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Question: If $\operatorname{Sin} X=\frac{-1}{2}$ and $X$ lies in Quadrant IV, find the values of (i) $\operatorname{Sin} \frac{X}{2}$ (ii) $\operatorname{Cos} \frac{X}{2}$ (iii) $\tan \frac{X}{2}$ Solution: Given: $\sin x=\frac{-1}{2}$ and $x$ lies in Quadrant IV. To Find: i) $\sin \frac{x}{2}$ ii) $\cos \frac{x}{2}$ iii) $\tan \frac{x}{2}$ Now, since $\sin x=\frac{-1}{2}$ We know that $\cos x=\pm \sqrt{1-\sin ^{2} x}$ $\cos x=\pm \sqrt{1-\left(\frac{-1}{2}\right)^{2}}$ $\cos x=\pm \sqrt{1-\frac...
Read More →Find the points on the curve
Question: Find the points on the curve $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ at which the tangents are (i) parallel to $x$-axis (ii) parallel to $y$-axis. Solution: (i) The slope of the $x$-axis is 0 . Now, let $\left(x_{1}, y_{1}\right)$ be the required point. Since, the point lies on the curve. Hence, $\frac{x_{1}{ }^{2}}{9}+\frac{y_{1}{ }^{2}}{16}=1 \quad \ldots(1)$ Now, $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ $\Rightarrow \frac{2 x}{9}+\frac{2 y}{16} \frac{d y}{d x}=0$ $\Rightarrow \frac{y}{16} ...
Read More →Solve this
Question: If $\cos x=\frac{-3}{5}$ and $\frac{\pi}{2}x\pi$, find the values of (i) $\sin \frac{x}{2}$ (ii) $\cos \frac{x}{2}$ (iii) $\tan \frac{x}{2}$ Solution: Given: $\cos x=-\frac{3}{5}$ and $\frac{\pi}{2}x\pi . i . e, x$ lies in $\|$ quadrant To Find: i) $\sin \frac{x}{2}$ ii) $\cos \frac{x}{2}$ iii) $\tan \frac{x}{2}$ i) $\sin \frac{x}{2}$ Formula used: $\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}$ Now, $\sin \frac{x}{2}=\pm \sqrt{\frac{1-\left(\frac{-3}{5}\right)}{2}}=\pm \sqrt{\frac{\f...
Read More →According to Stefan’s law of radiation,
Question: According to Stefans law of radiation, a black body radiates energy T4from its unit surface area every second where T is the surface temperature of the black body and = 5.67 10-8W/m2K4is known as Stefans constant. A nuclear weapon may be thought of as a ball of radius 0.5 m. When denoted, it reaches temperature of 106K and can be treated as a black body. (a) estimate the power it radiates (b) if surrounding has water at 30oC, how much water can 10% of the energy produced evaporate in 1...
Read More →Calculate the stress developed inside a tooth
Question: Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of 57oC is drunk. You can take body temperature to be 37oC and = 1.7 10-5oC, bulk modulus for copper = 140 109N/m2. Solution: Decrease in temperature, ∆t = 57 37 = 20oC Coefficient of linear expansion, = 1.7 10-5oC Bulk modulus for copper, B = 140 109N/m2 Coefficient of cubical expansion, = 3 = 5.1 10-5oC Increase in volume with increase in temperature is given as $\Delta V=\gamma V \Del...
Read More →We would like to prepare a scale whose length
Question: We would like to prepare a scale whose length does not change with temperature. It is proposed to prepare a unit scale of this type whose length remains, say 10 cm. We can use a bimetallic strip made of brass and iron each of different length whose length would change in such a way that difference between their lengths remain constant. If iron= 1.2 10-5/K and brass= 1.8 10-5/K, what should we take as length of each strip? Solution: Change in length of iron rod $\Delta L=\alpha_{\text {...
Read More →Find the points on the curve
Question: Find the points on the curve $x^{2}+y^{2}-2 x-3=0$ at which the tangents are parallel to the $x$-axis. Solution: Let $\left(x_{1}, y_{1}\right)$ be the required point. Since the point lie on the curve. Hence $x_{1}{ }^{2}+y_{1}{ }^{2}-2 x_{1}-3=0 \ldots$ (1) Now, $x^{2}+y^{2}-2 x-3=0$ $\Rightarrow 2 x+2 y \frac{d y}{d x}-2=0$ $\therefore \frac{d y}{d x}=\frac{2-2 x}{2 y}=\frac{1-x}{y}$ Now, Slope of the tangent $=\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}=\frac{1-x_{1}}{y...
Read More →One day in the morning, Ramesh filled up 1/3 bucket
Question: One day in the morning, Ramesh filled up 1/3 bucket of hot water from geyser, to take bath. Remaining 2/3 was to be filled by cold water to bring mixture to a comfortable temperature. Suddenly Ramesh had to attend something which would take some times, say 5-10 minutes before he could take bath. Now he has two options: (a) fill the remaining bucket completely by cold water and then attend the work (b) first attend to the work and fill the remaining bucket just before taking bath. Which...
Read More →100 g of water is supercooled to -10oC.
Question: 100 g of water is supercooled to -10oC. At this point, due to some disturbance mechanised or otherwise some of it suddenly freezes to ice. What will be the temperature of the resultant mixture and how much mass would freeze? Solution: Mass of water = 100 g At -10oC the mixture has water and ice Heat required by the mixture is ms∆t = (100)(1)(0-(-10) = 1000 Cal Therefore, the mass of the mixture, m = Q/L = 12.5 g...
Read More →Solve this
Question: If $\sin x=\frac{\sqrt{5}}{3}$ and $\frac{\pi}{2}x\pi$, find the values of (i) $\sin \frac{x}{2}$ (ii) $\cos \frac{x}{2}$ (iii) $\tan \frac{\mathrm{x}}{2}$ Solution: Given: $\sin x=\frac{\sqrt{5}}{3}$ and $\frac{\pi}{2}x\pi$ i.e, $x$ lies in the Quadrant II . To Find: i) $\sin \frac{x}{2}$ ii) $\cos \frac{x}{2}$ iii) $\tan \frac{x}{2}$ Now, since $\sin x=\frac{\sqrt{5}}{3}$ We know that $\cos x=\pm \sqrt{1-\sin ^{2} x}$ $\cos x=\pm \sqrt{1-\left(\frac{\sqrt{5}}{3}\right)^{2}}$ $\cos x=...
Read More →Calculate the temperature which has same
Question: Calculate the temperature which has same numerical value on Celsius and Fahrenheit scale. Solution: Consider two fixed points for the construction of the scale of temperature. The first fixed point is for the freezing point which is known as lower fixed point. The second fixed point is for the boiling point which is known as upper fixed point. Following is the identity that is used for the conversion: $\frac{\text { reading on any scale }-(L F P)}{(U F P)-(L F P)}=$ constant for all sc...
Read More →Find the points on the curve
Question: Find the points on the curve $\frac{x^{2}}{4}+\frac{y^{2}}{25}=1$ at which the tangents are parallel to the (i) $x$-axis (ii) $y$-axis. Solution: (i) The slope of the $x$-axis is 0 . Now, let $\left(x_{1}, y_{1}\right)$ be the required point. Since, the point lies on the curve. Hence, $\frac{x_{1}{ }^{2}}{4}+\frac{y_{1}{ }^{2}}{25}=1 \quad \ldots(1)$ Now, $\frac{x^{2}}{4}+\frac{y^{2}}{25}=1$ $\therefore \frac{2 x}{4}+\frac{2 y}{25} \frac{d y}{d x}=0$ $\Rightarrow \frac{2 y}{25} \frac{d...
Read More →Why does a metal bar appear hotter
Question: Why does a metal bar appear hotter than a wooden bar at the same temperature? Equivalently it also appears cooler than wooden bar if they are both colder than room temperature. Solution: A metal bar appears to be hotter than a wooden bar at the same temperature because the rate of transfer of heat in a metal is faster than in the wood. Similarly, metal bar appears to be colder than the wooden bar as the specific heat of the metal is very low than the wood....
Read More →A student records the initial length l,
Question: A student records the initial length l, change in temperature ∆T and change in length ∆l of a rod as follows: If the first observation is correct, what can you say about observations 2, 3, and 4. Solution: It is given that the first observation is correct which means that the coefficient of linear expansion is $\alpha=\frac{\Delta l}{l \times \Delta T}=\frac{4 \times 10^{-4}}{2 \times 10}=2 \times 10^{-50} \mathrm{C}^{-1}$ For second observation, $\Delta l=\alpha l \Delta T=2 \times 10...
Read More →Is the bulb of a thermometer
Question: Is the bulb of a thermometer made of diathermic or adiabatic wall? Solution: The bulb of a thermometer is made of diathermic walls as they allow the conduction of the heat....
Read More →A glass full of hot milk is poured
Question: A glass full of hot milk is poured on the table. It begins to cool gradually. Which of the following is correct? (a) the rate of cooling is constant till milk attains the temperature of the surrounding (b) the temperature of milk falls off exponentially with time (c) while cooling, there is a flow of heat from milk to the surrounding as well as from surrounding to the milk but the net flow of heat is from milk to the surrounding and that is why it cools (d) all three phenomena, conduct...
Read More →‘Gulab Jamuns’ (assumed to be spherical)
Question: Gulab Jamuns (assumed to be spherical) are to be heated in an oven. They are available in two sizes, one twice bigger than the other. Pizzas (assumed to be discs) are also to be heated in the oven. They are also in two sizes, one twice big in radius than the other. All four are put together to be heated to oven temperature. Choose the correct option from the following: (a) both size gulab jamuns will get heated at the same time (b) smaller gulab jamuns are heated before bigger ones (c)...
Read More →Mark the correct options:
Question: Mark the correct options: (a) A system X is in thermal equilibrium with Y but not with Z. System Y and Z may be in thermal equilibrium with each other (b) A system X is in thermal equilibrium with Y but not with Z. Systems Y and Z are not in thermal equilibrium with each other (c) A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z must be in thermal equilibrium with each other (d) A system X is neither in thermal equilibrium with Y nor with Z. The syste...
Read More →Mark the correct options:
Question: Mark the correct options: (a) A system X is in thermal equilibrium with Y but not with Z. System Y and Z may be in thermal equilibrium with each other (b) A system X is in thermal equilibrium with Y but not with Z. Systems Y and Z are not in thermal equilibrium with each other (c) A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z must be in thermal equilibrium with each other (d) A system X is neither in thermal equilibrium with Y nor with Z. The syste...
Read More →A sphere, a cube, and a thin circular plate,
Question: A sphere, a cube, and a thin circular plate, all of the same material and same mass are initially heated to same high temperature. (a) plate will cool fastest and cube the slowest (b) sphere will cool fastest and cube the slowest (c) plate will cool fastest and sphere the slowest (d) cube will cool fastest and plate the slowest Solution: The correct answer is (c) plate will cool fastest and sphere the slowest...
Read More →At what points on the curve
Question: At what points on the curve $y=x^{2}-4 x+5$ is the tangent perpendicular to the line $2 y+x=7 ?$ Solution: Let $\left(x_{1}, y_{1}\right)$ be the required point. Slope of the given line $=\frac{-1}{2}$ Slope of the line perpendicular to this line $=2$ Since, the point lies on the curve. Hence, $y_{1}=x_{1}{ }^{2}-4 x_{1}+5 \quad \ldots(1)$ Now, $y=x^{2}-4 x+5$ $\therefore \frac{d y}{d x}=2 x-4$ Now, Slope of the tangent at $\left(x_{1}, y_{1}\right)=\left(\frac{d y}{d x}\right)_{\left(...
Read More →The radius of a metal sphere at room
Question: The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is . The sphere is heated a little by a temperature ∆T so that its new Temperature is T + ∆T. The increase in the volume of the sphere is approximately (a) $2 \pi R \alpha \Delta T$ (b) $\pi R^{2} \alpha \Delta T$ (c) $4 \pi R^{3} \alpha \Delta T / 3$ (d) $4 \pi R^{3} \alpha \Delta T$ Solution: The correct answer is (d)$4 \pi R^{3} \alpha \Delta T$...
Read More →Heat is associated with
Question: Heat is associated with (a) kinetic energy of random motion of molecules (b) kinetic energy of orderly motion of molecules (c) total kinetic energy of random and orderly motion of molecules (d) kinetic energy of random motion in some cases and kinetic energy of orderly motion in other Solution: The correct answer is (a) kinetic energy of random motion of molecules...
Read More →Find the points on the curve
Question: Find the points on the curve $2 a^{2} y=x^{3}-3 a x^{2}$ where the tangent is parallel to $x$-axis. Solution: Let $\left(x_{1}, y_{1}\right)$ represent the required points. The slope of the $x$-axis is 0 . Here, $2 a^{2} y=x^{3}-3 a x^{2}$ Since, the point lies on the curve. Hence, $2 a^{2} y_{1}=x_{1}^{3}-3 a x_{1}^{2} \quad \ldots$ (1) Now, $2 a^{2} y=x^{3}-3 a x^{2}$ On differentiating both sides w.r.t. $x$, we get $2 a^{2} \frac{d y}{d x}=3 x^{2}-6 a x$ $\Rightarrow \frac{d y}{d x}...
Read More →As the temperature is increased,
Question: As the temperature is increased, the time period of a pendulum (a) increases as its effective length increases even though its centre of mass still remains at the centre of the bob (b) decreases as its effective length increases even though its centre of mass still remains at the centre of the bob (c) increases as its effective length increases due to shifting of the centre of mass below the centre of the bob (d) decreases as its effective length remains the same but the centre of mass...
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