Date & Time: 21st March 2019, 10:30 am

Duration: 2h30m

1 . All questions are compulsory.

2 . Use of calaculators is not permitted. You may ask for logarithmic tables, if required.

3 . There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.

Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .

Chapter: [4.01] Three - Dimensional Geometry

Find the value of p for which the following lines are perpendicular :

`(1-x)/3 = (2y-14)/(2p) = (z-3)/2 ; (1-x)/(3p) = (y-5)/1 = (6-z)/5`

Chapter: [4.01] Three - Dimensional Geometry

Find the integerating factor of the differential equation `xdy/dx - 2y = 2x^2` .

Chapter: [3.04] Differential Equations

If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| .

Chapter: [2.02] Matrices

If `y = sin^-1 x + cos^-1 x , "find" dy/dx`

Chapter: [3.01] Continuity and Differentiability

If A = `[[3,9,0] ,[1,8,-2], [7,5,4]]` and B =`[[4,0,2],[7,1,4],[2,2,6]]` , then find the matrix `B'A'` .

Chapter: [2.02] Matrices

Form the differential equation representing the family of curves `y^{2} = m(a^{2} - x^{2}) by eliminating the arbitrary constants 'm' and 'a'.

Chapter: [3.04] Differential Equations

Find :

`∫ sin(x-a)/sin(x+a)dx`

Chapter: [3.05] Integrals

Find :

`∫(log x)^2 dx`

Chapter: [3.05] Integrals

Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk` and `vecb = 3hati - 2hatj + 2hatk` .

Chapter: [4.02] Vectors

Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.

Chapter: [4.02] Vectors

Mother, father and son line up at random for a family photo. If A and B are two events given by

A = Son on one end, B = Father in the middle, find P(B / A).

Chapter: [6.01] Probability

Let X be a random variable which assumes values x_{1} , x_{2}, x_{3} , x_{4} such that 2P (X = x_{1}) = 3P (X = x_{2}) = P (X = x_{3}) = 5P (X = x_{4}). the probability distribution of X.

Chapter: [6.01] Probability

A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads.

Chapter: [6.01] Probability

If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .

Chapter: [1.02] Relations and Functions

Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.

Chapter: [1.01] Inverse Trigonometric Functions

If e^{y} ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`

Chapter: [3.01] Continuity and Differentiability

Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`

Chapter: [3.01] Continuity and Differentiability

Find the intervals in which function f given by f(x) = 4x^{3} - 6x^{2} - 72x + 30 is (a) strictly increasing, (b) strictly decresing .

Chapter: [3.02] Applications of Derivatives

Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.

Chapter: [1.02] Relations and Functions

If f (x) = `(4x + 3)/(6x - 4) , x ≠ 2/3`, show that fof (x) = x for all ` x ≠ 2/3` . Also, find the inverse of f.

Chapter: [1.02] Relations and Functions

Using properties of determinants, prove that

`|[b+c , a ,a ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc

Chapter: [2.01] Determinants

If y = (sec^{-1} x )^{2} , x > 0, show that

`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`

Chapter: [3.01] Continuity and Differentiability

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .

Chapter: [3.05] Integrals

Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

Chapter: [3.05] Integrals

Find the value of λ for which the following lines are perpendicular to each other:

`(x - 5)/(5 lambda + 2 ) = ( 2 - y )/5 = (1 - z ) /-1 ; x /1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

Chapter: [4.01] Three - Dimensional Geometry

Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.

Chapter: [4.02] Vectors

Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`

Chapter: [3.04] Differential Equations

Solve the differential equation: (1 +x^{2 }) dy + 2xy dx = cot x dx

Chapter: [3.04] Differential Equations

Find the area of the region.

{(x,y) : 0 ≤ y ≤ x^{2 }, 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .

Chapter: [3.03] Applications of the Integrals

Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.

Chapter: [3.05] Integrals

Find the mean and variance of the random variable X which denotes the number of doublets in four throws of a pair of dice.

Chapter: [6.01] Probability

Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).

Chapter: [4.01] Three - Dimensional Geometry

Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.

Chapter: [4.01] Three - Dimensional Geometry

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

Chapter: [3.02] Applications of Derivatives

If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A^{-1}Hence, solve the system of equations:

x +y + z = 6

y + 3z = 11

and x -2y +z = 0

Chapter: [2.01] Determinants

Find the inverse of the following matrix, using elementary transformations:

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

Chapter: [2.01] Determinants

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A

require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit.

Chapter: [5.01] Linear Programming

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