Recent questions tagged sec-c

Find the equation of the plane which contains the line of intersection of the planes $ \overrightarrow r. (\hat i + 2\hat j + 3\hat k ) -4 = 0 \overrightarrow r. (2\hat i + \hat j - \hat k ) + 5 = 0$ and which is perpendicular to the plane $\overrightarrow r. (5\hat i + 3\hat j - 6\hat k ) + 8 = 0 $ .

asked Jan 7, 2013 by thanvigandhi_1

1 answer

Solve the following linear programming problem graphically Minimize $ Z = x-5y+20$. Subject of the constraints : $ \begin{array} xx-y \geq 0 \\ -x+2y \geq 2 \\ x \geq 3 \\ y \leq 4 \\ x,y \geq 0. \end{array} $

asked Jan 7, 2013 by thanvigandhi_1

0 answers

Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black. One ball is transferred from Bag I to II, then a ball is drawn from Bag II, the ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

asked Jan 6, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of the linear equations : $\begin{array} xx - y = 4 \\ 2x + 3y + 4z = 17 \\ y +2z = 7 \end{array} $

asked Jan 4, 2013 by thanvigandhi_1

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Given : $ A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}. B = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix} $ Find AB and use this result in solving the following system of equations : $ \begin{array} xx - y + z = 4 \\ x - 2y - 2z = 9 \\ 2x + y +3z = 1 \end{array} $

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Prove that the height of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius R is $ \large\frac{2R}{\sqrt 3}.$ Also find the maximum volume.

asked Jan 4, 2013 by thanvigandhi_1

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Find the point on the curve $ y^2= 4x$ which is nearest to the point ( 2, -8 ).

asked Jan 4, 2013 by thanvigandhi_1

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Find the area of the region included between the parabola $ 4y = 3x^2$ and the lines $ 3x-2y+12 = 0$.

asked Jan 4, 2013 by thanvigandhi_1

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

asked Jan 4, 2013 by thanvigandhi_1

0 answers

If $ l_1, m_1, n_1 \: and \: l_2, m_2, n_2 $ are direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $ m_1n_2-m_2n_1, n_1l_2-n_2l_1, l_1m_2-l_2m_1.$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

The area of the quadilateral $ABCD$ where $A(0,4,1),B(2,3,-1),C(4,5,0)$ and $D(2,6,2)$ is equal to

asked Jan 4, 2013 by sreemathi.v

1 answer

If $l_1,m_1,n_1,l_2,m_2,n_2,l_3,m_3,n_3$ are direction cosines of three mutually perpendicular lines,prove that the line whose direction cosines are proportional to $l_1+l_2+l_3,m_1+m_2+m_3,n_1+n_2+n_3$ make equal angles with them.

asked Jan 4, 2013 by sreemathi.v

1 answer

Show that the straight lines whose direction cosines are given by $2l+2m-n=0\;and\;mn+nl+lm=0$ are at right angles.

asked Jan 4, 2013 by sreemathi.v

1 answer

$\overrightarrow{AB}=3 \hat i-\hat j+\hat k \;and\; \overrightarrow{CD}=-3\hat i+2 \hat j+4\hat k$ are two vectors. The position vectors of the points $A$ and $C$ are $6\hat i+7\hat j+4\hat k\;and\;-9\hat j+2\hat k,$ respectively.Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{AB}\;and\;\overrightarrow{CD}$ both.

asked Jan 4, 2013 by sreemathi.v

1 answer

Show that the points $ (\hat i-\hat j+3 \hat k)\; and\;3(\hat i+\hat j+\hat k)$ are equidistant from the plane $ \overrightarrow{r}.(5 \hat i+2 \hat j-7 \hat k)+9=0$ and lies on opposite side of it.

asked Jan 4, 2013 by sreemathi.v

1 answer

Find the equation of the plane through the intersection of the planes $ \overrightarrow{r}.\quad(\hat i+3 \hat j)-6=0\; and\;\overrightarrow{r}.(3\hat i-\hat j-4 \hat k)=0,$ whose perpendicular distance from origin is unity.

asked Jan 4, 2013 by sreemathi.v

1 answer

The plane ax+by=0 is rotated about its line of intersection with the plane z=0 through an angle $\alpha$.Prove that the equation of the plane in its new position is $ax+by\pm \bigg(\sqrt {a^2+b^2}\tan \alpha\bigg)z=0$.

asked Jan 4, 2013 by sreemathi.v

1 answer

A toy company manufactures two types of dolls, A and B. Markets tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is almost half of that for dolls of type A. Further the production level of dolls of type A can exceed three times the production of dolls of other type by almost 600 units. If the company makes profit of Rs. 12 and Rs.16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

asked Jan 4, 2013 by thanvigandhi_1

0 answers

An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of an accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04, respectively. One of the insured vehicles meets with an accident. Find the probability that (i) it is a car (ii) it is a truck.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane which is perpendicular to the plane $5x+3y+6z+8=0$ and which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0.$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the shortest distance between the lines given by $\overrightarrow{r}=(8+3\lambda)\hat i-(9+16\lambda)\hat j+(10+7\lambda)\hat k\; and\; \overrightarrow{r}=15\hat i+29\hat j+5\hat k+\mu(3\hat i+8\hat j-5\hat k)$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the equation of the plane through the points $(2,1,-1)$ and $(-1,3,4)$ and perpendicular to the plane $x-2y+4z=10.$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the equations of the line passing through the point $(3,0,1)$ and parallel to the planes $x+2y=0$ and $3y-z=0.$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the length and the foot of perpendicular from the point$\bigg(1,\large\frac{3}{2}$$,2\bigg)$ to the plane 2x-2y+4z+5=0

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the distance of a point(2,4,-1) from the line $\large \frac{x-5}{1}\;=\frac{y-3}{4}\;=\frac{z-6}{-9}$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the foot of perpendicular from the point(2,3,-8)to the line $\Large \frac{4-x}{2}\;=\frac{y}{6}\;=\frac{1-z}{3}$.Also,find the perpendicular distance from the given point to the line.

asked Jan 3, 2013 by sreemathi.v

1 answer

Show that the function $ f : R \rightarrow { x \in R : -1 < x < 1 }$ defined by $ f(x) =\large \frac{x}{1+|x|}$$, x \in r$ is one-one and onto.

asked Jan 3, 2013 by thanvigandhi_1

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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 Jan 3, 2013 by thanvigandhi_1

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A rectangle is inscribed in a semi-circle of radius r with one of its sides on the diameter of the sei-circle. Find the maximum area of the rectangle.

asked Jan 3, 2013 by thanvigandhi_1

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Using integration, find the area of region enclosed between the circles $ x^2+y^2=4 \: and \: (x-2)^2+y^2=4.$

asked Jan 3, 2013 by thanvigandhi_1

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Find the vector equation of the line through the point ( 1, 2, -4 ) and perpendicular to the two lines $\large \frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7} $ and, $ \large\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5} $

asked Jan 3, 2013 by thanvigandhi_1

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Show that the lines : $ \frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \: and \: \frac{x-2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}. $

asked Jan 3, 2013 by thanvigandhi_1

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A company manufacture two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minute each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minute each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit Rs. 5 each for type A and Rs.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profits.

asked Jan 3, 2013 by thanvigandhi_1

0 answers

If a variable line in two adjacent positions has direction cosines l,m,n and $l+\delta l,m+\delta m,n+\delta n$,show that the small angle $\delta\theta$ between the two positions is given by \[\delta\theta^2=\delta l^2+\delta m^2+\delta n^2\]

asked Jan 3, 2013 by sreemathi.v

1 answer

Prove that the line through $A(0,-1,-1)$ and $B(4,5,1)$ intersects the line through $C(3,9,4)$ and $D(-4,4,4)$

asked Jan 3, 2013 by sreemathi.v

1 answer

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