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

Duration: 3h

Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b

Chapter: [0.03] Matrices

Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1

Chapter: [0.1] Vectors

The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A

Chapter: [0.1] Vectors

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is `2hati-3hatj+6hatk`

Chapter: [0.11] Three - Dimensional Geometry

Find the maximum value of `|(1,1,1),(1,1+sintheta,1),(1,1,1+costheta)|`

Chapter: [0.03] 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.

Chapter: [0.03] Matrices

Show that the equation of normal at any point t on the curve *x* = 3 cos t – cos^{3}t and y = 3 sin t – sin^{3}t is 4 (y cos^{3}t – sin^{3}t) = 3 sin 4t

Chapter: [0.06] Applications of Derivatives

Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`

Chapter: [0.07] Integrals

Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`

Chapter: [0.07] Integrals

Find `intsqrtx/sqrt(a^3-x^3)dx`

Chapter: [0.07] Integrals

Evaluate `int_(-1)^2|x^3-x|dx`

Chapter: [0.07] Integrals

Find the particular solution of the differential equation

(1 – *y*^{2}) (1 + log *x*) *dx* + 2*xy* *dy* = 0, given that *y *= 0 when *x* = 1.

Chapter: [0.09] Differential Equations

Find the general solution of the following differential equation :

`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`

Chapter: [0.09] Differential Equations

Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.

Chapter: [0.1] Vectors

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines.

`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`

Chapter: [0.11] Three - Dimensional Geometry

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C

Chapter: [0.13] Probability

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins

Chapter: [0.13] Probability

Prove that:

`tan^(-1)""1/5+tan^(-1)""1/7+tan^(-1)""1/3+tan^(-1)""1/8=pi/4`

Chapter: [0.02] Inverse Trigonometric Functions

Solve for *x*:

`2tan^(-1)(cosx)=tan^(-1)(2"cosec" x)`

Chapter: [0.02] Inverse Trigonometric Functions

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Chapter: [0.04] Determinants

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.06] Applications of Derivatives

Find the values of p and q for which

f(x) = `{((1-sin^3x)/(3cos^2x),`

is continuous at *x* = *π*/2.

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.06] Applications of Derivatives

Find the intervals in which f(*x*) = sin 3*x* – cos 3*x*, 0 < *x* < *π*, is strictly increasing or strictly decreasing.

Chapter: [0.06] Applications of Derivatives

Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.

Chapter: [0.08] Applications of the Integrals

Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).

Also, find the ratio in which P divides the line segment AB.

Chapter: [0.1] Vectors

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

Chapter: [0.13] Probability

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Chapter: [0.12] Linear Programming

Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).

Chapter: [0.01] Relations and Functions

Prove that `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (*x* + *y* + *z*) and hence find the quotient.

Chapter: [0.03] Matrices [0.04] Determinants

Using elementary transformations, find the inverse of the matrix A = `((8,4,3),(2,1,1),(1,2,2))`and use it to solve the following system of linear equations :

8*x* + 4*y* + 3*z* = 19

2*x* + *y* + *z* = 5*x* + 2*y* + 2*z* = 7

Chapter: [0.03] Matrices [0.04] Determinants

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