The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4

Concept: Types of Relations

if A =`((5,a),(b,0))` is symmetric matrix show that a = b

Concept: Symmetric and Skew Symmetric Matrices

Solve `3tan^(-1)x + cot^(-1) x = pi`

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch

Without expanding at any stage, find the value of:

`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`

Concept: Determinant of a Matrix of Order 3 × 3

Find the value of constant ‘k’ so that the function f (x) defined as

f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`

is continous at x = -1

Concept: Concept of Continuity

Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.

Concept: Approximations

Evaluate `int(x^3+5x^2 + 4x + 1)/x^2 dx`

Concept: Integration as an Inverse Process of Differentiation

Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`

Concept: General and Particular Solutions of a Differential Equation

If A and B are events such as that P(A) = `1/2`, P(B) = `1/3` and P(A ∩ B) = `1/4`, then find

1) P(A / B)

2) P(B / A)

Concept: Conditional Probability

In a race, the probabilities of A and B winning the race are `1/3` and `1/6` respectively. Find the probability of neither of them winning the race.

Concept: Independent Events

If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`

Concept: Types of Functions

if `tan^(-1) a + tan^(-1) b + tan^(-1) x = pi`, prove that a + b + c = abc

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch

Use properties of determinants to solve for x:

`|(x+a, b, c),(c, x+b, a),(a,b,x+c)| = 0` and `x != 0`

Concept: Determinant of a Matrix of Order 3 × 3

Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.

Concept: Concept of Continuity

Verify Rolle’s theorem for the following function:

`f(x) = e^(-x) sinx " on" [0, pi]`

Concept: Mean Value Theorem

if `x = tan(1/a log y)`, prove that `(1+x^2) (d^2y)/(dx^2) + (2x + a) (dy)/(dx) = 0`

Concept: Derivatives of Inverse Trigonometric Functions

Evaluate `int tan^(-1) sqrtx dx`

Concept: Integration as an Inverse Process of Differentiation

Find the points on the curve y = `4x^3 - 3x + 5` at which the equation of the tangent is parallel to the x-axis.

Concept: Tangents and Normals

Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Concept: Increasing and Decreasing Functions

Solve `sin x dy/dx - y = sin x.tan x/2`

Concept: Solutions of Linear Differential Equation

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.

Concept: General and Particular Solutions of a Differential Equation

Using matrices, solve the following system of equations :

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

Concept: Minors and Co-factors

Using elementary transformation, find the inverse of the matrix :

`[(0,1,2),(1,2,3),(3,1,1)]`

Concept: Matrices - Proof of the Uniqueness of Inverse

A speaks the truth in 60% of the cases, while B is 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

Concept: Independent Events

A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height

Concept: Maximum and Minimum Values of a Function in a Closed Interval

Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`

Concept: Methods of Integration - Integration by Substitution

Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`

Concept: Properties of Definite Integrals

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample.

If the sample is drawn without replacement, find :

1) The probability distribution of X

2) Mean of X

3) Variance of X

Concept: Probability Distribution Function

Find `lambda` if the scalar projection of `vec a = lambda hat i + hat j + 4 hat k` on `vec b = 2hati + 6hatj + 3hatk` is 4 units

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors

The Cartesian equation of the line is 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, –5, 0) and parallel to the given line.

Concept: Vector and Cartesian Equation of a Plane

Find the equation of the plane through the intersection of the planes `vecr.(hati + 3hatj - hatk) = 9` and `vecr.(2hati - hatj + hatj) = 3` and passing through the origin.

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

If A, B, C are three non- collinear points with position vectors `vec a, vec b, vec c`, respectively, then show that the length of the perpendicular from Con AB is `|(vec a xx vec b)+(vec b xx vec c) + (vec b xx vec a)|/|(vec b - vec a)|`

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

Show that the four points A, B, C and D with position vectors `4hati + 5hatj + hatk`, `-hatj-hatk`, `3hati + 9hatj + 4hatk` and `4(-hati + hatj + hatk)` respectively are coplanar

Concept: Scalar Triple Product of Vectors

Draw a rough sketch of the curve and find the area of the region bounded by curve y^{2} = 8x and the line x =2.

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

Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.

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

Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`

Concept: Vector and Cartesian Equation of a Plane