^{if `A=[[2,0,0],[0,2,0],[0,0,2]]`} then A^{6}=^{ ......................}

(a) 6A

(b) 12A

(c) 16A

(d) 32A

Concept: Operations on Matrices - Addition of Matrices

The principal solution of `cos^-1(-1/2)` is :

(a) π/3

(b) π/6

(c) 2π/3

(d) 3π/2

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

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then.........................

(a) fg = ch (b) gh = cf

(c) Jh = cg (d) hf= - eg

Concept: Pair of Straight Lines - Condition for Parallel Lines

Write the converse and contrapositive of the statement — “If two triangles are congruent, then their areas are equal.”

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence

Find ‘k' if the sum of slopes of lines represented by equation x^{2}+ kxy - 3y^{2} = 0 is twice their product.

Concept: Acute Angle Between the Lines

Find the angle between the planes `bar r.(2bar i+barj-bark)=3 and bar r.(hati+2hatj+hatk)=1`

Concept: Angle Between Two Planes

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

Concept: Equation of a Line in Space

If `bara=bari+2barj, barb=-2bari+barj,barc=4bari+3barj`, find x and y such that `barc=xbara+ybarb`

Concept: Vectors - Linear Combination of Vectors

If A, B, C, D are (1, 1, 1), (2, I, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Concept: Scalar Triple Product of Vectors

Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q

Concept: Mathematical Logic - Truth Tables of Compound Statements

If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`

Concept: Section formula

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1

Concept: Direction Cosines and Direction Ratios of a Line

In any ΔABC if a^{2} , b^{2} , c^{2} are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

Concept: Trigonometric Functions - Solution of a Triangle

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

Concept: Elementary Operation (Transformation) of a Matrix

If θ is the acute angle between the lines represented by equation ax^{2} + 2hxy + by^{2} = 0 then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|, a+b!=0`

Concept: Acute Angle Between the Lines

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Concept: Pair of Straight Lines - Point of Intersection of Two Lines

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

Find the general solution of : cos x - sin x = 1.

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

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).

Concept: Plane - Equation of a Plane Passing Through Three Non Collinear Points

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F_{ 1} and F_{2} are available. Food F_{1} costs Rs. 6 per unit and food F_{2} costs Rs. 10 per unit. One unit of food F_{1} contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F_{2} contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

Concept: Different Types of Linear Programming Problems

A random variable X has the following probability distribution:

then E(X)=....................

(a) 0.8

(b) 0.9

(c) 0.7

(d) 1.1

Concept: Random Variables and Its Probability Distributions

If `int_0^alpha3x^2dx=8` then the value of α is :

(a) 0

(b) -2

(c) 2

(d) ±2

Concept: Properties of Definite Integrals

The differential equation of y=c/x+c^{2} is :

(a)`x^4(dy/dx)^2-xdy/dx=y`

(b)`(d^2y)/dx^2+xdy/dx+y=0`

(c)`x^3(dy/dx)^2+xdy/dx=y`

(d)`(d^2y)/dx^2+dy/dx-y=0`

Concept: General and Particular Solutions of a Differential Equation

Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`

Concept: Properties of Definite Integrals

If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`

Concept: General and Particular Solutions of a Differential Equation

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

Concept: Evaluation of Definite Integrals by Substitution

If y=e^{ax },show that `xdy/dx=ylogy`

Concept: Derivatives of Implicit Functions

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

Concept: Conditional Probability

Integrate : sec^{3} x w. r. t. x.

Concept: Methods of Integration - Integration by Parts

If y = (tan^{-1} x)^{2}, show that `(1+x^2)^2(d^2y)/dx^2+2x(1+x^2)dy/dx-2=0`

Concept: Differential Equations - Linear Differential Equation

If `f(x)=[tan(pi/4+x)]^(1/x), `

= k ,for x=0

is continuous at x=0 , find k.

Concept: Continuity - Continuity of a Function at a Point

Find the co-ordinates of the points on the curve y=x-(4/x) where the tangents are parallel to the line y=2x

Concept: Conics - Tangents from a Point Outside Conics

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

Concept: Methods of Integration - Integration by Parts

Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`

Concept: Properties of Definite Integrals

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

If `log_10((x^3-y^3)/(x^3+Y^3))=2 `

Concept: Derivatives of Functions in Parametric Forms

Let the p. m. f. (probability mass function) of random variable x be

`p(x)=(4/x)(5/9)^x(4/9)^(4-x), x=0, 1, 2, 3, 4`

=0 otherwise

find E(x) and var (x)

Concept: Probability Distribution - Probability Mass Function (P.M.F.)

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

Solve the differential equation (x^{2} + y^{2})dx- 2xydy = 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

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.)