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Recent questions tagged 2012

Prove that $\large \frac{9\pi}{8}-\frac{9}{4}$$\sin^{-1}\large\frac{1}{3}=\large\frac{9}{4}$$\sin^{-1}\large\frac{2\sqrt2}{3} $

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Express $ \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} $ as the sum of a symmetic and skew-symmetric matrix.

asked Jan 28, 2013 by thanvigandhi_1

1 answer

If \( A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix},\) show that \( A^2-7A+10I=0.\) Hence , find \( A^{-1} \)

asked Jan 28, 2013 by thanvigandhi_1

1 answer

For what value of k, is the following continuous at x = 0? $ f(x) = \left\{ \begin{array}{l l}\large\frac{\sqrt{1+x}-\sqrt{1-x}}{sin^{-1}x}, & \quad { x \neq 0}\\ k, & \quad { x = 0} \\ \end{array}. \right.$

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Differentiate \( \sin(x^x)+(\sin x)^x,\) with respect to x.

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Find the equations of tangent and normal to curve \( y=x-\sin x\cos x\) at \( x=\large\frac{\pi}{2}.\)

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \sqrt{\large\frac{1-\sqrt x}{1+\sqrt x}}$$dx. $

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int_{-1}^{\Large\frac{3}{2}} |x\sin\: \pi\:$$x|dx$

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Show that $ x\cos \: \bigg(\large\frac{y}{x} \bigg) \large\frac{dy}{dx}=$$y\cos \bigg(\large \frac{y}{x} \bigg) +x$ is homogeneous and solve it.

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Solve the differential equation : $ y-x\large\frac{dy}{dx}=$$a \bigg( y^2+x^2\large\frac{dy}{dx} \bigg) $ where $x=a, y=a.$

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Solve the differential equation: $\large \frac{dy}{dx}=\frac{y}{x}$$+\log x$

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Let \( \overrightarrow a = \hat i + \hat j,\overrightarrow b = \hat i - \hat j \: and \: \overrightarrow c = \hat i + 2\hat j + 3\hat k,\) if \( n \) is unit vector such that \( \overrightarrow a.\overrightarrow n=0 \: and \: \overrightarrow b.\overrightarrow n=0,\) then show that \( \overrightarrow c.\overrightarrow n=3.\)

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Show that the points with position vectors are given by \( 2\hat i - \hat j + \hat k, \hat i - 3\hat j - 5\hat k\) and \( 3\hat i - 4\hat j - 4\hat k\) from a right-angled triangle.

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Find the equation of plane through the points (2,2,1), (9,3,6) and perpendicular to the plane \( 2x+6y+6z=9.\)

asked Jan 28, 2013 by thanvigandhi_1

0 answers

In a village of 50 people, 20 people have their fast on 2nd october. Two persons are selected at random out of them. Write the probability distribution for selected persond who are on fast. Also, find the mean of the distribution.

asked Jan 28, 2013 by thanvigandhi_1

0 answers

For the matrix \( A= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix} \), show that \( A^3-6A^2+5A+11I=0.\) Hence, find \(A^{-1}.\)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

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

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the maximum area of an isosceles triangle inscribed in the ellipse $ \large\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with its vertex at one end of the major axis.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the area of the region bounded by the parabola \( y=x^2+1\) and the lines \( y=x,x=0 \: and \: x=2.\)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Prove that $ \int_0^{\large\frac{\pi}{2}} ( \sqrt{\tan\:x}+ \sqrt{\cot\:x})dx= \sqrt2 \pi $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the equation of plane through the intersection of the plane \( x+y+z=9 \: and \: 2x+3y+4z+5=0 \) and passing through the point (1,1,1).

asked Jan 27, 2013 by thanvigandhi_1

0 answers

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 line of shortest distance.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If a young man rides his motorcycle at 25 km/hour, he had to spend Rs.2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs.5 per km. He has Rs.100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as L.P.P. and solve it graphically.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Assume that the chances of a patient having a heart attack is 40%. It is assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time, a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

asked Jan 27, 2013 by thanvigandhi_1

1 answer

If \( f(x) = \cos \: x \: and \: g(x) = x^3,\) find \( fog \:and \:gof. \)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If \( sin \bigg( \sin^{-1} \frac{1}{5} + \cos^{-1} x \bigg) = 1,\) find x.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If \( A = \begin{bmatrix} 6 & -5 \\ 2 & 3 \end{bmatrix} \), then show that A-A' is skew matrix.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

The point (3,-2),(x,2),(8,8) are collinear, find x using determinants.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int\large\frac{x^2+\cos 4x}{4x^3+3\sin 4x}$$dx $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If \( \int_0^a 3x^2dx=27\), find the value of a.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If \( | \overrightarrow a | = 10, | \overrightarrow b | = 15 \) and the angle between the vector \( \overrightarrow a \: and \: \overrightarrow b \: is \:45^{\circ}, \) then find \( \overrightarrow a.\overrightarrow b\).

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find a vector in the direction of \( -\hat i + \sqrt3 \hat k,\) whose magnitude is 3.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the angle between the lines $\large \frac{x+2}{1}=\frac{y}{2}=\frac{z-1}{0}$ and $\large\frac{x-1}{3}=\frac{y-4}{2}=\frac{z+5}{1} $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the value of x, such that \( A = \begin{bmatrix} x & 5 \\ 4 & 2x \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ 16 & 4 \end{bmatrix} \)

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Show that the R defined on the set A of all triangles as $R = \{(T_1, T_2): T_1$ is similar to $T_2 \}$ is an equivalence relation. Consider three right-angled triangles \( T_1\) with sides 3, 4, 5, \( T_2\) with sides 5, 12, 13 and \( T_3\) with sides 6, 8, 10. Which triangles among \( T_1, T_2 and T_3\) are related?

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Solve for x : $ \tan^{-1}(x+1)+\tan^{-1}(x-1)=\tan^{-1}\large\frac{8}{31} $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If $ \cos^{-1}\large\frac{x}{a}$$+\cos^{-1}\large\frac{y}{b}= \alpha $, prove that $\large \frac{x^2}{a^2}-\frac{2xy}{ab}$$\cos \alpha+\large\frac{y^2}{b^2}$$=\sin^2\alpha. $

asked Jan 27, 2013 by thanvigandhi_1

1 answer

(i)A sum of rupees ten thousand is put into three investments at the rate of 10%, 12% and 15% p.a. The combined income of first and second investment is Rs. 620 more of the income of third investment. If the combined income is Rs. 2420, then find the investment in each using matrices.(ii)"One can buy many things with very little savings".Explain the importance of savings in student's life.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If \( f(x) = \left\{ \begin{array}{l l}x^2, & \quad if\; { x \leq 1}\\ ax+b, & \quad if\; { x > 1} \\ \end{array}. \right.\) is differentiable at x = 1. Find the value of a and b.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Verify Lagrange's mean value theorem for the following function : \( f(x) = \sin \: x - \sin\: 2x \: in \: [0, \pi ] \)

asked Jan 27, 2013 by thanvigandhi_1

1 answer

If \( x\sqrt{1+y} +y\sqrt{1+x}=0, \: find \:\large \frac{dy}{dx} \)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

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

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Prove that : $\int_{-a}^a \sqrt{\large\frac{a-x}{a+x}}dx=a\pi$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Show that the curve \( 4x=y^2\: and \: 4xy=k\) cut at right angles, if \( k^2=512\)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation : $ \large\frac{dy}{dx}$$+2y=xe^{4x} $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the differential equation of all circles which pass through origin and whose centre lies on y - axis.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Solve the differential equation \( (x+y)dy+(x-y)dx=0\)

asked Jan 27, 2013 by thanvigandhi_1

1 answer

Find a vector of magnitude 15 which is perpendicular to both the vectors \( 4\hat i - \hat j + 8\hat k \) and \( - \hat j + \hat k\)

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the equation of the line through A(5,-3,2) and through the intersection $\large \frac{x-2}{1} = \frac{y-3}{5} = \frac{z-4}{4}$ and $ \large\frac{x-4}{3} = \frac{y-2}{4} = \frac{z+3}{-3} $

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Arjun throw a die again and again until three fours are obtained. Find the probability of obtaining the third four in the fourth throw of a die.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

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