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Recent questions tagged sec-b

x and y are the sides of two squares such that $y=x-x^2$.Find the rate of change of the area of second square with respect to the area of first square.

asked Jan 11, 2013 by sreemathi.v

1 answer

The volume of a cube increases at a constant rate.Prove that the increase in its surface area varies inversely as the length of the side.

asked Jan 11, 2013 by sreemathi.v

1 answer

A swimming pool is to be drained for cleaning .If L represents the number of litres of water in the pool seconds after the pool has been plugged off to drain and $L=200(10-t)^2$.How fast is the water running out at the end of 5 seconds?What is the average rate at which the water flows out during the first 5 seconds?

asked Jan 11, 2013 by sreemathi.v

1 answer

A man, 2m tall, walks at the rate of $1\frac{2}{3}m/s$ towards a street light which is $5\frac{1}{3}m$ above the ground.At what rate is the tip of his shadow moving?At what rate is the length of the shadow changing when he is $3\frac{1}{3}m$ from the base of the light?

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3cm and 3.0005cm,respectively.

asked Jan 11, 2013 by sreemathi.v

1 answer

Two men A and B start with velocities v at the same time from the junction of two roads inclined at $45^\circ$ to each other.If they travel by different roads,find the rate at which they are being separated.

asked Jan 10, 2013 by sreemathi.v

1 answer

A kite is moving horizontally at a height of 151.5 meters If the speed of kite is 10m/s,how fast is the string being let out,when the kite is 250m away from the boy who is flying the kite?The height of boy is 1.5m

asked Jan 10, 2013 by sreemathi.v

1 answer

Show that the relation R in the set \( A = { x \in Z:0 \leq x \leq 12},\) given by (i) R = { a, b : |a-b| is a multiple of 4 } (ii) R = {a, b ; a = b}. is equivalence relation. Find the set of all eleents related to 1 in each case.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that : $ \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}{2} \bigg). $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : \( \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{bmatrix} = (a-b)(b-c)(c-a). \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : \( \begin{bmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{bmatrix} = 2(a+b+c)^3. \)

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find \(\large \frac{dy}{dx} \) of following : $ y=(\sin \: x)^x+\sin^{-1} \sqrt x. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using Mathematical Induction prove that $ \large\frac{d}{dx}(x^n)$$=nx^{n-1}, \forall \: n \: \in \: N$ for all positive integers \( n \).

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by \( f(x)=2x^3-3x^2-36x+7\) is : \[ \begin{array} ((i)\: strictly\: increasing \quad & (ii)\: strictly\: decreasing \end{array} \]

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that the curves \( x=y^2\) and \( xy=k\) cut at right angles if \( 8k^2 = 1. \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{2\sin\: x+3\cos \: x}{3\sin \: x + 4 \cos \: x} $$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \ (x+3) \sqrt {3-4x-x^2} dx. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int\large \frac{1}{5cos\: x-12sin\: x}$$dx$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find \( | \overrightarrow a | \: and \: | \overrightarrow b |, \) if \( ( \overrightarrow a + \overrightarrow b).( \overrightarrow a - \overrightarrow b)=8 \: and \: | \overrightarrow a |=8| \overrightarrow b|.\)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the points (2, 3, 2), (2, -2, -1) and (7, 0, 6).

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines $\large \frac{x}{1}=\large\frac{y}{2}=\large\frac{z}{3} $ and $\large\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

There are two bags I and II. Bag I contains 3 white and 4 black balls and Bag II contains 5 white and 6 black balls. One ball is drawn at random from one of the bags and is found to be white. Find the probability that it was drawn from bag I.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

In each of the follwing cases, state whether the function is one-one, onto or bijective. Justify your answer. $\begin{array}{1 1} (i) \: f: R \rightarrow R\: defined\: by \ f(x)=3-4x \\ (ii)\: f: R \rightarrow R\: defined\: by \ f(x)=1+x^2 .\end{array} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Solve for x : $ \tan^{-1} (2x)+\tan^{-1}(3x)=\large\frac{\pi}{4}. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants show that: \( \begin{bmatrix} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{bmatrix} = (x+y+z)(x-z)^2.. \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If \( a, b, c\) are in A.P., show the following: \( \begin{bmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{bmatrix} = 0. \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg]; $$0 < x <\large \frac{\pi}{2} = \large\frac{1}{2} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Differentiate \( (x^2-5x+8)(x^2+7x+9)\) in three ways mentioned : (i) by using product rule (ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation. Do they all give the same answer?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If \( u, v \: and \ w \) are function of \( x\) , then show that $ \large\frac{d}{dx}(u.v.w) = \frac{du}{dx}.v.w+u.\frac{dv}{dx}.w+u.v.\frac{dw}{dx}$In two ways first by repeated application or product rule, secondly by logatithmic differentiation.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate $\int\large \frac{\sin(x-\alpha)}{\sin(x+\alpha)}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Verify that \( y = e^x.\cos\: x\) is a solution of the differential equation : $\large \frac{d^2y}{dx^2}$$-2\large\frac{dy}{dx}$$+2y=0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Two cards are drawn successively without replacement from a well shuffled deck of cards.Find the mean and standard variation of the random variable X where X is the number of aces.

asked Jan 9, 2013 by sreemathi.v

1 answer

A bag contains (2n+1) coins.It is known that n of these coins have a head on both sides where as the rest of the coins are fair.A coin is picked up at random from the bag and is tossed.If the probability that the toss results in a head is $\Large \frac{31}{42}$,determine the value of n.

asked Jan 9, 2013 by sreemathi.v

1 answer

The probability distribution of a discrete random variable X is given as under:

asked Jan 9, 2013 by sreemathi.v

1 answer

An item is manufactured by three machines A,B and C.Out of the total number of items manufactured during a specified period,50% are manufactured on A,30% on B and 20% on C.2% of the items produced on A and 2% of items produced on B are defective,and 3% of these produced on C are defective.All the items are stored at one godown.One item is drawn at random and is found to be defective.What is the probability that it was manufactured on machine A?

asked Jan 9, 2013 by sreemathi.v

1 answer

By examining the chest X ray,the probability that TB is detected when a person is actually suffering is 0.99.The probability of an healthy person diagnosed to have TB Is 0.001.In a certain city,1 in 1000 people suffers from TB.A person is selected at random and is diagnosed to have TB.What is the probability that he actually has TB?

asked Jan 9, 2013 by sreemathi.v

1 answer

Three bags contain a number of red and white balls as follows:

asked Jan 9, 2013 by sreemathi.v

1 answer

Find the probability distribution of the maximum of two scores obtained when a die is thrown twice .Determine also the mean of the distribution.

asked Jan 9, 2013 by sreemathi.v

1 answer

Solve the following differential equation: $(x^2+3xy+y^2)dx-x^2\: dy = 0. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the points A, B and C with position vectors \( \overrightarrow a = 3\hat i - 4\hat j - 4\hat k, \overrightarrow b = 2\hat i - \hat j + 4\hat k \: and \: \overrightarrow c = \hat i - 3\hat j - 5\hat k,\) respectively, form the vertices of a right angled triangle.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Solve the equation $ 2x^2\large\frac{dy}{dx}$$- 2xy+y^2=0,y(e)=e.$

asked Jan 9, 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 Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

A man is known to speak the truth 3 out of 4 times. He throws a dice and reports that it is a six. Find the probability that is actually a six.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the relation R, defined in the set A of all triangles as \( R = (T_1, T_2):T_2\) is similar to \( T_2\)}, is equivalence relation. Consider three right triangles \( T_1\) , with sides 3, 4, 5; \( T_2\) with sides 5, 12, 13 and \( T_3\) with sides 6, 8, 10. Which triangle among \( T_1, T_2 \: and T_3\) are related?

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \sin \bigg[ \frac{\large \pi}{\large 3}-$$\sin^{-1} \bigg( -\frac{\large 1}{\large 2} \bigg) \bigg]. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using properties of determinants prove that \( \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{bmatrix} = (a-b)(b-c)(c-a)(a+b+c). \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that \( \begin{bmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{bmatrix} = (5x+4)(4-x)^2. \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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