Recent questions in Model Papers

Evaluate : $\int_0^1 \mathrm \: \large\frac{dx}{1+x^2} $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find a unit vector in the direction of $ \overrightarrow a = 3\hat i - 2\hat j + 6\hat k $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the angle between the vectors : $ \overrightarrow a = \hat i - \hat j + \hat k $ and $ \overrightarrow b = \hat i + \hat j - \hat k $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

For what value of $ \lambda$ are the vectors $ \overrightarrow a = 2\hat i + \lambda\hat j + \hat k $ and $ \overrightarrow b = \hat i - 2\hat j + 3\hat k $ perpendicular to each other?

asked Dec 20, 2012 by thanvigandhi_1

0 answers

(i) Is the binary operation * , defined on set N, given by $ab =\large \frac{a+b}{2}$ for all $ a,b \in\: N$, commutative? \[ \] (ii) Is the above binary operation associative?

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Prove the following : $ \tan^{-1}\large\frac{1}{3}$$ + \tan^{-1}\large\frac{1}{5}$$+ \tan^{-1}\large\frac{1}{7}$$+\tan^{-1}\large\frac{1}{8} = \frac{\pi}{4} $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Let $ A = \begin{bmatrix} 3 & 2 & 5 \\[0.3em] 4 & 1 & 3 \\[0.3em] 0 & 6 & 7 \end{bmatrix}$ Express $A$ as sum of two matrices such that one is symmetric and the other is skew symmetric.

asked Dec 20, 2012 by thanvigandhi_1

1 answer

For what value of $k$ is the following function continuous at $ x=2$? $ f(x) = \left\{ \begin{array}{l l} 2x+1\: \: \: ; & \quad \text{ $x$ < 2}\\ k\: \: \: \: \: \: \: \: \: \:\: \: \: \: ; & \quad \text{ $x$ =2}\\ 3x-1\: \: \: ; & \quad \text{ $x$ >2} \end{array} \right.$

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Differentiate the following with respect to $ x $ : $\tan^-1 \bigg(\large \frac{\sqrt {1+x} - \sqrt {1-x}}{\sqrt {1+x} + \sqrt {1-x}}\bigg) $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the equation of tangent to the curve $ x = \sin 3t, y = \cos 2t, at\: t = \large\frac{\pi}{4} $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^\pi \mathrm \:\large\frac{x\sin x}{1+\cos^2x}$$ dx $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Solve the following differential equation : $ (x^2-y^2) dx +2 \: xy \: dy = 0 $ given that $ y = 1 $ when $ x = 1 $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

solve the following differential equation : $ cos^2x\:\large \frac{dy}{dx} $$+ y = tan \: x $

asked Dec 20, 2012 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 Dec 20, 2012 by thanvigandhi_1

0 answers

Find the shortest distance between the following lines : \[ \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1} \: and \: \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1} \]

asked Dec 19, 2012 by thanvigandhi_1

0 answers

A pair of dice is thrown four times. If getting a doublet is considered a success, find the probability distribution of number of successes.

asked Dec 19, 2012 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $ \begin{bmatrix} \alpha & \beta & \gamma \\[0.3em] \alpha^2 & \beta^2 & \gamma^2 \\[0.3em] \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{bmatrix} = ( \alpha - \beta ) (\beta - \gamma) (\gamma - \alpha) ( \alpha + \beta + \gamma ) $

asked Dec 19, 2012 by thanvigandhi_1

0 answers

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

asked Dec 19, 2012 by thanvigandhi_1

0 answers

Using integration find the area of the region bounded by the parabola $ y^2 = 4x $ and the circle $ 4x^2 + 4y^2 = 9 $.

asked Dec 19, 2012 by thanvigandhi_1

0 answers

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

asked Dec 19, 2012 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the point ( -1, -1, 2 ) and perpendicular to each of the following planes : $2x + 3y - 3z = 2 \: and \: 5x - 4y + z = 6 $

asked Dec 19, 2012 by thanvigandhi_1

0 answers

A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are listed below. He has maximum area of 9000 $ m^2 $ available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output.

asked Dec 19, 2012 by thanvigandhi_1

0 answers

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truckare 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.

asked Dec 19, 2012 by thanvigandhi_1

0 answers

By using the properties of definite integrals,evaluate the integral $\begin{align}\int\limits_0^4\mid x-1\mid dx \end{align}$

asked Dec 3, 2012 by sreemathi.v

1 answer

Ask Questions, Get Answers

Evaluate : $\int_0^1 \mathrm \: \large\frac{dx}{1+x^2} $

Find a unit vector in the direction of \( \overrightarrow a = 3\hat i - 2\hat j + 6\hat k \)

Find the angle between the vectors : \( \overrightarrow a = \hat i - \hat j + \hat k \) and \( \overrightarrow b = \hat i + \hat j - \hat k \)

For what value of \( \lambda\) are the vectors \( \overrightarrow a = 2\hat i + \lambda\hat j + \hat k \) and \( \overrightarrow b = \hat i - 2\hat j + 3\hat k \) perpendicular to each other?

(i) Is the binary operation * , defined on set N, given by \(ab =\large \frac{a+b}{2}\) for all \( a,b \in\: N\), commutative? \[ \] (ii) Is the above binary operation associative?

Prove the following : $ \tan^{-1}\large\frac{1}{3}$$ + \tan^{-1}\large\frac{1}{5}$$+ \tan^{-1}\large\frac{1}{7}$$+\tan^{-1}\large\frac{1}{8} = \frac{\pi}{4} $

Let $ A = \begin{bmatrix} 3 & 2 & 5 \\[0.3em] 4 & 1 & 3 \\[0.3em] 0 & 6 & 7 \end{bmatrix}$ Express $A$ as sum of two matrices such that one is symmetric and the other is skew symmetric.

For what value of \(k\) is the following function continuous at \( x=2\)? $ f(x) = \left\{ \begin{array}{l l} 2x+1\: \: \: ; & \quad \text{ $x$ < 2}\\ k\: \: \: \: \: \: \: \: \: \:\: \: \: \: ; & \quad \text{ $x$ =2}\\ 3x-1\: \: \: ; & \quad \text{ $x$ >2} \end{array} \right.$

Differentiate the following with respect to \( x \) : $\tan^-1 \bigg(\large \frac{\sqrt {1+x} - \sqrt {1-x}}{\sqrt {1+x} + \sqrt {1-x}}\bigg) $

Find the equation of tangent to the curve \( x = \sin 3t, y = \cos 2t, at\: t = \large\frac{\pi}{4} \)

Evaluate : $ \int_0^\pi \mathrm \:\large\frac{x\sin x}{1+\cos^2x}$$ dx $

Solve the following differential equation : $ (x^2-y^2) dx +2 \: xy \: dy = 0 $ given that \( y = 1 \) when \( x = 1 \)

solve the following differential equation : $ cos^2x\:\large \frac{dy}{dx} $$+ y = tan \: x $

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 \( \overrightarrow a . \overrightarrow c = 3\)

Find the shortest distance between the following lines : \[ \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1} \: and \: \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1} \]

A pair of dice is thrown four times. If getting a doublet is considered a success, find the probability distribution of number of successes.

Using properties of determinants, prove the following : $ \begin{bmatrix} \alpha & \beta & \gamma \\[0.3em] \alpha^2 & \beta^2 & \gamma^2 \\[0.3em] \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{bmatrix} = ( \alpha - \beta ) (\beta - \gamma) (\gamma - \alpha) ( \alpha + \beta + \gamma ) $

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

Using integration find the area of the region bounded by the parabola \( y^2 = 4x \) and the circle \( 4x^2 + 4y^2 = 9 \).

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

Find the equation of the plane passing through the point ( -1, -1, 2 ) and perpendicular to each of the following planes : $2x + 3y - 3z = 2 \: and \: 5x - 4y + z = 6 $

By using the properties of definite integrals,evaluate the integral $\begin{align}\int\limits_0^4\mid x-1\mid dx \end{align}$