Date: March 2013

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

`pi/6`

`pi/3`

`(5pi)/6`

`(-5pi)/6`

Chapter: [3] Trigonometric Functions

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

Chapter: [7] Vectors

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

7

-7

`+-sqrt7`

`+-sqrt19`

Chapter: [6] Conics

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

Chapter: [1] Mathematical Logic

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

Chapter: [2] Matrices

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

Chapter: [9] Line

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

Chapter: [5] Circle

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

Chapter: [3] Trigonometric Functions

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

Chapter: [1] Mathematical Logic

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`

Chapter: [4] Pair of Straight Lines

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

Chapter: [6] Conics

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.

Chapter: [2] Matrices

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

Chapter: [3] Trigonometric Functions

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

Chapter: [7] Vectors

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

Chapter: [1] Mathematical Logic

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.

Chapter: [5] 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.

Chapter: [4] Pair of Straight 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.

Chapter: [6] Conics

Solve the following L.P.P graphically:

Maximize :Z = 10x + 25y

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

Chapter: [11] 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)

Chapter: [10] Plane

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

(A) 0

(B)-3/2

(C) 1

(D)3/2

Chapter: [14] Applications of Derivative

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

(A)log(log x)+ c

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

(C) 2log x + c

(D) log x + c

Chapter: [15] Integration

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

Chapter: [17] Differential Equation

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

Chapter: [13] Differentiation

Find the approximate value of ` sqrt8.95 `

Chapter: [14] Applications of Derivative

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

Chapter: [16] Applications of Definite Integral

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

Chapter: [18] Statistics

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.

Chapter: [16] Applications of Definite Integral

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.

Chapter: [12] Continuity

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

Chapter: [13] Differentiation

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

Chapter: [15] Integration

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.

Chapter: [19] Probability Distribution

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?

Chapter: [14] Applications of Derivative

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.

Chapter: [17] Differential Equation

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

Chapter: [15] Integration

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"`

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

Chapter: [19] Probability Distribution

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

Chapter: [19] Probability Distribution

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

Chapter: [13] Differentiation

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

Chapter: [16] Applications of Definite Integral

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.

Chapter: [18] Statistics

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