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Mathematics All India Set 3 2016-2017 CBSE (Commerce) Class 12 Question Paper Solution

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Mathematics [All India Set 3]
Marks: 100 Academic Year: 2016-2017
Date & Time: 19th March 2017, 12:30 pm
Duration: 3h
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[1] 1

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):}`

Concept: Concept of Continuity
Chapter: [0.05] Continuity and Differentiability
[1] 2

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

Concept: Types of Matrices
Chapter: [0.03] Matrices
[1] 3

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

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[1] 4

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

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
[2] 5

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

Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[2] 6

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

Concept: Linear Programming Problem and Its Mathematical Formulation
Chapter: [0.12] Linear Programming
[2] 7

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.

Concept: Independent Events
Chapter: [0.13] Probability
[8] 8

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

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
[2] 9

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

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
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[2] 10

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

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[2] 11

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

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.03] Matrices
[2] 12

The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

Concept: Rate of Change of Bodies Or Quantities
Chapter: [0.06] Applications of Derivatives
[4] 13

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.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
[4] 14

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

Concept: Introduction of Product of Two Vectors
Chapter: [0.1] Vectors
[4] 15

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

Concept: Bayes’ Theorem
Chapter: [0.13] Probability
[4] 16

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

Concept: Indefinite Integral by Inspection
Chapter: [0.07] Integrals
[4] 17 | 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`

Concept: Properties of Determinants
Chapter: [0.04] Determinants

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

Concept: Order of a Matrix
Chapter: [0.03] Matrices
[4] 18 | Attempt Any One

if xx+xy+yx=ab, then find `dy/dx`.

Concept: Logarithmic Differentiation
Chapter: [0.05] Continuity and Differentiability

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

Concept: Second Order Derivative
Chapter: [0.05] Continuity and Differentiability
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[4] 19 | Attempt Any One

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

Concept: Definite Integral as the Limit of a Sum
Chapter: [0.07] Integrals

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

Concept: Properties of Definite Integrals
Chapter: [0.07] Integrals
[4] 20

Solve the following linear programming problem graphically :

Maximise Z = 7x + 10y subject to the constraints

4x + 6y ≤ 240

6x + 3y ≤ 240

x ≥ 10

x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[4] 21

Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.07] Integrals
[4] 22

if `veca = 2hati - hatj - 2hatk " and " vecb = 7hati + 2hatj - 3hatk`, , then express `vecb` in the form of `vecb = vec(b_1) + vec(b_2)`, where `vec(b_1)`  is parallel to `veca` and `vec(b_2)` is perpendicular to `veca`

Concept: Vectors and Their Types
Chapter: [0.1] Vectors
[4] 23

Find the general solution of the differential equation `dy/dx - y = sin x`

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [0.09] Differential Equations
[6] 24 | 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).

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals

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

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 25

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

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.09] Differential Equations
[6] 26 | 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)

Concept: Plane - Equation of a Plane in Normal Form
Chapter: [0.11] Three - Dimensional Geometry

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`

Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
[6] 27 | 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`

Concept: Inverse of a Function
Chapter: [0.01] Relations and Functions

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.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 28

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations

2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3

Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[6] 29

A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives

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