Recent questions tagged medium

Find the equation of the plane passing through the line of intersection of the planes $ 2x+y-z=3 \: and \: 5x-3y+4z+9=0$ and parallel to the line $\large \frac{x-1}{2}=\frac{y-3}{4}=\frac{z-5}{5}. $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^{\large\frac{\pi}{4}} \log(1+\tan x)dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $xdy-(y+2x^2)dx=0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the line of intersection of the planes $\overrightarrow r. (\hat i + \hat j + \hat k) = 1 \: and \: \overrightarrow r. (2\hat i + 3\hat j - \hat k) +4 = 0$ and parallel to x - axis.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Using vector method, find the angle between two diagonals of a cube?

asked Feb 5, 2013 by meena.p

1 answer

Using vector method prove that : $a)\;\sin (A+B)=\sin A\cos B+\cos A\sin B$

asked Feb 5, 2013 by meena.p

1 answer

A binary operation * on the set (0,1,2,3,4,5} is defined as : $ f(x) = \left\{ \begin{array}{l l}a+b, & \quad if { a+b < 6} \\ a+b-6, & \quad if { a+b \geq 6} \end{array} \right. $ Show that zero is the identity for this operation and each element 'a' of the set is invertible with 6-a, being the inverse of 'a'.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

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

asked Feb 5, 2013 by thanvigandhi_1

1 answer

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

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equations : $ xdy-ydx=\sqrt{x^2+y^2}dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $(y+3x^2)\large\frac{dx}{dy}$$=x.$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Find the distance of the point (-1,-5,-10) from the point of intersection of the line $ \overrightarrow r = (2\hat i - \hat j + 2\hat k)+\lambda (3\hat i + 4\hat j + 2\hat k) $ and the plane $ \overrightarrow r = (\hat i - \hat j + \hat k)=5.$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Let $ \bar{a},\bar{b},\;and\;\bar{c}$ be three vectors such that $|\bar{a}|=3,|\bar{b}|=4,|\bar{c}|=5$ and each one of them being perpendicular to sum of the other two, find $ |\bar{a}+\bar{b}+\bar{c}|$

asked Feb 4, 2013 by meena.p

1 answer

Prove that angle in a semi-circle is a right angle.

asked Feb 4, 2013 by meena.p

1 answer

Prove the triangle inequality $\bigg|\overrightarrow{a}+\overrightarrow{b}\bigg|\leq\bigg|\overrightarrow{a}\bigg|+\bigg|\overrightarrow{b}\bigg|$

asked Feb 4, 2013 by meena.p

1 answer

If $ \overrightarrow {a}+ \overrightarrow {b}+ \overrightarrow {c}=\overrightarrow 0$ show that $ \overrightarrow {a} \times \overrightarrow {b}=\overrightarrow {b} \times \overrightarrow {c}=\overrightarrow {c} \times \overrightarrow {a}$

asked Feb 4, 2013 by meena.p

1 answer

If the function f(x) given by $ f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if { x > 0} \\ 11, & \quad if { x = 1} \\ 5ax-2b, & \quad if { x < 1 } \end{array} \right. $ is continuous at x = 1, find the values of a and b.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove the following : $ \begin{vmatrix} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{vmatrix} = xyz(x-y)(y-z)(z-x) $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Evaluate:\[\int \limits _{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt {\tan x}}\]

asked Feb 4, 2013 by meena.p

1 answer

Evaluate:\[\int\limits_0^{\pi/2} \frac {\sqrt {\tan x}}{1+\sqrt {\tan x}}dx\]

asked Feb 4, 2013 by meena.p

1 answer

Prove the following : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}} \bigg] = \frac{x}{2}, x \in \bigg( 0, \frac{\pi}{4} \bigg). $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Find the value of $ \tan{-1} \bigg(\large \frac{x}{y} \bigg)-\tan^{-1} \bigg(\large \frac{x-y}{x+y} \bigg). $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove that $\begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ca & cb & -c^2 \end{vmatrix} = 4a^2b^2c^2 $

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find the value of 'a' for which the function f defined as $ f(x) = \left\{ \begin{array}{l l}asin\frac{\pi}{2}(x+1), & \quad { x \leq 0} \\ \large \frac{\tan\: x-\sin x}{x^3}, & \quad { x > 0} \end{array} \right. $ is continuous at x = 0.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Evaluate $\int \limits_0^ 1 x(1-x)^n dx$

asked Feb 4, 2013 by meena.p

1 answer

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

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Solve $ \tan^{-1}(x+1)+tan^{-1}(x-1)=tan^{-1}\large\frac{8}{31} $.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Solve for x : $ 2\tan^{-1}(\sin x)=\tan^{-1} (2\sec x),0 < x < \large\frac{\pi}{2} $

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Simplify : $ \cos^{-1} \bigg(\large \frac{3}{5}$$ \cos \theta +\large \frac{4}{5}$$ sin \theta \bigg) $

asked Feb 2, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}$, find $x$ and $y$ such that $ A^2-xA+yI=0. $

asked Feb 2, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} cos\: \alpha & sin\: \alpha \\ -sin\: \alpha & cos\: \alpha \end{bmatrix} $, then show that AA' = I.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Using elementary transformation, find the inverse of the matrix $\begin{bmatrix} 3 & 10 \\ 2 & 7 \end{bmatrix} $

asked Jan 30, 2013 by thanvigandhi_1

1 answer

If $ y=\sqrt{x^2-1} - log \bigg(\Large\frac{1}{x}+\sqrt{1+\frac{1}{x^2}}\bigg),\;\normalsize find\;\Large\frac{dy}{dx}$

asked Jan 28, 2013 by meena.p

1 answer

If $ x=\cos \theta + log \tan \Large\frac{\theta }{2},\normalsize y=\sin \theta \;find\; \Large\frac {d^2y}{dx^2}\normalsize \;at\;\theta =\Large\frac{\pi}{4}$

asked Jan 28, 2013 by meena.p

1 answer

If $ y=|\cos x|+|\sin x|,\;find\; \Large\frac{dy}{dx}\; \normalsize at\;x=\Large\frac{2\pi}{3}$

asked Jan 28, 2013 by meena.p

1 answer

If $ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $, prove that $A^n = \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} $, for $ n \in N.$

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Let $y= \tan^{-1}\bigg[\Large\frac{4x}{1+5x^2}\bigg] \normalsize +\tan^{-1}\bigg[\Large\frac{2+3x}{3-2x}\bigg],$ show that $ \Large\frac{dy}{dy}=\frac{5}{1+25x^2}$

asked Jan 25, 2013 by meena.p

1 answer

Find $\Large\frac{dy}{dx},$ when $y=\sin^{-1}\bigg[x\sqrt{1-x}-\sqrt x \sqrt{1-x^2}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

Differentiate w.r.t. x, $ y=\tan^{-1}\bigg[\Large\frac{a\cos x-b\sin x}{b\cos x+a \sin x}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

Let $\phi(x)\;andh(x)$ be derivable at x = c. Show that necessary and sufficient condition for the function defined as :\[f(x)= \left\{ \begin{array}{1 1} \phi(x), & \quad x \leq c \\ h(x), & \quad x>c \end{array} \right. \]to be derivable at x= c are $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

asked Jan 25, 2013 by meena.p

1 answer

Let $f(x) = \left\{ \begin{array}{l l} \Large\frac{1-sin^3x}{3cos^2x}, & \quad if\; { x < \Large\frac{\pi}{2}}\\ a, & \quad if\; { x = \Large\frac{\pi}{2}} \\ \Large\frac{b(1-sin x)}{(\pi - 2x)^2}, & \quad if \;{ x > \Large\frac{\pi}{2}} \end{array}. \right.$ If f(x) is continuous function at x $ x = \Large\frac{\pi}{2} $, find a and b.

asked Jan 25, 2013 by meena.p

1 answer

Discuss the continuity of the following function at x = 0 $f(x)= \left\{ \begin{array}{1 1} \Large\frac{x^4+x^3+2x^2}{\tan ^{-1}x} & \quad ,\;x\neq0 \\ 10 & \quad ,\;x=0 \end{array} \right. $

asked Jan 25, 2013 by meena.p

1 answer

Let $ f(x)= \left\{ \begin{array}{1 1} |x|\cos \frac{1}{x} & \quad if\;x\neq0 \\ 0 & \quad if\;x=0 \end{array} \right. $ then discuss the continuity of f(x) at x = 0

asked Jan 25, 2013 by meena.p

1 answer

The function $ \Large\frac{log(1+ax)-log(1-bx)}{x}$ is not defined at x = 0. Find the value of f(x) so that f(x) is continuous at x = 0.

asked Jan 25, 2013 by meena.p

1 answer

Show that the function $f(x) = | \sin x+\cos x| $ is continuous at $ x =\Large\pi$

asked Jan 25, 2013 by meena.p

1 answer

Determine the values of a, b and c for which the function :$f(x)= \left\{ \begin{array}{1 1} \Large\frac{\sin (a+1)x+\sin x}{x} & \quad ,\;x<0 \\ c & \quad ,\;x=0 \\ \Large\frac{\sqrt{x+bx^2}-\sqrt x}{b\sqrt{x^3}} & \quad ,\;x>0 \end{array} \right. $ may continuous at x = 0

asked Jan 25, 2013 by meena.p

1 answer

Given that $ f(x)= \left\{ \begin{array}{1 1} \Large\frac{1- \cos 4x}{x^2}, & \quad if\;x<0 \\ a & \quad ,if\;x=0 \\ \Large\frac{\sqrt x}{\sqrt{16+\sqrt{x}}-4}, & \quad if\;x>0 \end{array}. \right. $ If f(x) is continuous at x = 0, find the value of a.

asked Jan 25, 2013 by meena.p

1 answer

Prove that $ 2\tan^{-1} \large\frac{1}{5}$$ + \tan^{-1} \large\frac{1}{8}$$ = \tan^{-1}\large \frac{4}{7} $

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If $ \cos^{-1}\Large\frac{x}{2}+\normalsize\cos^{-1}\large\frac{y}{3}=\theta ,\;\normalsize prove\;that\;9x^2-12xy\cos\theta +4y^2=36\sin^2\theta $

asked Jan 24, 2013 by meena.p

1 answer

Solve for x:$\tan^{-1}(x+1)+\tan^{-1}x+\tan^{-1}(x-1)=\tan^{-1}3.$

asked Jan 24, 2013 by meena.p

1 answer

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