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Recent questions tagged sec-b

Find the equation of the plane through the intersection of the planes $3x -y + 2z -4 = 0$ and $x + y + z -2 = 0$ and the point $(2, 2, 1).$

asked Dec 15, 2012 by thanvigandhi_1

1 answer

Find the vector equation of the plane passing through the intersection of the planes $ \overrightarrow r .(2\hat i + 2 \hat j - 3\hat k) = 7, \overrightarrow r .(2\hat i + 5\hat j + 3\hat k ) = 9$ and through the point $(2, 1, 3).$

asked Dec 15, 2012 by thanvigandhi_1

1 answer

Find the equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane$x -y + z = 0.$

asked Dec 15, 2012 by thanvigandhi_1

1 answer

Find the angle between the planes whose vector equations are $ \overrightarrow r. ( 2\hat i + 2\hat j - 3\hat k ) = 5 \: and\: \overrightarrow r. ( 3\hat i - 3\hat j + 5\hat k ) = 3. $

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the angle between the following pair of lines: $ \frac{\large x-2}{\large 2} = \frac{\large y-1}{\large 5} = \frac{\large z + 3}{\large -3} \: and\: \frac{\large x+2}{\large -1} = \frac{\large y-4}{\large 8} = \frac{\large z - 5}{\large 4} $

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the shortest distance between the lines whose vector equations are $ \overrightarrow r = ( 1 - t ) \hat i + ( t - 2 )\hat j + ( 3 - 2t )\hat k$ and $ \overrightarrow r = ( s + 1 )\hat i + ( 2s - 1 )\hat j - (2s + 1)\hat k $

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the shortest distance between the lines: $ \frac{ \large x + 1 }{ \large 7} = \frac{ \large y + 1}{ \large -6} = \frac{ \large z + 1}{ \large 1}$ and $ \frac{ \large x - 3 }{ \large 1} = \frac{ \large y - 5}{ \large -2} = \frac{ \large z - 7}{ \large 1}$

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the shortest distance between the lines $ \overrightarrow r = ( \hat i + 2\hat j + \hat k) + \lambda ( \hat i - \hat j + \hat k )$ and $ \overrightarrow r = 2 \hat i - \hat j - \hat k + \mu (2\hat i + \hat j + 2\hat k )$

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Show that the lines $ \frac{ \large x - 5 }{ \large 7} = \frac{ \large y + 2}{ \large -5} = \frac{ \large z}{ \large 1}$ and $ \frac{ \large x }{ \large 1} = \frac{ \large y}{ \large 2} = \frac{ \large z}{ \large 3}$ are perpendicular to each other.

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the values of p so that the lines $ \frac{ \large 1 - x }{ \large 3} = \frac{ \large 7y - 14}{ \large 2p} = \frac{ \large z - 3}{ \large 2}\: and \: \frac{ \large 7 - 7x }{ \large 3 \: p} = \frac{ \large y - 5}{ \large 1} = \frac{ \large 6 - z}{ \large 5}$ are at right angles.

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the line of intersection of the planes \(\overrightarrow r. (\hat i + \hat j + \hat k)=1\) and \( \overrightarrow r. (2\hat i + 3\hat j - \hat k) + 4 = 0\) and parallel to \(x\) - axis.

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Find the angle between the pairs of lines: $ \overrightarrow r = 2\hat i - 5 \hat j + \hat k + \lambda (3\hat i + 2\hat j + 6\hat k)\: and \: \overrightarrow r = 7\hat i - 6 \hat k+ \mu (\hat i + 2\hat j + 2\hat k) $

asked Dec 14, 2012 by thanvigandhi_1

1 answer

For the differential equations, find the particular solution satisfying the given condition: \[ \bigg[ x \: sin^2 \bigg( \frac{x}{y} - y \bigg) \bigg] dx + xdy = 0; y \frac{\large \pi}{\large 4} when \: x = 1\]

asked Dec 12, 2012 by thanvigandhi_1

1 answer

Show that the differential equation is homogeneous and solve :$ \bigg( 1 + e^{\Large\frac{x}{y}} \bigg) dx + e^{\Large\frac{x}{y}} \bigg( 1 -\large \frac{x}{y} \bigg) $$dy= 0 $

asked Dec 12, 2012 by thanvigandhi_1

1 answer

Prove the following: \[ 2 \: tan^{-1} \frac{1}{2} + tan ^{-1} \frac{1}{7} = tan^{-1} \frac{31}{17} \]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following function in the simplest form: \[ tan^{-1} \frac{\sqrt {1 + x^2} - 1}{x}, x \neq 0\]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following function in the simplest form: \[ tan^{-1} \frac{1}{\sqrt{x^2 - 1}}, | x | > 1 \]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following function in the simplest form: \[ tan^{-1} \bigg( \sqrt {\frac{1-cos \: x}{1 + cos \: x}} \bigg), x < \pi\]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following function in the simplest form: \[ tan^{-1} \bigg( \frac{cos\: x - sin\: x}{cos\: x + sin\: x} \bigg) , 0 < x < \pi \]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following in the simplest form: $ \tan^{-1} \large \frac{x}{\sqrt {a^2 - x^2}},$$ | x | < a $

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Write the following function in the simplest form:\[ tan^{-1} \bigg( \frac{3a^2x-x^3}{a^3-3ax^2} \bigg), a > 0 ; \frac{-a}{\sqrt 3} \leq x \leq \frac{a}{\sqrt 3} \]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Find the value of $ \tan^{-1} \bigg[ 2\cos \bigg(2\sin^{-1}\frac{1}{2}\bigg) \bigg] $

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Find the value of \[ tan\frac{1}{2} \bigg[ sin^{-1} \frac{2x}{1+x^2} + cos^{-1} \frac{1-y^2}{1+y^2} \bigg], | x | < 1, y > 0 \: and \: xy < 1\]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

If $ \tan^{-1} \frac{\large x-1}{\large x-2} + \tan^{-1} \frac{\large x+1}{\large x+2} = \frac{\large \pi}{\large 4} $ then find the value of $x$

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Find the values of the expression $ \tan \bigg( \sin^{-1} \frac{3}{5} + \cot^{-1} \frac{3}{2} \bigg) $

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Let $f:\;N\;\to N\;be\;defined\;by\;f(n)= \left\{ \begin{array}{1 1} \frac{n+1}{2}, & if\;n\;is\;odd\\ \frac{n}{2}, & if\;is\;even\end{array} \right. \qquad for\;all\;n \in N$ \[\text{state whether the function f is bijective.Justify your answer.} \]

asked Dec 10, 2012 by sreemathi.v

1 answer

What does $ 2\sin^{-1} \frac{3}{5}$ reduce to

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Prove the following\[3\sin^{-1}x=\sin^{-1}(3x-4x^3)\;x\in\bigg[-\frac{1}{2},\frac{1}{2}\bigg]\]

asked Dec 6, 2012 by sreemathi.v

1 answer

Prove the following\[\tan^{-1}\frac{2}{11}+\tan^{-1}\frac{7}{24}=\tan^{-1}\frac{1}{2}\]

asked Dec 6, 2012 by sreemathi.v

1 answer

Show that the vector $ \hat i + \hat j + \hat k$ is equally inclined to the axes OX, OY and OZ.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $ \hat i + 2\hat j − \hat k$ and $ – \hat i + \hat j + \hat k$ respectively, in the ratio $2 : 1$ internally

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that the points A, B and C with position vectors, \( \overrightarrow a = 3\hat i − 4\hat j − 4\hat k, \overrightarrow b = 2\hat i − \hat j + \hat k\) and \( \overrightarrow c = \hat i − 3\hat j − 5\hat k\) , respectively form the vertices of a right angled triangle.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that each of the given three vectors is a unit vector: $ \large\frac{1}{7}$$ ( 2\hat i + 3\hat j + 6\hat k),\large\frac{1}{7}$$ ( 3\hat i - 6\hat j + 2\hat k), \large\frac{1}{7}$$ ( 6\hat i + 2\hat j - 3\hat k) $ Also, show that they are mutually perpendicular to each other.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Find $| \overrightarrow a |$ and $| \overrightarrow b |$ , if $(\overrightarrow a + \overrightarrow b) ⋅ (\overrightarrow a − \overrightarrow b) = 8$ and $| \overrightarrow a |= 8 | \overrightarrow b |.$

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a = 2\hat i + 2\hat j + 3\hat k,\: \overrightarrow b = − \hat i + 2\hat j + \hat k$ and $ \overrightarrow c = 3\hat i + \hat j$ are such that $\overrightarrow a + λ\overrightarrow b $ is perpendicular to $ \overrightarrow c $ , then find the value of $λ.$

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that $| \overrightarrow a | \overrightarrow b + | \overrightarrow b | \overrightarrow a$ is perpendicular to $| \overrightarrow a | \overrightarrow b - | \overrightarrow b | \overrightarrow a$, for any two nonzero vectors $\overrightarrow a$ and $\overrightarrow b$.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If $ \overrightarrow a, \overrightarrow b, \overrightarrow c, $are unit vectors such that $ \overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0$, find the value of $ \overrightarrow a ⋅ \overrightarrow b + \overrightarrow b ⋅ \overrightarrow c + \overrightarrow c ⋅ \overrightarrow a$.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If the vertices $A, B, C$ of a triangle $ABC$ are $(1, 2, 3), (-1, 0, 0), (0, 1, 2)$ respectively, then find $ \angle ABC. \angle ABC$ is the angle between the vectors $ \overline {BA}$ and $\overline {BC}$.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that the vectors \(2\hat i - \hat j + \hat k, \: \hat i - 3\hat j - 5\hat k\) and \(3\hat i - 4\hat j - 4\hat k\) form the vertices of a right angled triangle.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Integrate the function $\frac{\large x}{\large \sqrt{x+4}},x\;>\;0$

asked Dec 6, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{\large x}{\large 9-4x^2}$

asked Dec 6, 2012 by sreemathi.v

1 answer

Integrate the function $\tan^2(2x-3)$

asked Dec 6, 2012 by sreemathi.v

1 answer

Integrate the function: $\int \frac{\large 2cos\: x-3sin\: x}{\large 6cos\: x+4sin\: x}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{\sin x}{1+\cos x}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{1}{1+\cot x}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{1}{1-\ tanx}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{\sqrt{\tan x}}{\sin x\cos x}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Integrate the function $\frac{x^3\sin\bigg(\tan^{-1}x^4\bigg)}{1+x^8}$

asked Dec 5, 2012 by sreemathi.v

1 answer

Find the integral of the function: $\int sin(3x). cos(4x) dx$

asked Dec 5, 2012 by sreemathi.v

1 answer

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