If the points A(2, 1, 1), B(0, -1, 4) and C(k, 3, -2) are collinear, then k

(A) 0

(B) 1

(C) 4

(D) -4

Concept: Relation Between Direction Ratio and Direction Cosines

The inverse of the matrix `[[-1,5],[-3,2]]` is_______

(A) 1/13`[[2,-5],[3,-1]]`

(B) 1/13 `[[-1,5],[-3,2]]`

(C) 1/13 `[[-1,-3],[5,2]]`

(D) 1/13 `[[1,5],[3,-2]]`

Concept: Matrices - Inverse of a Matrix Existance

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) `1/5`

(B) `sqrt(1/5)`

(C) `4/5`

(D) `2/5`

Concept: Trigonometric Functions - Solution of a Triangle

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Concept: Scalar Triple Product of Vectors

In Δ ABC, prove that, a (b cos C - c cos B) = b^{2} - c^{2}.

Concept: Trigonometric Functions - Trigonometric equations

If from a point Q (a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

Concept: Trigonometric Functions - Trigonometric equations

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation

Write the following statement in symbolic form and find its truth value:

∀ n ∈ N, n^{2} + n is an even number and n^{2} - n is an odd number.

Concept: Mathematical Logic - Truth Value of Statement in Logic

Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.

Concept: Mathematical Logic - Truth Value of Statement in Logic

Find the shortest distance between the lines `(x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5`

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type

Solve the following equations by method of reduction:

x-y + z = 4,

2x + y - 3z = 0,

x + y + z = 2

Concept: Matrices - Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method

If θ is the measure of acute angle between the pair of lines given by `ax^2+2hxy+by^2=0,` then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0`

Concept: Acute Angle Between the Lines

Using vector method, find incentre of the triangle whoose vertices are P(0, 4, 0), Q(0, 0, 3)

and R(0, 4, 3).

Concept: Vectors - Application of Vectors to Geometry

Construct the switching circuit for the statement (p ∧ q) ∨ (~ p) ∨ (p ∧ ~ q).

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x^{2} +2xy- 3y^{2} = 0.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Concept: Basic Concepts of Trigonometric Functions

If l, m, n are the direction cosines of a line, then prove that l^{2} + m^{2} + n^{2} = 1. Hence find the

direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.

Concept: Direction Cosines and Direction Ratios of a Line

Find the vector and cartesian equations of the plane passing through the points A( 1, 1, -2), B(1, 2, 1) and C(2, -1, 1).

Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤ 3, -2x + y ≤ 1, x ≥ 0, y ≥ 0.

Also find maximum value of Z.

Concept: Graphical Method of Solving Linear Programming Problems

Derivatives of tan^{3}θ with respect to sec^{3}θ at θ=π/3 is

(A)` 3/2`

(B) `sqrt3/2`

(C) `1/2`

(D) `-sqrt3/2`

Concept: Derivatives of Functions in Parametric Forms

The equation of tangent to the curve y = 3x^{2} - x + 1 at the point (1, 3) is

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

Concept: Conics - Tangents and normals - equations of tangent and normal at a point

The expected value of the number of heads obtained when three fair coins are tossed simultaneously is

(A) 1

(B) 1.5

(C) 0

(D) -1

Concept: Probability Distribution - Expected Value, Variance and Standard Deviation of a Discrete Random Variable

Find dy/dx if x sin y + y sin x = 0.

Concept: Derivatives of Implicit Functions

Test whether the function, f(x) = x -1/x, x ∈ R, x ≠ 0, is increasing or decreasing

Concept: Increasing and Decreasing Functions

Evaluate: `intsinsqrtx/sqrtxdx`

Concept: Evaluation of Definite Integrals by Substitution

Form the differential equation by eliminating arbitrary constants from the relation `y=Ae^(5x)+Be^(-5x)`

Concept: Formation of Differential Equation by Eliminating Arbitary Constant

The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.

Concept: Bernoulli Trials and Binomial Distribution

Solve: dy/dx = cos(x + y)

Concept: Methods of Integration - Integration by Substitution

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Concept: Methods of Integration - Integration by Parts

If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).

Concept: Continuity - Continuity of a Function at a Point

If y = f(x) is a differentiable function of x such that inverse function x = f^{–1} (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`

Concept: Derivative - Derivative of Inverse Function

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

Concept: Maxima and Minima

Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`

Concept: Properties of Definite Integrals

Discuss the continuity of the following function, at x = 0.

`f(x)=x/|x|,for x ne0`

`=1,`for `x=0`

Concept: Continuity - Continuity of a Function at a Point

If the population of a country doubles in 60 years, in how many years will it be triple under

the assumption that the rate of increase in proportional to the number of inhabitants?

[Given : log 2 = 0.6912 and log 3 = 1.0986.]

Concept: Differential Equations - Applications of Differential Equation

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.

Concept: Conditional Probability

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Concept: Conditional Probability

Find: `I=intdx/(sinx+sin2x)`

Concept: Methods of Integration - Integration Using Partial Fractions

Find the area of the region lying between the parabolas y^{2} = 4ax and x^{2} = 4ay.

Concept: Area Between Two Curves

Given the p. d. f. (probability density function) of a continuous random variable x as :

`f(x)=x^2/3, -1`

= 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)

Concept: Probability Distribution - Probability Density Function (P.D.F.)