.
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 66
Out of these values of Z, the minimum value of Z is 18,60,000 at x = 80 and y = 60.
Since the feasible region is unbounded so we draw the graph of inequality
12000 x + 15000 y < 1860000
i. e., 4x + 5y ≤ 620
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 67
We observe that open half one represented by L have no point common with feasible region.
Z = 12000 × 80 + 15000 × 60
= ₹ 18,60,000.
Hence, the factories I and II work for 80 and 60 number of days respectively.

Question 26.
In a factory which manufactures bolts, machines A, B and C manufacture respectively 30%, 50% and 20% of the bolts. Of their outputs 3, 4 and 1 percent respectively are defective bolts. A bolt is drawn at random from the product and is found to be defective. Find the probability that this is not manufactured by machine B. [6]
Solution:
Let E1, E2, E3 and A be the events defined as below
E1 = the bolt is manufactured by machine A.
E2 = the bolt is manufactured by machine B.
E3 = the bolt is manufactured by machine C.
A = the bolt is defective.
then, P(E1) = Probability that the bolt drawn is manufactured by machine A = \frac{30}{100}
P(E2) = Probability that the bolt drawn is manufactured by machine B = \frac{50}{100}
P(E3) = Probability that the bolt drawn is manufactured by machine C = \frac{20}{100}
P(A/E1) = Probability that the bolt drawn is defective given that it is manufactured by machine A.
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 68
CBSE Previous Year Question Papers Class 12 Maths 2015 Outside Delhi 69

All questions are same in Outside Delhi Set II and Set III

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