Recent questions tagged 2012

Find the coordinates of the foot of the perpendicular drawn from the point $(0,2,3)$ on the line $\large\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3} $. Also, find length of perpendicular.

asked Feb 10, 2013 by thanvigandhi_1

1 answer

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items. A fan costs Rs. 360 and a sewing machine cost Rs. 240. He can sell a fan at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise the profit? Translate the problem as an LPP and solve it graphically.

asked Feb 10, 2013 by thanvigandhi_1

1 answer

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver.

asked Feb 10, 2013 by thanvigandhi_1

0 answers

Find the principal value of the following \[ tan^{-1} \bigg( -\frac{1}{\sqrt 3} \bigg) \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the principal value of $ tan^{-1} \sqrt 3 - sec^{-1}(-2) $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If $A = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix} \: and \: B = \begin{bmatrix} -2 & -1 & -4 \end{bmatrix} $verify that $ (AB)^T=B^TA^T $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \begin{vmatrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{vmatrix} $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the value of x: $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} x & 3 \\ 2x & 5 \end{bmatrix}$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Differentiate following w.r.t. $ x \cos ( \sqrt 3 x )$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Determine order and degree of $ (y^{'''})^2+(y^{''})^3+(y^{'})^4+y^5=0 $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the projection of $ \overrightarrow a = 2\hat i +3\hat j + 2\hat k\: on \: \overrightarrow b = \hat i + 2\hat i - \hat j + \hat k$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If $ | \overrightarrow a |=13 | \overrightarrow b |=5\: and \: \overrightarrow a.\overrightarrow b=60$ then find $ |\overrightarrow a$ x $ \overrightarrow b|$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the direction cosines of the line passing through the following points (-2,4,-5),(1,2,3)

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that $\bigg( tan \bigg( \frac{\pi}{4}+\frac{1}{2} cos^{-1}\frac{a}{b} \bigg)+tan \bigg(\frac{\pi}{4}-\frac{1}{2} cos^{-1}\frac{a}{b} \bigg) = \frac{2b}{a} \bigg)$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that the relation R on the set A = {1,2,3,4,5} given by R = { (a,b) : | a - b | is even } is an equivalence relation.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that $ \begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c).$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Examine the continuity of the function $ f(x) = \left\{ \begin{array}{l l}\large\frac{| sin\: x |}{x}, & \quad if \;x \; \neq\; 0 \\ 1, & \quad if \;x = 0 \end{array} \right. $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Determine the value of k so that the function $ f(x) = \left\{ \begin{array}{l l}kx^2, & \quad if { x \leq 2 } \\ 3, & \quad if { x > 2 } \end{array} \right. $ is continuous

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If $ y= [ x + \sqrt{x^2+a^2}]^n. $ then prove that : $\large\frac{dy}{dx}=\frac{ny}{\sqrt{x^2+a^2}}$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Differentiate $ \tan^{-1} \{ \large\frac{\sqrt{1+x^2}-1}{x} \}$ w.r.t. $ x$.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int x\: tan^{-1}x\: dx$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_{\Large\frac{\pi}{6}}^{\Large\frac{\pi}{3}}\large\frac{\sin\: x + \cos \: x}{\sqrt{\sin\: 2x}}$$ dx$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_1^4 (x^2-x)dx$ as limit sums.

asked Feb 9, 2013 by thanvigandhi_1

0 answers

Find the angle between the vectors $( \overrightarrow a + \overrightarrow b)\: and \: ( \overrightarrow a - \overrightarrow b)\: if \: \overrightarrow a = (2i-j+3k)\: and\: \overrightarrow b = (3\hat i + \hat j -2\hat k).$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the shortest distance between the following pairs of lines whose vector equation are $ \overrightarrow r = 3\hat i + 8\hat j + 3\hat k + \lambda(3\hat i - \hat j + \hat k)\: and \: \overrightarrow r = -3\hat i - 7\hat j + 6\hat k + \mu(-3\hat i + 2\hat j + 4\hat k) $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red?

asked Feb 9, 2013 by thanvigandhi_1

1 answer

A small firm manufactures items A and B. The total number of items that it can manufacture in a day is at the most 24 items. A takes one hour to make while item B takes only half an hour. The maximum time available per dat is 16 hours. If the profit on one unit of item A be Rs. 300 and that of one unit of item B be Rs. 160, how many of each type of item should be produced to maximize the profit? Solve the problem graphically.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Solve the following systems of equations using matrix method : \[ x-y=3;\: 2x+3y+4x=17; \: y+2z=7\]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

The slope of the tangent of any point on a curve is $ \lambda $ times the slope of the line joining the point of contact to the origin. Formulate the differential equation and hence find the equation of the curve.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $ \frac{2R}{\sqrt 3} $. Find the volume of the largest cylinder inscribed in a sphere of radius R.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that the area of the right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the area of the smaller region bounded by the ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}=$$1 $ and the straight line $\large\frac{x}{a}+\frac{y}{b}$$=1.$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the co-ordinates of the foot of perpendicular drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

An insurance company insured 3000 scooter drivers, 5000 car drivers and 7000 truck drivers. The probabilities of their meeting with an accident respectively are 0.04, 0.05 and 0.15. One of the insured persons meets with an accident. Find the probability that he is a car driver.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Let * ba a binary operation on set Q of rational defined as $ ab=\large\frac{ab}{5}.$ Write the identity for , if any.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Using the principal values, evaluate the following \[ tan^{-1}\frac{1}{\sqrt{3}}+sin^{-1} \bigg( -\frac{1}{2} \bigg) \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} $, find A + A', where A' is the transpose of matrix A.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If $\begin{bmatrix} 15 & x+y \\ 2 & y \end{bmatrix} = \begin{bmatrix} 15 & 8 \\ x-y & 3 \end{bmatrix} $find the value of x and y

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ 2\begin{vmatrix} 7 & -2 \\ -10 & 5 \end{vmatrix} $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^{\large\frac{\pi}{2}} \large\frac{\sin\: x-\cos\: x}{1+\sin\: x.\cos\:x}$$dx $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int (cosec^2x-\cot\: x)e^x\: dx $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find $ \overrightarrow a.\overrightarrow b\: if \: \overrightarrow a = 3\hat i - \hat j + 2\hat k\: and \: \overrightarrow b = 2\hat i + 3\hat j + 3\hat k$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Write the direction cosines of a line parallel to the line $ \large\frac{3-x}{3} = \frac{y+2}{-2} = \frac{z+2}{6}.$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If the function $( f : R \rightarrow R )$ is given by $( f(x) =\large \frac{x+3}{2} )$ and $( g : R \rightarrow R)$ is given by $( g(x)=2x-3,)$ find (i) fog and (ii) gof. Is $( f^{-1}=g)$?

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Prove that : \[ 2tan^{-1} \bigg( \frac{1}{2} \bigg) + tan^{-1} \bigg( \frac{1}{7} \bigg) = tan^{-1} \bigg( \frac{31}{17} \bigg) \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Solve the following for x : \[ tan^{-1} \bigg( \frac{x-1}{x-2} \bigg) + tan^{-1} \bigg( \frac{x+1}{x+2} \bigg) = \frac{\pi}{4} \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Using the properties of determinants, prove the following : \[ \begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix} = 2 \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix} \]

asked Feb 9, 2013 by thanvigandhi_1

1 answer

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