If A = `([cos alpha, sin alpha],[-sinalpha, cos alpha])` , find α satisfying 0 < α < `pi/r`when `A+A^T=sqrt2I_2` where A^{T} is transpose of A.

Concept: Operations on Matrices - Addition of Matrices

If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k.

Concept: Order of a Matrix

For what values of k, the system of linear equations

x + y + z = 2

2x + y – z = 3

3x + 2y + kz = 4

has a unique solution?

Concept: Elementary Transformations

Write the sum of intercepts cut off by the plane `vecr.(2hati+hatj-k)-5=0` on the three axes

Concept: Plane - Intercept Form of the Equation of a Plane

Find λ and μ if

`(hati+3hatj+9k)xx(3hati-lambdahatj+muk)=0`

Concept: Determinant of a Square Matrix

If `veca=4hati-hatj+hatk` then find a unit vector parallel to the vector `veca+vecb`

Concept: Components of a Vector

Solve for x : tan^{-1} (x - 1) + tan^{-1}x + tan^{-1} (x + 1) = tan^{-1} 3x

Concept: Properties of Inverse Trigonometric Functions

Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`

Concept: Properties of Inverse Trigonometric Functions

A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

Concept: Inverse of a Matrix by Elementary Operations

If f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`

is continuous at x = 0, then find the values of a and b.

Concept: Continuous Function of Point

If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`

Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`

Concept: Second Order Derivative

if `y = sin^(-1)[(6x-4sqrt(1-4x^2))/5]` Find `dy/dx `.

Concept: Derivatives of Inverse Trigonometric Functions

Find the equation of tangents to the curve y= x^{3} + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.

Concept: Tangents and Normals

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Concept: Methods of Integration - Integration by Substitution

Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`

Concept: Integration as an Inverse Process of Differentiation

Evaluate `int_(-2)^2x^2/(1+5^x)dx`

Concept: Properties of Definite Integrals

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Concept: Methods of Integration - Integration by Substitution

Find the particular solution of differential equation:

`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`

Concept: General and Particular Solutions of a Differential Equation

Find the particular solution of the differential equation

2y e^{x/y} dx + (y - 2x e^{x/y}) dy = 0

given that x = 0 when y = 1.

Concept: General and Particular Solutions of a Differential Equation

Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are coplanar.

Concept: Shortest Distance Between Two Lines

Find the coordinates of the foot of perpendicular drawn from the point A (-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

Concept: Three - Dimensional Geometry Examples and Solutions

A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y.

Concept: Conditional Probability

A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first

Concept: Probability Examples and Solutions

Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X.Also, find the mean and variance of the distribution.

Concept: Mean of a Random Variable

LetA= R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) ε A.

Concept: Concept of Binary Operations

Prove that `y=(4sintheta)/(2+costheta)-theta `

Concept: Simple Problems on Applications of Derivatives

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`

Concept: Simple Problems on Applications of Derivatives

Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

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

Find the equation of the plane which contains the line of intersection of the planes

`vecr.(hati-2hatj+3hatk)-4=0" and"`

`vecr.(-2hati+hatj+hatk)+5=0`

and whose intercept on x-axis is equal to that of on y-axis.

Concept: Vector and Cartesian Equation of a Plane

A retired person wants to invest an amount of Rs. 50, 000. His broker recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximise his returns.

Concept: Graphical Method of Solving Linear Programming Problems

Using properties of determinants, prove that

`|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3`

Concept: Properties of Determinants

If A= `((1,0,2),(0,2,1),(2,0,3))` and A^{3} - 6A^{2} +7A + kI_{3} = O find k.

Concept: Introduction of Operations on Matrices