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Recent questions tagged sec-c

Obtain the inverse of the following matrix, using elementary operations $ A = \begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix} $

asked Feb 6, 2013 by thanvigandhi_1

1 answer

A given quantity of metal is to be cast into a solid half cicular cylinder (i.e., with rectangular base and semicircular ends). Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its circular ends is \( \pi : ( \pi + 2). \)

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find all the local maximum values and local minimum values of the function \( f(x)=\sin 2x-x, -\large\frac{\pi}{2} < x < \large\frac{\pi}{2} \)

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Prove that the image of the point (3,-2,1) in the plane \( 3x-y+4z=2\) lies on the plane \( x+y+z+4=0\).

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{x^4dx}{(x-1)(x^2+1)} $

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Two dice are thrown simultaneously. Let X denotes the number of sixes. Find the probability distribution of X. Also find the mean, variance and standard deviation of X using probability distribution table.

asked Feb 6, 2013 by thanvigandhi_1

0 answers

A catering agency has two kitchen to prepare food at two places A and B. From these places 'Mid-day Meal' is to be supplied to three different schools situated at P, Q , R. The monthly requirement of the school are respectively 40,40 and 70 food packets. A packet contains lunch for 1000 students. The capacity of kitchen A and B are 60 and 70 packets per month respectively. The transportation cost per packets from the kitchens to school is given below : How many packets from each kitchen should be transported to school so that the cost of transportation is minimum? Also find the minimum cost.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes : $ x+2y+3z-7=0 \: and \: 2x-3y+4z=0 $

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the vector equation of the plane, passing through the points A(2,2,-1), B(3,4,2) and C(7,0,6). Also, find the cartesian equation of the plane.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

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

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Using elementary transformations, find the inverse of the matrix : $ \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Use product of AB where $A= \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \; and \;B=\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}$ <br> to solve the system of equations : $ \\ x-y+2z=1 \\ 2y-3z=1 \\ 3x-2y+4z=2 $

asked Feb 5, 2013 by thanvigandhi_1

1 answer

A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10 metres. Find the dimensions of the rectangle so as to admit maximum light through the whole opening.

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Using the method of integration, find the area of the region bounded by the lines : $ 2x+y=4 :3x-2y=6;x-3y+5=0$

asked Feb 5, 2013 by thanvigandhi_1

0 answers

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

asked Feb 5, 2013 by thanvigandhi_1

0 answers

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

0 answers

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

0 answers

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

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

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

If a, b & c are the lengths of the sides of a triangle, using vector method, show that its area is $\sqrt {s(s-a)(s-b)(s-c)}$ where $2s=a+b+c$

asked Feb 5, 2013 by meena.p

1 answer

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

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

Newtons law of cooling states that the rate of change of the temperature T of an object is proportional to the difference between T and the (constant) temperature t of the surrounding medium. We can write it as \[\frac{dT}{dt}=-K(T-t),K>0\; constant.\]A cup of coffee is served at $ 185^\circ F$ in a room where the temperature is $65^\circ F .$ Two minutes later the temperature of the coffee has dropped to $155^\circ F.$(log 3/4=0.144, log3 = 1.09872). Find the time required for coffee to have $105^\circ F$ temperature.

asked Feb 4, 2013 by meena.p

0 answers

When the interest is compounded continuously, the amount of money invested increases at a rate proportional to its size. If Rs.1000 is invested at 10% compounded continuously, in how many years will the original investment double itself?

asked Feb 4, 2013 by meena.p

0 answers

Solve: \[\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}\] (Hint : take $ x=r\cos \theta,y=r \sin \theta,$ so that $x^2+y^2=r^2)$

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy},\;where\; y(1)=1\]

asked Feb 4, 2013 by meena.p

0 answers

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Show that of all the rectangles of given area, the square has the smallest perimeter.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using matrix method, solve the following system of equations : $\large\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4,\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1, \frac{6}{x}+\frac{9}{y}-\frac{20}{z}$$=2; x,y,z \neq 0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Using elementary transformations, find the inverse of the matrix $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^{\large\frac{\pi}{2}} 2\sin x\cos x\tan^{-1}(\sin x)dx $

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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 Feb 4, 2013 by thanvigandhi_1

0 answers

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time. If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the number of tennis rackets and crickets bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve it graphically.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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