Inverse of the statement pattern (p ∨ q) → (p ∧ q) is

(A) (p ∧ q) → (p ∨ q)

(B) ∼ (p ∨ q) → (p ∧ q)

(C) (∼ p ∨ ∼ q) → (∼ p ∧ ∼ q)

(D) (∼ p ∧ ∼ q) → (∼ p ∨ ∼ q)

Concept: Mathematical Logic - Sentences and Statement in Logic

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

Concept: Vectors - Collinearity and Coplanarity of Vectors

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`

Concept: Trigonometric Functions - Solution of a Triangle

Find the angle between the lines `barr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk)` and `barr=5 hati-2hatk+mu(3hati+2hatj+6hatk)`

Concept: Acute Angle Between the Lines

If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p

Concept: Mathematical Logic - Truth Value of Statement in Logic

If `A =[[2,-3],[3,5]]` then find A^{-1} by adjoint method.

Concept: Determinants - Adjoint Method

By vector method show that the quadrilateral with vertices A (1, 2, –1), B (8, –3, –4), C (5, –4, 1), D (–2, 1, 4) is a parallelogram.

Concept: Vectors - Diagonals of a Parallelogram Bisect Each Other and Converse

Find the general solution of the equation sin x = tan x.

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

Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax^{2}+ 2hxy + by^{2}= 0

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

Find the principal value of `sin^-1(1/sqrt2)`

Concept: Basic Concepts of Trigonometric Functions

Find the cartesian form of the equation of the plane `bar r=(hati+hatj)+s(hati-hatj+2hatk)+t(hati+2hatj+hatj)`

Concept: Vector and Cartesian Equation of a Plane

Simplify the following circuit so that new circuit has minimum number of switches. Also draw simplified circuit.

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of magnitude 5 along the direction of line.

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

Maximize:

z = 3x + 5y

Subject to: x + 4y ≤ 24

3x + y ≤ 21

x + y ≤ 9

x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems

Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`

Concept: Shortest Distance Between Two Lines

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

Concept: Distance of a Point from a Plane

In any triangle ABC with usual notations prove c = a cos B + b cos A

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

Find p and k if the equation px^{2} – 8xy + 3y^{2 }+14x + 2y + k = 0 represents a pair of perpendicular lines.

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

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Concept: Elementary Operation (Transformation) of a Matrix

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

Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Concept: Order and Degree of a Differential Equation

`∫_4^9 1/sqrtxdx=`_____

(A) 1

(B) –2

(C) 2

(D) –1

Concept: Properties of Definite Integrals

If the p.d.f. of a continuous random variable X is given as

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

=0 otherwise

then c.d.f. fo X is

(A) `x^3/9+1/9`

(B) `x^3/9-1/9`

(C) `x^2/4+1/4`

(D) `1/(9x^3)+1/9`

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

If `y = sec sqrtx` then find dy/dx.

Concept: Derivative - Derivative of Functions in Product of Function Form

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Concept: Methods of Integration - Integration Using Partial Fractions

Find the area of the region lying in the first quandrant bounded by the curve y^{2}= 4x, X axis and the lines x = 1, x = 4

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

For the differential equations find the general solution:

sec^{2} x tan y dx + sec^{2} y tan x dy = 0

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

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

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))` for x ≠ 0 is continuous at x = 0, find f (0).

Concept: Continuity of Some Standard Functions - Trigonometric Function

Evaluate : `∫1/(3+2sinx+cosx)dx`

Concept: Methods of Integration - Integration by Substitution

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 point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Concept: Rate of Change of Bodies Or Quantities

The probability mass function for X = number of major defects in a randomly selected

appliance of a certain type is

X = x | 0 | 1 | 2 | 3 | 4 |

P(X = x) | 0.08 | 0.15 | 0.45 | 0.27 | 0.05 |

Find the expected value and variance of X.

Concept: Variance of Binomial Distribution (P.M.F.)

Prove that `int_0^af(x)dx=int_0^af(a-x) dx`

hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`

Concept: Properties of Definite Integrals

If y = e^{tan x}+ (log x)^{tan x }then find dy/dx

Concept: General and Particular Solutions of a Differential Equation

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)^{19} = 0⋅1348]

Concept: Random Variables and Its Probability Distributions

Find a and b, so that the function f(x) defined by

f(x)=-2sin x, for -π≤ x ≤ -π/2

=a sin x+b, for -π/2≤ x ≤ π/2

=cos x, for π/2≤ x ≤ π

is continuous on [- π, π]

Concept: Continuity - Continuity of a Function at a Point

Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

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

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

Find the approximate value of log10 (1016) given that log_{10}^{e }= 0⋅4343.

Concept: Approximations