Date: March 2014

Let R = {(a, a^{3}) : a is a prime number less than 5} be a relation. Find the range of R.

Chapter: [1.02] Relations and Functions

Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .

Chapter: [1.01] Inverse Trigonometric Functions

Use elementary column operations \[C_2 \to C_2 - 2 C_1\] in the matrix equation \[\begin{pmatrix}4 & 2 \\ 3 & 3\end{pmatrix} = \begin{pmatrix}1 & 2 \\ 0 & 3\end{pmatrix}\begin{pmatrix}2 & 0 \\ 1 & 1\end{pmatrix}\] .

Chapter: [2.02] Matrices

If \[\begin{pmatrix}a + 4 & 3b \\ 8 & - 6\end{pmatrix} = \begin{pmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{pmatrix},\] ,write the value of a − 2b.

Chapter: [2.02] Matrices

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

Chapter: [2.02] Matrices

Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .

Chapter: [3.05] Integrals

Evaluate : \[\int\limits_0^\frac{\pi}{4} \tan x dx\] .

Chapter: [3.05] Integrals

Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\]

Chapter: [4.02] Vectors

Find a vector in the direction of vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] which has magnitude 21 units.

Chapter: [4.02] Vectors

Find the angle between the lines

\[\vec{r} = \left( 2 \hat{i} - 5 \hat{j} + \hat{k} \right) + \lambda\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)\] and \[\vec{r} = 7 \hat{i} - 6 \hat{k} + \mu\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right)\]

Chapter: [4.01] Three - Dimensional Geometry

Let f : W → W be defined as f(x) = x − 1 if x is odd and f(x) = x + 1 if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers.

Chapter: [1.02] Relations and Functions

Solve for *x* : \[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .

Chapter: [1.01] Inverse Trigonometric Functions

Prove that:

cot^{−1} 7 + cot^{−1} 8 + cot^{−1} 18 = cot^{−1} 3 .

Chapter: [1.01] Inverse Trigonometric Functions

Using properties of determinants, prove that \[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\] .

Chapter: [2.01] Determinants

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`

Chapter: [3.01] Continuity and Differentiability

If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]

Chapter: [3.04] Differential Equations

Find the approximate value of *f*(3.02), up to 2 places of decimal, where *f*(*x*) = 3*x*^{2} + 5*x* + 3.

Chapter: [3.02] Applications of Derivatives

Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is

(a) strictly increasing

(b) strictly decreasing

Chapter: [3.02] Applications of Derivatives

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

Chapter: [3.05] Integrals

Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .

Chapter: [3.05] Integrals

Solve the differential equation (x^{2} − yx^{2}) dy + (y^{2} + x^{2}y^{2}) dx = 0, given that y = 1, when x = 1.

Chapter: [3.04] Differential Equations

Solve the differential equation \[\frac{dy}{dx}\] + *y* cot *x* = 2 cos *x*, given that *y* = 0 when x = \[\frac{\pi}{2}\] .

Chapter: [3.04] Differential Equations

Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] are coplanar if and only if** **\[\vec{a} + \vec{b}\], \[\vec{b} + \vec{c}\] and \[\vec{c} + \vec{a}\] are coplanar.

Chapter: [4.02] Vectors

Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].

Chapter: [4.02] Vectors

Find the shortest distance between the following pairs of lines whose vector are: \[\overrightarrow{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( 2 \hat{i} - \hat{j} + \hat{k} \right) \text{ and } , \overrightarrow{r} = 2 \hat{i} + \hat{j} - \hat{k} + \mu\left( 3 \hat{i} - 5 \hat{j} + 2 \hat{k} \right)\]

Chapter: [4.01] Three - Dimensional Geometry

Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution.

Chapter: [6.01] Probability

Two schools P and Q want to award their selected students on the values of tolerance, kindness and leadership. School P wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively, with a total award money of Rs 2,200. School Q wants to spend Rs 3,100 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is Rs 1,200, using matrices, find the award money for each value.

Chapter: [2.02] Matrices

Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.

Chapter: [3.02] Applications of Derivatives

Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x}dx\] .

Chapter: [3.05] Integrals

Find the area of the smaller region bounded by the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1\] and the line \[\frac{x}{3} + \frac{y}{2} = 1 .\]

Chapter: [3.03] Applications of the Integrals

Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence, find the distance of point P (–2, 5, 5) from the plane obtained

Chapter: [4.01] Three - Dimensional Geometry

Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.

Chapter: [4.01] Three - Dimensional Geometry

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 25 and that from a shade is Rs 15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Formulate an LPP and solve it graphically.

Chapter: [5.01] Linear Programming

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver?

Chapter: [6.01] Probability

Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that

(i) all the five cards diamonds

(ii) only 3 cards are diamonds

(iii) none is a diamond

Chapter: [6.01] Probability

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