Recent questions in Model Papers

In a class, having 60% boys, 5% of the boys and 10% of the girls have an I.Q. of more than 150. A student is selected at random and found to have an I.Q. of more than 150. Find the probability that the selected student is a boy.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Show that the function $ f : R \rightarrow { x \in R : -1 < x < 1 }$ defined by $ f(x) =\large \frac{x}{1+|x|}$$, x \in r$ is one-one and onto.

asked Jan 3, 2013 by thanvigandhi_1

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Show that the right circular cone of least curved surface and given volume has an altitude equal to $ \sqrt 2$ times the radius of the base.

asked Jan 3, 2013 by thanvigandhi_1

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A rectangle is inscribed in a semi-circle of radius r with one of its sides on the diameter of the sei-circle. Find the maximum area of the rectangle.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Using integration, find the area of region enclosed between the circles $ x^2+y^2=4 \: and \: (x-2)^2+y^2=4.$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Find the vector equation of the line through the point ( 1, 2, -4 ) and perpendicular to the two lines $\large \frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7} $ and, $ \large\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5} $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Show that the lines : $ \frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \: and \: \frac{x-2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}. $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

A company manufacture two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minute each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minute each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit Rs. 5 each for type A and Rs.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profits.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Compute the following : $ \begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix} $

asked Jan 3, 2013 by thanvigandhi_1

1 answer

If $ A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix} $ and $ B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix} $ then find $( A + 2B)'.$

asked Jan 3, 2013 by thanvigandhi_1

1 answer

A matrix A, of order 3 x 3, has determinant 4 find the value of |3A|.

asked Jan 3, 2013 by thanvigandhi_1

1 answer

If $y = x +\tan x$, prove that : $\cos^2x. \large\frac{d^2y}{dx^2}$$-2y+2x = 0. $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

The total revenue received from the sale of x units of a product is given by $ R(x) = 13x^2 + 26x + 15 $ Find the marginal revenue when x = 7.

asked Jan 3, 2013 by thanvigandhi_1

1 answer

$ \int \: \frac{\large x^3-x^2+x-1}{\large x-1}dx $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Find the differential equation $y = x \sin x.$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Compute the magnitude of the following vectors : $ \overrightarrow a = \hat i + \hat j + \hat k, \overrightarrow b = 2\hat i - 7\hat j - 3\hat k\: and \: \overrightarrow c = \frac{1}{\sqrt 3}\hat i + \frac{1}{\sqrt 3}\hat j - \frac{1}{\sqrt 3}\hat k $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

If A and B one the events associated with a random experiment and $ P(A) = 0.2, P(B) = 0.3, P (A \cup B ) = 0.4$, find $ (i) P( A \cap B ) and (ii) P( A / B ) $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

In a class having 70% boys, 20% of the boys and 10% of the girls are players. A student is selected at random from the class and is found to be a player. Find the probability that the selected student is a girl.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Show that the relation R defined by (a, b ) R ( c, d ) $ \Rightarrow $ a + b + c on the set N x N is an equivalence relation.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $ \begin{bmatrix} \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{bmatrix} $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Prove, using the properties of determinants $ \begin{bmatrix} a+b & b+c & c+a \\ b+c & c+a & a+b \\ c+a & a+b & b+c \end{bmatrix} = 2 \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix} $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

If $ y = 3e^{2x} + 2e^{3x} $ prove that, $ \large\frac{d^2y}{dx^2} = $$5\large\frac{dy}{dx}$$+6y = 0.$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Differentiate following function w.r.t. $ x$ $ x^{\large \sin x} + ( \sin x )^{\large \cos x} $

asked Jan 3, 2013 by thanvigandhi_1

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Find the equation of tangent to the curve $ y = \sqrt {3x-2}$ which is parallel to the line $ 4x-2y+5=0$.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by $ f(x) = x^3 + \large\frac{1}{x^3}$$, x \neq 0 $ is, (i) increasing, (ii) decreasing.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Evaluate : \[ (i) \int \: \sqrt {x^2+4x+8} dx \] \[ (ii) \int \: \sqrt {(x-5)(7-x)} dx \] \[ (iii) \int \: \sqrt {14x-20-2x^2} dx \] \[ (iv) \int \: \sqrt {4ax-x^2} dx \]

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Evaluate $ \int\large\frac{dx}{x[6(\log x)^2+7(\log x)+2]} $

asked Jan 3, 2013 by thanvigandhi_1

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Evaluate $\int\: tan^{-1} \sqrt {\large\frac{1-sinx}{1+sinx}}$$dx.$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation : $ \frac{dy}{dx} = \frac{x(2y-x)}{x(2y+x)}, $ if y = 1 when x = 1.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Find the particular solution, satisfying the given condition for the differential equation $ \large\frac{dy}{dx} - \frac{y}{x} +$$ cosec \bigg(\large \frac{y}{x} \bigg)$$ = 0, y = 0\: when \: x = 1$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

If $ \overrightarrow a = \hat i + \hat j + \hat k \: and \: \overrightarrow b = \hat j - \hat k, $ Find a vector $ \overrightarrow c $ such that $ \overrightarrow a $ x $ \overrightarrow c = \overrightarrow b $ and $ \overrightarrow a . \overrightarrow c = 3$.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

If $ \overrightarrow a + \overrightarrow b + \overrightarrow c = 0\: and\: | \overrightarrow a | = 3, |\overrightarrow b | = 5\: and \: |\overrightarrow c | = 7 .$Show that the angle between $ \overrightarrow a \: and \overrightarrow b$ is $ 60^{\circ} $

asked Jan 3, 2013 by thanvigandhi_1

0 answers

There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

asked Jan 3, 2013 by thanvigandhi_1

1 answer

Find $ \mu$ variance for the following distribution :

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Give example of relations which are : \[ \begin{array} (i)\: Symmetric \: but \: neither \: reflexive \: nor\: transitive \\ (ii) \: Transitive\: but\: neither\: reflexive\: nor\: symmetric \\ (iii)\: Reflexive \: and\: symmetric\: but\: not \: transitive \\ (iv)\: Reflexive \: and transitive\: but\: not\: symmetric \\ (v)\: Symmetric\: and\: transitive\: but \: not\: reflexive \end{array}\]

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Let $ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $ , show that $ (aI + bA)^n = a^nI+na^{n-1}bA, $ where I is the identity matrix of order zero $ n \in N. $ If $ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} $ prove that : $ \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}, n \in N. $

asked Jan 3, 2013 by thanvigandhi_1

1 answer

If A and B are square matrices of the same order such that AB = BA, then proved by induction that $ AB^n = ^nA.$ Further, prove that $ (AB)^n = A^nB^n$ for all $ n \in N.$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Show that the rectangle of maximum, area that can be inscribed in a circle is a square.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the equation of the tangent to the curve $ y = \sqrt {3x-2} $ which is parallel to the line $ 4x-2y+5 = 0$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using integration, find the area lying above x - axis included, between the circle $ x^2+y^2 = 8x$ and the parabola $y^2 = 4x$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the distance of the point ( -2, 3, -4 ) from the line $\large \frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}$, measured parallel to the plane $ 4x+12y-3z+1=0$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the co-ordinates of the image of the point ( 1, 3, 4 ) in the plane $ 2x-y+z+3=0$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60/kg and food Q costs Rs. 80/kg. Food P contains 3 units/kg of vitamin A and 5 units of vitamin B, while food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Show that the relation R in the set R of real numbers, defined as $ R = {( a, b ) : a \leq b^2}$, is neither reflexive, nor symmetric , nor transitive.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the value of $ cosec^{-1} (2) $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Find the value of y if $\begin{bmatrix} x-y & 2 \\ x & 5 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

In the matrix $A=\begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt 3 & 1 & -5 & 17 \end{bmatrix} $Write (i) The order of the matrix .

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Write the value of following determinants : $ \begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & 2a-b & b-c \end{vmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \: \large\frac{cos\sqrt x}{\sqrt x}$$dx $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

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