Date & Time: 14th March 2016, 10:30 am

Duration: 3h

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.

Chapter: [2.02] Matrices

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

Chapter: [2.02] Matrices

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?

Chapter: [2.01] Determinants

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

Chapter: [4.01] Three - Dimensional Geometry

Find λ and μ if

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

Chapter: [2.01] Determinants

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

Chapter: [4.02] Vectors

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

Chapter: [1.01] 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`

Chapter: [1.01] 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?

Chapter: [2.02] Matrices

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.

Chapter: [3.01] Continuity and Differentiability

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`

Chapter: [3.01] Continuity and Differentiability

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

Chapter: [3.01] Continuity and Differentiability

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

Chapter: [3.02] Applications of Derivatives

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

Chapter: [3.05] Integrals

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

Chapter: [3.05] Integrals

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

Chapter: [3.05] Integrals

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

Chapter: [3.05] Integrals

Find the particular solution of differential equation:

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

Chapter: [3.04] Differential Equations

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.

Chapter: [3.04] Differential Equations

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

Chapter: [4.01] Three - Dimensional Geometry

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.

Chapter: [4.01] Three - Dimensional Geometry

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.

Chapter: [6.01] 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

Chapter: [6.01] Probability

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.

Chapter: [6.01] Probability

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.

Chapter: [1.02] Relations and Functions

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

Chapter: [3.02] 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)`

Chapter: [3.02] 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).

Chapter: [3.03] Applications of the Integrals

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.

Chapter: [4.01] Three - Dimensional Geometry

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.

Chapter: [5.01] Linear Programming

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`

Chapter: [2.01] Determinants

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

Chapter: [2.02] Matrices

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