Date: July 2016

Inverse of the statement pattern (p ∨ q) → (p ∧ q) is

(A) (p ∧ q) → (p ∨ q)

(B) ∼ (p ∨ q) → (p ∧ q)

(C) (∼ p ∨ ∼ q) → (∼ p ∧ ∼ q)

(D) (∼ p ∧ ∼ q) → (∼ p ∨ ∼ q)

Chapter: [0.01] Mathematical Logic

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

Chapter: [0.07] Vectors

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`

Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

Find the angle between the lines `barr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk)` and `barr=5 hati-2hatk+mu(3hati+2hatj+6hatk)`

Chapter: [0.04] Pair of Straight Lines

If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

If `A =[[2,-3],[3,5]]` then find A^{-1} by adjoint method.

Chapter: [0.02] Matrices

By vector method show that the quadrilateral with vertices A (1, 2, –1), B (8, –3, –4), C (5, –4, 1), D (–2, 1, 4) is a parallelogram.

Chapter: [0.07] Vectors

Find the general solution of the equation sin x = tan x.

Chapter: [0.03] Trigonometric Functions

Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax^{2}+ 2hxy + by^{2}= 0

Chapter: [0.04] Pair of Straight Lines

Find the principal value of `sin^-1(1/sqrt2)`

Chapter: [0.03] Trigonometric Functions

Find the cartesian form of the equation of the plane `bar r=(hati+hatj)+s(hati-hatj+2hatk)+t(hati+2hatj+hatj)`

Chapter: [0.1] Plane

Simplify the following circuit so that new circuit has minimum number of switches. Also draw simplified circuit.

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of magnitude 5 along the direction of line.

Chapter: [0.09] Line

**Solve the following LPP by graphical method:**

Maximize: z = 3x + 5y

Subject to: x + 4y ≤ 24

3x + y ≤ 21

x + y ≤ 9

x ≥ 0, y ≥ 0

Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems

Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`

Chapter: [0.09] Line

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

Chapter: [0.016] Line and Plane [0.1] Plane

In any triangle ABC with usual notations prove c = a cos B + b cos A

Chapter: [0.03] Trigonometric Functions

Find p and k if the equation px^{2} – 8xy + 3y^{2 }+14x + 2y + k = 0 represents a pair of perpendicular lines.

Chapter: [0.09] Line

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Chapter: [0.02] Matrices

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Chapter: [0.015] Vectors [0.07] Vectors

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: [0.026000000000000002] Differential Equations [0.17] Differential Equation

`∫_4^9 1/sqrtxdx=`_____

(A) 1

(B) –2

(C) 2

(D) –1

Chapter: [0.15] Integration

If the p.d.f. of a continuous random variable X is given as

`f(x)=x^2/3` for -1< x<2

=0 otherwise

then c.d.f. fo X is

(A) `x^3/9+1/9`

(B) `x^3/9-1/9`

(C) `x^2/4+1/4`

(D) `1/(9x^3)+1/9`

Chapter: [0.19] Probability Distribution

If `y = sec sqrtx` then find dy/dx.

Chapter: [0.13] Differentiation

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find the area of the region lying in the first quandrant bounded by the curve y^{2}= 4x, X axis and the lines x = 1, x = 4

Chapter: [0.16] Applications of Definite Integral

For the differential equations find the general solution:

sec^{2} x tan y dx + sec^{2} y tan x dy = 0

Chapter: [0.17] Differential Equation

Given is X ~ B (n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Chapter: [0.2] Bernoulli Trials and Binomial Distribution

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))` for x ≠ 0 is continuous at x = 0, find f (0).

Chapter: [0.12] Continuity

Evaluate : `∫1/(3+2sinx+cosx)dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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: [0.13] Differentiation

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Chapter: [0.14] Applications of Derivative

The probability mass function for X = number of major defects in a randomly selected

appliance of a certain type is

X = x | 0 | 1 | 2 | 3 | 4 |

P(X = x) | 0.08 | 0.15 | 0.45 | 0.27 | 0.05 |

Find the expected value and variance of X.

Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution

Prove that `int_0^af(x)dx=int_0^af(a-x) dx`

hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`

Chapter: [0.15] Integration

If y = e^{tan x}+ (log x)^{tan x }then find dy/dx

Chapter: [0.17] Differential Equation

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)^{19} = 0⋅1348]

Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

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]

Chapter: [0.12] Continuity

Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

Chapter: [0.16] Applications of Definite Integral

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

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

Hence evaluate, `int xe^xdx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find the approximate value of log10 (1016) given that log_{10}^{e }= 0⋅4343.

Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

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