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

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

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

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

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

If A is a square matrix such that A^{2} = I, then find the simplified value of (A – I)^{3} + (A + I)^{3} – 7A.

Concept: Types of Matrices

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 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

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

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 *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

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

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

Concept: Properties of Inverse Trigonometric Functions

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

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

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

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`

Concept: Second Order Derivative

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

Concept: Derivatives of Inverse Trigonometric Functions

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

Concept: Derivatives of Implicit Functions

If *y* = *x ^{x}*, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

Concept: Simple Problems on Applications of Derivatives