q^()_(p)(cos ax-sin bx)^(2) (where a, b are integers) = -Turito

General

Easy

Maths-

$Q with dot on top subscript p superscript p left parenthesis cos invisible function application a x minus sin invisible function application b x right parenthesis squared$ (where a, b are integers) =

Maths-General

$π$
$- π$
0
2 $π$

Answer:The correct answer is: 2 $pi$ The centre of the sphere is (1, 2, –3) so if other extremity of diameter is (x₁, y₁, z₁), then
$fraction numerator x subscript 1 end subscript plus 2 over denominator 2 end fraction$ = 1, $fraction numerator y subscript 1 end subscript minus 1 over denominator 2 end fraction$ = 2, $fraction numerator z subscript 1 end subscript plus 1 over denominator 2 end fraction$ = –3
\ Required point is (0, 5, 7).
Hence (c) is the correct answer.

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General

maths-

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

Let direction ratios of the required line be <a, b, c>
Therefore a - 2 b - 2 c = 0
And 2 b + c = 0
Þ c = - 2 b
a - 2 b + 4b = 0 Þ a = - 2 b
Therefore direction ratios of the required line are <- 2b, b, - 2b> = <2, - 1, 2>
direction cosines of the required line

open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses

open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

maths-General

open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses

open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses

General

maths-

$0 with hat on top subscript 1 superscript 1 fraction numerator sin invisible function application x minus x squared over denominator 3 minus vertical line x vertical line end fraction d x equals$

maths-General

General

maths-

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

Let direction cosines of straight line be l, m, n \ 4l + m + n = 0 l – 2m + n = 0 Þ

\frac{l}{3} = \frac{m}{- 3} = \frac{n}{- 9}

\frac{l}{- 1} = \frac{m}{+ 1} = \frac{n}{3}

\ Equation of straight line is

\frac{x - 2}{- 1} = \frac{y + 1}{1} = \frac{z + 1}{3}

. Hence (c) is the correct choice.

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

maths-General

Let direction cosines of straight line be l, m, n \ 4l + m + n = 0 l – 2m + n = 0 Þ

\frac{l}{3} = \frac{m}{- 3} = \frac{n}{- 9}

\frac{l}{- 1} = \frac{m}{+ 1} = \frac{n}{3}

\ Equation of straight line is

\frac{x - 2}{- 1} = \frac{y + 1}{1} = \frac{z + 1}{3}

. Hence (c) is the correct choice.

General

maths-

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

Since these two lines are intersecting so shortest distance between the lines will be 0. Hence (c) is the correct answer.

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

maths-General

Since these two lines are intersecting so shortest distance between the lines will be 0. Hence (c) is the correct answer.

General

maths-

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

maths-General

Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.

General

maths-

The value of $\int_{0}^{100} {\sqrt{x}} d x$ (where {x} is the fractional part of x) is

maths-General

General

maths-

The shortest distance between the two straight line $\frac{x - 4 / 3}{2} = \frac{y + 6 / 5}{3} = \frac{z - 3 / 2}{4}$ and $\frac{5 y + 6}{8} = \frac{2 z - 3}{9} = \frac{3 x - 4}{5}$ is

maths-General

General

physics-

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

V to the power of 2 end exponent equals U to the power of 2 end exponent minus 2 g left parenthesis L minus L cos invisible function application theta right parenthesis

fraction numerator 5 g L over denominator 4 end fraction equals 5 g L minus 2 g L left parenthesis 1 minus cos invisible function application theta right parenthesis

5 equals 20 minus 8 plus 8 cos invisible function application theta

cos invisible function application theta equals negative fraction numerator 7 over denominator 8 end fraction

fraction numerator 3 pi over denominator 4 end fraction less than theta less than pi

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

physics-General

V to the power of 2 end exponent equals U to the power of 2 end exponent minus 2 g left parenthesis L minus L cos invisible function application theta right parenthesis

fraction numerator 5 g L over denominator 4 end fraction equals 5 g L minus 2 g L left parenthesis 1 minus cos invisible function application theta right parenthesis

5 equals 20 minus 8 plus 8 cos invisible function application theta

cos invisible function application theta equals negative fraction numerator 7 over denominator 8 end fraction

fraction numerator 3 pi over denominator 4 end fraction less than theta less than pi

General

physics-

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies

Velocity of the bob at the point A

v = \sqrt{5 g L}

(i)

{(\frac{v}{2})}^{2} = v^{2} - 2 g h (i i)

h = L (1 - \cos θ) (i i i)

S o l v i n g E q s . (i), (i i) a n d (i i i), w e g e t

\cos θ = - \frac{7}{8}

o r θ = {c o s}^{- 1} (- \frac{7}{8}) = 151 °

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies

physics-General

Velocity of the bob at the point A

v = \sqrt{5 g L}

(i)

{(\frac{v}{2})}^{2} = v^{2} - 2 g h (i i)

h = L (1 - \cos θ) (i i i)

S o l v i n g E q s . (i), (i i) a n d (i i i), w e g e t

\cos θ = - \frac{7}{8}

o r θ = {c o s}^{- 1} (- \frac{7}{8}) = 151 °

General

physics-

A piece of wire is bent in the shape of a parabola $y = k x^{2} (y$ -axis vertical) with a bead of mass $m$ on it. The bead can side on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the $x$ -axis with a constant acceleration $a$ . The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the $y$ -axis is

m a \cos θ = m g \cos (90 - θ)

\Rightarrow \frac{a}{g} = \tan θ \Rightarrow \frac{a}{g} = \frac{d y}{d x}

\Rightarrow \frac{d}{d x} {(k x)}^{2} = \frac{a}{g} \Rightarrow x = \frac{a}{2 g k}

A piece of wire is bent in the shape of a parabola $y = k x^{2} (y$ -axis vertical) with a bead of mass $m$ on it. The bead can side on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the $x$ -axis with a constant acceleration $a$ . The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the $y$ -axis is

physics-General

m a \cos θ = m g \cos (90 - θ)

\Rightarrow \frac{a}{g} = \tan θ \Rightarrow \frac{a}{g} = \frac{d y}{d x}

\Rightarrow \frac{d}{d x} {(k x)}^{2} = \frac{a}{g} \Rightarrow x = \frac{a}{2 g k}

General

physics-

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

As the prisms Q and R are of the same material and have identical shape they combine to form a slab with parallel faces. Such a slab does not cause any deviation.

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

physics-General

As the prisms Q and R are of the same material and have identical shape they combine to form a slab with parallel faces. Such a slab does not cause any deviation.

General

physics-

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length $s equals t to the power of 3 end exponent plus 5 comma$ where $s$ is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

G i v e n comma blank s equals t to the power of 3 end exponent plus 5

S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent

a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t

therefore T a n g e n t i a l blank a c c e l e r a t i o n blank a t blank t equals 2 blank s comma

a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent

a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent

therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent

therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length $s equals t to the power of 3 end exponent plus 5 comma$ where $s$ is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

physics-General

G i v e n comma blank s equals t to the power of 3 end exponent plus 5

S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent

a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t

therefore T a n g e n t i a l blank a c c e l e r a t i o n blank a t blank t equals 2 blank s comma

a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent

a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent

therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent

therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent

General

physics-

In the given figure, what is the angle of prism

Angle of prism is the angle between incident and emergent surfaces.

In the given figure, what is the angle of prism

physics-General

Angle of prism is the angle between incident and emergent surfaces.

General

physics-

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

In minimum deviation position refracted ray inside the prism is parallel to the base of the prism

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

physics-General

In minimum deviation position refracted ray inside the prism is parallel to the base of the prism

General

maths-

The equation of the plane containing the line $fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text is end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0$ where al + bm + cn is equal to

Since these two lines are intersecting so shortest distance between the lines will be 0.
Hence (c) is the correct answer.

The equation of the plane containing the line $fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text is end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0$ where al + bm + cn is equal to

maths-General

Since these two lines are intersecting so shortest distance between the lines will be 0.
Hence (c) is the correct answer.

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$Q with dot on top subscript p superscript p left parenthesis cos invisible function application a x minus sin invisible function application b x right parenthesis squared$ (where a, b are integers) =

$π$
$- π$
0
2 $π$

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The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

$0 with hat on top subscript 1 superscript 1 fraction numerator sin invisible function application x minus x squared over denominator 3 minus vertical line x vertical line end fraction d x equals$

$0 with hat on top subscript 1 superscript 1 fraction numerator sin invisible function application x minus x squared over denominator 3 minus vertical line x vertical line end fraction d x equals$

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

The value of $\int_{0}^{100} {\sqrt{x}} d x$ (where {x} is the fractional part of x) is

The value of $\int_{0}^{100} {\sqrt{x}} d x$ (where {x} is the fractional part of x) is

The shortest distance between the two straight line $\frac{x - 4 / 3}{2} = \frac{y + 6 / 5}{3} = \frac{z - 3 / 2}{4}$ and $\frac{5 y + 6}{8} = \frac{2 z - 3}{9} = \frac{3 x - 4}{5}$ is

The shortest distance between the two straight line $\frac{x - 4 / 3}{2} = \frac{y + 6 / 5}{3} = \frac{z - 3 / 2}{4}$ and $\frac{5 y + 6}{8} = \frac{2 z - 3}{9} = \frac{3 x - 4}{5}$ is

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

In the given figure, what is the angle of prism

In the given figure, what is the angle of prism

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

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(where a, b are integers) =

0 2

Answer:The correct answer is: 2The centre of the sphere is (1, 2, –3) so if other extremity of diameter is (x1, y1, z1), then = 1, = 2, = –3\ Required point is (0, 5, 7).Hence (c) is the correct answer.

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The foot of the perpendicular on the line drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

The foot of the perpendicular on the line drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

∫−π/4π/4 ex(xsin⁡x)e2x−1dx is equal to

∫−π/4π/4 ex(xsin⁡x)e2x−1dx is equal to

Let f:R→R,f(x)=|x−[x]|,[x]|x−[x+1]|,[x] is odd 1 is even where [.] denotes greatest integer function, then ∫−24 f(x)dx is equal to

Let f:R→R,f(x)=|x−[x]|,[x]|x−[x+1]|,[x] is odd 1 is even where [.] denotes greatest integer function, then ∫−24 f(x)dx is equal to

The value of ∫01 |sin 2πx|∣dx is equal to

The value of ∫01 |sin 2πx|∣dx is equal to

The value of ∫0100 {x}dx (where {x} is the fractional part of x) is

The value of ∫0100 {x}dx (where {x} is the fractional part of x) is

The shortest distance between the two straight linex−4/32=y+6/53=z−3/24 and 5y+68=2z−39=3x−45 is

The shortest distance between the two straight linex−4/32=y+6/53=z−3/24 and 5y+68=2z−39=3x−45 is

A bob of mass is suspended by a massless string of length . The horizontal velocity at position is just sufficient to make it reach the point . The angle at which the speed of the bob is half of that at , satisfies

A bob of mass is suspended by a massless string of length . The horizontal velocity at position is just sufficient to make it reach the point . The angle at which the speed of the bob is half of that at , satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length where is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length where is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

In the given figure, what is the angle of prism

In the given figure, what is the angle of prism

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

The equation of the plane containing the line where al + bm + cn is equal to

The equation of the plane containing the line where al + bm + cn is equal to

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$Q with dot on top subscript p superscript p left parenthesis cos invisible function application a x minus sin invisible function application b x right parenthesis squared$ (where a, b are integers) =

$π$
$- π$
0
2 $π$

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

$\int_{- π / 4}^{π / 4} \frac{e^{x} (x \sin x)}{e^{2 x} - 1} d x$ is equal to

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

Let $f : R \to R, f (x) = \{\begin{matrix} | x - [x] |, [x] \\ | x - [x + 1] |, [x] \end{matrix}$ $\begin{aligned} is odd \\ 1 is even where [.] \end{aligned}$ denotes greatest integer function, then $\int_{- 2}^{4} f (x) d x$ is equal to

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

The value of $\int_{0}^{1} | \sin 2 π x | ∣ d x$ is equal to

The value of $\int_{0}^{100} {\sqrt{x}} d x$ (where {x} is the fractional part of x) is

The value of $\int_{0}^{100} {\sqrt{x}} d x$ (where {x} is the fractional part of x) is

The shortest distance between the two straight line $\frac{x - 4 / 3}{2} = \frac{y + 6 / 5}{3} = \frac{z - 3 / 2}{4}$ and $\frac{5 y + 6}{8} = \frac{2 z - 3}{9} = \frac{3 x - 4}{5}$ is

The shortest distance between the two straight line $\frac{x - 4 / 3}{2} = \frac{y + 6 / 5}{3} = \frac{z - 3 / 2}{4}$ and $\frac{5 y + 6}{8} = \frac{2 z - 3}{9} = \frac{3 x - 4}{5}$ is

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies

A bob of mass M is suspended by a massless string of length L. The horizontal velocity $v$ at position A is just sufficient to make it reach the point B. The angle $θ$ at which the speed of the bob is half of that at A, satisfies