#### Alternate Sets

If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.

Concept: Concept of Binary Operations

Find the magnitude of each of two vectors `veca` and `vecb` having the same magnitude such that the angle between them is 60° and their scalar product is `9/2`

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

if the matrix A =`[(0,a,-3),(2,0,-1),(b,1,0)]` is skew symmetric, Find the value of 'a' and 'b'

Concept: Types of Matrices

Find the value of `tan^(-1) sqrt3 - cot^(-1) (-sqrt3)`

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

The total cost C(*x*) associated with the production of *x* units of an item is given by C(*x*) = 0.005*x*^{3} – 0.02*x*^{2} + 30*x* + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Concept: Rate of Change of Bodies Or Quantities

Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x

Concept: Derivatives of Inverse Trigonometric Functions

Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`

Concept: Types of Matrices

Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`

Concept: Properties of Inverse Trigonometric Functions

A black and a red dice are rolled. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Concept: Conditional Probability

If *θ* is the angle between two vectors `hati - 2hatj + `3hatk`and 3hati - 2hatj + hatk` find `sin theta`

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

Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where *a* and *b* are arbitrary constants.

Concept: Formation of a Differential Equation Whose General Solution is Given

Evaluate `int (cos 2x + 2sin^2x)/(cos^2x) dx`

Concept: Properties of Indefinite Integral

If *y* = sin (sin *x*), prove that `(d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0`

Concept: Higher Order Derivative

Find the particular solution of the differential equation e^{x} tan y dx + (2 – e^{x}) sec^{2} y dy = 0, give that `y = pi/4` when x = 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable

Find the particular solution of the differential equation `dy/dx + 2y tan x = sin x` given that y = 0 when x = `pi/3`

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable

Find the shortest distance between the lines `vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk)` and `vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)`

Concept: Shortest Distance Between Two Lines

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X

Concept: Random Variables and Its Probability Distributions

Using properties of determinants, prove that `|(1,1,1+3x),(1+3y, 1,1),(1,1+3z,1)| = 9(3xyz + xy + yz+ zx)`

Concept: Properties of Determinants

Find the equations of the tangent and the normal, to the curve 16x^{2} + 9y^{2} = 145 at the point (x_{1}, y_{1}), where x_{1} = 2 and y_{1} > 0.

Concept: Tangents and Normals

Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing

Concept: Increasing and Decreasing Functions

Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`

Concept: Methods of Integration - Integration Using Partial Fractions

Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the ride?

Concept: Baye'S Theorem

Let `veca = 4hati + 5hatj - hatk`, `vecb = hati - 4hatj + 5hatk` and `vecc = 3hati + hatj - hatk`. Find a vector `vecd` which is perpendicular to both `vecc` and `vecb and vecd.veca = 21`

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

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?

Concept: Maxima and Minima

if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`

Concept: Derivatives of Implicit Functions

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`

Concept: Derivatives of Functions in Parametric Forms

Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`

Concept: Evaluation of Definite Integrals by Substitution

Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.

Concept: Definite Integral as the Limit of a Sum

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Concept: Different Types of Linear Programming Problems

Let A = {*x* ∈ Z : 0 ≤ *x* ≤ 12}. Show that R = {(*a*, *b*) : *a*, *b *∈ A, |*a* – *b*| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]

Concept: Types of Relations

Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)

Concept: Types of Functions

Find the area of the region in the first quadrant enclosed by the *x*-axis, the line *y* = *x* and the circle *x*^{2} + *y*^{2} = 32.

Concept: Area Under Simple Curves

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find *A*^{−1}. Using A^{−1} solve the system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = – 5

x + y – 2z = – 3

Concept: Applications of Determinants and Matrices

Using elementary row transformations, find the inverse of the matrix A = `[(1,2,3),(2,5,7),(-2,-4,-5)]`

Concept: Elementary Operation (Transformation) of a Matrix

Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

Concept: Three - Dimensional Geometry Examples and Solutions