Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

Concept: Direction Cosines and Direction Ratios of a Line

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

(a) 2

(b) -2

(c) 10

(d) -10

Concept: Determinants - Adjoint Method

The general solution of the trigonometric equation tan^{2} θ = 1 is..........................

(a)`theta =npi+-(pi/3),n in z`

(b)`theta =npi+-pi/6, n in z`

(c)`theta=npi+-pi/4, n in z`

(d) `0=npi, n in z`

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

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.

Concept: Basic Concepts of Vector Algebra

The Cartestation equation of line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Concept: Equation of a Line in Space

Equation of a plane is `vecr (3hati-4hatj+12hatk)=8`. Find the length of the perpendicular from the origin to the plane.

Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

Concept: Angle Between Two Lines

Write the dual of the following statements: (p ∨ q) ∧ T

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

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

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb +z barc=0`

Concept: Vectors - Conditions of Coplanarity of Three Vectors

Using truth table prove that :

`~p^^q-=(p vv q)^^~p`

Concept: Mathematical Logic - Truth Tables of Compound Statements

In any ΔABC, with usual notations, prove that `b^2=c^2+a^2-2ca cosB`.

Concept: Trigonometric Functions - Solution of a Triangle

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

Concept: Acute Angle Between the Lines

Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1

Concept: Elementary Operation (Transformation) of a Matrix

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.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Homogenous Equation

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Concept: Mathematical Logic - Application - Introduction to Switching Circuits

if A, B, C, D are (1, i, I), (2, l ,3), (3; 2, 2) and (3, 3, 4) respetivly., then find the volume of the parallepiped with AB, AC and AD as concurrent edges

Concept: Scalar Triple Product of Vectors

Find the equation of the plane passing through the line of intersection of planes 2x - y + z = 3 and 4x- 3y + 5z + 9 = 0 and parallel to the line

` (x+1)/2=(y+3)/4=(z-3)/5`

Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4

Concept: Graphical Method of Solving Linear Programming Problems

Show that:

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

Concept: Basic Concepts of Trigonometric Functions

If y =1-cosθ , x = 1-sinθ , then ` dy/dx at " "0 =pi/4` is ............

Concept: Derivatives of Functions in Parametric Forms

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

Concept: Differential Equations - Linear Differential Equation

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

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

Examine the continuity of the function

f(x) =sin x- cos x, for x ≠ 0

=- 1 ,forx=0

at the poinl x = 0

Concept: Introduction of Continuity

Verify Rolle's theorem for the function

f(x)=x^{2}-5x+9 on [1,4]

Concept: Mean Value Theorem

Evaluate : `intsec^nxtanxdx`

Concept: Properties of Definite Integrals

The probability mass function (p.m.f.) of X is given below:

X=x | 1 | 2 | 3 |

P (X= x) | 1/5 | 2/5 | 2/5 |

find E(X^{2})

Concept: Probability Distribution - Probability Mass Function (P.M.F.)

Given that X~ B(n = 10, p), if E(X) = 8. find the value of p.

Concept: Statistics - Bivariate Frequency Distribution

Ify y=f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f (g(x)) is a differentiable function of x and

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

Concept: Derivative - Every Differentiable Function is Continuous but Converse is Not True

Obtain the differential equation by eliminating arbitrary constants A, B from the equation -

y = A cos (log x) + B sin (log x)

Concept: Formation of Differential Equation by Eliminating Arbitary Constant

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`

Concept: Methods of Integration - Integration Using Partial Fractions

An open box is to be made out of a piece of a square card board of sides 18 cms. by cutting off equal squares from the comers and tumi11g up the sides. Find the maximum volume of the box.

Concept: Maxima and Minima

Prove that :

`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`

Concept: Evaluation of Definite Integrals by Substitution

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Concept: Continuity - Continuity in Interval - Definition

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Concept: General and Particular Solutions of a Differential Equation

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

Concept: Conditional Probability

If x^{p}y^{q}=(x+y)^{p+q} then Prove that `dy/dx=y/x`

Concept: Exponential and Logarithmic Functions

Find the area of the sector of a circle bounded by the circle x^{2} + y^{2} = 16 and the line y = x in the ftrst quadrant.

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

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

Concept: Methods of Integration - Integration by Parts

A random variable X has the following probability distribution :

X=x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

P[X=x] | k | 3k | 5k | 7k | 9k | 11k | 13k |

(a) Find k

(b) find P(O <X< 4)

(c) Obtain cumulative distribution function (c. d. f.) of X.

Concept: Probability Distribution - Cumulative Probability Distribution of a Discrete Random Variable