Recent questions tagged 2012

There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

asked Jan 3, 2013 by thanvigandhi_1

1 answer

Find $ \mu$ variance for the following distribution :

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Give example of relations which are : \[ \begin{array} (i)\: Symmetric \: but \: neither \: reflexive \: nor\: transitive \\ (ii) \: Transitive\: but\: neither\: reflexive\: nor\: symmetric \\ (iii)\: Reflexive \: and\: symmetric\: but\: not \: transitive \\ (iv)\: Reflexive \: and transitive\: but\: not\: symmetric \\ (v)\: Symmetric\: and\: transitive\: but \: not\: reflexive \end{array}\]

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Let $ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $ , show that $ (aI + bA)^n = a^nI+na^{n-1}bA, $ where I is the identity matrix of order zero $ n \in N. $ If $ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} $ prove that : $ \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}, n \in N. $

asked Jan 3, 2013 by thanvigandhi_1

1 answer

If A and B are square matrices of the same order such that AB = BA, then proved by induction that $ AB^n = ^nA.$ Further, prove that $ (AB)^n = A^nB^n$ for all $ n \in N.$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Show that the rectangle of maximum, area that can be inscribed in a circle is a square.

asked Jan 2, 2013 by thanvigandhi_1

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Find the equation of the tangent to the curve $ y = \sqrt {3x-2} $ which is parallel to the line $ 4x-2y+5 = 0$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using integration, find the area lying above x - axis included, between the circle $ x^2+y^2 = 8x$ and the parabola $y^2 = 4x$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the distance of the point ( -2, 3, -4 ) from the line $\large \frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}$, measured parallel to the plane $ 4x+12y-3z+1=0$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the co-ordinates of the image of the point ( 1, 3, 4 ) in the plane $ 2x-y+z+3=0$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60/kg and food Q costs Rs. 80/kg. Food P contains 3 units/kg of vitamin A and 5 units of vitamin B, while food Q contains 4 units/kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Show that the relation R in the set R of real numbers, defined as $ R = {( a, b ) : a \leq b^2}$, is neither reflexive, nor symmetric , nor transitive.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the value of $ cosec^{-1} (2) $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Find the value of y if $\begin{bmatrix} x-y & 2 \\ x & 5 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

In the matrix $A=\begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt 3 & 1 & -5 & 17 \end{bmatrix} $Write (i) The order of the matrix .

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Write the value of following determinants : $ \begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & 2a-b & b-c \end{vmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \: \large\frac{cos\sqrt x}{\sqrt x}$$dx $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Show that the vectors : $ \overrightarrow a = -4\hat i - 6\hat j - 2\hat k, \overrightarrow b = -\hat i + 4\hat j + 3\hat k \: and \: \overrightarrow c = -8\hat i - \hat j + 3\hat k $ are coplanar.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the sum of the vectors : $ \overrightarrow a = \hat i - 2\hat j + \hat k, \overrightarrow b = -2\hat i + 4\hat j + 5\hat k \: and \: \overrightarrow c = \hat i - 6\hat j - 7\hat k $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the point ( 1, -1, 2 ) having ( 2, 3, 2 ) as direction ratios of normal to the plane?

asked Jan 2, 2013 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} \forall \: n \: \in\: N. \right.$ Find whether the function f is bijective.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Prove that : $ tan^{-1}\bigg(\frac{1}{3} \bigg)+tan^{-1}\bigg(\frac{1}{5}\bigg)+tan^{-1}\bigg(\frac{1}{7}\bigg)+tan^{-1}\bigg(\frac{1}{8}\bigg)=\large\frac{\pi}{4} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, show that $\begin{bmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{bmatrix} = (a + b + c )^3 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, show that $ \begin{bmatrix} b+c & a & b \\ c+a & c & a \\ a+b & b & c \end{bmatrix} = (a + b + c )(a-c)^2 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

If f(x) , defined by the following, is continuous at x = 0 , find the value of a, b and c. $f(x)=\left \{\begin{array}{1 1}\large\frac{\sin(a+1)x+\sin x}{x}, & if\;x< 0\\c, & if\;x= 0\\ \large\frac{\sqrt{x+bx^2}-\sqrt x}{bx^{\large\frac{3}{2}}}, & if\;x>0\end{array}\right.$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

If $ x = a \bigg( \cos \theta +\ log \tan\large \frac{\theta}{2} \bigg)$ and $ y = a\sin\: \theta$, find the value of $ \large\frac{dy}{dx}$ at $ \theta = \large\frac{\pi}{4} $.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the function $ f(x) = \large\frac{4x^2+1}{x} $ is, (i) increasing, (ii) decreasing.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the equation of the tangent and the normal to the curve $ y = x^2+4x+1 $ at the point whose x - coordinate is 3.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Evaluate : \[ (i) \int \: (x-3) \sqrt {x^2+4x+3} dx \] \[ (ii) \int \: (3x+5) \sqrt {2x^2+3x+7} dx \] \[ (iii) \int \: (x-4) \sqrt {4+x-x^2} dx \] \[ (iv) \int \: (5x-1) \sqrt {6+5x-2x^2} dx \]

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \: \large\frac{1}{\cos(x+a)\cos(x+b)}$$dx $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

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

asked Jan 2, 2013 by thanvigandhi_1

0 answers

For any three vectors : $ \overrightarrow a, \overrightarrow b \: and \: \overrightarrow c $ prove that $ \bigg[ \overrightarrow a + \overrightarrow b \: \: \overrightarrow b + \overrightarrow c \: \overrightarrow c + \overrightarrow a \bigg] = 2 \bigg[ \overrightarrow a \: \overrightarrow b \: \overrightarrow c \bigg]. $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the vector and cartesian equation of the sphere described on the join of the points (2, -3, 4 ) and (-5, 6, -7 ) as the extremities of a diameter.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the root of the perpendicular drawn from the point P(1, 6, 3) on the line $\large \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}. $ Also, find its distance from P.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

12 cards, numbered 1 to 12, are placed in a box, mixed up throughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations : $ 2x-3y+5y = 11 ; 3x+2y-4z = -5 ; x+y-2z = -3 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using elementary transformations, find the inverse of the following matrix : $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5 \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is $ 8m^3$ . If building of tank cost Rs. 70 per sq. metre for the base and Rs. 45 per sq. metre for sides, what is the cost of least expensive tank?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using integration, find the area of the triangular region whose sides have the equation $ y = 2x+1, y = 3x+1\: and \: x = 4 $.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Prove that $ x^2-y^2 = c ( x^2-y^2 )^2 $ is the general solution of differential equation $( x^3-3xy^2) dx = (y^3-3x^2y) dx$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

From the point P(1, 2, 4 ) a perpendicular is drawn on the plane $ 2x+y-2z+3 = 0 $. Find the equation of the length and coordinates of the foot of the $ \bot $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

A farmer mixes two brands P and Q of cattle feed. Brand P costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C . The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units, respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Relation R in the set of Z of all integers defined as R = {(x, y) : x-y is an integer }.

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Using principle value, evaluate the following : $ \cos^{-1} \bigg( \cos\large\frac{2\pi}{3} \bigg)$$ + \sin^{-1} \bigg( \sin\large\frac{2\pi}{3} \bigg) .$

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Find the value of x and y if : $2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

If matrix A = [ 1 2 3 ] Write AA' where A' is the transpose of matrix A.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Write the value of the determinant : $\begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

The radius of a circle is increasing uniformly at the rate of 3 cm per second. Find the rate in which the area of the circle is increasing when the radius is 10 cm.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

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