Recent questions tagged sec-b

The number of real solutions of the equation$\sqrt{1+\cos2x}=\sqrt 2\cos^{-1}(\cos x)\;in\begin{bmatrix}\frac{\Large \pi}{2},\pi\end{bmatrix}$

asked Dec 21, 2012 by sreemathi.v

1 answer

If $\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi$,then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals\[(A)\;0\quad(B)\quad 1\quad(C)\quad 6\quad(D)\quad 12\]

asked Dec 21, 2012 by sreemathi.v

1 answer

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When $ x = 8 cm $ and $ y = 6 cm $ , find the rate of a change of (a) the perimeter, (b) the area of the rectangle.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

If $ \sin\: y = x\sin (a+y)$, prove that $ \large\frac{dy}{dx} = \frac{\large \sin^2(a+y)}{\large \sin a}. $

asked Dec 21, 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.$ Find whether the function f is bijective.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

If $\sin^{-1}\frac{2a}{1+a^2} +\cos^{-1}\frac{1-a^2}{1+a^2}=\tan^{-1}\frac{2x}{1-x^2},where\; a,x,\in\;[0,1],$then the value of x is

asked Dec 21, 2012 by sreemathi.v

1 answer

Evaluate : $ \int \mathrm \: \large\frac{dx}{ \sqrt{5-4x-2x^2}} $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

If $ y = \frac{\large sin^{-1}x}{\large \sqrt {1-x^2}} $, show that $ (1 - x^2) \large\frac{d^2y}{dx^2} - 3x \large\frac{dy}{dx} $$- y = 0 $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

On a multiple choice examination with three possible answers ( out of which only one is correct ) for each of the five questions, what is the probability that the candidate would get four or more correct answers just by guessing?

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin {bmatrix} 1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 1 + 3p + 2q \\ 3 & 6 + 3p & 1 + 6p + 3q \end{bmatrix} = 1 $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

The value of $\sin(\tan^{-1}(.75))$ is equal to

asked Dec 20, 2012 by sreemathi.v

1 answer

Solve the following differential equation : \[ x \frac{dy}{dx} = y - x \: tan \bigg(\frac{y}{x} \bigg) \]

asked Dec 20, 2012 by thanvigandhi_1

0 answers

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

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the shortest distance between the following two lines : $ \overrightarrow r = (1+\lambda)\hat i + (2-\lambda)\hat j + (\lambda+1)\hat k ; \overrightarrow r = ( 2\hat i - \hat j - \hat k ) + \mu \: ( 2\hat i + \hat j + 2\hat k ). $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

prove the following : $ \cot^{-1} \bigg(\large \frac{\sqrt {1+\sin x} + \sqrt {1-\sin x}}{\sqrt {1+\sin x} - \sqrt {1-\sin x}} \bigg) = \large\frac{x}{2},$$ x \in \: \bigg(0,\large \frac{\pi}{4} \bigg) $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

The scalar product of the vector $ \hat i + \hat j + \hat k $ with the unit vector along the sum of the vectors $ 2\hat i + 4\hat j - 5\hat k $ and $ \lambda \hat i + 2\hat j + 3\hat k$ is equal to one. Find the value of $ \lambda$

asked Dec 20, 2012 by thanvigandhi_1

0 answers

What is the value of $\tan({\frac{1}{2}}\sin^{-1}{\frac{3}{4}})$?

asked Dec 20, 2012 by sreemathi.v

1 answer

What does $\tan^{-1}{\frac{1}{4}}+\tan^{-1}{\frac{2}{9}}$ reduce to?

asked Dec 20, 2012 by sreemathi.v

1 answer

Show that $\sin^{-1}{\frac{5}{13}}+\cos^{-1}{\frac{3}{5}}=\tan^{-1}{\frac{63}{16}}$

asked Dec 20, 2012 by sreemathi.v

1 answer

Prove that $ \sin^{-1}{\frac{8}{17}}+\sin^{-1}{\frac{3}{5}}=\sin^{-1}{\frac{77}{85}}$

asked Dec 20, 2012 by sreemathi.v

1 answer

Solve the following equation $\cos\big(\tan^{-1}x\big)=\sin\bigg(\cot^{-1}{\frac{3}{4}}\bigg)$

asked Dec 20, 2012 by sreemathi.v

1 answer

Show that $\cos\bigg(2\tan^{-1}{\frac{1}{7}}\bigg)=\sin\bigg(4\tan^{-1}{\frac{1}{3}}\bigg)$

asked Dec 20, 2012 by sreemathi.v

1 answer

If $2\tan^{-1}(\cos x)=\tan^{-1}(2cosecx)$,then what is the value of $x$?

asked Dec 20, 2012 by sreemathi.v

1 answer

Find the value of the expression $\sin \bigg(2\tan^{-1}{\frac{1}{3}}\bigg)+\cos \big(\tan^{-1}2\sqrt 2\big)$

asked Dec 20, 2012 by sreemathi.v

1 answer

Find the real solution of the equation <BR>$\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{2}$

asked Dec 20, 2012 by sreemathi.v

1 answer

(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

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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

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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

Let us define a relation R in R as aRb if $a \geq b$.Then R is

asked Dec 19, 2012 by sreemathi.v

1 answer

If a relation R on the set (1,2,3) be defined by R={(1,2)}, then R is

asked Dec 19, 2012 by sreemathi.v

1 answer

The maximum number of equivalence relations on the set A={1,2,3} are

asked Dec 19, 2012 by sreemathi.v

1 answer

Let * be binary operation defined on R by $ ab=1+ab, a,b \in R.$ Then the operation is

asked Dec 19, 2012 by sreemathi.v

1 answer

Let A={1,2,3....9} and R be the relation in AxA defined by (a,b)R(c,d) if a+d=b+c for (a,b),(c,d) in AxA.Prove that R is an equivalence relation and also obtain the equivalent class[(2,5)].

asked Dec 19, 2012 by sreemathi.v

1 answer

Let $A=[-1,1].$ Then,dicuss whether the following functions defined on $A$ are one-one,onto or bijective:$\; f(x)\;=\;\frac{x}{2} $

asked Dec 18, 2012 by sreemathi.v

1 answer

Let A=R-{3},B=R-{1}.Let $f \;:\;A \rightarrow B$ be defined by $f(x)=\Large{\frac{x-2}{x-3}}\normalsize \forall x \in A$.Then show that f is bijective.

asked Dec 18, 2012 by sreemathi.v

1 answer

Given A={2,3,4},B={2,5,6,7}.Construct an example of each of the following:\begin{array}{1 1}(a)\quad an\;injective\;mapping\;from\;A\;to\;B\\(b)\quad a\; mapping\;from\;A\;to\;B\;which\;is\;not\;injective\\(c)\quad a\;mapping\;from\;B\;to\;A \end{array}

asked Dec 18, 2012 by sreemathi.v

1 answer

Let $R$ be relation defined on the set of natural number $N$ as follows:\[R=\{(x,y):x\;\;\;N,y\;\;\;N,2x+y=41\}.\]Find the domain and range of the relation R.Also verify whether R is reflexive,symmetric and transitive.

asked Dec 18, 2012 by sreemathi.v

1 answer

If A={1,2,3,4},define relations on A which have properties of being:\begin{array}{1 1}(a)\quad reflexive,transitive\;but\;not\;symmetric & \;\\(b)\quad symmetric\; but \;neither\;reflexive\;nor\;transitive & \;\\(c)\quad reflexive,symmetric\;and\;transitive\end{array}

asked Dec 18, 2012 by sreemathi.v

1 answer

Let f : $R \to R$ be the function defined by $f(x)=2x-3$ $\forall x \in R$.Write $f^{-1}.$

asked Dec 18, 2012 by sreemathi.v

1 answer

There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?

asked Dec 18, 2012 by thanvigandhi_1

1 answer

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3\hat i + 5\hat j - 6 \hat k.$

asked Dec 15, 2012 by thanvigandhi_1

1 answer

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