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

Duration: 3h

If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.

Chapter: [0.03] Matrices

If `veca","vecb","vecc`are unit vectors such that `veca+vecb+vecc=0`, then write the value of `veca.vecb+vecb.vecc+vecc.veca`

Chapter: [0.1] Vectors

if `|vecaxxvecb|^2+|veca.vecb|^2=400 ` and `|vec a| = 5` , then write the value of `|vecb|`

Chapter: [0.1] Vectors

Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.

Chapter: [0.11] Three - Dimensional Geometry

If `[2 1 3]([-1,0,-1],[-1,1,0],[0,1,1])([1],[0],[-1])=A` , then write the order of matrix A.

Chapter: [0.03] Matrices

If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.

Chapter: [0.04] Determinants

Find the values of *a* and *b*, if the function *f* defined by

*x*= 1.

Chapter: [0.05] Continuity and Differentiability

Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if *x* ∈ (–1, 1) .

Chapter: [0.06] Applications of Derivatives

If *x* = sin *t* and* y *= sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .

Chapter: [0.06] Applications of Derivatives

Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .

Chapter: [0.06] Applications of Derivatives

Evaluate : \[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\] .

Chapter: [0.07] Integrals

Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .

Chapter: [0.07] Integrals

Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .

Chapter: [0.07] Integrals

Find : \[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\] .

Chapter: [0.07] Integrals

Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .

Chapter: [0.09] Differential Equations

Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .

Chapter: [0.09] Differential Equations

If \[\vec{a} \times \vec{b} = \vec{c} \times \vec{d} \text { and } \vec{a} \times \vec{c} = \vec{b} \times \vec{d}\] , show that \[\vec{a} - \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .

Chapter: [0.1] Vectors

Prove that the lines through *A* (0, −1, −1) and *B* (4, 5, 1) intersects the line through *C* (3, 9, 4) and *D* (−4, 4, 4). Also, find their point of intersection.

Chapter: [0.11] Three - Dimensional Geometry

A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.

Chapter: [0.13] Probability

Let, X denote the number of colleges where you will apply after your results and P(X = *x*) denotes your probability of getting admission in *x *number of colleges. It is given that

where *k* is a positive constant. Find the value of *k*. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Chapter: [0.13] Probability

Prove that : \[\cot^{- 1} \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} = \frac{x}{2}, 0 < x < \frac{\pi}{2}\] .

Chapter: [0.02] Inverse Trigonometric Functions

Solve for *x* : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .

Chapter: [0.02] Inverse Trigonometric Functions

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?

Chapter: [0.03] Matrices

Using integration find the area of the region bounded by the curves \[y = \sqrt{4 - x^2}, x^2 + y^2 - 4x = 0\] and the x-axis.

Chapter: [0.08] Applications of the Integrals

Find the equation of the plane which contains the line of intersection of the planes \[x + 2y + 3z - 4 = 0 \text { and } 2x + y - z + 5 = 0\] and whose *x*-intercept is twice its *z*-intercept.

Chapter: [0.11] Three - Dimensional Geometry

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B.

Chapter: [0.13] Probability

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :

Tablets |
Iron |
Calcium |
Vitamin |

x | 6 | 3 | 2 |

y | 2 | 3 | 4 |

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Chapter: [0.12] Linear Programming

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

Chapter: [0.01] Relations and Functions

If \[a, b\] and c are all non-zero and

Chapter: [0.04] Determinants

If \[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] ,find adj·A and verify that A(adj·A) = (adj·A)A = |A| I_{3}.

Chapter: [0.03] Matrices

The sum of the surface areas of a cuboid with sides *x*, 2*x* and \[\frac{x}{3}\] and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if *x* is equal to three times the radius of sphere. Also find the minimum value of the sum of their volumes.

Chapter: [0.06] Applications of Derivatives

Find the equation of tangents to the curve *y* = cos(*x *+* y*), –2π ≤ *x* ≤ 2π that are parallel to the line *x *+ 2*y* = 0.

Chapter: [0.06] Applications of Derivatives

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