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# Question Paper Solutions - Mathematics 2016 - 2017-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education)

SubjectMathematics
Year2016 - 2017 (March)

#### Alternate Sets

Marks: 100
[1]1

If for any 2 x 2 square matrix A, A(adj A) = [(8,0), (0,8)], then write the value of |A|

Chapter: [2.01] Matrices and Determinants
Concept: Types of Matrices
[1]2

Determine the value of 'k' for which the follwoing function is continuous at x = 3

f(x) = {(((x+3)^2-36)/(x-3),  x != 3), (k,  x =3):}

Chapter: [3.01] Continuity, Differentiability and Differentiation
Concept: Concept of Continuity
[1]3

Find int (sin^2 x - cos^2 x)/(sin xcosx) dx

Chapter: [3.03] Integrals
Concept: Indefinite Integral by Inspection
[1]4

Find the distance between the planes 2x - y +  2z = 5 and 5x - 2.5y + 5z = 20

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Shortest Distance Between Two Lines
[2]5

If A is a skew symmetric matric of order 3, then prove that det A  = 0

Chapter: [11] Matrices
Concept: Symmetric and Skew Symmetric Matrices
[2]6

Find the value of c in Rolle's theorem for the function f(x) = x^3 - 3x " in " (-sqrt3, 0)

Chapter: [3.02] Applications of Derivatives
Concept: Increasing and Decreasing Functions
[2]7

The Volume of cube is increasing at the rate of 9 cm 3/s. How fast is its surfacee area increasing when the length of an edge is 10 cm?

Chapter: [3.02] Applications of Derivatives
Concept: Rate of Change of Bodies Or Quantities
[2]8

Show that the function f(x) = x^3 - 3x^2 + 6x - 100 is increasing on R

Chapter: [3.02] Applications of Derivatives
Concept: Increasing and Decreasing Functions
[2]9

The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate

Chapter: [4.02] Three - Dimensional Geometry
Concept: Vector and Cartesian Equation of a Plane
[2]10

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

Chapter: [22] Probability
Concept: Independent Events
[2]11

Two tailors, A and B, earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP

Chapter: [5.01] Linear Programming
Concept: Linear Programming Problem and Its Mathematical Formulation
[2]12

Find int dx/(5 - 8x - x^2)

Chapter: [3.03] Integrals
Concept: Properties of Indefinite Integral
[4]13

If tan^(-1)  (x- 3)/(x - 4) + tan^(-1)  (x +3)/(x + 4) = pi/4, then find the value of x.

Chapter: [3.03] Integrals
Concept: Indefinite Integral by Inspection
[4]14 | Attempt Any One

Using properties of determinants, prove that

|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3

Chapter: [2.01] Matrices and Determinants
Concept: Properties of Determinants

Find matrix A such that ((2,-1),(1,0),(-3,4))A = ((-1, -8),(1, -2),(9,22))

Chapter: [2.01] Matrices
Concept: Order of a Matrix
[4]15 | Attempt Any One

if x^y + y^x = a^bthen Find dy/dx

Chapter: [4] Differentiation
Concept: Derivatives of Implicit Functions

If ey (x + 1) = 1, show that  (d^2y)/(dx^2) =((dy)/(dx))^2

Chapter: [3.01] Continuity and Differentiability
Concept: Second Order Derivative
[4]16

Find int (cos theta)/((4 + sin^2 theta)(5 - 4 cos^2 theta)) d theta

Chapter: [3.03] Integrals
Concept: Properties of Indefinite Integral
[4]17 | Attempt Any One

Evaluate the definite integrals int_0^pi (x tan x)/(sec x + tan x) dx

Chapter: [3.03] Integrals
Concept: Definite Integral as the Limit of a Sum

Evaluate: int_1^4 {|x -1|+|x - 2|+|x - 4|}dx

Chapter: [3.03] Integrals
Concept: Properties of Definite Integrals
[4]18

Solve the differential equation (tan^(-1) x- y) dx = (1 + x^2) dy

Chapter: [3.05] Differential Equations
Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
[4]19

Show that the points A, B, C with position vectors 2hati- hatj + hatk, hati - 3hatj - 5hatk and 3hati - 4hatj - 4hatk respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle

Chapter: [4.01] Vectors
Concept: Introduction of Product of Two Vectors
[4]20

Find the value of λ, if four points with position vectors 3hati + 6hatj+9hatk, hati + 2hatj + 3hatk,2hati + 3hatj + hatk and 4hati + 6hatj + lambdahatk are coplanar.

Chapter: [4.01] Vectors
Concept: Scalar Triple Product of Vectors
[4]21

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean 'and variance of X.

Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions
[4]22

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chos~n at random from the school and he was found ·to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer

Chapter: [22] Probability
Concept: Baye'S Theorem
[4]23

Maximise Z = x + 2y subject to the constraints

x + 2y >= 100

2x - y <= 0

2x + y <= 200

Solve the above LPP graphically

Chapter: [10] Linear Programming (Section C)
Concept: Graphical Method of Solving Linear Programming Problems
[6]24

Determine the product [(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)] and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.

Chapter: [2.01] Matrices and Determinants
Concept: Types of Matrices
[6]25 | Attempt Any One

Consider f:R - {-4/3} -> R - {4/3} given by f(x) = (4x + 3)/(3x + 4). Show that f is bijective. Find the inverse of f and hence find f^(-1) (0) and X such that f^(-1) (x) = 2

Chapter: [1] Relations and Functions (Section A)
Concept: Inverse of a Function

Let A = Q x Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∈ A. Determine, whether * is commutative and associative. Then, with respect to * on A

1) Find the identity element in A

2) Find the invertible elements of A.

Chapter: [12.03] Functions
Concept: Concept of Binary Operations
[6]26

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

Chapter: [3.02] Applications of Derivatives
Concept: Maxima and Minima
[6]27 | Attempt Any One

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4 , 1), B (6, 6) and C (8, 4).

Chapter: [7] Application of Integrals (Section B)
Concept: Area Under Simple Curves

Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x - 2y + 12 = 0.

Chapter: [7] Application of Integrals (Section B)
Concept: Area Under Simple Curves
[6]28

Find the particular solution of the differential equation (x - y) dy/dx = (x + 2y) given that y = 0 when x = 1.

Chapter: [3.05] Differential Equations
Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
[6]29 | Attempt Any One

Find the coordinates of the point where the line through the points (3, - 4, - 5) and (2, - 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2,- 3) and (0, 4, 3)

Chapter: [4.02] Three - Dimensional Geometry
Concept: Plane - Equation of a Plane in Normal Form

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is 1/x^2 + 1/y^2 + 1/z^2 = 1/p^2

Chapter: [4.02] Three - Dimensional Geometry
Concept: Plane - Intercept Form of the Equation of a Plane
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