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Recent questions tagged sec-b

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

Prove that $\bigg( tan \bigg( \frac{\pi}{4}+\frac{1}{2} cos^{-1}\frac{a}{b} \bigg)+tan \bigg(\frac{\pi}{4}-\frac{1}{2} cos^{-1}\frac{a}{b} \bigg) = \frac{2b}{a} \bigg)$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that the relation R on the set A = {1,2,3,4,5} given by R = { (a,b) : | a - b | is even } is an equivalence relation.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that $ \begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c).$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Examine the continuity of the function $ f(x) = \left\{ \begin{array}{l l}\large\frac{| sin\: x |}{x}, & \quad if \;x \; \neq\; 0 \\ 1, & \quad if \;x = 0 \end{array} \right. $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Determine the value of k so that the function $ f(x) = \left\{ \begin{array}{l l}kx^2, & \quad if { x \leq 2 } \\ 3, & \quad if { x > 2 } \end{array} \right. $ is continuous

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If \( y= [ x + \sqrt{x^2+a^2}]^n. \) then prove that : $\large\frac{dy}{dx}=\frac{ny}{\sqrt{x^2+a^2}}$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Differentiate \( \tan^{-1} \{ \large\frac{\sqrt{1+x^2}-1}{x} \}\) w.r.t. \( x\).

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If \( y = \tan^{-1}x\) show that $(1+x^2)\large\frac{d^2y}{dx^2}$$+2x\large\frac{dy}{dx}$$=0$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_{\Large\frac{\pi}{6}}^{\Large\frac{\pi}{3}}\large\frac{\sin\: x + \cos \: x}{\sqrt{\sin\: 2x}}$$ dx$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : \( \int_1^4 (x^2-x)dx\) as limit sums.

asked Feb 9, 2013 by thanvigandhi_1

0 answers

Find the angle between the vectors \(( \overrightarrow a + \overrightarrow b)\: and \: ( \overrightarrow a - \overrightarrow b)\: if \: \overrightarrow a = (2i-j+3k)\: and\: \overrightarrow b = (3\hat i + \hat j -2\hat k).\)

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the shortest distance between the following pairs of lines whose vector equation are $ \overrightarrow r = 3\hat i + 8\hat j + 3\hat k + \lambda(3\hat i - \hat j + \hat k)\: and \: \overrightarrow r = -3\hat i - 7\hat j + 6\hat k + \mu(-3\hat i + 2\hat j + 4\hat k) $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red?

asked Feb 9, 2013 by thanvigandhi_1

1 answer

The slope of the tangent of any point on a curve is \( \lambda \) times the slope of the line joining the point of contact to the origin. Formulate the differential equation and hence find the equation of the curve.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If the function $( f : R \rightarrow R )$ is given by $( f(x) =\large \frac{x+3}{2} )$ and $( g : R \rightarrow R)$ is given by $( g(x)=2x-3,)$ find (i) fog and (ii) gof. Is $( f^{-1}=g)$?

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that : \[ 2tan^{-1} \bigg( \frac{1}{2} \bigg) + tan^{-1} \bigg( \frac{1}{7} \bigg) = tan^{-1} \bigg( \frac{31}{17} \bigg) \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Solve the following for x : \[ tan^{-1} \bigg( \frac{x-1}{x-2} \bigg) + tan^{-1} \bigg( \frac{x+1}{x+2} \bigg) = \frac{\pi}{4} \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Using the properties of determinants, prove the following : \[ \begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix} = 2 \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix} \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If the function defined by $ f(x) = \left\{ \begin{array}{l l}2x-1, & \quad if { x < 2 } \\ a, & \quad if { x = 2 } \\ x+1, & \quad if { x > 2 } \end{array} \right. $ is continuous at x = 2, find the value of a. Also discuss the continuity of f(x) at x = 3.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the intervals in which the following function is (a) increasing (b) decreasing : $ f(x) = x^3-12x^2+36x+17 $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the equation of the tangent to the curve \( x^2+3y=3\), which is parallel to the line $ y-4x+5=0$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{2x+5}{\sqrt{7-6x-x^2}}$$dx$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Using properties of definite integrals, evaluate the following : $ \int_0^{\pi}\large\frac{x\: sin\: x}{1+cos^2x}$$dx $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Form the differential equation representing the family of ellipses having foci on x - axis and center at the origin.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Show that the points \( A(-2\hat i + 3\hat j + 5\hat k), B(i+2j+3k) \: and \: C(7\hat i - \hat k) \) are collinear.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane that contains the point (1,-1,2) and is perpendicular to each of the planes \( 2x=3y-2z=5\: and \: x+2y-3z=8.\)

asked Feb 8, 2013 by thanvigandhi_1

0 answers

Three cards are drawn successively with replacement from a well shuffled deck of 52 playing cards. If getting a card of spade is considered a success, find the probability distribution of the number of success.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Prove that \( tan^{-1}\sqrt x = \frac{1}{2}cos^{-1} \bigg(\frac{1-x}{1+x} \bigg), x \in [0,1].\)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve the equation \( sin^{-1}6x+sin^{-1}6\sqrt 3 x = -\frac{\pi}{2}. \)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If $ f(x)=\sqrt x (x \geq 0)\: and \: g(x) = x^2-1$ are two real functions, find \( fog\: and \: gof.\: Is\: fog = gof. \)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

prove that : $ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc \bigg( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 \bigg) = (bc+ca+ab+abc).$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Show that the function = 2x - | x | is continuous at x = 0.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

For what value of k, is the following function continuous at x = 0 ? $ f(x) = \left\{ \begin{array}{l l}\large\frac{1-\cos4x}{8x^2}, & \quad if { x \neq 0 } \\ k, & \quad if { x = 0 } \end{array} \right. $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Differentiate the following function w.r.t. \( x.\) $ (sin\: x)^{\large cos^{-1}x} $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If $ y=tan^{-1}\left \{\large\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right \} $,find $\large\frac{dy}{dx}.$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the angle of intersection of the curve $( y^2=x \: and \: x^2=y)$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{dx}{\sqrt{8+2x-x^2}} $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve : $ (x^2+xy)dy=(x^2+y^2)dx $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve : $\large \frac{dy}{dx}+\frac{y}{x}=e^x$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

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