Recent questions tagged 2012

Evaluate : $ \int_0^{\large\frac{\pi}{4}} \large\frac{sin\: x + cos \: x}{9+16 sin 2x}$$dx $

asked Feb 5, 2013 by thanvigandhi_1

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Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes : $ x+2y+3z=5 \: and \: 3x+3y+z=0 $

asked Feb 5, 2013 by thanvigandhi_1

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A factory has two machines A and B. Past record shows that machineA produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Write the angle between two vectors $ \overrightarrow a \: and \: \overrightarrow b$ with magnitudes $ \sqrt 3 $ and 2 respectively having $ \overrightarrow a.\overrightarrow b=\sqrt 6 $.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Prove the following : $ \tan^{-1} \bigg(\large\frac{1}{2} \bigg) + \tan^{-1} \bigg( \large\frac{1}{5} \bigg) + \tan^{-1} \bigg( \frac{1}{8} \bigg)=\large\frac{\pi}{4} $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, solve the following for x : $\begin{vmatrix} x+a & x & x \\ x & x+a & x \\ x & x & x+a \end{vmatrix} = 0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^1 \log \bigg(\large\frac{1}{x}-1\bigg)$$dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $ xdy-(y-x^3)dx=0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

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

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the line of intersection of the planes $ 2x+y-z=3 \: and \: 5x-3y+4z+9=0$ and parallel to the line $\large \frac{x-1}{2}=\frac{y-3}{4}=\frac{z-5}{5}. $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Write the unit vector in the direction of the vector $ \overrightarrow a = 2\hat i + \hat j + 2\hat k $.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Prove the following : $ 2tan^{-1} \bigg(\large \frac{1}{2} \bigg)$$ +\tan^{-1} \bigg( \large\frac{1}{7} \bigg)$$ = \tan^{-1} \bigg( \large\frac{31}{17} \bigg) $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, solve the following for x : $ \begin{vmatrix} a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{vmatrix} = 0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^{\large\frac{\pi}{4}} \log(1+\tan x)dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $xdy-(y+2x^2)dx=0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Using matrices, solve the following system of equations : $ x+2y+z=7, x+3z=11 \: and \: 2x-3y=1 $

asked Feb 5, 2013 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 Feb 5, 2013 by thanvigandhi_1

1 answer

Let A = {1,2,3}, B = {4,5,6,7} and let f = {(1,4), (2,5), (3,6)} be a function from A to B. State whether f is one-one or not.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

What is the principal value of $\cos^{-1} \bigg( \cos\large\frac{2\pi}{3} \bigg) $$+ \sin^{-1} \bigg( \sin\large\frac{2\pi}{3} \bigg) $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $\begin{vmatrix} cos 15^{\circ} & sin 15^{\circ} \\ sin 75^{\circ} & cos 75^{\circ} \end{vmatrix} $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix} $ write $ A^{-1} $ in terms of A.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

If a matrix has 5 elements, write all possible orders it can have.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int (ax+b)^3 dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{dx}{\sqrt{1-x^2} } $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Write the direction - cosines of the line joining the points (1,0,0) and (0,1,1).

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Write the projection of the vector $ \hat i - \hat j $ on the vector $ \hat i + \hat j $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Write the vector equation of the line given by $\large \frac{x-5}{3} = \frac{y+4}{7}=\frac{z-6}{2} $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

A binary operation * on the set (0,1,2,3,4,5} is defined as : $ f(x) = \left\{ \begin{array}{l l}a+b, & \quad if { a+b < 6} \\ a+b-6, & \quad if { a+b \geq 6} \end{array} \right. $ Show that zero is the identity for this operation and each element 'a' of the set is invertible with 6-a, being the inverse of 'a'.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, solve the following for x : $ \begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ x-4 & 2x-9 & 3x-16 \\ x-8 & 2x-27 & 3x-64 \end{vmatrix} = 0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Find the relationship between 'a' and 'b' so that the function 'f' defined by $ f(x) = \left\{ \begin{array}{l l}ax+1, & \quad if { x \leq 3} \\ bx+3, & \quad if { x > 3} \end{array} \right. $ is continuous at x = 3.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

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

asked Feb 5, 2013 by thanvigandhi_1

1 answer

If $ x=\tan\bigg(\large\frac{1}{a}$$\ log \: y \bigg) $ show that $ (1+x^2)\large\frac{d^2y}{dx^2}$$+(2x-a)\large\frac{dy}{dx}=0 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

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

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equations : $ xdy-ydx=\sqrt{x^2+y^2}dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $(y+3x^2)\large\frac{dx}{dy}$$=x.$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Find the shortest distance between the following lines whose vector equations are $ \overrightarrow r = (1-t)\hat i +(t-2)\hat j + (3-2t)\hat k \: and \: \overrightarrow r = (s+1)\hat i +(2s-1)\hat j - (2s+1)\hat k .$

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Using vectors, find the area of the triangle with vertices A(1,1,2), B(2,3,5) and C(1,5,5).

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

asked Feb 5, 2013 by thanvigandhi_1

0 answers

A random variable X has the following probability distribution : Determine : \begin{bmatrix} (i)\: K & (ii) \: P(X<3) & (iii)\: P(X>6) & (iv)\: P(0<X<3) \end{bmatrix}

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations: $ 4x+3y+3z=60, x+2y+3z=45 \: and \: 6x+2y+3z=70 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Show that the right-circular cone of least curved surface and given volume has an altitude equal to $ \sqrt 2 $ times the radius of the base.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_{\Large\frac{\pi}{6}}^{\Large\frac{\pi}{3}}\large \frac{dx}{1+\sqrt{tan \: x}} $

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{6x+7}{\sqrt{(x-5)(x-4)}}$$dx $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Stetch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x - axis and between x = -6 to x = 0.

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Find the distance of the point (-1,-5,-10) from the point of intersection of the line $ \overrightarrow r = (2\hat i - \hat j + 2\hat k)+\lambda (3\hat i + 4\hat j + 2\hat k) $ and the plane $ \overrightarrow r = (\hat i - \hat j + \hat k)=5.$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Given three identical boxes I, II and III each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $ \cos^{-1} \bigg( \cos\large\frac{7\pi}{6} \bigg). $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\int \large\frac{2-3sin \: x}{cos^2x}$$dx $

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin{vmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{vmatrix}=(5x+4)(4-x)^2 $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

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