Recent questions tagged 2009

Evaluate : $ \int \mathrm \: \frac{\large \cos \sqrt x}{\large \sqrt x} dx $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

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

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Write the value of the following determinant : $ \begin {vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end {vmatrix} $

asked Dec 21, 2012 by thanvigandhi_1

1 answer

Find the value of x from the following : $\begin {bmatrix} x & 4 \\ 2 & 2x \end {bmatrix} = 0 $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Find the value of p if $ ( 2\hat i + 6\hat j + 27\hat k )$ x $ ( \hat i + 3\hat j + p\hat k ) = \overrightarrow 0. $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Write the direction cosines of a line equally inclined to the three coordinate axes.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

If $ \overrightarrow P $ is a unit vector and $ ( \overrightarrow x - \overrightarrow p ). ( \overrightarrow x + \overrightarrow p ) = 80$, then find $ | \overrightarrow x | $.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When $ x = 8 cm $ and $ y = 6 cm $ , find the rate of a change of (a) the perimeter, (b) the area of the rectangle.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

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

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Let f : N $ \rightarrow $ N be defined by : $ f(n) = \left\{ \begin{array}{l l} \large\frac{n+1}{2} & \quad \text{if $n$ is odd}\\ \large\frac{n}{2} & \quad \text{ if $n$ is even} \end{array} for \: all\: n\: \in \: N. \right.$ Find whether the function f is bijective.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Evaluate : $ \int \mathrm \: \large\frac{dx}{ \sqrt{5-4x-2x^2}} $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

If $ y = \frac{\large sin^{-1}x}{\large \sqrt {1-x^2}} $, show that $ (1 - x^2) \large\frac{d^2y}{dx^2} - 3x \large\frac{dy}{dx} $$- y = 0 $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

On a multiple choice examination with three possible answers ( out of which only one is correct ) for each of the five questions, what is the probability that the candidate would get four or more correct answers just by guessing?

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin {bmatrix} 1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 1 + 3p + 2q \\ 3 & 6 + 3p & 1 + 6p + 3q \end{bmatrix} = 1 $

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Solve the following differential equation : \[ x \frac{dy}{dx} = y - x \: tan \bigg(\frac{y}{x} \bigg) \]

asked Dec 20, 2012 by thanvigandhi_1

0 answers

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

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the shortest distance between the following two lines : $ \overrightarrow r = (1+\lambda)\hat i + (2-\lambda)\hat j + (\lambda+1)\hat k ; \overrightarrow r = ( 2\hat i - \hat j - \hat k ) + \mu \: ( 2\hat i + \hat j + 2\hat k ). $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

prove the following : $ \cot^{-1} \bigg(\large \frac{\sqrt {1+\sin x} + \sqrt {1-\sin x}}{\sqrt {1+\sin x} - \sqrt {1-\sin x}} \bigg) = \large\frac{x}{2},$$ x \in \: \bigg(0,\large \frac{\pi}{4} \bigg) $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

The scalar product of the vector $ \hat i + \hat j + \hat k $ with the unit vector along the sum of the vectors $ 2\hat i + 4\hat j - 5\hat k $ and $ \lambda \hat i + 2\hat j + 3\hat k$ is equal to one. Find the value of $ \lambda$

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the equation of the plane determined by the points A( 3, -1, 2 ) , B(5, 2, 4) and C(-1, -1, 6) . Also find the distance of the point P(6, 5, 9) from the plane.

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the area of the region included between the parabola $ y^2 = x $ and the line $ x + y = 2$.

asked Dec 20, 2012 by thanvigandhi_1

0 answers

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

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations : $ \begin{array} xx + y + z = 6 \\ x + 2z = 7 \\ 3x + y + z = 12 \end{array} $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Coloured balls are distributed in three bags as shown in the table below, a bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. what is the probability that they came from bag 1 ?

asked Dec 20, 2012 by thanvigandhi_1

1 answer

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760, to invest and has a space for at most 20 items. A fan cost him Rs.360 and a sewing machine Rs.240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit ? Formulate this as a linear programming problem and solve it graphically.

asked Dec 20, 2012 by thanvigandhi_1

0 answers

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is $ \frac{\pi}{3} $

asked Dec 20, 2012 by thanvigandhi_1

0 answers

Find the value of x, if $\begin {bmatrix} 3x+y & -y \\ 2y-x & 3 \end{bmatrix} = \begin {bmatrix} 1 & 2 \\ -5 & 3 \end{bmatrix} $

asked Dec 20, 2012 by thanvigandhi_1

1 answer

Ask Questions, Get Answers

Evaluate : $ \int \mathrm \: \frac{\large \cos \sqrt x}{\large \sqrt x} dx $

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

Write the value of the following determinant : $ \begin {vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end {vmatrix} $

Find the value of x from the following : $\begin {bmatrix} x & 4 \\ 2 & 2x \end {bmatrix} = 0 $

Find the value of p if \( ( 2\hat i + 6\hat j + 27\hat k )\) x \( ( \hat i + 3\hat j + p\hat k ) = \overrightarrow 0. \)

Write the direction cosines of a line equally inclined to the three coordinate axes.

If \( \overrightarrow P \) is a unit vector and \( ( \overrightarrow x - \overrightarrow p ). ( \overrightarrow x + \overrightarrow p ) = 80\), then find \( | \overrightarrow x | \).

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When \( x = 8 cm \) and \( y = 6 cm \) , find the rate of a change of (a) the perimeter, (b) the area of the rectangle.

If \( \sin\: y = x\sin (a+y)\), prove that \( \large\frac{dy}{dx} = \frac{\large \sin^2(a+y)}{\large \sin a}. \)

Let f : N \( \rightarrow \) N be defined by : $ f(n) = \left\{ \begin{array}{l l} \large\frac{n+1}{2} & \quad \text{if $n$ is odd}\\ \large\frac{n}{2} & \quad \text{ if $n$ is even} \end{array} for \: all\: n\: \in \: N. \right.$ Find whether the function f is bijective.

Evaluate : $ \int \mathrm \: \large\frac{dx}{ \sqrt{5-4x-2x^2}} $

If \( y = \frac{\large sin^{-1}x}{\large \sqrt {1-x^2}} \), show that $ (1 - x^2) \large\frac{d^2y}{dx^2} - 3x \large\frac{dy}{dx} $$- y = 0 $

On a multiple choice examination with three possible answers ( out of which only one is correct ) for each of the five questions, what is the probability that the candidate would get four or more correct answers just by guessing?

Using properties of determinants, prove the following : $\begin {bmatrix} 1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 1 + 3p + 2q \\ 3 & 6 + 3p & 1 + 6p + 3q \end{bmatrix} = 1 $

Solve the following differential equation : \[ x \frac{dy}{dx} = y - x \: tan \bigg(\frac{y}{x} \bigg) \]

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

Find the shortest distance between the following two lines : $ \overrightarrow r = (1+\lambda)\hat i + (2-\lambda)\hat j + (\lambda+1)\hat k ; \overrightarrow r = ( 2\hat i - \hat j - \hat k ) + \mu \: ( 2\hat i + \hat j + 2\hat k ). $

prove the following : $ \cot^{-1} \bigg(\large \frac{\sqrt {1+\sin x} + \sqrt {1-\sin x}}{\sqrt {1+\sin x} - \sqrt {1-\sin x}} \bigg) = \large\frac{x}{2},$$ x \in \: \bigg(0,\large \frac{\pi}{4} \bigg) $

The scalar product of the vector \( \hat i + \hat j + \hat k \) with the unit vector along the sum of the vectors \( 2\hat i + 4\hat j - 5\hat k \) and \( \lambda \hat i + 2\hat j + 3\hat k\) is equal to one. Find the value of \( \lambda\)

Find the equation of the plane determined by the points A( 3, -1, 2 ) , B(5, 2, 4) and C(-1, -1, 6) . Also find the distance of the point P(6, 5, 9) from the plane.

Find the area of the region included between the parabola \( y^2 = x \) and the line \( x + y = 2\).

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

Using matrices, solve the following system of equations : $ \begin{array} xx + y + z = 6 \\ x + 2z = 7 \\ 3x + y + z = 12 \end{array} $

Coloured balls are distributed in three bags as shown in the table below, a bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. what is the probability that they came from bag 1 ?

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is \( \frac{\pi}{3} \)

Find the value of x, if $\begin {bmatrix} 3x+y & -y \\ 2y-x & 3 \end{bmatrix} = \begin {bmatrix} 1 & 2 \\ -5 & 3 \end{bmatrix} $