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CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi

CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi

Time allowed: 3 hours
Maximun 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 Outside Delhi Set I

Section – A

Question 1.
If A is a square matrix satisfying A’ A = I, write the value of |A|. [1]
Solution:
Given, A’ A= I
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 1

Question 2.
If y = x | x |, find \frac{d y}{d x}for x < 0. [1]
Solution:
If y = x | x |
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 2

Question 3.
Find the order and degree (if defined) of the differential equation [1]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 3
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 4
Order of this equation is 2.
Degree of this equation is not defined.

Question 4.
Find the direction consines of a line which makes equal angles with the coordinate axes. [1]
Solution :
Let the direction cosines of the line make an angle a with each of the coordinate axes and direction cosines be l, m and n.
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 5
A line passes through the point with position vector 2 \hat{i}-\hat{j}+4 \hat{k}and is in the direction of the vector \hat{i}+\hat{j}-2 \hat{k}. Find the equation of the line in cartesian form.
Solution:
The line passes through a point (2, -1, 4) and has direction ratios proportional to (1, 1, – 2).
Cartesian equation of the line
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 6

Section – B

Question 5.
Examine whether the operation * defined on R, the set of all real numbers, by a * b=\sqrt{a^{2}+b^{2}}is a binary operation or not, and if it is a binary operation, find whether it is associative or not.** [2]

Question 6.
If A = \left[\begin{array}{cc}{4} & {2} \\ {-1} & {1}\end{array}\right], show that (A – 2I)(A – 3I) = 0. [2]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 7

Question 7.
Find: [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 8
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 9
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 10

Question 8.
Find: [2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 11
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 12
OR
Find:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 13
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 14

Question 9.
Find the differential equation of the family of curves y = Ac2x + Be-2x, where A and B are arbitrary constants. [2]
Solution:
Given, y = Ae2x + Be-2x (i)
On differentiating equation (i) w.r.t. x, we get
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 15

Question 10.
If |\vec{a}|=2,|\vec{b}|and \vec{a} \times \vec{b}=3 \hat{i}+2 \hat{j}+6 \hat{k}, find the angle between \vec{a}and \vec{b}. [2]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 16
Or
Find the volume of a cuboid whose edges are given by -3 \hat{i}+7 \hat{j}+5 \hat{k},-5 \hat{i}+7 \hat{j}-3 \hat{k}and 7 \hat{i}-5 \hat{j}-3 \hat{k}.
Solution:
If a, b, c are edges of a cuboid.
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 17
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 18

Question 11.
If P(not A) = 0·7, P(B) = 0·7 and P(B/A) = 0·5, then find P(A∆B). [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 19

Question 12.
A coin is tossed 5 times. What is the probability of getting (i) 3 heads, (ii) at most 3 heads? [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 20
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 21
OR
Find the probabilty distribution of X, the number of heads in a simultaneous toss of two coins.
Solution:
If we toss two coins simultaneously then sample space is given by (HH, HT, TH, IT)
Then probability distribution is,
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 22

Section – C

Question 13.
Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. [4]
Solution:
Here, R = {(a, b): b = a +1}
∴ R = {(a, a + 1): a, a + 1 ϵ (1, 2, 3, 4, 5 ,6)}
⇒ R = {(1, 2,) (2, 3), (3, 4), (4, 5), (5, 6)}
(i) R is not reflexive as {a, a} ∉ R∀a
(ii) R is not symmetric as (1, 2) ϵ R but (2, 1) ∉ R
(iii) R is not transitive as (1, 2) ϵ R, (2, 3) ϵ R but (1, 3) ϵ R
OR
Let f: N → Y be a function defined as
f(x) = 4x + 3,
where Y = {y ϵ N : y = 4x + 3, for some x ϵ N}.
Show that f is invertible. Find its inverse.
Solution:
Consider an arbitrary element of Y. By the definition of y, y = 4x + 3, for some x in the domain N.
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 23

Question 14.
Find the value \left(\cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{2}{3}\right)[4]
Solution:
\left(\cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{2}{3}\right)
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 24
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 25

Question 15.
Using properties of determinants, show that [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 26
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 27
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 28

Question 16.
If x \sqrt{1+y}+y \sqrt{1+x}= 0 and x ≠ y, prove that \frac{d y}{d x}=-\frac{1}{(x+1)^{2}}[4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 29
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 30
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 31
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 32

Question 17.
If If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that \frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3 / 2}}{\frac{d^{2} y}{d x^{2}}}is a constant independent of a and b. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 33
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 34
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 35
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 36

Question 18.
Find the equation of the normal to the curve x2 = 4y which passes through the point (-1, 4). [4]
Solution:
Suppose the normal at P(x1, y1) on the parabola x2 = 4y passes through (-1, 4)
Since, P(x1, y1) lies on x2 = 4y
x_{1}^{2}= 4y1
The equation of curve is x2 = 4y
Differentiating with respect to x, we have
2x = 4 \frac{d y}{d x}
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 37
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 38

Question 19.
Find: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 39
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 40
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 41

Question 20.
Prove that : \int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d xand hence evaluate \int_{0}^{\pi / 2} \frac{x}{\sin x+\cos x} d x. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 42
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 43
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 44
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 45
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 46

Question 21.
Solve the differential equation: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 47
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 48
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 49
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 50

Question 22.
The scalar product of the vector \vec{a}=\hat{i}+\hat{j}+\hat{k}with a unit vector along the sum of the vectors \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}and \vec{c}=\lambda \hat{i}+2 \hat{j}+3 \hat{k}is equal to 1. Find the value of λ and hence find the unit vector along \vec{b}+\vec{c}. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 51
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 52

Question 23.
If the lines \frac{x-1}{-3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}and \frac{x-1}{3 \lambda}=\frac{y-1}{2}=\frac{z-6}{-5}are perpendicular, find the value of λ. Hence find whether the lines are intersecting or not. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 53
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 54
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 55

Section – D

Question 24.
If A = \left[\begin{array}{lll}{1} & {3} & {4} \\ {2} & {1} & {2} \\ {5} & {1} & {1}\end{array}\right], find A-1 [6]
Hence solve the system of equations
x + 3y + 4z = 8
2x + y + 2z = 5
and 5x + y + z = 7
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 56
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 57
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 58
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 59
OR
Find the inverse of the following matrix, using elementary transformation:
A = \left[\begin{array}{ccc}{2} & {0} & {-1} \\ {5} & {1} & {0} \\ {0} & {1} & {3}\end{array}\right]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 60
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 61

Question 25.
Show that the height the cylinder of maximum volume that can inscribed in a sphere of radius R is \frac{2 R}{\sqrt{3}}. Also find the maximum volume. [6]
Solution:
Let, ‘x’ be the diameter of the base of the cylinder and let ‘h’ be height of the cylinder.
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 62
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 63
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 64
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 65
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 66

Question 26.
Using method of integration, find the area of the triangle whose vertices are (1, 0), (2, 2) and (3, 1). [6]
Solution:
A(1, 0), B(2, 2) and C(3, 1)
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 67
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 68
OR
Using method of integration, find the area of the region enclosed between two circles x2 + y2 = 4 and (x – 2)2 + y2 = 4.
Solution:
Equations of the given circles are,
x2 + y2 = 4 …(i)
(x – 2)2 + y2 = 4 …(ii)
Equation (i) is a circle with centre O at the origin and radius 2. Equation (ii) is a circle with centre C (2, 0) and radius 2.
Solving equation (i) and (ii) we have
(x – 2)2 + y2 = x2 + y2
or x2 – 4x + 4 + y2 = x2 + y2 or x = 1 which gives y = ±\sqrt{3}
Thus, the points of intersection of the given circles are A (1, \sqrt{3}) and A'(1, \sqrt{3})
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 69
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 70
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 71

Question 27.
Find the vector and cartesian equations of the plane passing through the points having position vectors \hat{i}+\hat{j}-2 \hat{k}, 2 \hat{i}-\hat{j}+\hat{k}and \hat{i}+2 \hat{j}+\hat{k}. Write the equation of a plane passing through a point (2, 3, 7) and parallel to the plane obtained above. Hence, find the distance between the two parallel planes. [6]
Solution :
Let A, B, C be the points with position
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 72
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 73
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 74
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 75
OR
Find the equation of the line pasing through (2, -1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5) and (3, 2, 4). Also, find their point of intersection.
Solution:
We know that the equation of line passing through points (x1, y1, z1) and(x2, y2, z2) is given by
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 76
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 77

Question 28.
There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is the two-headed coin? [6]
Solution:
Given, there are three coins.
Let,
E1 = coin is two headed
E2 = biased coin
E3 = unbiased coin
A = shows only head
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 78

Question 29.
A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requries 3g of silver and 1 g of gold while that of type B requires 1 g of silver and 2g of gold. The company can use at the most of 9 g of silver and 8 of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, find the number of units of each type that the company should produce to maximize profit. Formulate the above LPP and solve it graphically and also find the maximum profit [6]
Solution:
There are two types of goods, A and B and let units of type A be x and units of type B be y.
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 79
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 80
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 81

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

Section – A

Question 1.
Find | AB |, if A = \left[\begin{array}{rr}{0} & {-1} \\ {0} & {2}\end{array}\right]anf B = \left[\begin{array}{ll}{3} & {5} \\ {0} & {0}\end{array}\right]. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 82

Question 2.
Differentiate e^{\sqrt{3 x}}, with respect to x. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 83

Section – B

Question 6.
If A = \left[\begin{array}{ll}{p} & {2} \\ {2} & {p}\end{array}\right]and | A3| = 125, the find the value of p. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 84

Question 12.
Find the general solution of the differential equation \frac{d y}{d x}=e^{x+y}. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 85

Section – C

Question 21.
If (a + bx)ey/x = x, then prove that [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 86
Solution:
We have
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 87
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 88

Question 22.
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of its edge is 12 cm? [4]
Solution:
Let x be the length of side, V be the volume and S be the surface area of cube.
Then, V = x3 and S = 6x2, where x is a function of time t
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 89
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 90

Question 23.
Find the cartesian and vector equations of the plane passing through the point A(2, 5, – 3), B(- 2, – 3, 5) and C(5, 3, – 3). [4]
Solution:
We know that the general equation of the plane passing through three points (x1, y1, z1) (x1, y1, Z1), (x3, y3, z3)
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 91
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 92
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 93

Section – D

Question 24.
Find the point on the curve y2 = 4x, which is nearest to the point (2, – 8). [6]
Solution:
Given curve is of the form, y2 = Ax and let p(x, y) is a point on the curve which is nearest to the point (2, – 8).
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 94
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 95
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 96

Question 25.
Find \int_{1}^{3}\left(x^{2}+2+e^{2 x}\right)as the limit of sums. [6]
Solution:
We have
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 97
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 98
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 99
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 100
OR
Using integration, find the area of the triangular region whose sides have the equation y = 2x + 1, y = 3x + 1 and x = 4.
Solution:
The equations of sides of triangle are
y = 2x + 1, …(i)
y = 3x + 1 …(ii)
and x = 4 …(iii)
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 101
The equation y = 2x + 1 meets x and y axes at \left(-\frac{1}{2}, 0\right)and (0, 1). By joining these two points we obtain the graph of x + 2y = 2. Similarly, graphs of other equations are drawn.
Solving equation (i), (ii) and (iii) in pairs, we obtain the coordinates of verticies of ∆ABC are A(0, 1), B (4, 13) and C (4, 9).
Then, area of ∆ABC = Area (OLBAO) – Area (OLCAO)
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 102
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 103

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

Section – A

Question 1.
Find the differenital equation representing the family of curves y = ae2x + 5, where a is an arbitrary constant. [1]
Solution:
Given, y = ae2x + 5
On differentiating w.r.t. x, we get
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 104

Question 2.
If y = cos\sqrt{3 x}, then find \frac{d y}{d x}. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 105

Section – B

Question 5.
Show that the points \mathbf{A}(-2 \hat{i}+3 \hat{j}+5 \hat{k}), \mathbf{B}(\hat{i}+2 \hat{j}+3 \hat{k})and \mathbf{c}(7 \hat{i}-\hat{k})are collinear. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 106
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 107
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 108

Question 6.
Find:[2]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 109
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 110

Section – C

Question 13.
Solve for x: tan-1 + tan-1(x – 1) = tan-1\left(\frac{8}{31}\right). [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 111
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 112

Question 14.
If x = aet (sin t + cost t) and y = aet (sin t – cos t), then prove that \frac{d y}{d x}=\frac{x+y}{x-y}. [4]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 113
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 114
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 115
OR
Differentiate xsinx + (sinx)cosx with respect to x.
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 116
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 117

Question 15.
Find: [4]
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 118
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 119
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 120
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 121

Section – D

Question 24.
Show that for the matrix A = \left[\begin{array}{rrr}{1} & {1} & {1} \\ {1} & {2} & {-3} \\ {2} & {-1} & {3}\end{array}\right], A3 – 6A2 + 5A + 111 = 0. Hence, find A -1.
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 122
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 123
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 124
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 125
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 126
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 127
OR
Using matrix method, solve the following system of equations:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
Solution:
The given equations are
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
These equations can be written in the form AX = B, where
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 128
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 129

Question 26.
A bag contains 5 red and 4 black balls, a second bag contains 3 red and 6 black balls. One of the two bags is selected at random and two balls are drawn at random (without replacement) both of which are found to be red. Find the probability that the balls are drawn from the second bag. [6]
Solution:
Let E1 be the event of choosing the bag I, E2 be the event of choosing the bag II and A be the event of drawing a red ball.
Then,
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 130
CBSE Previous Year Question Papers Class 12 Maths 2019 Outside Delhi 131

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