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Recent questions tagged 2012

If the function defined by $ f(x) = \left\{ \begin{array}{l l}2x-1, & \quad if { x < 2 } \\ a, & \quad if { x = 2 } \\ x+1, & \quad if { x > 2 } \end{array} \right. $ is continuous at x = 2, find the value of a. Also discuss the continuity of f(x) at x = 3.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If \( y=(\log\: x)^x+(x)^{\large \cos\:x},\: find\: \large\frac{dy}{dx} \).

asked Feb 9, 2013 by thanvigandhi_1

1 answer

If \( x=a \bigg( \cos\: \theta +\log \tan\large \frac{\theta}{2} \bigg) \) and \( y = a\sin\: \theta,\) find the value of $ \large\frac{d^2y}{dx^2}$$\: at \: \theta = \large\frac{\pi}{4}.$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the intervals in which the following function is (a) increasing (b) decreasing : $ f(x) = x^3-12x^2+36x+17 $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the equation of the tangent to the curve \( x^2+3y=3\), which is parallel to the line $ y-4x+5=0$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{2x+5}{\sqrt{7-6x-x^2}}$$dx$

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Using properties of definite integrals, evaluate the following : $ \int_0^{\pi}\large\frac{x\: sin\: x}{1+cos^2x}$$dx $

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Form the differential equation representing the family of ellipses having foci on x - axis and center at the origin.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Show that the points \( A(-2\hat i + 3\hat j + 5\hat k), B(i+2j+3k) \: and \: C(7\hat i - \hat k) \) are collinear.

asked Feb 9, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane that contains the point (1,-1,2) and is perpendicular to each of the planes \( 2x=3y-2z=5\: and \: x+2y-3z=8.\)

asked Feb 8, 2013 by thanvigandhi_1

0 answers

Three cards are drawn successively with replacement from a well shuffled deck of 52 playing cards. If getting a card of spade is considered a success, find the probability distribution of the number of success.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Using matrices, solve the following system of equation \[ 2x+8y+5z=5\]\[x+y+z=-2\]\[x+2y-z=2\]

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Show that the cylinder of given volume open at the top has minimum total surface area if its height is equal to radius of the base.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : \( \int_1^3 (2x^2+3)dx\) as the limit of sums.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : $\begin{align} \int \frac{tan \: x+tan^3x}{1+tan^3x} \end{align}$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the area of the origin enclosed between the two curves \( (x-6)^2+y^2=36\: and \: x^2+y^2=36\)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Using integration find the area of the region bounded by the triangle whose vertices are (1,3)(2,5) and (3,4).

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the perpendicular distance of the points (2,3,4) from the line \( \large\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}\), Also, find the coordinates of the foot of the perpendicular.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 per day. How many days shall each work, if it is desired to produce at least 60 shirts and 32 pants at a minimum labour cost? Make it as a LPP and solve the problem graphically.

asked Feb 8, 2013 by thanvigandhi_1

0 answers

A bag contains 4 red balls and 4 black balls. Another bag contains 2 red and 6 black. One of the bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If \( f(x)=x+7\: and \: g(x)=x-7, x \in R\) find fog(7).

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find principal value of the expression \( cos^{-1}(cos(-680^{\circ})).\)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix} $ show that $A^{-1} = \frac{1}{19}A.$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If \( \begin{bmatrix} xy & 4 \\ z+6 & x+y \end{bmatrix} = \begin{bmatrix} 8 & w \\ 0 & 6 \end{bmatrix} \), then find values of x,y,z and w.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the value of x if $ \begin{bmatrix} x & 2 \\ 18 & x \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 18 & 6 \end{bmatrix}$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

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

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int_{-\Large\frac{\pi}{2}}^{\Large\frac{\pi}{2}} \sin^7x\: dx$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the value of \( \lambda \) for which the vectors \( \overrightarrow \alpha = 3\hat i + \hat j - 2\hat k \: and \: \overrightarrow b = \hat i + \lambda\hat j - 3\hat k \) are perpendicular to each other.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If a line makes angles \( 90^{\circ}, 135^{\circ}, 45^{\circ}\) with the x, y and z axes respectively, find its direction cosines.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Compute the magnitude of the following vector $\large\frac{1}{\sqrt 3}\hat i + \frac{1}{\sqrt 3}\hat j - \frac{1}{\sqrt 3}\hat k. $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Prove that \( tan^{-1}\sqrt x = \frac{1}{2}cos^{-1} \bigg(\frac{1-x}{1+x} \bigg), x \in [0,1].\)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve the equation \( sin^{-1}6x+sin^{-1}6\sqrt 3 x = -\frac{\pi}{2}. \)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If $ f(x)=\sqrt x (x \geq 0)\: and \: g(x) = x^2-1$ are two real functions, find \( fog\: and \: gof.\: Is\: fog = gof. \)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

prove that : $ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc \bigg( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 \bigg) = (bc+ca+ab+abc).$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Show that the function = 2x - | x | is continuous at x = 0.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

For what value of k, is the following function continuous at x = 0 ? $ f(x) = \left\{ \begin{array}{l l}\large\frac{1-\cos4x}{8x^2}, & \quad if { x \neq 0 } \\ k, & \quad if { x = 0 } \end{array} \right. $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Differentiate the following function w.r.t. \( x.\) $ (sin\: x)^{\large cos^{-1}x} $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If $ y=tan^{-1}\left \{\large\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right \} $,find $\large\frac{dy}{dx}.$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the angle of intersection of the curve $( y^2=x \: and \: x^2=y)$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{dx}{\sqrt{8+2x-x^2}} $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve : $ (x^2+xy)dy=(x^2+y^2)dx $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve : $\large \frac{dy}{dx}+\frac{y}{x}=e^x$

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Solve the differential equation $ x.\cos\bigg(\large\frac{y}{x} \bigg).\frac{dy}{dx}$$=y\cos \bigg( \large\frac{y}{x} \bigg) $$+x $

asked Feb 8, 2013 by thanvigandhi_1

1 answer

If \( \overrightarrow a = \hat i + \hat j + \hat k \: and \: \overrightarrow b = \hat j - \hat k\), then find a vector \( \overrightarrow c\) such that \( \overrightarrow a \) x \( \overrightarrow c = \overrightarrow b\: and \: \overrightarrow a.\overrightarrow c = 3 \)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is \( \frac{5}{6} \), What is the probability that he will knock down fewer than 2 hurdles.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Find the vector equation of the line passing through (1,2,3) and perpendicular to the plane \( \overrightarrow r.(\hat i + 2\hat j - 5\hat k)+9=0\)

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Show that the height of a closed cylinder of given volume and the least surface area is equal to its diameter.

asked Feb 8, 2013 by thanvigandhi_1

0 answers

An open box with a square base is to be made out of a given quantity of cardboard of area \( c^2\) square units. Show that the maximum volume of the box is \(\large\frac{c^3}{6\sqrt 3}\) cubic units.

asked Feb 8, 2013 by thanvigandhi_1

1 answer

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

asked Feb 8, 2013 by thanvigandhi_1

1 answer

Evaluate : \[ \int_1^4 [ |x-1|+|x-2|+|x-3| ]dx \]

asked Feb 8, 2013 by thanvigandhi_1

0 answers

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