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Recent questions tagged 2012

If \( x\sqrt{1+y}+y\sqrt{1+x}=0\:, find \: \large\frac{dy}{dx}\)

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Show that $ \int_0^{\large\frac{\pi}{2}} \sqrt{\tan\: x}+\sqrt{\cot x}=\sqrt {2\pi} $

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( y=e^{\large \tan\: x},\) prove that $ \cos^2x\large\frac{d^2y}{dx}$$-(1+\sin\: 2x)\large\frac{dy}{dx}$$=0$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( y=(x)^x+(\cos x)^{2x} \: find \: \large\frac{dy}{dx}\)

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate : $\begin{align} \int \limits_{0}^{2}(x^2+3)dx \end{align}$ as limit of sum.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{8\: dx}{(x+2)(x^2+4)}$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( \hat i+ \hat j+\hat k,\: 2\hat i+5\hat j,\: 3\hat i+2\hat j=3\hat k\: and \: \hat i-6\hat j-\hat k\) are the position vectorsof the points A,B,C and D, find the angle between \( \overrightarrow{AB}\: and \: \overrightarrow{CD}\). Deduce that \( \overrightarrow{AB}\: and \: \overrightarrow{CD}\) are collinear.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find the equation of the line passing through the point P(4,6,2) and the point of intersection of the line \( \frac{x-1}{3}=\frac{y}{2}=\frac{z+1}{7} \) and the plane \( x+y-z=8\).

asked Feb 11, 2013 by thanvigandhi_1

0 answers

A and B throw a pair of die turn by turn. The first to throw 9 is awarded a prize. If A starts the game, What is the probability of A getting the prize?

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Find the maximum area of the isosceles triangle inscribed in the ellipse $ \large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$ with its vertex at one end of major axis.

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Show that the semi-vertical angle of the right circular cone of given total surface area and maximum volume is \( sin^{-1} \frac{1}{3} \).

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Using matrices, solve the following system of equation $ x+y+z=3; 2x-y+z=2 ; x-2y+3z=2$

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Using elementary transformations, find the inverse of the following matrix : $ \begin{bmatrix} 2 & -1 & 4 \\ 4 & 0 & 2 \\ 3 & -2 & 7 \end{bmatrix} $

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Find the area of the part of the circle $ x^2+y^2=16$ which is exterior to the parabola $y^2=6x$.

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $\large\frac{dy}{dx}$$+y\cot\: x = x^2\cot x+2x $

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from the point (2,-1,5) to the line \( \large\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11} \).

asked Feb 11, 2013 by thanvigandhi_1

1 answer

An aerolpane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each second class ticket. The airline reserves atleast 20 seats for the first class. However, at least 4 times as many passengers prefer to travel by second class, than by the first class. Determine how many tickets of each type must be sold, inorder to maximise the profit for the airline. What is the maximum profit? Make an L.P.P. and solve it graphically.

asked Feb 11, 2013 by thanvigandhi_1

1 answer

A man is known to speak truth 3 out of 4 times. He throws a die and report that it is a 6. Find the probability that it is actually 6.

asked Feb 11, 2013 by thanvigandhi_1

1 answer

If \( f(x) = x+7\: and \: g(x) = x-7, x \in R, \: find \: (fog) \: (7) \)

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Evaluate ; \[ sin \bigg[ \frac{\pi}{3}-sin^{-1} \bigg( -\frac{1}{2} \bigg) \bigg] \]

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Find the value of x and y if $ 2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} =\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix} $

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Compute $ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Find the cofactor of \( a_{12}\) in the following \[ \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}\]

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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 Feb 11, 2013 by thanvigandhi_1

1 answer

Prove the following : \[ cos^{-1} \bigg( \frac{4}{5} \bigg)+ cos^{-1} \bigg( \frac{12}{13} \bigg)= cos^{-1} \bigg( \frac{33}{65} \bigg) \]

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Solve the following for x \[ tan^{-1}x+2cot^{-1}x=\frac{2\pi}{3} \]

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Using the properties of determinants, prove the following : \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c).\]

asked Feb 11, 2013 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 if { x \: < \: 2 } \\ k, & \quad if { x = 2 } \\ 3x-1, & \quad if { x > 2 } \end{array} \right. $

asked Feb 11, 2013 by thanvigandhi_1

1 answer

If \( y = 3\cos (\log\: x)+4\sin (\log\: x),\) show that $ x^2\large\frac{d^2y}{dx^2}$$+x\large\frac{dy}{dx}$$+y=0$

asked Feb 11, 2013 by thanvigandhi_1

1 answer

If \( y=x^{sin\: x}+(log\:x)^x,\: find\: \large\frac{dy}{dx}.\)

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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 Feb 11, 2013 by thanvigandhi_1

1 answer

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 Feb 11, 2013 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 Feb 11, 2013 by thanvigandhi_1

1 answer

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes.

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Using matrices, solve the following system of equation \[ x+y-z=3\] \[ 2x+3y+z=10 \] \[ 3x-y-7z=1\]

asked Feb 11, 2013 by thanvigandhi_1

1 answer

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

asked Feb 11, 2013 by thanvigandhi_1

1 answer

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height \( h \: is \: \frac{1}{3}h\)

asked Feb 11, 2013 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 Feb 10, 2013 by thanvigandhi_1

1 answer

Evaluate : \[ \int_{-a}^a \sqrt{\frac{a-x}{a+x}}dx \]

asked Feb 10, 2013 by thanvigandhi_1

1 answer

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 Feb 10, 2013 by thanvigandhi_1

1 answer

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