Class 11th

A man starts walking from the point P(-3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then 50(PR)² + (RQ)² is equal to__________.

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Contributor-Level 10

$\Rightarrow \frac{6}{3} = \frac{2}{- t} \Rightarrow t = - 1$

->R (-1,0)

$P R^{2} + R Q^{2} = 20 + 5 = 25$

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Class 11th Physics

A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02R. The percentage difference in the time periods of the two satellites is:

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Time period of Satellite = T = $\frac{2 π R}{v} = {(\frac{4 π^{2} R^{3}}{G M})}^{\frac{1}{2}}$

$\Rightarrow T α R^{\frac{3}{2}} \Rightarrow \frac{T_{2}}{T_{1}} = \frac{{(1.02 R)}^{\frac{3}{2}}}{R^{\frac{3}{2}}} = {(1.02)}^{\frac{3}{2}} = {(1 + 0.02)}^{\frac{3}{2}}$

The percentage difference in the time periods of the two satellites = $\frac{Δ T}{T_{1}}$ * 100

= (0.03) * 100 = 3%

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Class 11th Physics

A particle is making simple harmonic motion along the X-axis. If at a distances x₁ and x₂ from the mean position the velocities of the particle are $υ_{1} a n d υ_{2}$ respectively. The time period of its oscillation is given as:

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As we know that $υ^{2} = ω^{2} (A^{2} - x^{2})$ $^{}^{}^{}^{}$ for SHM, so

$υ_{1}^{2} = ω^{2} (A^{2} - x_{1}^{2}) . . . . . . . (i), a n d υ_{2}^{2} = ω^{2} (A^{2} - x_{2}^{2}) . . . . . . . . . (i i)$

Subtracting equation (ii) from equation (i), we have

$υ_{1}^{2} - υ_{2}^{2} = ω^{2} (x_{2}^{2} - x_{1}^{2}) \Rightarrow ω = \frac{2 π}{T} = \sqrt[]{\frac{υ_{1}^{2} - υ_{2}^{2}}{x_{2}^{2} - x_{1}^{2}}} \Rightarrow T = 2 π \sqrt[]{\frac{x_{2}^{2} - x_{1}^{2}}{υ_{1}^{2} - υ_{2}^{2}}}$

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All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is________.

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Class 11th Relations and Functions

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Let [t] denote the greatest integer $\leq t .$ The number of points where the function f(x) = $[x] | x^{2} - 1 |$ + sin $(\frac{π}{[x] + 3}) - [x + 1], x \in (- 2, 2)$ is not continuous is__________.

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$f (x) = {\begin{cases} - 2 x^{2} + 3, - 2 < x < - 1 \\ x^{2}, - 1 \leq x < 0 \\ \frac{\sqrt[]{3}}{2} - 1, 0 \leq x < 1 \\ x^{2} - 3 + \frac{1}{\sqrt[]{2}}, 1 \leq x < 2 \end{cases}$

$f (- 1^{-}) = f (- 1^{+}) = f (- 1) = 1$

$f (1^{+}) = f (1) = \frac{1}{\sqrt[]{2}} - 2$

Points of discontinuity

x = 0, 1

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Class 11th Physics

Which of the following graphs represent the behaviour of an ideal gas? Symbol have their usual meaning.

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Class 11th Work, Energy and Power

Contributor-Level 10

According to ideal gas law we know that

PV = nRT

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A body at rest is moved along a horizontal straight line by a machine delivering a constant power. The distance moved by the body in time 't' is proportional to:

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Class 11th Mechanical Properties of Solids

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According to question, we can write

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The length of a metal wire is $l_{1},$ when the tension in its is T₁ and is $l_{2}$ when the tension is T₂. The natural length of the wire is:

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Contributor-Level 10

Assuming Tension in metal wire suspended from roof varies linearly and 0 and T0 (developed due its own weight) are tensions are ends of wire of length $l .$ So

$T_{1} α (l_{1} - l), a n d T_{2} α (l_{2} - l)$

$\frac{T_{1}}{T_{2}} = \frac{l_{1} - l}{l_{2} - l} \Rightarrow l = \frac{T_{2} l_{1} - T_{1} l_{2}}{T_{2} - T_{1}}$

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Let the points of intersection of the lines x – y + 1 = 0, x – 2y + 3 = 0 and 2x – 5y + 11 = 0 are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is_____.

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Contributor-Level 10

ar (ABC) = 4ar (DEF)

$= 4 * \frac{1}{2} * | 2 (2 - 5) + 1 (5 - 3) + 7 (3 - 2) | = 2 | - 6 + 2 + 7 | = 6$

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