#### Alternate Sets

If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.

Concept: Types of Relations

If tan^{-1}x+tan^{-1}y=π/4,xy<1, then write the value of x+y+xy.

Concept: Inverse Trigonometric Functions (Simplification and Examples)

If A is a square matrix, such that A^{2}=A, then write the value of 7A−(I+A)^{3}, where I is an identity matrix.

Concept: Types of Matrices

If `[[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]` find the value of x+y.

Concept: Equality of Matrices

If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x

Concept: Introduction of Operations on Matrices

If `f(x) =∫_0^xt sin t dt` , then write the value of *f *' (x).

Concept: Integration as an Inverse Process of Differentiation

find `∫_2^4 x/(x^2 + 1)dx`

Concept: Evaluation of Definite Integrals by Substitution

Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel

Concept: Basic Concepts of Vector Algebra

Find `veca.(vecbxxvecc), " if " veca=2hati+hatj+3hatk, vecb=-hati+2hatj+hatk " and " vecc=3hati+hatj+2hatk`

Concept: Vectors Examples and Solutions

If the Cartesian equations of a line are ` (3-x)/5=(y+4)/7=(2z-6)/4` , write the vector equation for the line.

Concept: Equation of a Line in Space

If the function* f* : R → R be given by *f*[*x*] = *x*^{2} + 2 and *g* : R → R be given by `g(x)=x/(x−1)` , x≠1, find fog and gof and hence find fog (2) and gof (−3).

Concept: Inverse of a Function

Prove that

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

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

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Using properties of determinants, prove that

`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`

Concept: Properties of Determinants

Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`

Concept: Derivatives of Functions in Parametric Forms

If y = P e^{ax} + Q e^{bx}, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Concept: General and Particular Solutions of a Differential Equation

Find the value(s) of x for which y = [x(x − 2)]^{2} is an increasing function.

Concept: Increasing and Decreasing Functions

Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .

Concept: Tangents and Normals

Evaluate :

`∫_0^π(4x sin x)/(1+cos^2 x) dx`

Concept: Evaluation of Definite Integrals by Substitution

Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`

Concept: Methods of Integration - Integration by Substitution

Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.

Concept: General and Particular Solutions of a Differential Equation

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

Concept: Solutions of Linear Differential Equation

Show that four points A, B, C and D whose position vectors are

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Concept: Coplanarity of Two Lines

The scalar product of the vector `veca=hati+hatj+hatk` with a unit vector along the sum of vectors `vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk` is equal to one. Find the value of λ and hence, find the unit vector along `vecb +vecc`

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

A line passes through (2, −1, 3) and is perpendicular to the lines `vecr=(hati+hatj-hatk)+lambda(2hati-2hatj+hatk) and vecr=(2hati-hatj-3hatk)+mu(hati+2hatj+2hatk)` . Obtain its equation in vector and Cartesian from.

Concept: Equation of a Line in Space

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Concept: Probability Examples and Solutions

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.

Concept: Adjoint and Inverse of a Matrix

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius *r* is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Concept: Maxima and Minima

Evaluate:

`∫1/(cos^4x+sin^4x)dx`

Concept: Methods of Integration - Integration by Substitution

Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).

Concept: Area Between Two Curves

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Concept: Distance of a Point from a Plane

Find the distance of the point (2, 12, 5) from the point of intersection of the line

`vecr=2hati-4hat+2hatk+lambda(3hati+4hatj+2hatk) `

Concept: Three - Dimensional Geometry Examples and Solutions

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30, respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Concept: Graphical Method of Solving Linear Programming Problems

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?

Concept: Baye'S Theorem

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

Concept: Mean of a Random Variable