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CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi

CBSE Previous Year Question Papers Class 12 Maths 2019 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 2019 Delhi Set I

Section – A

Question 1.
If A and B are square matrices of the same order 3, such that | A | = 2 and AB = 2I, write the value of |B|. [1]
Solution:
Given, | A | = 2 and AB = 2I
∴ | A | =21
⇒ | A | | B | = 2I
⇒ 2|B | = 2I
⇒ | B | = I

Question 2.
If f(x) = x + 1, find \frac{d}{d x}(f o f)(x). [1]
Solution:
Given, f(x) = x + 1
Now, f0f(x) = f(f(x)
= f(x + 1)
= x + 1 + 1
= x + 2
\frac{d}{d x}(f o f)(x)= 1

Question 3.
Find the order and the degree of the differential equation x^{2} \frac{d^{2} y}{d x^{2}}=\left\{1+\left(\frac{d y}{d x}\right)^{2}\right\}^{4}. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 1

Question 4.
If a line makes angles 90°, 135°, 45° with x, y and z axes respectively, find its direction cosines. [1]
Solution:
Given, α = 90°, β = 135°, γ = 45°
So, l = cos 90° = 0
m = cos135° – cos(180° – 45°)
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 2
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector 2 \hat{i}+2 \hat{j}-3 \hat{k}. [1]
Solution:
Given, the line passes through the point (3, 4, 5).
and parallel to the vector 2 \hat{i}+2 \hat{j}-3 \hat{k}.
D. R. s of the given vector are < 2, 2, – 3 >.
∴ Vector equation of line,
\vec{r}=(3 \hat{i}+4 \hat{j}+5 \hat{k})+\lambda(2 \hat{i}+2 \hat{j}-3 \hat{k})

Section – B

Question 5.
Examine whether the operation * defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?** [2]

Question 6.
Find a matrix A such that 2A – 3B + 5C = 0, where B = \left[\begin{array}{rrr}{-2} & {2} & {0} \\ {3} & {1} & {4}\end{array}\right]and C = \left[\begin{array}{rrr}{2} & {0} & {-2} \\ {7} & {1} & {6}\end{array}\right]. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 3

Question 7.
Find: \int \frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}} d x. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 4
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 5

Question 8.
Find: [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 6
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 7
OR
Find: \int \sin ^{-1}(2 x) d x. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 8

Question 9.
Form the differential equation representing the family of cures y = e2x (a + bx), where ‘a’ and ‘b’ are arbitrary constants. [2]
Solution:
Given, y = e2x (a +bx) …(i)
Differentiating w.r.t. x, we get
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 9
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 10
This is the required differential equation.

Question 10.
If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is  \sqrt{{3}} . [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 11
OR
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 12[2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 13

Question 11.
A die marked 1,2,3 in red and 4,5,6 in green is tossed. Let A be the event “number is even”and B be the event “number is marked red”. Find whether the events A and B are independent or not. [2]
Solution:
Given, S = {1, 2, 3, 4, 5, 6 }
Let the two events be
A : The number is even
B : The number is red
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 14
Hence, A and B are ot independent

Question 12.
A die is thrown 6 times. If “getting an odd number” is a “success”, what is the probability of
(i) 5 successes?
(ii) atmost 5 success? [2]
OR
The random variable X has a probability distribution P(X) of the following form, where ‘k’ is some number
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 15
Determine the value of ‘k’. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 16

Section – C

Question 13.
Show that the relation R on R defined as R = [(a, b) : a ≤ b], is reflexive, and transitive but not symmetric. [2]
Solution:
Reflexive:
Let a ϵ R
∴ a ≤ a
So, (a, a) ϵ R
Hence, R is reflexive.

Symmetric:
Let (a, b) ϵ R
Then (b, a) ϵ R
Then, a ≤ b
⇒ b ≤ a
∴ (b, a) ∉ R
Hence, R is not symmetric.

Transitive:
Let, a, b, c ϵ R, such that (a, b) ϵ R and (b, c) ϵ R
Then, a ≤ b
and b ≤ c
⇒ a ≤ c
⇒ (a, c) ϵ R
Hence, R is transitive.
Hence, R is reflexive and transitive but not Symmetric. Hence Proved.
OR
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto.
Find inverse of f : N → S, where S is range of f.
Solution:
Given, f : N → N, f(x) = x2 + x +1
Let A be the set of natural number (domain),
and B be the set of natural number (co-domain).
For One-One:
Let x1, x2 ϵ A such that f(x1) = f(x2)
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 17
Not possible since domain = N
So, f is not onto.
Hence, f is one-one but not onto. Hence Proved
Now, f : N → S : f(x) = x2 + x + 1
where S = range (Given)
f : N → S is onto as co-domain = range.
Hence, f is invertible.
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 18

Question 14.
Solve : tan-14x + tan-1 6x = \frac{\pi}{4}[2]
Solution:
We have
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 19

Question 15.
Using properties of determinants, prove that [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 20
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 21

Question 16.
If log(x2 + y2) = 2 tan-2\left(\frac{y}{x}\right), show that [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 22
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 23
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 24
OR
If xy – yx = ab, find \frac{d y}{d x}
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 25
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 26
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 27

Question 17.
If y = (sin-1x)2, prove that [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 28
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 29

Question 18.
Find the equation of tangent to the curve y = \sqrt{3 x-2}which is parallel to the line 4x – 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 30
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 31
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 32

Question 19.
Find: \int \frac{3 x+5}{x^{2}+3 x-18} d x. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 33
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 34
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 35

Question 20.
Prove that \int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x, hence evaluate [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 36
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 37
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 38
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 39

Question 21.
Solve the differential equation xdy – ydx = \sqrt{x^{2}+y^{2}} d x, given that y = 0 when x = 1. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 40
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 41
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 42
OR
Solve the differential equation (1 + x2)\frac{d y}{d x}+ 2xy – 4x2 = 0, subject to the initial condition y(0) = 0. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 43
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 44
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 45

Question 22.
If \hat{i}+\hat{j}+\hat{k}, \quad 2 \hat{i}+5 \hat{j}, \quad 3 \hat{i}+2 \hat{j}-3 \hat{k}and \hat{i}-6 \hat{j}-\hat{k}respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether \overrightarrow{\mathrm{AB}} \text { and } \overrightarrow{\mathrm{CD}}are collinear or not. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 46

Question 23.
Find the value of λ, so that the line \frac{1-x}{3}=\frac{7 y-14}{\lambda}=\frac{z-3}{2} \text { and } \frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}are at right angles. Also, find whether the lines are intersecting or not. [2]
Solution:
Equation of 1st line,
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 47
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 48
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 49
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 50
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 51

Section – D

Question 24.
If A = \left[\begin{array}{lll}{1} & {1} & {1} \\ {1} & {0} & {2} \\ {3} & {1} & {1}\end{array}\right], find A-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12 [6
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 52
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 53
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 54
OR
Find the inverse of the following matrix using elementary operations.
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 55
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 56
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 57
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 58

Question 25.
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs ₹ 70 per square metre for the base and ₹ 45 per square metre for the sides, what is the cost of least expresive tank? [6]
Solution:
Let the length and breadth of the tank be x and y metres, respectively.
Given, Volume = 8 m3
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 59
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 60
Hence, the cost of least expensive tank is ₹ 1000.

Question 26.
Using integration, find the area of a triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2). [6]
Solution:
Given, A (2, 5), B (4, 7) and C (6, 2) be the vertices of a triangle.
The equation of side AB.
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 61
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 62
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 63
OR
Find the area of a region lying above x-axis and included between the circle x2 + y2 = 8x and inside of the parabola y2 = 4A:.
Solution:
Given, equation of circle is x2 + y2 = 8x can be expressed as
(x – 4)2 + y2 = 16 …(i)
Centre is (4, 0) and radius is 4 and equation of parabola is y2 = 4x …(ii)
Let Required Area = I
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 64
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 65

Question 27.
Find the vector and Cartesian equation of the plane passing through the points (2, 2, -1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above. [6]
Solution:
Let A (2, 2, -1), B (3, 4, 2,) and C (7, 0, 6)
The equation of plane passing through A(2, 2, -1),
a(x – 2) + b(y – 2) + c(z + 1) = 0 …(i)
Since (3, 4, 2) and (7, 0, 6) lies on plane
a + 2b + 3c = 0 …(ii)
and 5a – 2b + 7c = 0 …(iii)
Solving equation (ii) and (iii), we get
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 66
a = 20k, b = 8k and c = -12k
Putting the values of a, b and c in (i), we get
20k(x – 2) + 8k(y – 2) a – 12k (z + 1) = 0
⇒ 4k [5x – 10 + 2y – 4 – 3z – 3] = 0
⇒ 5x + 2y – 3z – 17 = 0
This is the required equation of the plane.
Now, the second plane passes through the points (4, 3, 1).
Since, this plane is parallel to the above plane,
∴ D. R/s of the second plane be < 5, 2, – 3 >
So, equation of second plane,
5(x – 4) + 2(y – 3) – 3(z – 1) = 0
⇒ 5x – 20 + 2y – 6 – 3z + 3 = 0
⇒ 5x + 2y – 3z – 23 = 0
OR
Find the vector equation of the plane that contains the lines \vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2 \hat{j}+\hat{k})and the point (-1, 3, – 4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane, thus obtained.
Solution:
Given, equation of the given line
\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2 \hat{j}+\hat{k})
The plane passes through the point (- 1, 3, -4).
Then the equation of the plane,
a(x + 1) + b (y – 3) + c (z + 4) = 0 …(i)
Since (1, 1) lies on the plane,
∴ 2a – 2b + 4c = 0 …(ii)
Also, (1, 2, -1) lies on the plane
∴ 2a – b + 3c = 0 …(iii)
Solving equations (ii) and (iii), we get
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 67
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 68

Question 28.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators B and C produces 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job 30% of the time and C on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random .from his and is found to be defective. What is the probability that it was produces by A? [6]
Solution: .
Let Hi be the event items produced by A
H2 be the event items produced by B
H3 be the event items produced by C
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 69

Question 29.
A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making two one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer’s profit on an item of model A is ₹15 and on an item of model B is ₹ 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit. [6]
Solution:
Let, x be the number of items of model A and y be the number of items of model B
Let Z be the required profit.
Subject to constraints:
2x + y ≤ 8 × 5
⇒ 2x + y ≤ 40
2x + 3y ≤ 8 × 10
⇒ 2x + 3y ≤ 80
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 70
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 71
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 72
Thus, the maximum profit is obtained when the manufacture produces 10 items of model A and 20 items of model B and the maximum profit ₹ 250.

CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi Set II

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

Section – A

Question 2.
If f(x) = x + 7 and gix) = x – 7, x ϵ R, then find \frac{d}{d x}(f \circ g)(x)/ [1]
Solution:
Given, f(x) = x + 7
and y(x) = x – 7
Given, (fog) (x) = f(g (x)
= f(x – 7)
= f(x – 7) + 7
= x
Now, Differentiating w.r.t, x, we get
\frac{d(f o g)}{d x}(x)=1

Question 3.
Find the value of x – y, if
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 73
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 74

Section – B

Question 6.
If A = \left[\begin{array}{ccc}{2} & {0} & {1} \\ {2} & {1} & {3} \\ {1} & {-1} & {0}\end{array}\right], then find (A2 – 5A). [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 75
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 76

Question 12.
Find: \int \frac{\tan ^{2} x \sec ^{2} x}{1-\tan ^{6} x} d x. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 77

Section – C

Question 13.
Solve for x: tan-1(2x) + tan-1(3x) = \frac{\pi}{4}. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 78

Question 18.
Using properties of determinants, prove the following: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 79
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 80
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 81

Question 19.
If x = cos t + log tan \left(\frac{t}{2}\right), y = sin t, then find the values of \frac{d^{2} y}{d t^{2}} \text { and } \frac{d^{2} y}{d x^{2}} \text { at } t=\frac{\pi}{4}. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 82
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 83
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 84

Section – D

Question 24.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of a radius r is \frac{4 r}{3}. Also find the maximum volume of cone. 3 [6]
Solution:
Let R be the radius of cone. Let OA = OB = r (radius of sphere)
AC = r + x …(i) (height of cone)
Let V be the volume of cone.
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 85
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 86
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 87
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 88

Question 25.
If A = \left[\begin{array}{ccc}{2} & {-3} & {5} \\ {3} & {2} & {-4} \\ {1} & {1} & {-2}\end{array}\right], then find A-1. Hence solve the following system of equations : 2x – 3y + 5z = 11, 3x + 2y – 4z = – 5, x + y – 2z = – 3. [6]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 89
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 90
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 91
OR
Obtain the inverse of the following matrix using elementary operations:
A = \left[\begin{array}{rrr}{-1} & {1} & {2} \\ {1} & {2} & {3} \\ {3} & {1} & {1}\end{array}\right]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 92
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 93
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 94
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 95
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 96

CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi Set III

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

Section – A

Question 1.
If 3A – B = \left[\begin{array}{ll}{5} & {0} \\ {1} & {1}\end{array}\right]and B = \left[\begin{array}{ll}{4} & {3} \\ {2} & {5}\end{array}\right], then find the matrix A. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 97
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 98

Question 2.
Write the order and the degree of the following differential equation: [1]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 99
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 100

Section – B

Question 5.
Find: \int \sin x \cdot \log \cos x d x[2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 101

Question 6.
Evaluate : \int_{-\pi}^{\pi}\left(1-x^{2}\right) \sin x \cos ^{2} x d x. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 102
OR
Evaluate: \int_{-1}^{2} \frac{|x|}{x} d x.
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 103
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 104

Question 8.
Find a matrix A such that 2A – 3B + 5C = 0, where [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 105
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 106
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 107

Section – C

Question 13.
Using propeties of determinants, prove the following: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 108
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 109
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 110
Hence Proved.

Question 20.
Find: \int \frac{\cos x}{(1+\sin x)(2+\sin x)} d x[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 111
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 112

Question 21.
Solve the differential equation: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 113
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 114
OR
Solve the differential equation: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 115
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 116
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 117

Section – D

Question 26.
Prove that the curves y2 = 4x and x2 = 4y divide the by of the squre bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. [6]
Solution:
Given, y2 = 4x …(i)
x2 = 4y …(ii)
x = 0, x = y, y = 0 and y = 4
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 118
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 119
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 120
The curves y2 = 4x and x2 = 4y divides the area of the square into three equal parts. Hence Proved.
OR
Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4).
Solution:
Given, the vertices of AABC, A(2, 3), B (3, 5) and C (4, 4).
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 121
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 122
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 123

Question 29.
Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean and variance of the number of kings. [4]
Solution:
Let S = Number of kings.
and F = number of non kings
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 124
CBSE Previous Year Question Papers Class 12 Maths 2019 Delhi 125

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