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

(A) `1/5[[3,-1],[-2,1]]`

(B) `1/5[[3,1],[-2,1]]`

(C) `1/5[[-3,1],[-2,1]]`

(D) `1/5[[3,-1],[2,-1]]`

Concept: Matrices - Inverse of a Matrix Existance

If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`

(A) 100

(B) 101

(C) 110

(D) 109

Concept: Scalar Triple Product of Vectors

If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.

(A) `0,1/sqrt2,-1/sqrt2`

(B) `0,-1/sqrt2,-1/sqrt2`

(C) `1,1/sqrt2,1/sqrt2`

(D) `0,-1/sqrt2,1/sqrt2`

Concept: Direction Cosines and Direction Ratios of a Line

`barr=(hati-2hatj+3hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `barr.(3hati-2hatj+phatk)=10`, find the value of p.

Concept: Plane - Equation of Plane Passing Through the Given Point and Parallel to Two Given Vectors

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

Concept: Angle Between Line and a Plane

Write the negations of the following statements:

a.`forall n in N, n+7>6`

b. The kitchen is neat and tidy.

Concept: Mathematical Logic - Sentences and Statement in Logic

Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.

Concept: Direction Cosines and Direction Ratios of a Line

If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.

Concept: Basic Concepts of Vector Algebra

If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.

Concept: Properties of Inverse Trigonometric Functions

Write the converse, inverse and contrapositive of the following statement.

“If it rains then the match will be cancelled.”

Concept: Mathematical Logic - Sentences and Statement in Logic

Find p and q, if the equation `px^2-8xy+3y^2+14x+2y+q=0` represents a pair of prependicular lines.

Concept: Pair of Straight Lines - Condition for Perpendicular Lines

Find the equation of the plane passing through the intersection of the planes 3x + 2y – z + 1 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

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

Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r=(mbarb+nbara)/(m+n)` . Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

Concept: Equation of a Line in Space

The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`

Concept: Trigonometric Functions - Solution of a Triangle

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

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

Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).

Concept: Vector and Cartesian Equation of a Plane

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

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

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=[[1,-1,2],[3,0,-2],[1,0,3]]` verify that A (adj A) = |A| I.

Concept: Determinants - Adjoint Method

A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.

If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

Concept: Graphical Method of Solving Linear Programming Problems

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`

Concept: Acute Angle Between the Lines

find the acute angle between the lines

x^{2} – 4xy + y^{2} = 0.

Concept: Acute Angle Between the Lines

Given f (x) = 2x, x < 0

= 0, x ≥ 0

then f (x) is _______.

(A) discontinuous and not differentiable at x = 0

(B) continuous and differentiable at x = 0

(C) discontinuous and differentiable at x = 0

(D) continuous and not differentiable at x = 0

Concept: Continuity - Discontinuity of a Function

If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`

(A) 1

(B) 2

(C) –1

(D) –2

Concept: Properties of Definite Integrals

The function f (x) = x^{3} – 3x^{2} + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing

Concept: Increasing and Decreasing Functions

Differentiate 3^{x} w.r.t. log_{3}x

Concept: Exponential and Logarithmic Functions

Check whether the conditions of Rolle’s theorem are satisfied by the function

f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]

Concept: Mean Value Theorem

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`

Concept: Methods of Integration - Integration by Substitution

Find the area of the region bounded by the curve x^{2} = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.

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

Given X ~ B (n, p)

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

Concept: Bernoulli Trials and Binomial Distribution

If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))` for x ≠ 0 is continuous at x = 0, find f (0).

Concept: Continuity - Continuity of a Function at a Point

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.

Concept: Variance of Binomial Distribution (P.M.F.)

Suppose that 80% of all families own a television set. If 5 families are interviewed at random, find the probability that

a. three families own a television set.

b. at least two families own a television set.

Concept: Conditional Probability

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

Concept: Approximations

The rate of growth of bacteria is proportional to the number present. If, initially, there were

1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½

hours.

[Take `sqrt2` = 1.414]

Concept: Rate of Change of Bodies Or Quantities

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.

Concept: Methods of Integration - Integration by Parts

If f (x) is continuous on [–4, 2] defined as

f (x) = 6b – 3ax, for -4 ≤ x < –2

= 4x + 1, for –2 ≤ x ≤ 2

Show that a + b =`-7/6`

Concept: Algebra of Continuous Functions

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

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

Concept: Methods of Integration - Integration by Parts

Probability distribution of X is given by

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

P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |

Find P(X ≥ 2) and obtain cumulative distribution function of X

Concept: Random Variables and Its Probability Distributions

Solve the differential equation `dy/dx -y =e^x`

Concept: General and Particular Solutions of a Differential Equation

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/dxne0`

Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`

then show that `dy/dx=1/sqrt(1-x^2)`, where `|x|<1`

Concept: Derivative - Derivative of Inverse Function

Evaluate: `∫8/((x+2)(x^2+4))dx`

Concept: Methods of Integration - Integration Using Partial Fractions