Date: March 2014

if `2[[3,4],[5,x]]+[[1,y],[0,1]]=[[7,0],[10,5]]` , find (x−y).

Chapter: [2.02] Matrices

Solve the following matrix equation for *x*: `[x 1] [[1,0],[−2,0]]=0`

Chapter: [2.02] Matrices

If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.

Chapter: [2.01] Determinants

Write the antiderivative of `(3sqrtx+1/sqrtx).`

Chapter: [3.05] Integrals

If `sin (sin^(−1)(1/5)+cos^(−1) x)=1`, then find the value of *x*.

Chapter: [1.01] Inverse Trigonometric Functions

Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]

Write the value of *x* given by 2 * (*x* * 5) = 10.

Chapter: [1.02] Relations and Functions

Find the projection of the vector `hati+3hatj+7hatk` on the vector `2hati-3hatj+6hatk`

Chapter: [4.02] Vectors

Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane `vec r.(hati+hatj+hatk)=2`

Chapter: [4.01] Three - Dimensional Geometry

Evaluate each of the following integral:

Chapter: [3.05] Integrals

Write a unit vector in the direction of the sum of the vectors \[\overrightarrow{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}\] and \[\overrightarrow{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}\].

Chapter: [4.02] Vectors

Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].

Chapter: [4.02] Vectors

Vectors `veca,vecb and vecc ` are such that `veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7 ` Find the angle between `veca and vecb`

Chapter: [4.02] Vectors

Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`

Chapter: [3.04] Differential Equations

Evaluate : `∫(sin^6x+cos^6x)/(sin^2x.cos^2x)dx`

Chapter: [3.05] Integrals

Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`

Chapter: [3.05] Integrals

Find the intervals in which the function f(*x*) = 3*x*^{4} − 4*x*^{3} − 12*x*^{2} + 5 is

(a) strictly increasing

(b) strictly decreasing

Chapter: [3.02] Applications of Derivatives

Find the equations of the tangent and normal to the curve *x* = a sin^{3}θ and y = a cos^{3}θ at θ=π/4.

Chapter: [3.02] Applications of Derivatives

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (*a*, *b*) R (*c*, *d*) if *a* + *d* = *b* + *c* for (*a*, *b*), (*c*, *d*) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].

Chapter: [1.02] Relations and Functions

Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `

Chapter: [1.01] Inverse Trigonometric Functions

Prove that

\[2 \tan^{- 1} \left( \frac{1}{5} \right) + \sec^{- 1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{- 1} \left( \frac{1}{8} \right) = \frac{\pi}{4}\] .

Chapter: [1.01] Inverse Trigonometric Functions

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

Chapter: [3.02] Applications of Derivatives

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Give that

(i) the youngest is a girl.

(ii) at least one is a girl.

Chapter: [6.01] Probability

Using properties of determinants, prove the following:

Chapter: [2.01] Determinants

Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?

Chapter: [3.02] Applications of Derivatives

Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that

*x*= 1.

Chapter: [3.04] Differential Equations

Prove that the line \[\vec{r} = \left( \hat{i }+ \hat{j }- \hat{k} \right) + \lambda\left( 3 \hat{i} - \hat{j} \right) \text{ and } \vec{r} = \left( 4 \hat{i} - \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{k} \right)\] intersect and find their point of intersection.

Chapter: [4.01] Three - Dimensional Geometry

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

Chapter: [5.01] Linear Programming

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

Chapter: [6.01] Probability

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

Chapter: [6.01] Probability

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x^{2} + y^{2}= 32.

Chapter: [3.03] Applications of the Integrals

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).

Chapter: [4.01] Three - Dimensional Geometry

Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

Chapter: [4.01] Three - Dimensional Geometry

Two schools P and Q want to award their selected students on the values of discipline, politeness and punctuality. The school P wants to award Rs *x* each, Rs y each and Rs z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs 1,000. School Q wants to spend Rs 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value.

Apart from the above three values, suggest one more value for awards.

Chapter: [2.02] Matrices

Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .

Chapter: [3.05] Integrals

Of all the closed right circular cylindrical cans of volume 128π cm^{3}, find the dimensions of the can which has minimum surface area.

Chapter: [3.02] Applications of Derivatives

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