Recent questions in Model Papers

If none of a,b and c is zero, using properties of determinants, prove that $\begin{vmatrix} -bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{vmatrix} = (bc=ca+ab)^3$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y),$ prove that $\large\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If $y= \bigg( x+\sqrt{x^2-1} \bigg)^m,$ then show that $(x^2+1)\large\frac{d^2y}{dx^2}$ $+x\large\frac{dy}{dx}$ $-m^2y=0$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find all the points of discontinuity of the function $f(x)=[x^2]$ on [1,2), where [.] denotes the greatest integer function.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Differentiate $\sin^{-1}(2x\sqrt{1-x^2})$ w.r.t. $\cos^{-1} \bigg( \large\frac{1-x^2}{1+x^2} \bigg)$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{1}{\cos(x-a)\cos(x-b)}$ $dx$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int x(\log x )^2.dx$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \frac{x}{x^3-1} dx$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Using properties of definite integrals, evaluate $\int_0^x \frac{xdx}{4-cos^2x}$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

The dot products of a vector with the vectors $\hat i - 3\hat k, \hat i - 2\hat k$ and $\hat i + \hat j + 4\hat k$ are 0,5 and 8 respectively. Find the vector.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the equation of plane passing through the point (1,2,1) and perpendicular to the line joining the points (1,4,2) and (2,3,5). Also, find the perpendicular distance of the plane from the origin.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the equation of the perpendicular drawn from the point p(2,4,-1) to the line $\frac{x+5}{1} = \frac{y+3}{4}=\frac{z-6}{-9}$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

A biased die is twice as likely to shown an even number as an odd number. The die is rolled three times. If occurance of an even number is considered a success, then write the probability distribution of number of successes. Also find the mean number of successes.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system, of equations : $\large\frac{1}{x}-\frac{1}{y}+\frac{1}{z}$ $=4: \: \large\frac{2}{x}+\frac{1}{y}-\frac{3}{z}$ $=0: \: \large\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ $=2 \qquad x \neq 0, y \neq 0, z \neq 0$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semivertical angle $\alpha, \: is \: \large\frac{4}{27}$ $\pi \: h^3\: tan^2\: \alpha$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Show that the normal at any point $\theta$ to the curve $x=a\cos\theta+a\theta\: \sin\theta$ and $y=a\sin\theta-a\theta \cos\theta$ is at a constant distance from the origin.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the area of the region: ${(x,y): 0 \leq y \leq x^2,0 \leq y \leq x+2:0 \leq x \leq 3}.$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the particular solution of the differential equation $(xdy-ydx)y.\sin \bigg(\large\frac{y}{x} \bigg)$ $=(ydx+xdy)x \cos\large\frac{y}{x},$ given that $y=\pi$ when $x = 3$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the point (1,1,1) and containing the line $\overrightarrow r = (3\hat i + \hat j + 5\hat k) + \lambda (3\hat i - \hat j - 5\hat k)$ . Also, show that the plane contains the line $\overrightarrow r = (-\hat i + 2\hat j + 5\hat k) + \lambda (\hat i - 2\hat j - 5\hat k)$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

A company sells two different products A and B. The products are produced in a common production process which has total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The demand in the market shows that the maximum number of units of A that can sold is 70 and that of B is 125. Profit on each unit of A is Rs. 20 and on B is Rs.15. How many units of A and B should be produced to maximise the profit? Form an L.P.P and solve it graphically.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

In an examination, 10 questions of true-false type are asked. A student toses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true' and if it falls tails, he answers 'false'. Show that the probability that he answers at most 7 questions correctly is $\frac{121}{128}$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Two bags A and B contain 4 white and 3 black balls and 2 white and 2 black balls respectively. From bag A, two balls are drawn at random and then transferred to bag B, A ball is then drawn from bag B and is found to be a black ball. What is the probability that the transferred balls were 1 white and 1 black?

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Using principal value, write the value of $\cos^{-1} \bigg( \large\frac{1}{2} \bigg)$ $+ 2\sin^{-1} \bigg(\large \frac{1}{2} \bigg)$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

If $\sin\: \{ \sin^{-1}\large\frac{1}{5}$ $+\cos^{-1}x \} = 1$ , then find the value of x.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Find the value of x, if : $\begin{bmatrix} 1 & x & 1 \end{bmatrix} \; \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix} \; \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix} = 0$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Find the values of x, if $\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} = \begin{bmatrix} 2x & 4 \\ 6 & x \end{bmatrix}$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Differentiate $e^{\large \sin\: x}$ w.r.t. $x$ .

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ and A + A' = I, then find the value of $\theta$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the degree of the differential equation $\bigg[ 1+ \bigg(\large\frac{dy}{dx} \bigg)^2 \bigg]^{\large\frac{3}{2}} =\large\frac{d^2y}{dx^2}$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the magnitude of two vectors $\overrightarrow a \: and \: \overrightarrow b$ having the same magnitude and such that angle between them is $30^{\circ}$ and their scalar product is 3.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If vectors $2\hat i-5\hat j + b\hat k$ and $2\hat i + a\hat j + 4\hat k$ are parallel, find the values of a and b.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the distance between the parallel planes $x+y-z=-4$ and $2x+2y-2z+10=0$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Let * be binary operation of N x N defined by (a,b)(c,d)=(a+c,b+d). Show that is commutative as well as associative. Find the identity element for the operation * on N x N, if any.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Given relation R is the set Z of all integers is given by $R= \{(a,b) \in Z \times Z, a-b$ is an integer }. Find whether R is an equivalence relation or not.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Prove that : $2\tan^{-1} \frac{1}{2} + \tan^{-1}\large \frac{1}{7}$ $= \tan^{-1} \large\frac{31}{17}$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Simplify : $\cos^{-1} \bigg(\large \frac{3}{5}$ $\cos \theta +\large \frac{4}{5}$ $sin \theta \bigg)$

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Principal and staff of a school decided to award prizes to their students for three values Honesty (x), cleanliness (y) and punctuality (z) in two groups. Group A (upto class VIII) and group B (IX-XII). It is decided that Rs. 12000 for three values (x,y,z) for 3,4,5 students of group A and Rs. 11040 for 5,4,3 students of group B respectively. If all the three prizes together amount to Rs. 2880 then, (i) Represent above situation by a matrix equation and form linear equation using matrices. (ii)Is it possible to solve the system of linear equations so obtained using matrices.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Find the point (s) of discountinuity of the function 'f' given by $f(x) = \left\{ \begin{array}{l l}|x|+3, & \quad { x \leq -3} \\ -2x, & \quad {-3 < x < 3} \\ 6x+2, & \quad { x > 3 } \end{array} \right.$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Differentiate : $\tan^{-1} \bigg( \large\frac{\sqrt{1-x^2}}{x} \bigg)$ w.r.t. $\cos^{-1} \bigg( 2x\sqrt{1-x^2} \bigg).$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

If $x= \bigg( \cos\theta + \log \tan\large\frac{\theta}{2} \bigg),$ $y=\sin\: \theta, \large\frac{d^2y}{dx^2}$ $\: at \: \theta=\large\frac{\pi}{4}$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

If $y=log \bigg( x+\sqrt{x^2+1} \bigg)$ , then prove that $(x^2+4)\large\frac{d^2y}{dx^2}$ $+x\large\frac{dy}{dx}=0$ .

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{2x}{(1+\sin x)(2+\sin\:x)}$ $dx$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{\sin \: 2x}{(1+x^2)(3+x^2)}$ $dx$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{\sin x\cos x}{a^2\sin^2x+b^2+\cos^2x}$ $dx$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate the following integral using limit of sums $\int_1^4 \: x^2-xdx$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int_0^a \large\frac{\sqrt x}{\sqrt x+\sqrt{a-x} }$ $dx$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Show that each of the given three vectors is a unit vector. $\frac{1}{7} (2\hat i + 3\hat j + 6\hat k), \frac{1}{7} (3\hat i - 6\hat j + 2\hat k), \frac{1}{7} (6\hat i + 2\hat j - 36\hat k).$ Also, show that they are mutually perpendicular to each other.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Find the equation of a line passing through the point (-1,-1,2) and perpendicular to the planes $3x+2y-3z=1 \: and \: 5x-4y+z=5$

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Find the equation of a line passing through the point (2,1,3) and perpendicular to the lines $\large\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$ and $\large\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$ .

asked Feb 2, 2013 by thanvigandhi_1

1 answer

In a group of student, selection of four students for cultural activity from 3 boys and 3 girls has to be done. Find the probability that either 3 boys and 1 girl or 3 girls and one boy are selected.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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