Recent questions tagged 2012

If $ \sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y),$ prove that $ \large\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}} $.

asked Jan 23, 2013 by thanvigandhi_1

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Find the intervals in which the functions $ f(x) = sin^4x+cos^4x, 0 \leq x \leq \large\frac{\pi}{2}$ is strictly increasing or decreasing.

asked Jan 23, 2013 by thanvigandhi_1

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Evaluate : $\large\frac{1-\cos x}{\cos x (1+\cos x)}$$dx $

asked Jan 23, 2013 by thanvigandhi_1

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Evaluate : $ \int e^{2x}\sec 2x(1+\tan2x)dx$

asked Jan 23, 2013 by thanvigandhi_1

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A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a number 6. Find the probability that it is actually a number 6.

asked Jan 23, 2013 by thanvigandhi_1

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A toy company manufactures two types of dolls, A and B. Each doll of type B takes twice as long as to produce as one of type A. If the company produces only type A. It can make a maximum of 2000 dolls per day. The supply of plastic is sufficient to produce 1500 dolls per day. Type B requires a fancy dress which cannot be available for more than 600 per day. If the company makes profit of Rs.3 and Rs.5 per doll respectively on doll A and doll B, how many of each should be produced per day in order to maximise the profit?

asked Jan 23, 2013 by thanvigandhi_1

1 answer

Compute the area bounded by the lines $ x+2y=2, y-x+ 1 \: and \: 2x+y=7.$

asked Jan 23, 2013 by thanvigandhi_1

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Using integration, find the area of the triangle ABC. Coordinates of whose vertices are A(2, 0), B(4, 5) and C(6 ; 3).

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^3 (x^2+2x) dx$ as limit of sum.

asked Jan 23, 2013 by thanvigandhi_1

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A variable plane, which remains at a constant distance '3p' from the origin cuts the coordinate axes at A, B and C. Show that locus of the centroid of the triangle is $\large \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = \frac{1}{p^2} .$

asked Jan 23, 2013 by thanvigandhi_1

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(i) A stone is dropped into a quite lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of circular wave is 8cm, how fast is the enclosed area increasing?

asked Jan 23, 2013 by thanvigandhi_1

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If $ A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix} $, find $ A^{-1} $ and hence solve the equation : $ x+2y+z=4, -x+y+z=0 ,x-3y+z = 2$

asked Jan 23, 2013 by thanvigandhi_1

1 answer

Write the following functions in the simplest form $\cos^{-1} (2x^2-1), 0 \leq x \leq 1.$

asked Jan 23, 2013 by thanvigandhi_1

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Find the principal value of $ \cos^{-1} \bigg( -\frac{1}{2} \bigg). $

asked Jan 23, 2013 by thanvigandhi_1

1 answer

Evaluate the determinant : $ \begin{bmatrix} log_ba & 1 \\ 1 & log_ab \end{bmatrix} .$

asked Jan 23, 2013 by thanvigandhi_1

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Find the values of x, y and z from the following equation: $ \begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix} .$

asked Jan 23, 2013 by thanvigandhi_1

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If $ \begin{vmatrix} 3x & 7 \\ 2 & 4 \end{vmatrix} $= 10, find x.

asked Jan 23, 2013 by thanvigandhi_1

1 answer

If $ f(x) = 2^{\sin^2x}, $ then find $\large \frac{dy}{dx} $.

asked Jan 23, 2013 by thanvigandhi_1

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What is the degree of the following differential equation ? $4 \bigg( \frac{dy}{dx} \bigg)^2 - 7y - \bigg( \frac{dy}{dx} \bigg)^3 + \sin x = 0 $

asked Jan 23, 2013 by thanvigandhi_1

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The coordinates of P and Q are (2, -3) and (1, -2) respectively. Find the coordinates of the position vector, equivalent to $ \overrightarrow {PQ} $.

asked Jan 23, 2013 by thanvigandhi_1

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If angle between the two vectors $ \overrightarrow a \: and \: \overrightarrow b$ of same magnitude is $ 60^{\circ} $ and $ \overrightarrow a. \overrightarrow b = 32,$ then find $ |\overrightarrow a | \: and \: | \overrightarrow b |. $

asked Jan 23, 2013 by thanvigandhi_1

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Find the vector equation of the line through the points $ 2\hat i - \hat j + \hat k $ and $ \hat i + 2\hat j - 3\hat k $.

asked Jan 23, 2013 by thanvigandhi_1

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Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset{1, 3, 5, 7} are related to each other and all elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7}} is related to any element of the subset {2, 4, 6}.

asked Jan 23, 2013 by thanvigandhi_1

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Solve for x : \[ tan^{-1} \bigg( \frac{2x}{1-x^2} \bigg) +cot^{-1} \bigg( \frac{1-x^2}{2x} \bigg) = \frac{\pi}{3}, x > 0\]

asked Jan 23, 2013 by thanvigandhi_1

1 answer

If $ \sin \bigg( \sin^{-1} \large\frac{1}{5}$$+\cos^{-1}x \bigg) = 1,$ then find the value of x.

asked Jan 23, 2013 by thanvigandhi_1

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Two firms X and Y decided to award prizes to their employees for three norms Obedience (O), Punctual (P), Quality of work (Q). Firm X decided to award Rs. 40000 for the three values to 2, 4, 7 employees respectively while Y decided to award Rs. 51000 to 4, 2, 8 employees respectively. If all the three prizes together amount to Rs. 8750, then (i) Represent the above situation by a matrix equation and form linear equation using matrix multiplication. (ii) Is it possible to solve the system of linear equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?

asked Jan 23, 2013 by thanvigandhi_1

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Solve the following differential equation : $ e^x \sqrt{1-y^2}dx+\large\frac{y}{x}$$dy=0$

asked Jan 23, 2013 by thanvigandhi_1

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If $ y = e^{a\: cos^{-1}x}, -1 \leq x \leq 1, $ show that : $ (1-x^2)\large\frac{d^2y}{dx^2}$$-x\large \frac{dy}{dx}$$-a^2y=0$

asked Jan 23, 2013 by thanvigandhi_1

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Find the values of a and b, so that the function f given by $ f(x) = \left\{ \begin{array}{l l} 1, & \quad if { x \leq 3}\\ ax+b, & \quad if { 3<x<5} \\ 7, & \quad if {x \geq 5} \end{array} \right. $ is continuous at x = 3 and x = 5.

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{\cos x}{(1+\sin x)(2+\sin \: x)} $$dx$

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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Using properties of definite integrals, evaluate : $ \int_0^1\large \frac{\log \: x}{\sqrt{1-x^2}} $$dx.$

asked Jan 23, 2013 by thanvigandhi_1

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If $ \overrightarrow a = 4\hat i + 5\hat j - \hat k, \overrightarrow b = \hat i - 4\hat j + 5\hat k, \overrightarrow c = 3\hat i + \hat j - \hat k, $ find a vector $ \overrightarrow d$ , which is perpendicular to both $ \overrightarrow a \: and \: \overrightarrow b \: and \: \overrightarrow d.\overrightarrow c = 21 $.

asked Jan 22, 2013 by thanvigandhi_1

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Find the vector equation of the line parallel to the line $ \large\frac{x-1}{5} = \frac{3-y}{2} = \frac{z+1}{4}$ and passing through (3, 0, -4). Also, find the distance between these two lines.

asked Jan 22, 2013 by thanvigandhi_1

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

asked Jan 22, 2013 by thanvigandhi_1

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In a school, 15 N.S.S. volunteer selected for cultural activities of which 3 are extras 3 volunteers are selected at random out of them. Let X denote the number of volunteer as extras. Find the probability distribution of X.

asked Jan 22, 2013 by thanvigandhi_1

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Use product $ \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix} $ to solve the system of equations : $ x-y+2z=1; 2y-3z=1; 3x-2y+4z = 2$

asked Jan 22, 2013 by thanvigandhi_1

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A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of hypotenuse is $ \bigg[ a^{\large\frac{2}{3}} + b^{\large\frac{2}{3}} \bigg]^{\large\frac{3}{2}} $.

asked Jan 22, 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 22, 2013 by thanvigandhi_1

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Find the area of the circle $ 4x^2+4y^2=9$ which is interior to the parabola $ x^2 = 4y$

asked Jan 22, 2013 by thanvigandhi_1

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Find the particular solution of the differential equation $ (x^2+1)\large\frac{dy}{dx}$$-2xy=(x^4+2x^2+1)\cos \: x ; y=0, \: when\: x = 0.$

asked Jan 22, 2013 by thanvigandhi_1

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Find the shortest distance between the lines $\large \frac{x-8}{3} = \frac{y+9}{-16} = \frac{z-10}{7}$$ \: and \: \large\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5} $ Also find equation of the shortest distance

asked Jan 22, 2013 by thanvigandhi_1

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A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B and C. Production of each chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximise profit? Formulate the problem as LPP and solve it graphically.

asked Jan 22, 2013 by thanvigandhi_1

1 answer

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both hearts. Find the probability of the lost card being a heart.

asked Jan 22, 2013 by thanvigandhi_1

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Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

asked Jan 22, 2013 by thanvigandhi_1

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

asked Jan 22, 2013 by thanvigandhi_1

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Evaluate : $ \sin \bigg[ \large\frac{\pi}{3} - $$\sin^{-1} \bigg( -\large\frac{1}{2} \bigg) \bigg] $

asked Jan 22, 2013 by thanvigandhi_1

0 answers

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