The negation of p ∧ (q → r) is

- p ∨ (~q ∨ r)
- ~p ∧ (q → r)
- ~p ∧ (~q → ~r)
- ~p ∨ (q ∧ ~r)

Concept: Mathematical Logic - Algebra of Statements

If `sin^-1(1-x) -2sin^-1x = pi/2` then x is

- -1/2
- 1
- 0
- 1/2

Concept: Basic Concepts of Trigonometric Functions

The joint equation of the pair of lines passing through (2,3) and parallel to the coordinate axes is

- xy -3x - 2y + 6 = 0
- xy +3x + 2y + 6 = 0
- xy = 0
- xy - 3x - 2y - 6 = 0

Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines

Find (AB)^{-1} if

`A=[(1,2,3),(1,-2,-3)], B=[(1,-1),(1,2),(1,-2)]`

Concept: Matrices - Inverse by Elementary Transformation

Find the vector equation of the plane passing through a point having position vector `3 hat i- 2 hat j + hat k` and perpendicular to the vector `4 hat i + 3 hat j + 2 hat k`

Concept: Vector and Cartesian Equation of a Plane

If `bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j + 2 hat k` are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1

Concept: Section formula

Find k, if one of the lines given by 6x^{2} + kxy + y^{2} = 0 is 2x + y = 0

Concept: Pair of Straight Lines - Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines

If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

Concept: Shortest Distance Between Two Lines

Examine whether the following logical statement pattern is tautology, contradiction or contingency.

[(p → q) ∧ q] → p

Concept: Mathematical Logic - Statement Patterns and Logical Equivalence

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

Concept: Vectors - Medians of a Triangle Are Concurrent

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

Concept: Shortest Distance Between Two Lines

In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`

Concept: Trigonometric Functions - Solution of a Triangle

Minimize `z=4x+5y ` subject to `2x+y>=7, 2x+3y<=15, x<=3,x>=0, y>=0` solve using graphical method.

Concept: Graphical Method of Solving Linear Programming Problems

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.

Concept: Elementary Operation (Transformation) of a Matrix

Find the volume of tetrahedron whose coterminus edges are `7hat i+hatk; 2hati+5hatj-3hatk and 4 hat i+3hatj+hat k`

Concept: Three Dimensional Geometry - Problems

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

Concept: Mathematical Logic - Algebra of Statements

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

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

Concept: Equation of a Line in Space

Find the vector equation of the plane passing through the points `hati +hatj-2hatk, hati+2hatj+hatk,2hati-hatj+hatk`. Hence find the cartesian equation of the plane.

Concept: Vector and Cartesian Equation of a Plane

Find the general solution of `sin x+sin3x+sin5x=0`

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

if the function

`f(x)=k+x, for x<1`

`=4x+3, for x>=1`

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

Concept: Continuity - Continuity of a Function at a Point

The equation of tangent to the curve y=`y=x^2+4x+1` at

(-1,-2) is...............

(a) 2x -y = 0 (b) 2x+y-5 = 0

(c) 2x-y-1=0 (d) x+y-1=0

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

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

p is ...........

(a) 0.6

(b) 0.7

(c) 0.8

(d) 0.4

Concept: Bernoulli Trials and Binomial Distribution

if `y=x^x` find `(dy)/(dx)`

Concept: Exponential and Logarithmic Functions

The displacement 's' of a moving particle at time 't' is given by s = 5 + 20t — 2t^{2}. Find its acceleration when the velocity is zero.

Concept: Maxima and Minima in Closed Interval

Find the area bounded by the curve y^{2} = 4ax, x-axis and the lines x = 0 and x = a.

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

The probability distribution of a discrete random variable X is:

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

P(X=x) | k | 2k | 3k | 4k | 5k |

find P(X≤4)

Concept: Probability Distribution of a Discrete Random Variable

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

Concept: Methods of Integration - Integration by Substitution

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

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Concept: Probability Distribution of a Discrete Random Variable

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Concept: Methods of Integration - Integration by Substitution

Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity

`f(x)=(4^x-e^x)/(6^x-1)` for x ≠ 0

`=log(2/3) ` for x=0

Concept: Concept of Continuity

Prove that : `int sqrt(a^2-x^2)dx=x/2sqrt(a^2-x^2)=a^2/2sin^-1(x/a)+c`

Concept: Evaluation of Definite Integrals by Substitution

A body is heated at 110°C and placed in air at 10°C. After 1 hour its temperature is 60°C. How much additional time is required for it to cool to 35°C?

Concept: Differential Equations - Applications of Differential Equation

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

Concept: Properties of Definite Integrals

Evaluate: `int (1+logx)/(x(2+logx)(3+logx))dx`

Concept: Evaluation of Definite Integrals by Substitution

If `y=cos^-1(2xsqrt(1-x^2))`, find dy/dx

Concept: Derivative - Derivative of Inverse Function

Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.

Concept: General and Particular Solutions of a Differential Equation

A wire of length l is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle and square is the least, if the radius of circle is half the side of the square.

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values

The following is the p.d.f. (ProbabiIity Density Function) of a continuous random variable X :

`f(x)=x/32,0<x<8`

= 0 otherwise

(a) Find the expression for c.d.f. (Cumulative Distribution Function) of X.

(b) Also find its value at x = 0.5 and 9.

Concept: Probability Distribution - Probability Density Function (P.D.F.)