Date & Time: 3rd March 2018, 11:00 am

Duration: 3h

If A = `[(2,-3),(4,1)]`, then adjoint of matrix A is

(A) `[(1,3),(-4,2)]`

(B) `[(1,-3),(-4,2)]`

(C) `[(1,3),(4,-2)]`

(D) `[(-1,-3),(-4,2)]`

Chapter: [2] Matrices

The principal solutions of sec x = `2/sqrt3` are _____

`pi/3,(11pi)/6`

`pi/6, (11pi)/6`

`pi/4,(11pi)/4`

`pi/6,(11pi)/4`

Chapter: [3] Trigonometric Functions

The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and –2, 1, 2 is ______.

Chapter: [8] Three Dimensional Geometry

Write the negations of the following statements :

1) All students of this college live in the hostel

2) 6 is an even number or 36 is a perfect square.

Chapter: [1] Mathematical Logic

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Chapter: [3] Trigonometric Functions

Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .

Chapter: [10] Plane

Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`

Chapter: [9] Line

if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`

Chapter: [7] Vectors

By vector method prove that the medians of a triangle are concurrent.

Chapter: [7] Vectors

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q )

Chapter: [1] Mathematical Logic

If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.

Chapter: [7] Vectors

Show that every homogeneous equation of degree two in x and y, i.e., ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through origin if h^{2}−ab≥0.

Chapter: [4] Pair of Straight Lines

In `triangle ABC` prove that `tan((C-A)/2) = ((c-a)/(c+a))cot B/2`

Chapter: [3] Trigonometric Functions

Find the inverse of the matrix `A = [(1,2,-2),(-1,3,0),(0,-2,1)]`using elementary row transformations.

Chapter: [2] Matrices

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

Find the angle between the lines `(x -1)/4 = (y - 3)/1 = z/8` and `(x-2)/2 = (y + 1)/2 = (z-4)/1`

Chapter: [9] Line

Write converse, inverse and contrapositive of the following conditional statement :

If an angle is a right angle then its measure is 90°.

Chapter: [1] Mathematical Logic

Prove that `sin^(-1) (3/5) + cos^(-1) (12/13) = sin^(-1) (56/65)`

Chapter: [3] Trigonometric Functions

Find the vector equation of the plane passing through the points A(1, 0, 1), B(1, –1, 1) and C(4, –3, 2).

Chapter: [10] Plane

Minimize Z = 7x + y subject to `5x + y >= 5, x + y >= 3, x>= 0, y >= 0`

Chapter: [11] Linear Programming Problems

Let the p. m. f. of a random variable X be __

P(x) = `(3-x)/10` for x = -1,0,1,2

= 0 otherwise

Then E(X ) is ________.

1

2

0

-1

Chapter: [19] Probability Distribution

if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.

(A) `1/2`

(B) `1/3`

(C) `1/4`

(D) `1/5`

Chapter: [15] Integration

Integrating factor of linear differential equation `x (dy)/(dx) + 2y =x^2 log x` is ____________

`1/x^2`

`1/x`

`x`

`x^2`

Chapter: [17] Differential Equation

Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`

Chapter: [15] Integration

if `y = tan^2(log x^3)`, find `(dy)/(dx)`

Chapter: [13] Differentiation

Find the area of ellipse `x^2/1 + y^2/4 = 1`

Chapter: [16] Applications of Definite Integral

Obtain the differential equation by eliminating the arbitrary constants from the following equation :

`y = c_1e^(2x) + c_2e^(-2x)`

Chapter: [17] Differential Equation

Given X ~ B (n, p)

If n = 10 and p = 0.4, find E(X) and var (X).

Chapter: [20] Bernoulli Trials and Binomial Distribution

Evaluate `int 1/(3+ 2 sinx + cosx) dx`

Chapter: [15] Integration

If `x = acos^3t`, `y = asin^3 t`,

Show that `(dy)/(dx) =- (y/x)^(1/3)`

Chapter: [13] Differentiation

Examine the continuity of the function:

f(x) = `(log100 + log(0.01+x))/"3x"," for "x != 0 = 100/3 `for x = 0; at x = 0.

Chapter: [12] Continuity

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).

Chapter: [14] Applications of Derivative

Prove that `int 1/(a^2 - x^2) dx = 1/"2a" log|(a +x)/(a-x)| + c`

Chapter: [15] Integration

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

= 0, if f (x) is an odd function.

Chapter: [15] Integration

if `f(x) = (x^2-9)/(x-3) + alpha` for x> 3

=5, for x = 3

`=2x^2+3x+beta`, for x < 3

is continuous at x = 3, find α and β.

Chapter: [12] Continuity

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

Chapter: [13] Differentiation

A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.

Chapter: [20] Bernoulli Trials and Binomial Distribution

Verify Rolle’s theorem for the following function:

f (x) = x^{2} - 4x + 10 on [0, 4]

Chapter: [14] Applications of Derivative

Find the particular solution of the differential equation:

`y(1+logx) dx/dy - xlogx = 0`

when y = e^{2} and x = e

Chapter: [17] Differential Equation

Find the variance and standard deviation of the random variable X whose probability distribution is given below :

x | 0 | 1 | 2 | 3 |

P(X = x) | `1/8` | `3/8` | `3/8` | `1/8` |

Chapter: [19] Probability Distribution

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