Recent questions in Model Papers

There are two bags I and II. Bag I contains 3 white and 4 red balls and Bag II contains 5 white and 6 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 Dec 31, 2012 by thanvigandhi_1

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Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $ \large\frac{1}{3} $ h.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find the point on the line $ \large\frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2} $ at a distance $ 3 \sqrt 2 $ from the point (1, 2, 3)

asked Dec 29, 2012 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 equal to $ 60^{\circ}$

asked Dec 29, 2012 by thanvigandhi_1

0 answers

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

asked Dec 29, 2012 by thanvigandhi_1

0 answers

If $ A = \begin{bmatrix} 1 & 2 & 2 \\[0.3em] 2 & 1 & 2 \\[0.3em] 2 & 2 & 1 \end{bmatrix}$ what is $ A^2-4A-5I=0$

asked Dec 29, 2012 by thanvigandhi_1

1 answer

A manufacturer can sell x items at a price of Rs. $ \bigg( 5 - \large \frac{x}{100} \bigg) $ each. The cost price of x items is Rs. $ \bigg( \large\frac{x}{5}$$ + 500 \bigg) $. Find the number of items he should sell to earn the maximum profit.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Solve for x : $ 2\tan^{-1} ( \cos x ) = \tan^{-1} (2 cosec x) $

asked Dec 29, 2012 by thanvigandhi_1

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Evaluate : $ \int \mathrm \: x \sin^{-1}x \: dx $

asked Dec 29, 2012 by thanvigandhi_1

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If $ (\cos x)^y = (\sin y)^x $, find $ \large\frac{dy}{dx} $

asked Dec 29, 2012 by thanvigandhi_1

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Find the intervals in which the function f given by $ f(x) = \sin x + \cos x, 0 \leq \: x \leq \: 2\pi, $is strictly increasing or strictly decreasing.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Prove that the curves $ y^2 = 4x$ and $ x^2 = 4y$ divide the area of the square bounded by $ x = 0, x = 4, y = 4$ and $y = 0$ into three equal parts.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

An open box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

asked Dec 29, 2012 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end {bmatrix} $, find $ A^{-1} $ . Using $ A^{-1} $ , solve the system of equations : $ 2x – 3y + 5z = 11 ,3x + 2y – 4z = –5 ,x + y – 2x = –3. $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

If a, b, c are the lengths of the sides respectively to the angles A, B and C of a $ \Delta$ABC, using vectors, prove that $ \cos C = \large\frac{a^2 + b^2 - c^2}{2ab} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the planes $ x + 2y + 3z = 5\: and\: 3x + 3y + z = 0$

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find the equation of the curve passing through the origin and satisfying the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy = 4x^2 $

asked Dec 29, 2012 by thanvigandhi_1

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Find \[ \int \mathrm \: \frac{x^2+1}{x^2-5x+6} dx \]

asked Dec 29, 2012 by thanvigandhi_1

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

asked Dec 29, 2012 by thanvigandhi_1

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Using elementary transformation, find the inverse of $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix} $

asked Dec 29, 2012 by thanvigandhi_1

1 answer

Evaluate :$ \int_1^4\: ( | x-1|+ | x-2 | + | x=3 | ) dx $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Let $ f : N \rightarrow N$ be defined by : $ f(n) = \left\{ \begin{array}{l l} \large \frac{n+1}{2} & \quad \text{if $n$ is odd}\\ \large \frac{n}{2} & \quad \text{ if $n$ is even} \end{array} for \: all\: n\: \in \: N. \right.$

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^1 \: \sin^{-1} (x \sqrt{1-x} - \sqrt x \sqrt {1-x^2} ) dx, 0 \leq \: x \leq \: 1. $

asked Dec 29, 2012 by thanvigandhi_1

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Using elementary transformations, find the inverse of the matrix : $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

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

asked Dec 29, 2012 by thanvigandhi_1

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For the curve $ y = 4x^3 - 2x^5 $, find all the points at which the tangent passes through origin.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

If $ x = a ( cos\: t + t\: sin \: t)$ and $ y = b ( sin\: t - t\: cos\: t )$, find $\large \frac{d^2y}{dx^2} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Solve for $ x : \sin^{-1} (1-x)-2 \sin^{-1}\: x = \large\frac{\pi}{2} $

asked Dec 29, 2012 by thanvigandhi_1

1 answer

Evaluate :$ \int \: \sqrt {\large\frac{1- \sqrt x}{1+ \sqrt x}}$$ dx $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

A helicopter is flying along the curve $ y = x^2 + 2$. A soldier is placed at the point (3, 2) . Find the nearest distance between the soldier and the helicopter.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Two equal sides of an isosceles triangle with fixed base 'a' are decreasing at the rate of 9 cm/second. How fast is the area of the triangle decreasing when the two sides are equal to 'a'.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Let f, g : R $ \rightarrow $ R be defined as $ f(x) = |x| \: and \: g(x) = [x]$ where [x] denotes the greatest integer less than or equal to x, find the fog $ \bigg(\large \frac{5}{2} \bigg)$ and gof $ (-\sqrt 2) $.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

A manufacturer can sell x items at a price of Rs. $ \bigg( 5-\large\frac{x}{100} \bigg) $ each. The cost price of x items is Rs. $ \bigg( \large\frac{x}{5} - \normalsize 500 \bigg) $. Find the number of items he should sell to earn maximum profit.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations : $ \begin{array} xx+y+z=6 & x+2z=7 & 3x+y+z = 12 \end{array} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of traingle is maximum when the angle between them is $ \large\frac{\pi}{3} $.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

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

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Show that the general solution of the differential equation $\large \frac{dy}{dx}+\frac{y^2+y+1}{x^2+x+1}\normalsize=0$ is given by $ x+y+1 = A(1-x-y-2xy)$, where A is parameter.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find the equation of plane determined by the points A(3, -1, 2), B(5, 2, 4) and C(-1, -1, 6). Also find the distance of the point P(6, 5, 9) from the plane.

asked Dec 29, 2012 by thanvigandhi_1

1 answer

A merchant plans to sell two types of personal computer - a desk top model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get a maximum profit if he does not want to invest more than 70 lakh and if his profit on the desktop model is Rs. 4,500 and the portable is Rs. 5,000.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Assume that the chance of a patient having a heart attack 40%. It is also assumed that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of a certain drug reduces its chance by 25%. At a time a patient can choose any one of the two option with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Write the number of all one-one functions from the set A = { a, b, c} to itself.

asked Dec 29, 2012 by thanvigandhi_1

1 answer

Find $ x \: if \;\tan^{-1} 4 + \cot^{-1} x =\large \frac{\pi}{2} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

What is the value of $ | 3\: I_3 | $ where $ I_3 $ is the identity matrix of order 3?

asked Dec 29, 2012 by thanvigandhi_1

0 answers

For what value of $ k $ , the matrix $ \begin{bmatrix} 2-k & 3 \\ -5 & 1 \end{bmatrix} $ is not invertible?

asked Dec 29, 2012 by thanvigandhi_1

0 answers

If A is a matrix of order 2 x 3 and B is a matrix of order 3 x 5, what is the order of matrix (AB)' or T?

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Write a value of $ \int \: \large\frac{dx}{\sqrt {4-x^2}} $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find f(x) satisfying the following : $ \int \: e^x ( sec^2 x + tan \: x ) dx = e^x\: f (x) +C $

asked Dec 29, 2012 by thanvigandhi_1

0 answers

In a triangle ABC, the sides AB and BC are represented by vectors $ 2\hat i - \hat j + 2\hat k, \hat i + 3\hat j + 5\hat k$ respectively. Find the vector representing CA.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

Find the value of $ \lambda $ for which the vector $ \overrightarrow a = 3\hat i + \hat j - 2\hat k $ and $ \overrightarrow b = \hat i + \lambda\hat j - 3\hat k $ are perpendicular to each other.

asked Dec 29, 2012 by thanvigandhi_1

0 answers

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