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Recent questions tagged 2012

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

The sum of three numbers is -1. If we multiply second number by 2, third number by 3 and add them, we get 5. If we subtract third number from the sum of first and second number, we get -1. Represent the information by the system of equation and find the number using inverse of a matrix.

asked Jan 26, 2013 by thanvigandhi_1

0 answers

A point on the hypotenuse of a right-angled triangle is at distances a and b from the sides. Show that length of the hypotenuse is at least \( (a^{\large\frac{2}{3}}+b^{\large\frac{2}{3}})^{\large\frac{3}{2}} \)

asked Jan 26, 2013 by thanvigandhi_1

0 answers

Show that semi-vertical angle of the right circular cone of given total surface area and maximum volume is \( sin^{-1}\frac{1}{3} \)

asked Jan 26, 2013 by thanvigandhi_1

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Find the area enclosed between the parabolas \( y^2=4ax\: and \: x^2=4ay\)

asked Jan 26, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^{\Large\frac{\pi}{4}} \large\frac{\sin x \: \cos x}{\cos^4x+sin^4x}$$dx $

asked Jan 26, 2013 by thanvigandhi_1

0 answers

From the distance of the point (2,3,4) from the line \( \large\frac{x+3}{3} = \frac{y-2}{6} = \frac{z}{2} \) measured parallel to the plane $3x+2y+2z+5=0.$

asked Jan 26, 2013 by thanvigandhi_1

0 answers

A company makes two kinds of belts, belt A is of pure leather and belt B is macro based. The respective profits are Rs.40 and Rs.30 per belt. Each belt of type A requires twice as much time as a belt of type B and if all the belts were of type B a company could make 1000 belts per day. The supply of leather and macro is sufficient for only 8000 belts per day (both A and B combined). Belt A requires a fancy buckle and only 4000 buckles are available. There are only 7000 buckles per day available for belt B. What should be the daily production of each belt to have maximum profit?

asked Jan 26, 2013 by thanvigandhi_1

1 answer

A card from a pack of 52 playing cards is lost. Two cards are drawn from the remaining cards and found to be both spade. (i) Find the probability of lost card being a spade card.(ii)While playing in ground you found a purse containing rupees and some other documents,what will you do?

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Write the number of one-one function from the set A = {1, 2, 3} to itself.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( \tan^{-1}2-\tan^{-1}1=\tan^{-1}x,\) then find x.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

What is the value of \( |4I_3|,\) where \( I_3\) is the identity matrix of order 3?

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( A = [ a_{ij}],\) where \( a_{ij}= \left\{ \begin{array}{l l}i+j, & \quad if\; { i \geq j}\\ i-j, & \quad if\; { i < j} \\ \end{array}. \right.\) construct a 3 x 2 matrix A.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If A is a matrix of order 3 x 4 and B is matrix of order 5 x 4, what is the order of matrix (AB)'.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{\sec^2x}{5+tan\: x}$$dx$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find f(x) satisfy the following : \( \int e^x (cosec^2x-\cot \: x)dx = e^x\: f(x)+C\)

asked Jan 25, 2013 by thanvigandhi_1

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For what value of k, the vectors \( 2\hat i - \hat j + \hat k\) and \( \hat i+\hat kj - 5\hat k \) are perpendicular to each other.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

For two vectors \( \overrightarrow a \: and \: \overrightarrow b,\) show that \( \overrightarrow a.\overrightarrow b \leq |\overrightarrow a| \: |\overrightarrow b|.\)

asked Jan 25, 2013 by thanvigandhi_1

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Find the value of p, such that the line \( \large\frac{x-3}{12}=\frac{y+1}{p}=\frac{z-3}{17} \) is perpendicular to 4x+y-3z=5.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If\( f : P \rightarrow R\) be function defined by \( f(x)=4x^3-7\), show that function f is a bijective function.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Let $ f(x) = \large\frac{ax}{x+1}$$, x \neq -1, \: if \: (fof) (x) = x,$ find the value of $a$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

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

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Represent the following problem by a system of linear equations. Rs.188000 is the cost of 3 cycles, 1 motorbike and 1 car, the cost of 2 cycles, 2 motorbikes and 1 car is Rs.228000. The cost of 4 cycles 2 cars and 1 motorbike is Rs. 330000. Use matrix to find the cost of each.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If \( x^y=e^{y-x}\), show that \( \large\frac{dy}{dx}=\frac{2-logx}{(1-log\:x)^2.} \)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Show that the function f(x) = |x-1| is continuous but not differentiable at x=1.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Verify Rolle's theorem for the function \( f(x) = e^x\cos \: x\: on \: \bigg[ -\large\frac{\pi}{2}, \frac{\pi}{2} \bigg] \).

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Assuming that petrol burnt per hour in driving a motorboat varies as the cube of its velocity. Show that the most economical speed when going against a current of p km/hour is \( \large\frac{3p}{2}\) km/hour.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_{-1}^{\large\frac{3}{2}} |x\sin\: \pi x |dx$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^{\large\frac{\pi}{2}} \sin2x\tan^{-1}(\sin x)dx $

asked Jan 25, 2013 by thanvigandhi_1

0 answers

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

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation $ (1+x^2)\large\frac{dy}{dx}$$+y =\ tan^{-1}x$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( \overrightarrow a \) x \( \overrightarrow b = \overrightarrow c \) x \( \overrightarrow d\) and \( \overrightarrow a\) x \( \overrightarrow c = \overrightarrow b \) x \( \overrightarrow d\) show that \( \overrightarrow a-\overrightarrow d\) is parallel to \( \overrightarrow b-\overrightarrow c\), where \( \overrightarrow a \neq \overrightarrow d\: and \: \overrightarrow b \neq \overrightarrow c.\)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find the value of \( \lambda \) so that the lines $ \large\frac{1-x}{3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\large\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5} $ are perpendicular.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Three balls are drawn one by one without replacement from a bag containing 5 white and 4 black balls. Find the probability of the number of red balls.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

A one hour routine check was made by security staff of MNC from \(\large \frac{1}{2} \) hour before office opening time. Total 80 cars are checked and result was 9 cars carrierd no official, 20 cars carried one official, 16 cars carried 2 officials, 20 cars carried 3 officials and 15 cars carried 4 officials. (i) Using this information, write down a row matrix and a column whose product gives us the total number of official carried by the 80 cars. (ii) What was the average number of official per car?

asked Jan 25, 2013 by thanvigandhi_1

0 answers

A wire of length 30 cm is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What could be the lengths of two peices, so that the combined area of the square and the circle is minimum?

asked Jan 25, 2013 by thanvigandhi_1

0 answers

An open box with a square base is to be made out of given quantity of sheet of area \( a^2 \). Show that maximum volume of the box is \( \large\frac{a^3}{6\sqrt3}.\)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Evalaute : $ \int \sqrt{\large\frac{sin(x-\alpha)}{sin(x+\alpha)}}$$dx $

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Evalaute : $\int \large\frac{1}{\sin^2x+\sin2x}$$dx. $

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find the area of the circle \( 4x^2+4y^2\) which is interior to the parabola \( x^2=4y \).

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Find the image of (1, 6, 3) in the line \( \large\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}. \)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Akshay throw a die again and again until three fours are obtained. Find the probability of obtaining the third four in the sixth throw of die by Akshay.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

A diet is to contain at least 80 units of minerals. Two foods \( F_1 \: and F_2\)are available. Food \( F_1 \) costs Rs.4 per unit food and \( F_2 \) costs Rs.6 per unit. A unit of food \( F_1 \) contains atleast 3 units of vitamin A and 4 units of minerals. A unit of food \( F_2 \)contains atleast 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods, and also meets the minimal nutritional requirements.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( f : R \rightarrow R\) is defined by \( f(x) = x^2-3x+2\) , find \( f (f(x)).\)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Show that \( sin^{-1} \bigg( 2x \sqrt{1-x^2} \bigg) = 2 sin^{-1}x.\)

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If \( \begin{bmatrix} 2x+5 & 3 \\ 5x+2 & 9 \end{bmatrix} = 0\), find x.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( \begin{bmatrix} x+3 & 4 \\ y-4 & x+y \end{bmatrix} = \begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix} \), find x and y.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int\large \frac{x^2+4x}{x^3+6x^2+5}$$dx$

asked Jan 25, 2013 by thanvigandhi_1

0 answers

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

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( | \overrightarrow a | = 2, |\overrightarrow b | = \sqrt 3\: and \: \overrightarrow a.\overrightarrow b = \sqrt 3.\) Find the angle between \( \overrightarrow a \: and \: \overrightarrow b.\)

asked Jan 25, 2013 by thanvigandhi_1

0 answers

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