Recent questions tagged 2012

$ \frac{d}{dx} ( log\: tan\: x)$ is equal to \[ \begin{array} ((A)\: 2\: sec\: 2x \quad & (B)\: 2\: cosec\: 2 x \\ (C)\: sec \: 2 x \quad & (D)\: cosec \: 2x \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The rate of change of the area of a circle with respect to its radius r when r = 6 is \[ \begin{array} ((A) \: 10 π \quad & (B) \: 12 π \\ (C) \: 8 π \quad & (D)\: 11 π \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The interval in which $y = x^2 e^{–x}$ is increasing is \[ \begin{array} ((A)\: (– ∞, ∞) \quad & (B)\: (–2, 0) \\ (C) \:(2, ∞) \quad & (D)\: (0, 2) \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

$ \int\: xsec^2xdx $ is equal to \[ \begin{array} ((A)\: tanx^2+C \quad & (B)\: tanx^2+C \\ (C)\: x\: tanx - log\: sin\: x \quad & (D)\: x\: tanx + log\: cos\: x+C \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

$ \int \: e^x \bigg(\frac{1}{x} - \frac{1}{x^2} \bigg) dx$ is equal to \[ \begin{array} ((A)\: \frac{-e^x}{x^2}+C \quad & (B)\: \frac{e^x}{x^2}+C \\ (C) \: \frac{-e^x}{x^2} \quad & (D) \: \frac{e^x}{x} \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Area bounded by the curve $ y = sinx\: between\: x = 0\: and\: x = 2π$ is \[ \begin{array} (A) 2sq. units \quad & (B) 4 sq. units\\ (C) 8sq. units \quad & (D) 16sq. units \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The general solution of the differential equation $ \frac{dy}{dx} = e^{x+y} $is \[ \begin{array} (A)\: e^x+e^{-y}=C \quad & (B)\: e^x+e^y=C \\ (C)\: e^{-x}+e^y=C \quad & (D)\: e^{-x}+e^{-y}=C \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The integrating factor of the differential equation $ x \frac{dy}{dx} -y = 2x^2 $is \[ \begin{array} (A)\: e^{-x} \quad & (B)\: e^{-y} \\ (C)\: \frac{1}{x} \quad & (D)\: x \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The vector $ 2\hat i + \propto\hat j + \hat k $ is perpendicular to the vector $ 2\hat i - \hat j - \hat k $ if \[ \begin{array} (A) \: \propto = 5 \quad & (B) \: \propto = -5 \\ (C) \: \propto = -3 \quad & (D) \: \propto = 3 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

If $ \overrightarrow u = 2\hat i + 2\hat j - \hat k $ and $ \overrightarrow v = 6\hat i - 3\hat j + 2\hat k $, then a unit vector perpendiculat to both $ \overrightarrow u $ and $ \overrightarrow v $ is \[ \begin{array} (A) \: \hat i - 10\hat j - 18\hat k \quad & (B) \: \frac{1}{\sqrt {17}} \bigg( \frac{1}{5}\hat i - 2\hat j -\frac{18}{5}\hat k \bigg) \\ (C)\: \frac{1}{\sqrt {473}} \bigg( 7\hat i - 10\hat j -18\hat k \bigg) \quad & (D)\: \frac{1}{\sqrt {425}} \bigg( \hat i - 10\hat j -18\hat k \bigg) \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

: If the given lines $ \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} $ and $ \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} $ are perpendicular, then k is \[ \begin{array} (A)\: -10 \quad & (B) \: \frac{10}{7} \\ (C)\: \frac{-10}{7} \quad & (D)\: \frac{-7}{10} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

The angle between the planes $ \overrightarrow r .(3\hat i − 4\hat j +5\hat k) = 0 \: and\: \overrightarrow r .(2\hat i − \hat j − 2\hat k) $is \[ \begin{array} (A)\: \frac{\pi}{3} \quad & (B) \: \frac{\pi}{2} \\ (C)\: \frac{\pi}{6} \quad & (D) \frac{\pi}{4} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

The probability of obtaining an even prime numbr on each die, when a pair of dice is rolled is \[ \begin{array} (A)\: 0 \quad & (B) \: \frac{1}{3} \\ (C)\: \frac{1}{12} \quad & (D) \: \frac{1}{36} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If R is the set of real numbers and f : R$ \rightarrow$R be the function defined by $f (x) = 1 (3 – x^3 )^{\frac{1}{3}}$ , then (fof) (x) is equal to \[ \begin{array} (A) \: x^{\frac{1}{3}} \quad & (B)\: x^3 \\ (C) \: x \quad & (D)\: (3 – x^3) \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Let R be the set of real numbers, f : R$ \rightarrow $R be the function defined by f (x) = 4x + 3, x ∈ R. Show that f is invertible. Find the inverse of f.

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Prove that \[ tan^{-1} \bigg( \frac{\sqrt {1+x}-\sqrt {1-x}}{\sqrt {1+x}+\sqrt {1-x}} \bigg) = \frac{\pi}{4} - \frac{1}{2} cos^{-1}x , \frac{-1}{\sqrt 2} \leq x \: \leq 1 \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If x, y, z are different numbers and \[ A = \begin{bmatrix} x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3 \end{bmatrix} = 0 \] , then show that 1 + x y z = 0

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Is the function f defined by \[ f (x) = \left\{ \begin{array}{l l} x+5, & \quad {if\: x\: \leq 1}\\ x-5 & \quad \text{ if $x$ > 1} \end{array} \right.\] a continuous function? Justify your answer.

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If $x = a (cos\: t + t\: sin \: t), y = a (sin\: t – t \: cos \: t)$. Find $ \frac{d^2y}{dx^2} $

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find \[ \int \mathrm \: ( sin^{-1} x)^2\: dx \]

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Evaluate \[ \int_{-2}^2 \mathrm \: | 1 - x^2 |dx \]

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find the solution of the differential equation $ (x^2 + y^2) dx = 2xy \: dy$

asked Dec 22, 2012 by thanvigandhi_1

0 answers

If the magnitude of the vectors, $ \overline {a} , \overline{b} , \overline{c} $ are 3, 4, 5 respectively and $ \overline{a} $ and $ \overline{b} + \overline{c}, \overline{b} $ and $ ( \overline{c} + \overline{a} ), \overline{c}\: and\: ( \overline{a} + \overline{b} )$ are mutually perpendicular, then find the magnitude of $( \overline{a} + \overline{b} + \overline{c} ) $

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find the equation of the straight line passing through (1, 2, 3) and perpendicular to the plane $ x + 2y – 5z + 9 = 0$

asked Dec 22, 2012 by thanvigandhi_1

0 answers

A man and his wife appear for an interview for two posts. The probability of husband’s selection is $ \frac{1}{7} $ and that of the wife’s selection is $ \frac{1}{5} $ . Find the probability that only one of them will be selected.

asked Dec 22, 2012 by thanvigandhi_1

1 answer

Show that the function f given by $ f (x) = tan^{–1} (sinx + cosx), x > 0$ is always an strictly increasing function in $ \bigg(0,\frac{\pi}{4} \bigg) $

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find the area of the region enclosed between the two circles $x^2+ y^2 = 4$ and $(x – 2)^2 + y^2 = 4$.

asked Dec 22, 2012 by thanvigandhi_1

0 answers

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol, and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X.

asked Dec 21, 2012 by balaji.thirumalai

0 answers

A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by any other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10} $ and $ \frac{2}{5} $ . The probabilities that he will be late are $ \frac{1}{4}, \frac{1}{3} $ and $ \frac{1}{12} $ if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he comes by train.

asked Dec 21, 2012 by thanvigandhi_1

1 answer

Find the distance between the point P (6, 5, 9) and the plane determined by points A (3, –1, 2), B (5, 2, 4) and C (–1, –1, 6).

asked Dec 21, 2012 by thanvigandhi_1

0 answers

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

Sketch the graph of y=| x+3 | and evaluate$\Large \int\limits_{-6}^0\normalsize| x+3 |dx$

asked Dec 17, 2012 by sreemathi.v

1 answer

Find the shortest distance between the lines whose vector equations are $ \overrightarrow r = ( \hat i + 2\hat j + 3\hat k) + \lambda ( \hat i - 3\hat j + 2\hat k )$ and $ \overrightarrow r = 4 \hat i + 5\hat j + 6\hat k + \mu (2\hat i + 3\hat j + \hat k )$

asked Dec 14, 2012 by thanvigandhi_1

1 answer

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

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the shortest distance between the lines $ \overrightarrow r = ( \hat i + 2\hat j + \hat k) + \lambda ( \hat i - \hat j + \hat k )$ and $ \overrightarrow r = 2 \hat i - \hat j - \hat k + \mu (2\hat i + \hat j + 2\hat k )$

asked Dec 14, 2012 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 Dec 14, 2012 by thanvigandhi_1

1 answer

Find the integral $\large \int \frac{\sec^2x}{\large cosec^2x}$$dx$

asked Dec 6, 2012 by sreemathi.v

1 answer

Integrate the function$\frac{(\large log \; x)^2}{\large x}$

asked Dec 6, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{(1+log\: x)^2}{x}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Find the area of the region ${(x, y) : y^2\: \leq \: 4x, 4x^2 + 4y^2\: \leq\: 9}$

asked Dec 4, 2012 by thanvigandhi_1

1 answer

By using the properties of definite integrals, evaluate the integral $\int\limits_0^\frac{\Large \pi}{\Large 2}(2 \log \sin x-\log \sin 2x)\;dx$

asked Dec 3, 2012 by sreemathi.v

1 answer

Evaluate the definite integrals: $\int\limits_1^4 [\; |x-1|+|x-2|+|x-3|\;]dx$

asked Dec 2, 2012 by sreemathi.v

1 answer

Find the particular solution satisfying the given condition $2xy+y^2-2x^2\large\frac{dy}{dx}$$=0;\;y=\;2\;when\;x=\;1$

asked Nov 29, 2012 by sreemathi.v

1 answer

Find the direction cosines of a line which makes equal angles with the coordinate axes.

asked Nov 29, 2012 by balaji.thirumalai

1 answer

prove that \[tan^{-1} \left ( \frac {{\sqrt {1+x}}-{\sqrt {1-x}}}{{\sqrt {1+x}}+{\sqrt {1-x}}} \right ) = \frac {\pi}{4} -\frac{1}{2} cos^{-1}x,-\frac{1}{\sqrt2} \leq x \leq 1 \]

asked Nov 29, 2012 by vaishali.a

1 answer

Find the particular solution of the differential equation $(1+e^{2x})dy + (1+y^2)e^xdx$ = 0, given that $y = 1$ when $x = 0$

asked Nov 29, 2012 by sreemathi.v

1 answer

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6 . What is the probability that he will knock down fewer than 2 hurdles?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

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