Recent questions tagged difficult

A company manufactures two types of sweaters :type A sweaters type B.It costs Rs 360 to make a type A sweater and Rs 120 to make a type B sweater.The company can make at most 300 sweaters and spend at most Rs72,000 a day.The number of sweaters of type B cannot exceed the number of sweaters of type A by more than 100.The company makes a profit of Rs 200 for each sweater of type A and Rs 120 for every sweater of type B.What is the maximum profit (in Rs.)?

asked Jan 7, 2013 by sreemathi.v

1 answer

A company manufactures two types of screws A and B.All the screws have to pass through a threading machine and a slotting machine.A box of type A screws requires 2 minutes on the threading machine and 3minutes on the slotting machine.A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine.In a week,each machine is available for 60 hours.On selling these screws,the company gets a profit of Rs100 per box on type A screws and Rs170 per box on type B screws.Solve the linear programming problem and determine the maximum profit to the manufacturer.

asked Jan 7, 2013 by sreemathi.v

1 answer

A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each.The cost for engaging each large van is Rs 400 and each small van is Rs 200.Not more than Rs 3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans.What will be the minimum cost?

asked Jan 7, 2013 by sreemathi.v

1 answer

A manufacture of electronic circuits has a stock of 200 resistors,120 transistors and 150 capacitors and is required to produce two types of circuits A and B .Type A requires 20 resistors,10 transistors and 10 capacitors .Type B requires 10 resistors,20 transistors and 30 capacitors.If the profit on type A circuit is Rs 50 and that an type B circuit is Rs 60,how many of circuits of type A and type B,should be produced by the manufacturer so as to maximize his profit?Determine the maximum profit.

asked Jan 7, 2013 by sreemathi.v

1 answer

True or False: The vector equation of the line $\Large \frac{x-5}{3}\;=\frac{y+4}{7}\;=\frac{z-6}{2}$ is\[\overrightarrow{r}=(5\hat i-4\hat j+6\hat k)+\lambda(3i\;+7j\;+2k).\]

asked Jan 7, 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

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.

asked Jan 4, 2013 by thanvigandhi_1

1 answer

Find the intervals in which the function f given by $ f(x) =\sin x + \cos x, 0 \leq x \leq 2\pi $ is strictly increasing or strictly decreasing.

asked Jan 4, 2013 by thanvigandhi_1

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

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

Find the angle between the lines whose direction cosines are given by the equations $l+m+n=0,l^2+m^2-n^2=0$

asked Jan 3, 2013 by sreemathi.v

1 answer

Find the equations of the two lines through the origin which intersect the line $\;\large\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\large \frac{\pi}{3}$ each.

asked Jan 3, 2013 by sreemathi.v

1 answer

Prove that the lines $x=py+q;z=ry+s$ and $x=p'y+q';z=r'y+s'$ are perpendicular if $pp'+rr'+1=0.$

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

If $y=y(x)$ satisfies $x\cos y + y\cos x = \pi$, then find $y''(0)$:

asked Jan 3, 2013 by sreemathi.v

1 answer

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

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

If $x=\sin t$ and $y=\sin pt$, prove that $(1-x^2)\large \frac{d^2y}{dx^2}$$-x\large\frac{dy}{dx}+\normalsize p^2y=0$

asked Jan 2, 2013 by sreemathi.v

1 answer

Verify mean value theorem for each of the function $f(x)=\large\frac{1}{4x-1}$ in $[1,4].$

asked Jan 2, 2013 by sreemathi.v

1 answer

If $y=\tan^{-1}x$, find $\large {\frac{d^2y}{dx^2}}$ in terms of y alone.

asked Jan 2, 2013 by sreemathi.v

1 answer

If $x\sin(a+y)+\sin a\cos(a+y)=0,prove\;that\;\large {\frac{dy}{dx}=\frac{\sin^2(a+y)}{\sin a}}$

asked Jan 2, 2013 by sreemathi.v

1 answer

If $x=a\sin 2t(1+\cos 2t)\;and\;y=b\cos 2t(1-\cos 2t),show\;that\bigg(\large \frac{dy}{dx}\bigg)_{at\;t=\frac{\pi}{4}}=\frac{-b}{a}$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find $\large \frac{dy}{dx}$ of the function expressed in parametric form $\sin x=\large\frac{2t}{1+t^2},$$\tan y=\large\frac{2t}{1-t^2}$

asked Dec 31, 2012 by sreemathi.v

1 answer

if $x=-9$ is a root of $\begin{vmatrix}x & 3 & 7\\2 & x & 2\\7 & 6 & x\end{vmatrix}=0$, then other roots are

asked Dec 28, 2012 by sreemathi.v

1 answer

If $x,y, z\in R,$ then the value of determinant $\small\begin{vmatrix}(2^x + 2^{-x})^2 & (2^x - 2^{-x})^2 & 1\\ (3^x + 3^{-x})^2 & (3^x - 3^{-x})^2 & 1\\(4^x + 4^{-x})^2 & (4^x - 4^{-x})^2 & 1\end{vmatrix}$ is

asked Dec 27, 2012 by sreemathi.v

1 answer

If $a+b+c\neq 0\;and\;\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}=0,then\;prove\;that\;a=b=c.$

asked Dec 26, 2012 by sreemathi.v

1 answer

Given $A=\begin{bmatrix}2 & 2 & 4\\4 & 2 & 4\\2 & 1 & 5\end{bmatrix},B=\begin{bmatrix}1 & 1 & 0\\2 & 3 & 4\\0 & 1 & 2\end{bmatrix}$,find BA and use this to solve the system of equations y+2z=7,x-y=3,2x+3y+4z=17

asked Dec 26, 2012 by sreemathi.v

1 answer

show that the $\Delta$ABC is an isosceles triangle if the determinant\[\Delta=\begin{bmatrix}1 & 1& 1\\1+\cos A & 1+\cos B & 1+\cos C\\\cos^2A+\cos A & \cos^2B+\cos B & \cos^2C+\cos C\end{bmatrix}=0\]

asked Dec 26, 2012 by sreemathi.v

1 answer

If $a_1,a_2,a_3,....,a_r$ are in G.P., then prove that the determinant $\begin{vmatrix} a_{r-1} & a_{r-5} & a_{r-9} \\ a_{r-7} & a_{r-11} & a_{r-15} \\ a_{r-13} & a_{r-17} & a_{r-21} \end{vmatrix} is\; independent\; of\; r$

asked Dec 24, 2012 by sreemathi.v

1 answer

Find the value of $\theta$ satisfying $\begin{bmatrix} 1 & 1 & \sin3\theta \\ -4 & 3 & \cos2\theta \\ 7 & -7 & -2 \end{bmatrix}\;=\;0$

asked Dec 24, 2012 by sreemathi.v

1 answer

If $a_1,a_2,a_3,......a_n$ is an arithmetic progression with common difference d,then evaluate the following expression $\tan \bigg( \tan^{-1}\bigg(\frac{d}{1+a_1a_2}\bigg)+\tan^{-1}\bigg(\frac{d}{1+a_2a_3}\bigg)+\tan^{-1}\bigg(\frac{d}{1+a_3a_4}\bigg)+........+\tan^{-1}\bigg(\frac{d}{1+a_{n-1}a_n}\bigg) \bigg) $

asked Dec 20, 2012 by sreemathi.v

1 answer

Find the value of $4\tan^{-1}{\large \frac{1}{5}}$$-\tan^{-1}{\large\frac{1}{239}}$

asked Dec 20, 2012 by sreemathi.v

1 answer

Find the real solution of the equation <BR>$\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{2}$

asked Dec 20, 2012 by sreemathi.v

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 Dec 14, 2012 by thanvigandhi_1

1 answer

For the differential equations, find the particular solution satisfying the given condition: \[ \bigg[ x \: sin^2 \bigg( \frac{x}{y} - y \bigg) \bigg] dx + xdy = 0; y \frac{\large \pi}{\large 4} when \: x = 1\]

asked Dec 12, 2012 by thanvigandhi_1

1 answer

Show that the differential equation is homogeneous and solve :$ \bigg( 1 + e^{\Large\frac{x}{y}} \bigg) dx + e^{\Large\frac{x}{y}} \bigg( 1 -\large \frac{x}{y} \bigg) $$dy= 0 $

asked Dec 12, 2012 by thanvigandhi_1

1 answer

Integrate the function $(x^3-1)^\frac{1}{3}\;x^5$

asked Dec 6, 2012 by sreemathi.v

1 answer