If p ˄ q = F, p → q = F, then the truth value of p and q is :

(A) T, T

(B) T, F

(C) F, T

(D) F, F

Concept: Mathematical Logic - Truth Value of Statement in Logic

If `A^-1=1/3[[1,4,-2],[-2,-5,4],[1,-2,1]]` and | A | = 3, then (adj. A) = _______

(A)`1/9[[1,4,-2],[-2,-5,4],[1,-2,1]]`

(B)`[[1,-2,1],[4,-5,-2],[-2,4,1]]`

(C)`[[1,4,-2],[-2,-5,4],[1,-2,1]]`

(D)`[[-1,-4,2],[2,5,-4],[1,-2,1]]`

Concept: Determinants - Adjoint Method

The slopes of the lines given by 12x^{2} + bxy + y^{2} = 0 differ by 7. Then the value of b is :

(A) 2

(B) ± 2

(C) ± 1

(D) 1

Concept: Acute Angle Between the Lines

In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`

Concept: Trigonometric Functions - Solution of a Triangle

Find ‘k’, if the equation kxy + 10x + 6y + 4 = 0 represents a pair of straight lines.

Concept: Acute Angle Between the Lines

If A, B, C, D are four non-collinear points in the plane such that `bar(AD)+bar( BD)+bar( CD)=bar O` then prove that point D is the centroid of the ΔABC.

Concept: Vectors - Centroid Formula for Vector

Find the direction cosines of the line

`(x=2)/2=(2y-5)/3; z=-1`

Concept: Direction Cosines and Direction Ratios of a Line

Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`

Concept: Distance of a Point from a Plane

Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).

Concept: Mathematical Logic - Truth Tables of Compound Statements

Show that every homogeneous equation of degree two in x and y, i.e., ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through origin if h^{2}−ab≥0.

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

Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `

Concept: Scalar Triple Product of Vectors

Find the inverse of the matrix, `A=[[1,3,3],[1,4,3],[1,3,4]]`by using column transformations.

Concept: Elementary Operation (Transformation) of a Matrix

In ΔABC, prove that : `tan((a-b)/2)=(a-b)/(a+b)cotC/2`

Concept: Trigonometric Functions - Solution of a Triangle

Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.

Concept: Coplanarity of Two Lines

Construct the simplified circuit for the following circuit:

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

Express `-bari-3barj+4bark ` as a linear combination of vectors `2bari+barj-4bark,2bari-barj+3bark`

Concept: Vectors - Linear Combination of Vectors

Find the length of the perpendicular from the point (3, 2, 1) to the line `(x-7)/2=(y-7)/2=(z-6)/3=lambda (say)`

Concept: Three - Dimensional Geometry - Condition for Perpendicular Lines

Show that the angle between any two diagonals of a cube is `cos^-1(1/3)`

Concept: Angle Between Line and a Plane

Minimize : Z = 6x + 4y

Subject to the conditions:

3x + 2y ≥ 12,

x + y ≥ 5,

0 ≤ x ≤ 4,

0 ≤ y ≤ 4

Concept: Graphical Method of Solving Linear Programming Problems

If `tan^-1((x-1)/(x-2))+cot^-1((x+2)/(x+1))=pi/4; `

Concept: Basic Concepts of Trigonometric Functions

If `y=sec^-1((sqrtx-1)/(x+sqrtx))+sin_1((x+sqrtx)/(sqrtx-1)), `

(A) x

(B) 1/x

(C) 1

(D) 0

Concept: Derivative - Derivative of Inverse Function

If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:

(A) 0

(B) π

(C) π/2

(D) π/4

Concept: Methods of Integration - Integration by Parts

The solution of the differential equation dy/dx = sec x – y tan x is:

(A) y sec x = tan x + c

(B) y sec x + tan x = c

(C) sec x = y tan x + c

(D) sec x + y tan x = c

Concept: General and Particular Solutions of a Differential Equation

Evaluate: `int1/(xlogxlog(logx))dx`

Concept: Evaluation of Definite Integrals by Substitution

Find the area bounded by the curve y^{2} = 4ax, x-axis and the lines x = 0 and x = a.

Concept: Area of the Region Bounded by a Curve and a Line

Find k, such that the function P(x)=k(4/x) ;x=0,1,2,3,4 k>0

=0 ,otherwise

Concept: Standard Deviation of Binomial Distribution (P.M.F.)

Given is X ~ B (n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Concept: Bernoulli Trials and Binomial Distribution - Calculation of Probabilities

Solve the differential equation `y-xdy/dx=0`

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable

Discuss the continuity of the function

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

** = 3, for x=π/2**

Concept: Continuity - Discontinuity of a Function

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f'(x).

Concept: Maxima and Minima

Differentiate `cos^-1((3cosx-2sinx)/sqrt13)` w. r. t. x.

Concept: Derivatives of Inverse Trigonometric Functions

Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

Concept: Methods of Integration - Integration by Substitution

A rectangle has area 50 cm^{2} . Find its dimensions when its perimeter is the least

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

= 0, if f (x) is an odd function.

Concept: Methods of Integration - Integration by Parts

If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`

Concept: Rate of Change of Bodies Or Quantities

Each of the total five questions in a multiple choice examination has four choices, only one of which is correct. A student is attempting to guess the answer. The random variable x is the number of questions answered correctly. What is the probability that the student will give atleast one correct answer?

Concept: Probability Distribution of a Discrete Random Variable

If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.

Concept: Continuity - Continuity of a Function at a Point

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

Concept: Maxima and Minima

Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.

Concept: General and Particular Solutions of a Differential Equation

Find the expected value, variance and standard deviation of random variable X whose probability mass function (p.m.f.) is given below:

X=x | 1 | 2 | 3 |

P(X=x) |
1/5 | 2/5 | 2/5 |

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