# Mathematics All India Set 3 2017-2018 Science (English Medium) Class 12 Question Paper Solution

Mathematics [All India Set 3]
Date: March 2018

[1] 1

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

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[1] 2

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

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [0.02] Inverse Trigonometric Functions
[1] 3

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.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[1] 4

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

Concept: Types of Matrices
Chapter: [0.03] Matrices
[2] 5

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.

Concept: Conditional Probability
Chapter: [0.13] Probability
[2] 6

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

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [0.1] Vectors
[2] 7

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

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
[2] 8

Evaluate : $\int\frac{\cos 2x + 2 \sin^2 x}{\cos^2 x}dx$ .

Concept: Indefinite Integral Problems
Chapter: [0.07] Integrals
[2] 9

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.

Concept: Rate of Change of Bodies or Quantities
Chapter: [0.06] Applications of Derivatives
[2] 10

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

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
[2] 11

Given $A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}$, compute A−1 and show that $2 A^{- 1} = 9I - A .$

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.04] Determinants
[2] 12

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

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 13

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

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

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?

Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[4] 15 | Attempt any one of the following
[4] 15.1

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.

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 15.2

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

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[4] 16 | Attempt any one of the following
[4] 16.1

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

Concept: Derivatives of Implicit Functions
Chapter: [0.05] Continuity and Differentiability
[4] 16.2

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find $\frac{dy}{dx}$ When  $\theta = \frac{\pi}{3}$ .

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 17

If y = sin (sin x), prove that $\frac{d^2 y}{d x^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0 .$

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 18 | Attempt any one of the following
[4] 18.1

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

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.09] Differential Equations
[4] 18.2

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

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.09] Differential Equations
[4] 19

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

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[4] 20

Find : $\int\frac{2 \cos x}{\left( 1 - \sin x \right) \left( 1 + \sin^2 x \right)}dx$ .

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 21

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 die ?

Concept: Bayes’ Theorem
Chapter: [0.13] Probability
[4] 22

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

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [0.1] Vectors
[4] 23

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

Concept: Properties of Determinants
Chapter: [0.04] Determinants
[6] 24

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.

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 25 | Attempt any one of the following
[6] 25.1

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]

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
[6] 25.2

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)

Concept: Types of Functions
Chapter: [0.01] Relations and Functions
[6] 26

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

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[6] 27

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.

Concept: Different Types of Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 28 | Attempt any one of the following
[6] 28.1

Evaluate : $\int\limits_0^\pi/4 \frac{\sin x + \cos x}{16 + 9 \sin 2x}dx$ .

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
[6] 28.2

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

Concept: Definite Integral as the Limit of a Sum
Chapter: [0.07] Integrals
[6] 29 | Attempt any one of the following
[6] 29.1

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

Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[6] 29.2

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

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants

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