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Recent questions tagged 2012

Write the value of $ \int_{\Large\frac{-\pi}{2}}^{\Large\frac{\pi}{2}} \sin^5 x dx $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Find the position vector of the midpoint of the line segment joining the points \( A ( 5\hat i + 3\hat j )\: and \: B ( 3\hat i - \hat j ) \).

asked Dec 26, 2012 by thanvigandhi_1

0 answers

If \( \overrightarrow a = 2\hat i - \hat j + 3\hat k \: and \: \overrightarrow b = (6\hat i + \lambda\hat j + 9\hat k )\) and \( \overrightarrow a | | \overrightarrow b, \) find the value of \( \lambda \).

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Find the distance of the point (a, b, c) from x - axis.

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Let N be the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d) if ad = bc. Show that R is an equivalence relation.

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Prove that \( tan^{-1} \bigg(\frac{\large 1}{\large 4} \bigg) + tan^{-1} \bigg( \frac{\large 2}{\large 9} \bigg ) = \frac{\large 1}{\large 2} cos^{-1} \bigg( \frac{3}{5} \bigg). \)

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Using properties of determinants, prove that : $ \begin{bmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{bmatrix} = (1+a^2+b^2+c^2) $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

For what values of a and b, the function f defined as : $ f(x) = \left\{ \begin{array}{l l} 3ax+b, & \quad \text{if $x$ < 1}\\ 11, & \quad \text{ if $x$ = 1} \\ 5ax-2b, & \quad \text{if $x$ > 1} \end{array} is \: continuous\: at\: x=1. \right.$

asked Dec 26, 2012 by thanvigandhi_1

0 answers

If \( x^y + y^x = a^b,\) find $\large\frac{dy}{dx} $

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the intervals inwhich the following function is strictly increasing or strictly decreasing \( f(x) = 20 - 9x + 6x^2 - x^3 \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Evaluate :$\int\: \large\frac{\sin x + \cos x}{\sqrt{\sin x - \cos x }} $$dx$

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Form the differeential equation of the family of circles having radii 3.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

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

asked Dec 25, 2012 by thanvigandhi_1

0 answers

If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is \( \sqrt 3\)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find whether the lines \( \overrightarrow r = ( \hat i - \hat j - \hat k ) + \lambda (2\hat i + \hat j ) \: and \: \overrightarrow r = ( 2\hat i - \hat j ) + \mu ( \hat i + \hat j - \hat k )\) intersect or not. If intersecting find their point of intersection.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Three balls are drawn one by one without replacement from a bag containing 5 white and 4 green balls. Find the probability distribution of number of green balls drawn.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

If \( A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix} \), find \( A^{-1} \) and hence solve the following system of equations : $\begin{array} 2x+y+3z=3 \\ 4x-y =3 \\ -7x+2y+z=2 \end{array} $

asked Dec 25, 2012 by thanvigandhi_1

0 answers

If the length of three sides of a trapezium, other tan the base are equal to 10 cm each, then find the area of trapezium when it is maximum.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Draw a rough sketch of the region enclosed between the circles \( x^2 + y^2 = 4 \) and \( (x-2)^2 + y^2 = 1\) . Using integration, find the area of the enclosed region.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Evaluate : \( \int_1^2 \: (x^2 + x + 2 )dx\) as a limit of sums

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the coordinates of the foot of the perpendicular distance of the poit (1, 3, 4) from the plane \( 2x - y + z + 3 = 0 \). Find also, the image of the point in the plane.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

An aerolpane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves atleast 20 seats for the executive class. However, at least 4 times as many passengers prefer to travel by ecenomy class, than by the executive class. Determine how many tickets of each type must be sold, inorder to maximise the profit for the airline. What is the maximum profit? Make an L.P.P. and solve it graphically.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Write the principal value of \( \cos^{-1} \bigg( \cos\large\frac{7\pi}{6} \bigg) \).

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Solve for $x$ and $y$ $\begin{bmatrix} 3 & \quad -4 \\ 1 & \quad 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3 \\ 11 \end{bmatrix} $

asked Dec 25, 2012 by thanvigandhi_1

1 answer

For what value of k , the matrix \( \begin{bmatrix} k & \quad 2 \\ 3 & \quad 4 \end{bmatrix} \) has no inverse.

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Evaluate : \( \begin{bmatrix} \sin10^{\circ} & \quad \cos10^{\circ} \\ \sin80^{\circ} & \quad \cos80^{\circ} \end{bmatrix} \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Evaluate : \( \int\:\large \frac{dx}{\sqrt x + x} \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Write the value of \( \int_{\Large\frac{-\pi}{2}}^{\Large\frac{\pi}{2}} \sin^5 xdx. \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find \( | \overrightarrow a \) x \( \overrightarrow b |\), if \( \overrightarrow a = \hat i - 7\hat j + 7\hat k\) and \( \overrightarrow b = 3\hat i - 2\hat j + 2\hat k \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the cartesian equation of line which passes through the point ( -2, 4, -5 ) and parallel to the line given by \( \large\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6} \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Consider \( f : R_+ \rightarrow [ -5, \infty ] \) given by \( f(x) = 9x^2 + 6x - 5\). Show that f is invertible with \( f^{-1} (y) = \bigg[\large \frac{\sqrt {y+6} - 1}{3} \bigg] \)

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Simplify\( 2\tan^{-1} \bigg(\large \frac{1}{5} \bigg) + \sec^{-1} \bigg( \large\frac{5\sqrt 2}{7} \bigg) + 2\: tan^{-1} \bigg( \frac{1}{8} \bigg) \)

asked Dec 25, 2012 by thanvigandhi_1

1 answer

Two farmers Mukul and Vipul cultivates only three crops potato, tomato and wheat. The annual sale of 2010, 2011 are given below : \[ Sale 2010 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 20000 & \quad 15000 & \quad 80000 & \quad Mukul \\ 30000 & \quad 10000 & \quad 90000 & \quad Vipul \end{bmatrix} \] \[ Sale 2011 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 30000 & \quad 15000 & \quad 60000 & \quad Mukul \\ 20000 & \quad 20000 & \quad 50000 & \quad Vipul \end{bmatrix} \]

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the value of 'a' for which the function f defined as $ f(x) = \left\{ \begin{array} {l l} a\sin \large\frac{\pi}{{2}(n+1)}, & \quad {x \: \leq \: 0} \\ \large\frac{tanx - sinx}{x^3}, & \quad {x \: > 0} \end{array} \right .$ is continuous at x=0. Differentiate \(( x \: cos \: x )^x + ( x\sin\: x )^{\frac{1}{x}}\) w.r.t. \( x \).

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Verify Rolle's Theorem for the following in the given interval : \( f(x) = (x^2 - 4x + 3 ) e^{2x} \)in [ 1, 3 ]

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by $ f(x) = \large\frac{4sin x - 2x - x\cos\: x}{2+ cos\: x} $ is $\text { (i) increasing (ii) decreasing } $

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Evaluate :$ \int_0^{\large\frac{\pi}{2}} \log\sin\: x\:dx $

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the particular solution of the differential equation $\large \frac{dy}{dx} - \frac{y}{x} $$+ cosec \bigg( \large\frac{y}{x} \bigg) = 0 $

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point ( x, y ) is equal to the sum of the coordinates of the point

asked Dec 24, 2012 by thanvigandhi_1

1 answer

If a,b,c are the lengths of the sides BC, CA and AB of a triangle ABC respectively, prove that \( \large\frac{a}{sinA} = \frac{b}{sin B} = \frac{c}{sinC} \)

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Show that points ( -6, 3, 2 ) ( 3, -2, 4 ) (5, 7, 3 ) and ( -13, 17, -1 ) are coplaner.

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Let * be a binary operation on set Q of rational numbers defined as ab = \( \large\frac{3ab}{7} \). Write the identity for , if any.

asked Dec 24, 2012 by thanvigandhi_1

1 answer

Let N be the set of natural numbers and R be the relation in N defined as \(R = {(a, b) : a = b – 2, b > 6}\). Then \[ \begin{array} ((A)\: (2, 4) ∈ R \quad & (B)\: (3, 8) ∈ R \\ (C) \: (6, 8) ∈ R \quad & (D)\: (8, 7) ∈ R. \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

1 answer

If \( sin^{–1} x = y\) , then \[ \begin{array} ((A)\: 0 ≤ y ≤ π \quad & (B) \: \frac{-\pi}{2} ≤ y ≤ \frac{\pi}{2} \\ (C)\: 0 < y < π \quad & (D) \: \frac{-\pi}{2} < y < \frac{\pi}{2} \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

\( sin \bigg (\frac{\pi}{3} - sin^{-1} (\frac{1}{2}) \bigg) \) is equal to \[ \begin{array} ((A) \: \frac{1}{2} \quad & (B)\: \frac{1}{3} \\ (C)\: \frac{1}{4} \quad & (D) \: 1 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

1 answer

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is \[ \begin{array} ((A)\: 27 \quad & (B)\: 18 \\ (C)\: 81 \quad & (D) \: 512 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Let A be a square matrix of order 3 × 3, then \( | kA | \) is equal to \[ \begin{array} ((A) \: k | A | \quad & (B)\: k^2 | A | \\ (C) \: k^3 | A | \quad & (D) \: 3k | A | \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

If \[ f(x) = \left\{ \begin{array}{l l} 1+x & \quad {when \: x\: \leq 2}\\ 5-x & \quad { when \: x\: \: \leq 3} \end{array} \right.\] , then \[ \begin{array} ((A)\: f\: is\: continuous \: at\: x = 2 \quad & (B)\: f\: is\: discontinuous \: at \: x = 2 \\(C) \: f\: is\: continuous\: at\: x = 3 \quad & (D)\: f\: is\: continuous\: at\: x = 3 \: and \: discontinuous \: at \: x=2 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

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