Prove that
Question: Prove that $8 \cos ^{3} 20^{\circ}-6 \cos 20^{\circ}=1$ Solution: To Prove: $8 \cos ^{3} 20^{\circ}-6 \cos 20^{\circ}=1$ Taking LHS, $=8 \cos ^{3} 20^{\circ}-6 \cos 20^{\circ}$ Taking 2 common, we get $=2\left(4 \cos ^{3} 20^{\circ}-3 \cos 20^{\circ}\right) \ldots$ (i) We know that, $\cos 3 x=4 \cos ^{3} x-3 \cos x$ Here, x = 20 So, eq. (i) becomes $=2\left[\cos 3\left(20^{\circ}\right)\right]$ $=2\left[\cos 60^{\circ}\right]$ $=2 \times \frac{1}{2}\left[\because \cos \left(60^{\circ}\...
Read More →If the tangent to the curve
Question: If the tangent to the curve $y=x^{3}+a x+b$ at $(1,-6)$ is parallel to the line $x-y+5=0$, find $a$ and $b$. Solution: Given: $x-y+5=0$ $\Rightarrow y=x+5$ $\Rightarrow \frac{d y}{d x}=1$ Now, $y=x^{3}+a x+b \quad \ldots(1)$ $\Rightarrow \frac{d y}{d x}=3 x^{2}+a$ Slope of the tangent at $(1,-6)=$ Slope of the given line $\Rightarrow\left(\frac{d y}{d x}\right)_{(1,-6)}=1$ $\Rightarrow 3+a=1$ $\Rightarrow a=-2$ On substituting $a=-2, x=1$ and $y=-6$ in eq. (1), we get $-6=1-2+b$ $\Righ...
Read More →If the tangent to the curve
Question: If the tangent to the curve $y=x^{3}+a x+b$ at $(1,-6)$ is parallel to the line $x-y+5=0$, find $a$ and $b$. Solution: Given: $x-y+5=0$ $\Rightarrow y=x+5$ $\Rightarrow \frac{d y}{d x}=1$\ Now, $y=x^{3}+a x+b \quad \ldots(1)$ $\Rightarrow \frac{d y}{d x}=3 x^{2}+a$ Slope of the tangent at $(1,-6)=$ Slope of the given line $\Rightarrow\left(\frac{d y}{d x}\right)_{(1,-6)}=1$ $\Rightarrow 3+a=1$ $\Rightarrow a=-2$ On substituting $a=-2, x=1$ and $y=-6$ in eq. (1), we get $-6=1-2+b$ $\Rig...
Read More →Consider two cylindrical rods of identical dimensions,
Question: Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass M is attached to each of the free ends at the centre of the rods. (a) both the rods will elongate but there shall be no perceptible change in shape (b) the steel rod will elongate and change shape but the rubber rod will only elongate (c) the steel rod will elongate without any perceptible change in shape, but the rubber rod will el...
Read More →A rigid bar of mass M is supported symmetrically
Question: A rigid bar of mass M is supported symmetrically by three wires each of length l. Those at each end are of copper and the middle one is of iron. The ratio of their diameter, if each is to have the same tension, is equal to (a) $Y_{\text {copper }} / Y_{\text {iron }}$ (b)$\sqrt{\frac{Y_{i r o n}}{Y_{\text {copper }}}}$ (c)$\frac{Y_{\text {iron }}^{2}}{Y_{\text {copper }}^{2}}$ (d)$\frac{Y_{\text {iron }}}{Y_{\text {copper }}}$ Solution: The correct answer is (b) $\sqrt{\frac{Y_{\text {...
Read More →Prove that
Question: Prove that $2 \cos ^{2} 15^{0}-1=\frac{\sqrt{3}}{2}$ Solution: To Prove: $2 \cos ^{2} 15^{\circ}-1=\frac{\sqrt{3}}{2}$ Taking LHS, $=2 \cos ^{2} 15^{\circ}-1 \ldots(\mathrm{i})$ We know that, $1+\cos 2 x=2 \cos ^{2} x$ Here, $x=15^{\circ}$ So, eq. (i) become $=\left[1+\cos 2\left(15^{\circ}\right)\right]-1$ $=1+\cos 30^{\circ}-1$ $=\cos 30^{\circ}\left[\because \cos \left(30^{\circ}\right)=\frac{\sqrt{3}}{2}\right]$ $=\frac{\sqrt{3}}{2}$ = RHS LHS = RHS Hence Proved...
Read More →A spring is stretched by applying
Question: A spring is stretched by applying a load to its free end. The strain produced in the spring is (a) volumetric (b) shear (c) longitudinal and shear (d) longitudinal Solution: The correct answer is (c) longitudinal and shear...
Read More →The temperature of a wire is doubled.
Question: The temperature of a wire is doubled. The Youngs modulus of elasticity (a) will also double (b) will become four times (c) will remain same (d) will decrease Solution: The correct answer is (d) will decrease...
Read More →Prove that
Question: Prove that $2 \sin 22 \frac{1^{0}}{2} \cos 22 \frac{1^{0}}{2}=\frac{1}{\sqrt{2}}$ Solution: To Prove: $2 \sin 22 \frac{1^{\circ}}{2} \cos 22 \frac{1^{\circ}}{2}=\frac{1}{\sqrt{2}}$ Taking LHS, $=2 \sin 22 \frac{1}{2}^{\circ} \cos 22 \frac{1}{2}^{\circ} \ldots$ (i) We know that, $2 \sin x \cos x=\sin 2 x$ Here, $\mathrm{x}=22 \frac{1}{2}=\frac{45}{2}$ So, eq. (i) become $=\sin 2\left(\frac{45}{2}\right)$ $=\sin 45^{\circ}$ $=\frac{1}{\sqrt{2}}\left[\because \sin \left(45^{\circ}\right)=...
Read More →The maximum load a wire can withstand
Question: The maximum load a wire can withstand without breaking when its length is reduced to half of its original length, will (a) be doubled (b) be half (c) be four times (d) remain same Solution: The correct answer is (d) remains same...
Read More →Find the values of a and b if the slope
Question: Find the values of $a$ and $b$ if the slope of the tangent to the curve $x y+a x+b y=2$ at $(1,1)$ is 2 . Solution: Given : $x y+a x+b y=2 \ldots$ .....(1) On differentiating both sides w.r.t. $x$, we get $x \frac{d y}{d x}+y+a+b \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}(x+b)=-a-y$ $\Rightarrow \frac{d y}{d x}=\frac{-a-y}{x+b}$ Now, $\left(\frac{d y}{d x}\right)_{(1,1)}=2$ $\Rightarrow \frac{-a-1}{1+b}=2$ $\Rightarrow-a-1=2+2 b$ $\Rightarrow-a=3+2 b$ $\Rightarrow a=-(3+2 b)$ On s...
Read More →Modulus of rigidity of ideal liquid is
Question: Modulus of rigidity of ideal liquid is (a) infinity (b) zero (c) unity (d) some finite small non-zero constant value Solution: The correct answer is (b) zero...
Read More →Prove that
Question: Prove that $\cos x \cos 2 x \cos 4 x \cos 8 x=\frac{\sin 16 x}{16 \sin x}$ Solution: To Prove: $\cos \mathrm{x} \cos 2 \mathrm{x} \cos 4 \mathrm{x} \cos 8 \mathrm{x}=\frac{\sin 16 \mathrm{x}}{16 \sin \mathrm{x}}$ Taking LHS, = cosx cos2x cos4x cos8x Multiply and divide by 2sinx, we get $=\frac{1}{2 \sin x}[2 \sin x \cos x \cos 2 x \cos 4 x \cos 8 x]$ $=\frac{1}{2 \sin x}[(2 \sin x \cos x) \cos 2 x \cos 4 x \cos 8 x]$ $=\frac{1}{2 \sin x}[\sin 2 x \cos 2 x \cos 4 x \cos 8 x][\because \s...
Read More →A satellite is to be placed in equatorial geostationary
Question: A satellite is to be placed in equatorial geostationary orbit around the earth for communication (a) calculate height of such a satellite (b) find out the minimum number of satellites that are needed to cover entire earth so that at least one satellites is visible from any point on the equator Solution: (a) Mass of the earth, M = 6 1024 kg Radius of the earth, R = 6.4 103 m Time period = 24 36 102 s G = 6.67 10-11 N.m2/kg2 Orbital radius = R + h Using orbital velocity, we can calculate...
Read More →Find the slopes of the tangent and the normal to the following curves at the indicted points:
Question: Find the slopes of the tangent and the normal to the following curves at the indicted points: (i) $y=\sqrt{x^{3}}$ at $x=4$ (ii) $y=\sqrt{x}$ at $x=9$ (iii) $y=x^{3}-x$ at $x=2$ (iv) $y=2 x^{2}+3 \sin x$ at $x=0$ (v) $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$ at $\theta=-\pi / 2$ (vi) $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$ at $\theta=\pi / 4$ (vii) $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$ at $\theta=\pi / 2$ (viii) $y=(\sin 2 x+\cot x+2)^{2}$ at $x=\pi / 2$ (ix) $x^{2}+3 ...
Read More →Find the slopes of the tangent and the normal to the following curves at the indicted points:
Question: Find the slopes of the tangent and the normal to the following curves at the indicted points: (i) $y=\sqrt{x^{3}}$ at $x=4$ (ii) $y=\sqrt{x}$ at $x=9$ (iii) $y=x^{3}-x$ at $x=2$ (iv) $y=2 x^{2}+3 \sin x$ at $x=0$ (v) $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$ at $\theta=-\pi / 2$ (vi) $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$ at $\theta=\pi / 4$ (vii) $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$ at $\theta=\pi / 2$ (viii) $y=(\sin 2 x+\cot x+2)^{2}$ at $x=\pi / 2$ (ix) $x^{2}+3 ...
Read More →An astronaut inside a small spaceship orbiting
Question: An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity? Solution: When the size of the space station orbiting around the earth is large, the astronaut will experience variation and this is because of acceleration due to gravity....
Read More →We can shield a charge from electric fields by
Question: We can shield a charge from electric fields by putting it side a hollow conductor. Can we shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means? Solution: No, it is possible to shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere as gravitation is independent of the medium....
Read More →How is the gravitational force between
Question: How is the gravitational force between two point masses affected when they are dipped in the water keeping the separation between them the same? Solution: The gravitational force is unaffected by the medium, and it remains the same between the point masses irrespective of their surrounding. Therefore, when the two-point masses are dipped in water, the gravitational force between them remains the same....
Read More →Give one example each of central
Question: Give one example each of central force and non-central force. Solution: Central force: Electrostatic force acting on the point charge Non-central force: Nuclear force between the atoms...
Read More →Molecules in the air in the atmosphere
Question: Molecules in the air in the atmosphere are attracted by the gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree. Solution: Molecules in the air in the atmosphere are attracted by the gravitational force of the earth but they do not fall into the earth because they are in random motion whereas apple has a downward motion....
Read More →The centre of mass of an extended body
Question: The centre of mass of an extended body on the surface of the earth and its centre of gravity (a) are always at the same point for any size of the body (b) are always at the same point only for spherical bodies (c) can never be at the same point (d) is close to each other for objects, say of sizes less than 100 m (e) both can change if the object is taken deep inside the earth Solution: The correct answer is (d) is close to each other for objects, say of sizes less than 100 m...
Read More →Which of the following are true?
Question: Which of the following are true? (a) a polar satellite goes around the earths pole in a north-south direction (b) a geostationary satellite goes around the earth in an east-west direction (c) a geostationary satellite goes around the earth in a west-east direction (d) a polar satellite goes around the earth in an east-west direction Solution: The correct answers are (a) a polar satellite goes around the earths pole in a north-south direction (c) a geostationary satellite goes around th...
Read More →Supposing Newton’s law of gravitation
Question: Supposing Newtons law of gravitation for gravitation forces F1 and F2 between two masses m1 and m2 at positions r1 and r2 read $F_{1}=-F_{2}=-\frac{r_{12}}{r_{12}^{3}} G M_{0}^{2}\left(\frac{m_{1} m_{2}}{M_{1}^{2}}\right)^{n} \quad$ where $\mathrm{Mo}$ is a constant of the dimension of mass, $\mathrm{r}_{12}=$ r1 r2and n is a number. In such a case, (a) the acceleration due to gravity on earth will be different for different object (b) none of the three laws of Kepler will be valid (c)...
Read More →Prove that
Question: Prove that $\frac{1-\cos 2 x+\sin x}{\sin 2 x+\cos x}=\tan x$ Solution: To prove: $\frac{1-\cos 2 x+\sin x}{\sin 2 x+\cos x}=\tan x$ Taking LHS, $=\frac{1-\cos 2 x+\sin x}{\sin 2 x+\cos x}$ $=\frac{(1-\cos 2 x)+\sin x}{\sin 2 x+\cos x}$ We know that, $1-\cos 2 x=2 \sin ^{2} x \ \sin 2 x=2 \sin x \cos x$ $=\frac{2 \sin ^{2} x+\sin x}{2 \sin x \cos x+\cos x}$ Taking sinx common from the numerator and cosx from the denominator $=\frac{\sin x(2 \sin x+1)}{\cos x(2 \sin x+1)}$ $=\frac{\sin ...
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