Date & Time: 6th March 2017, 11:00 am

Duration: 3h

If the points A(2, 1, 1), B(0, -1, 4) and C(k, 3, -2) are collinear, then k

(A) 0

(B) 1

(C) 4

(D) -4

Chapter: [8] Three Dimensional Geometry

The inverse of the matrix `[[-1,5],[-3,2]]` is _______

1/13`[[2,-5],[3,-1]]`

1/13 `[[-1,5],[-3,2]]`

1/13 `[[-1,-3],[5,2]]`

1/13 `[[1,5],[3,-2]]`

Chapter: [2] Matrices

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) `1/5`

(B) `sqrt(1/5)`

(C) `4/5`

(D) `2/5`

Chapter: [3] Trigonometric Functions

Find the volume of the parallelopiped whose coterminus edges are given by vectors

`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Chapter: [7] Vectors

In Δ ABC, prove that, a (b cos C - c cos B) = b^{2} - c^{2}.

Chapter: [3] Trigonometric Functions

If from a point Q (a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

Chapter: [3] Trigonometric Functions

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Chapter: [4] Pair of Straight Lines

Write the following statement in symbolic form and find its truth value:

∀ n ∈ N, n^{2} + n is an even number and n^{2} - n is an odd number.

Chapter: [1] Mathematical Logic

Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.

Chapter: [1] Mathematical Logic

Find the shortest distance between the lines `(x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5`

Chapter: [4] Pair of Straight Lines

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Chapter: [3] Trigonometric Functions

Solve the following equations by method of reduction:

x-y + z = 4,

2x + y - 3z = 0,

x + y + z = 2

Chapter: [2] Matrices

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

Using vector method, find incentre of the triangle whoose vertices are P(0, 4, 0), Q(0, 0, 3)

and R(0, 4, 3).

Chapter: [7] Vectors

Construct the switching circuit for the statement (p ∧ q) ∨ (~ p) ∨ (p ∧ ~ q).

Chapter: [1] Mathematical Logic

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.

Chapter: [4] Pair of Straight Lines

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Chapter: [3] Trigonometric Functions

If l, m, n are the direction cosines of a line, then prove that l^{2} + m^{2} + n^{2} = 1. Hence find the

direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.

Chapter: [8] Three Dimensional Geometry

Find the vector and cartesian equations of the plane passing through the points A( 1, 1, -2), B(1, 2, 1) and C(2, -1, 1).

Chapter: [10] Plane

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤ 3, -2x + y ≤ 1, x ≥ 0, y ≥ 0.

Also find maximum value of Z.

Chapter: [11] Linear Programming Problems

Derivatives of tan^{3}θ with respect to sec^{3}θ at θ=π/3 is

(A)` 3/2`

(B) `sqrt3/2`

(C) `1/2`

(D) `-sqrt3/2`

Chapter: [13] Differentiation

The equation of tangent to the curve y = 3x^{2} - x + 1 at the point (1, 3) is

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

Chapter: [6] Conics

The expected value of the number of heads obtained when three fair coins are tossed simultaneously is

(A) 1

(B) 1.5

(C) 0

(D) -1

Chapter: [19] Probability Distribution

Find dy/dx if x sin y + y sin x = 0.

Chapter: [13] Differentiation

Test whether the function, f(x) = x -1/x, x ∈ R, x ≠ 0, is increasing or decreasing

Chapter: [14] Applications of Derivative

Evaluate: `intsinsqrtx/sqrtxdx`

Chapter: [15] Integration

Form the differential equation by eliminating arbitrary constants from the relation `y=Ae^(5x)+Be^(-5x)`

Chapter: [17] Differential Equation

The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.

Chapter: [20] Bernoulli Trials and Binomial Distribution

Solve: dy/dx = cos(x + y)

Chapter: [15] Integration

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

If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).

Chapter: [12] Continuity

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`

Chapter: [13] Differentiation

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

Chapter: [14] Applications of Derivative

Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`

Chapter: [15] Integration

Discuss the continuity of the following function, at x = 0.

`f(x)=x/|x|,for x ne0`

`=1,`for `x=0`

Chapter: [12] Continuity

If the population of a country doubles in 60 years, in how many years will it be triple under

the assumption that the rate of increase in proportional to the number of inhabitants?

[Given : log 2 = 0.6912 and log 3 = 1.0986.]

Chapter: [17] Differential Equation

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.

Chapter: [19] Probability Distribution

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Chapter: [19] Probability Distribution

Find: `I=intdx/(sinx+sin2x)`

Chapter: [15] Integration

Find the area of the region lying between the parabolas y^{2} = 4ax and x^{2} = 4ay.

Chapter: [16] Applications of Definite Integral

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)

Chapter: [19] Probability Distribution

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