If`[bar a barb barc]!=0 ` then `bar a . barp +bar b . bar q+bar c. bar r ` is equal to

(a) 0

(b) 1

(c) 2

(d) 3

Concept: Vectors - Linear Combination of Vectors

The inverse of the matrix `[[2,0,0],[0,1,0],[0,0,-1]]`is --------

(a) `[[1/2,0,0],[0,1,0],[0,0,-1]]`

(b) `[[-1/2,0,0],[0,-1,0],[0,0,1]]`

(c) `[[-1,0,0],[0,-1/2,0],[0,0,1/2]]`

(d) `1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]`

Concept: Matrices - Inverse of a Matrix Existance

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

(a) `+-1/sqrt51,+-5/sqrt51,+-1/sqrt51`

(b) `+-5/sqrt51,+-5/sqrt51,+-5/sqrt51`

(c) `+-sqrt5,+-1,+-5`

(d) `+-sqrt51,+-sqrt51+-sqrt51`

Concept: Direction Cosines and Direction Ratios of a Line

Write truth values of the following statements :`sqrt5` is an irrational number but 3 +`sqrt 5` is a complex number.

Concept: Mathematical Logic - Truth Value of Statement in Logic

Write truth values of the following statements : ∃ n ∈ N such that n + 5 > 10.

Concept: Mathematical Logic - Truth Value of Statement in Logic

If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`

Concept: Scalar Triple Product of Vectors

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hati + hatj + 2hatk.`

Concept: Vector and Cartesian Equation of a Plane

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

Concept: Equation of a Line in Space

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

Concept: Basic Concepts of Vector Algebra

Using truth table, prove the following logical equivalence :

(p ∧ q)→r = p → (q→r)

Concept: Mathematical Logic - Truth Tables of Compound Statements

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

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

Show that: `2sin^-1(3/5)=tan^-1(24/7)`

Concept: Basic Concepts of Trigonometric Functions

Solve the following equations by the method of reduction :

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

Concept: Elementary Operation (Transformation) of a Matrix

Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `

Concept: Scalar Triple Product of Vectors

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.

Concept: Graphical Method of Solving Linear Programming Problems

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)

Concept: Trigonometric Functions - Solution of a Triangle

If p : It is a day time, q : It is warm, write the compound statements in verbal form

denoted by -

(a) p ∧ ~ q

(b) ~ p → q

(c) q ↔ p

Concept: Mathematical Logic - Compound Statement in Logic

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

Parametric form of the equation of the plane is `bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk)` λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

Concept: Vector and Cartesian Equation of a Plane

If the angle between the lines represented by ax^{2} + 2hxy + by^{2} = 0 is equal to the angle between the lines 2x^{2} - 5xy + 3y^{2} =0,

then show that 100(h^{2} - ab) = (a + b)^{2}

Concept: Angle Between Two Lines

Find the general solution of: sinx · tanx = tanx- sinx+ 1

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

The differential equation of the family of curves y=c_{1}e^{x}+c_{2}e^{-x} is......

(a)`(d^2y)/dx^2+y=0`

(b)`(d^2y)/dx^2-y=0`

(c)`(d^2y)/dx^2+1=0`

(d)`(d^2y)/dx^2-1=0`

Concept: General and Particular Solutions of a Differential Equation

If X is a random variable with probability mass function

P(x) = kx , x=1,2,3

= 0 , otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

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

If `sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........`

(a) y

(b) x

(c) y/x

(d) 0

Concept: Derivatives of Inverse Trigonometric Functions

If `y=sin^-1(3x)+sec^-1(1/(3x)), ` find dy/dx

Concept: Derivatives of Inverse Trigonometric Functions

Evaluate :`intxlogxdx`

Concept: Methods of Integration - Integration by Substitution

If `int_0^h1/(2+8x^2)dx=pi/16 `then find the value of h.

Concept: Fundamental Theorem of Calculus

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

Concept: Conditional Probability

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

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

Examine the continuity of the following function at given point:

`f(x)=(logx-log8)/(x-8) , `

` =8, `

Concept: Continuity - Discontinuity of a Function

If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`

Concept: General and Particular Solutions of a Differential Equation

Solve : 3e^{x} tanydx + (1 +e^{x}) sec^{2} ydy = 0

Also, find the particular solution when x = 0 and y = π.

Concept: General and Particular Solutions of a Differential Equation

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.

Concept: Rate of Change of Bodies Or Quantities

Evaluate : `intlogx/(1+logx)^2dx`

Concept: Properties of Definite Integrals

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint

Concept: Derivatives of Functions in Parametric Forms

Show that the function defined by f(x) =|cosx| is continuous function.

Concept: Introduction of Continuity

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

Concept: General and Particular Solutions of a Differential Equation

Given X ~ B(n, p). If n = 20, E(X) = 10, find p_{,} Var. (X) and S.D. (X).

Concept: Bernoulli Trials and Binomial Distribution - Normal Distribution (P.D.F)

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Concept: Statistics - Bivariate Frequency Distribution

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Concept: Mean Value Theorem

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

Concept: Methods of Integration - Integration by Substitution