If A = `[(2,-3),(4,1)]`, then adjoint of matrix A is

(A) `[(1,3),(-4,2)]`

(B) `[(1,-3),(-4,2)]`

(C) `[(1,3),(4,-2)]`

(D) `[(-1,-3),(-4,2)]`

Concept: Determinants - Adjoint Method

The principal solutions of sec x = `2/sqrt3` are _____

(A) `pi/3,(11pi)/6`

(B) `pi/6, (11pi)/6`

(C)`pi/4,(11pi)/4`

(D) `pi/6,(11pi)/4`

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

The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and –2, 1, 2 is ______.

Concept: Angle Between Two Lines

Write the negations of the following statements :

1) All students of this college live in the hostel

2) 6 is an even number or 36 is a perfect square.

Concept: Mathematical Logic - Sentences and Statement in Logic

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Concept: Angle Between Line and a Plane

Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .

Concept: Distance of a Point from a Plane

Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`

Concept: Equation of a Line in Space

if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`

Concept: Scalar Triple Product of Vectors

By vector method prove that the medians of a triangle are concurrent.

Concept: Vectors - Medians of a Triangle Are Concurrent

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q )

Concept: Mathematical Logic - Truth Tables of Compound Statements

If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and R(–5, 1, r), then find the values of p, q, r.

Concept: Section formula

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

In `triangle ABC` prove that `tan((C-A)/2) = ((c-a)/(c+a))cot B/2`

Concept: Trigonometric Functions - Trigonometric equations

Find the inverse of the matrix `A = [(1,2,-2),(-1,3,0),(0,-2,1)]`using elementary row transformations.

Concept: Matrices - Inverse of a Matrix Existance

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

Find the angle between the lines `(x -1)/4 = (y - 3)/1 = z/8` and `(x-2)/2 = (y + 1)/2 = (z-4)/1`

Concept: Line - Equation of Line Passing Through Given Point and Parallel to Given Vector

Write converse, inverse and contrapositive of the following conditional statement :

If an angle is a right angle then its measure is 90°.

Concept: Mathematical Logic - Difference Between Converse, Contrapositive, Contradiction

Prove that `sin^(-1) (3/5) + cos^(-1) (12/13) = sin^(-1) (56/65)`

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch

Find the vector equation of the plane passing through the points A(1, 0, 1), B(1, –1, 1) and C(4, –3, 2).

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

Minimize Z = 7x + y subject to `5x + y >= 5, x + y >= 3, x>= 0, y >= 0`

Concept: Graphical Method of Solving Linear Programming Problems

Let the p. m. f. of a random variable X be __

P(x) = `(3-x)/10` for x = -1,0,1,2

= 0 otherwise

Then E(X ) is ________.

(A) 1

(B) 2

(C) 0

(D) – 1

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

if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.

(A) `1/2`

(B) `1/3`

(C) `1/4`

(D) `1/5`

Concept: Definite Integral as the Limit of a Sum

Integrating factor of linear differential equation `x (dy)/(dx) + 2y =x^2 log x` is ____________

(A) `1/x^2`

(B) `1/x`

(C) x

(D) `x^2`

Concept: Differential Equations - Linear Differential Equation

Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`

Concept: Properties of Definite Integrals

if `y = tan^2(log x^3)`, find `(dy)/(dx)`

Concept: Derivatives of Composite Functions - Chain Rule

Find the area of ellipse `x^2/1 + y^2/4 = 1`

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

Obtain the differential equation by eliminating the arbitrary constants from the following equation :

`y = c_1e^(2x) + c_2e^(-2x)`

Concept: Formation of Differential Equation by Eliminating Arbitary Constant

Given X ~ B (n, p)

If n = 10 and p = 0.4, find E(X) and var (X).

Concept: Bernoulli Trials and Binomial Distribution

Evaluate `int 1/(3+ 2 sinx + cosx) dx`

Concept: Methods of Integration - Integration by Substitution

If `x = acos^3t`, `y = asin^3 t`,

Show that `(dy)/(dx) =- (y/x)^(1/3)`

Concept: Derivatives of Functions in Parametric Forms

Examine the continuity of the function:

f(x) = `(log100 + log(0.01+x))/"3x"`

= 100/3 for x = 0; at x = 0

Concept: Continuity - Defination of Continuity of a Function at a Point

Examine the maxima and minima of the function f(x) = 2x^{3} - 21x^{2} + 36x - 20 . Also, find the maximum and minimum values of f(x).

Concept: Maxima and Minima

Prove that `int 1/(a^2 - x^2) dx = 1/"2a" log|(a +x)/(a-x)| + c`

Concept: Fundamental Theorem of Calculus

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 `f(x) = (x^2-9)/(x-3) + alpha` for x> 3

=5, for x = 3

`=2x^2+3x+beta`, for x < 3

is continuous at x = 3, find α and β.

Concept: Continuity - Continuity of a Function at a Point

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

Concept: Derivative - Derivative of Inverse Function

A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.

Concept: Bernoulli Trials and Binomial Distribution

Verify Rolle’s theorem for the following function:

f (x) = x^{2} - 4x + 10 on [0, 4]

Concept: Mean Value Theorem

Find the particular solution of the differential equation:

`y(1+logx) dx/dy - xlogx = 0`

when y = e^{2} and x = e

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

Find the variance and standard deviation of the random variable X whose probability distribution is given below :

x | 0 | 1 | 2 | 3 |

P(X = x) | `1/8` | `3/8` | `3/8` | `1/8` |

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