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# Question Paper Solutions - Mathematics 2017 - 2018-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education)

SubjectMathematics
Year2017 - 2018 (March)

#### Alternate Sets

Marks: 100
[1]1

If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.

Chapter: [12.03] Functions
Concept: Concept of Binary Operations
[1]2

Find the magnitude of each of two vectors veca and vecb having the same magnitude such that the angle between them is 60° and their scalar product is 9/2

Chapter: [5] Vectors (Section B)
Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
[1]3

if the matrix A =[(0,a,-3),(2,0,-1),(b,1,0)] is skew symmetric, Find the value of 'a' and 'b'

Chapter: [11] Matrices
Concept: Types of Matrices
[1]4

Find the value of tan^(-1) sqrt3 - cot^(-1) (-sqrt3)

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
[2]5

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Chapter: [3.02] Applications of Derivatives
Concept: Rate of Change of Bodies Or Quantities
[2]6

Differentiate tan^(-1) ((1+cosx)/(sin x)) with respect to x

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[2]7

Given A = [(2,-3),(-4,7)] compute A^(-1) and show that 2A^(-1) = 9I - A

Chapter: [11] Matrices
Concept: Types of Matrices
[2]8

Prove that 3sin^(-1)x = sin^(-1) (3x - 4x^3), x in [-1/2, 1/2]

Chapter: [3] Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions
[2]9

A black and a red dice are rolled.  Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Chapter: [22] Probability
Concept: Conditional Probability
[2]10

If θ is the angle between two vectors hati - 2hatj + 3hatkand 3hati - 2hatj + hatk find sin theta

Chapter: [5] Vectors (Section B)
Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
[2]11

Find the differential equation representing the family of curves y = ae^(bx + 5). where a and b are arbitrary constants.

Chapter: [3.04] Differential Equations
Concept: Formation of a Differential Equation Whose General Solution is Given
[2]12

Evaluate int (cos 2x + 2sin^2x)/(cos^2x) dx

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

If y = sin (sin x), prove that (d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0

Chapter: [3.01] Continuity and Differentiability
Concept: Higher Order Derivative
[4]14
[4]14.1

Find the particular solution of the differential equation ex tan y dx + (2 – ex) sec2 y dy = 0, give that y = pi/4 when x = 0

Chapter: [3.05] Differential Equations
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable
[4]14.2

Find the particular solution of the differential equation dy/dx + 2y tan x = sin x given that y = 0 when x =  pi/3

Chapter: [3.05] Differential Equations
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable
[4]15

Find the shortest distance between the lines vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk) and vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)

Chapter: [4.02] Three - Dimensional Geometry
Concept: Shortest Distance Between Two Lines
[4]16

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X

Chapter: [14] Random Variable and Probability Distribution
Concept: Random Variables and Its Probability Distributions
[4]17

Using properties of determinants, prove that |(1,1,1+3x),(1+3y, 1,1),(1,1+3z,1)| = 9(3xyz + xy +  yz+ zx)

Chapter: [2.02] Determinants
Concept: Properties of Determinants
[4]18
[4]18.1

Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.

Chapter: [14] Applications of Derivative
Concept: Tangents and Normals
[4]18.2

Find the intervals in which the function f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12  is (a) strictly increasing, (b) strictly decreasing

Chapter: [5] Applications of Derivative
Concept: Increasing and Decreasing Functions
[4]19

Find int (2cos x)/((1-sinx)(1+sin^2 x)) dx

Chapter: [15] Integration
Concept: Methods of Integration - Integration Using Partial Fractions
[4]20

Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the ride?

Chapter: [22] Probability
Concept: Baye'S Theorem
[4]21

Let veca = 4hati + 5hatj - hatk, vecb  = hati - 4hatj + 5hatk and vecc = 3hati + hatj - hatk. Find a vector vecd which is perpendicular to both vecc and vecb and vecd.veca = 21

Chapter: [5] Vectors (Section B)
Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
[4]22

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?

Chapter: [14] Applications of Derivative
Concept: Maxima and Minima
[4]23
[4]23.1

if (x^2 + y^2)^2 = xy find (dy)/(dx)

Chapter: [3.01] Continuity and Differentiability
Concept: Derivatives of Implicit Functions
[4]23.2

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find dy/dx when theta = pi/3

Chapter: [4] Differentiation
Concept: Derivatives of Functions in Parametric Forms
[6]24
[6]24.1

Evaluate int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[6]24.2

Evaluate : int_1^3 (x^2 + 3x + e^x) dx as the limit of the sum.

Chapter: [15] Integration
Concept: Definite Integral as the Limit of a Sum
[6]25

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Chapter: [10] Linear Programming (Section C)
Concept: Different Types of Linear Programming Problems
[6]26
[6]26.1

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(ab) : a∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]

Chapter: [1] Relations and Functions (Section A)
Concept: Types of Relations
[6]26.2

Show that the function f: ℝ → ℝ defined by f(x) = x/(x^2 + 1), ∀x in Ris neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)

Chapter: [12.03] Functions
Concept: Types of Functions
[6]27

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.

Chapter: [3.04] Applications of the Integrals
Concept: Area Under Simple Curves
[6]28
[6]28.1

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

Chapter: [2.01] Matrices and Determinants
Concept: Applications of Determinants and Matrices
[6]28.2

Using elementary row transformations, find the inverse of the matrix A = [(1,2,3),(2,5,7),(-2,-4,-5)]

Chapter: [2.01] Matrices and Determinants
Concept: Elementary Operation (Transformation) of a Matrix
[6]29

Find the distance of the point (−1, −5, −10) from the point of intersection of the line vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk)  and the plane vec r (hati-hatj+hatk)=5

Chapter: [4.02] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions
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