The principal solution of the equation cot x=`-sqrt 3 ` is

`(A) pi/6`

`(B) pi/3`

`(C)(5pi)/6`

`(D)(-5pi)/6`

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

If the vectors `-3hati+4hatj-2hatk, hati+2hatk, hati-phatj` are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

Concept: Vectors - Collinearity and Coplanarity of Vectors

If the line y =x+k touches the hyperbola 9x^{2} -16y^{2} =144, then k = .............

(A) 7

(B) -7

(C)`+-sqrt7`

(D) `+-sqrt19`

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

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.

(B) Price increases and demand falls

Concept: Mathematical Logic - Logical Connectives

If `A=[[2,-2],[4,3]]` then find `A^-1` by adjoint method.

Concept: Determinants - Adjoint Method

Find the separate equations of the lines represented by the equation ` 3x^2-10xy-8y^2=0`

Concept: Equation of a Line in Space

Find the equation of the director circle of a circle ` x^2 + y^2 =100.`

Concept: Circle - Director circle

Find the general solution of the equation `4cos^2x=1`

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

Without using truth table show that P↔q ≡(P ∧ q) ∨ (~ p ∧ ~ q)

Concept: Mathematical Logic - Algebra of Statements

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

Show that the line x+ 2y + 8 = 0 is tangent to the parabola y^{2} = 8x. Hence find the point of contact

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

The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.

Concept: Elementary Operation (Transformation) of a Matrix

Find the general solution of cos x +sin x =1.

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

If `bar a and bar b` are any two non-zero and non-collinear vectors then prove that any vector `bar r ` coplanar with `bar a and bar b` can be uniquely expressed as `bar r=t_1bara+t_2barb` , where ` t_1 and t_2` are scalars

Concept: Vectors - Collinearity and Coplanarity of Vectors

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence

Find k if the length of the tangent segment from (8,-3) to the circle ` x^2+y^2-2x+ky-23=0` is `sqrt10 ` units.

Concept: Circle - Length of Tangent Segments to Circle

Show that the lines given by `(x+1)/-10=(y+3)/-1=(z-4)/1` and `(x+10)/-1=(y+1)/-3=(z-1)/4` intersect. Also find the co-ordinates of the point of intersection.

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

Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola `x^2/7-y^2/5=1` such that the sum of the cubes of their slopes is 8.

Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular

Solve the following L.P.P graphically:

Maximize :Z = 10x + 25y

Subject to : x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2 units from the point (1,1, 2)

Concept: Distance of a Point from a Plane

Function ` f (x)= x^2 - 3x +4` has minimum value at

(A) 0

(B)-3/2

(C) 1

(D)3/2

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

`int1/xlogxdx=...............`

(A)log(log x)+ c

(B) 1/2 (logx )^{2}+c

(C) 2log x + c

(D) log x + c

Concept: Methods of Integration - Integration by Parts

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

If x=at^{2}, y= 2at , then find dy/dx.

Concept: Derivatives of Functions in Parametric Forms

Find the approximate value of ` sqrt8.95 `

Concept: Approximations

Find the area of the region bounded by the parabola y^{2} = 16x and the line x = 3.

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

For the bivariate data r = 0.3, cov(X, Y) = 18, σ_{x} = 3, find σ_{y} .

Concept: Statistics - Bivariate Frequency Distribution

triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.

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

A function f (x) is defined as

f (x) = x + a, x < 0

= x, 0 ≤x ≤ 1

= b- x, x ≥1

is continuous in its domain.

Find a + b.

Concept: Algebra of Continuous Functions

If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`

Concept: Derivatives of Functions in Parametric Forms

Evaluate : `int1/(3+5cosx)dx`

Concept: Evaluation of Definite Integrals by Substitution

An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.

Concept: Conditional Probability

The surface area of a spherical balloon is increasing at the rate of 2cm^{2} / sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?

Concept: Rate of Change of Bodies Or Quantities

The slope of the tangent to the curve at any point is equal to y+ 2x. Find the equation of the curve passing through the origin.

Concept: Differential Equations - Linear Differential Equation

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

The time ( in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable X taking values between 25 and 35 minutes with p.d.f

`f(x)=1/10,25<=x<=35=0`

` =0 " " otherwise`

What is the probability that preparation time exceeds 33 minutes? Also find the c.d.f. of X.

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

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

Concept: Conditional Probability

If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`

Concept: Derivatives of Functions in Parametric Forms

Find the area of the region common to the circle x^{2} + y^{2} =9 and the parabola y^{2} =8x

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

For 10 pairs of observations on two variables X and Y, the following data are available:

`sum(x-2)=30, sum(y-5)=40, sum(x-2)^2=900, sum(y-5)^2=800, sum(x-2)(y-5)=480`

Find the correlation coefficient between X and Y.

Concept: Statistics - Karl Pearsonâ€™s Coefficient of Correlation