Prove that
Question: Prove that $\frac{\sin x+\sin 3 x+\sin 5 x}{\cos x+\cos 3 x+\cos 5 x}=\tan 3 x$ Solution: $=\frac{\sin x+\sin 3 x+\sin 5 x}{\cos x+\cos 3 x+\cos 5 x}$ $=\frac{(\sin 5 x+\sin x)+\sin 3 x}{(\cos 5 x+\cos x)+\cos 3 x}$ $=\frac{2 \sin \frac{5 x+x}{2} \cos \frac{5 x-x}{2}+\sin 3 x}{2 \cos \frac{5 x^{2}+x}{2} \cos \frac{5 x-x}{2}+\cos 3 x}$ $=\frac{2 \sin 3 x \cos x+\sin 3 x}{2 \cos 3 x \cos x+\cos 3 x}$ $=\frac{\sin 3 x(2 \cos x+1)}{\cos 3 x(2 \cos x+1)}$ $=\tan 3 x$ Using the formula, $\si...
Read More →In a two dimensional motion,
Question: In a two dimensional motion, instantaneous speed v0is a positive constant. Then which of the following are necessarily true? (a) the acceleration of the particle is zero (b) the acceleration of the particle is bounded (c) the acceleration of the particle is necessarily in the plane of motion (d) the particle must be undergoing a uniform circular motion Solution: The correct answer is (d) the particle must be undergoing a uniform circular motion...
Read More →State Lagrange's mean value theorem.
Question: State Lagrange's mean value theorem. Solution: Lagrange's Mean Value Theorem: Let $f(x)$ be a function defined on $[a, b]$ such that (i) it is continuous on $[a, b]$ and (ii) it is differentiable on $(a, b)$. Then, there exists a real number $c \in(a, b)$ such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$....
Read More →Prove that
Question: Prove that $\frac{\cos 9 x-\cos 5 x}{\cos 17 x-\sin 3 x}=\frac{-\sin 2 x}{\cos 10 x}$ Solution: $=\frac{\cos 9 x-\cos 5 x}{\sin 17 x-\sin 3 x}$ $=\frac{-2 \sin \frac{9 x+5 x}{2} \sin \frac{9 x-5 x}{2}}{2 \cos \frac{17 x+3 x}{2} \sin \frac{17 x-3 x}{2}}$ $=\frac{-2 \sin 7 x \sin 2 x}{2 \cos 10 x \sin 7 x}$ $=\frac{-\sin 2 x}{\cos 10 x}$ Using the formula, $\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}$ $\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}$...
Read More →State Lagrange's mean value theorem.
Question: State Lagrange's mean value theorem. Solution: Lagrange's Mean Value Theorem: Let $f(x)$ be a function defined on $[a, b]$ such that (i) it is continuous on $[a, b]$ and (ii) it is differentiable on $(a, b)$. Then, there exists a real number $c \in(a, b)$ such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$....
Read More →In a two dimensional motion,
Question: In a two dimensional motion, instantaneous speed v0 is a positive constant. Then which of the following are necessarily true? (a) the average velocity is not zero at any time (b) average acceleration must always vanish (c) displacements in equal time intervals are equal (d) equal path lengths are traversed in equal intervals Solution: The correct answer is (d) equal path lengths are traversed in equal intervals...
Read More →State Rolle's theorem.
Question: State Rolle's theorem. Solution: Let $f$ be a real valued function defined on the closed interval $[a, b]$ such that (i) it is continuous on the closed interval $[a, b]$, (ii) it is differentiable on the open interval $(a, b)$, and (iii) $f(a)=f(b)$ Then, there exists a real number $c \in(a, b)$ such that $f^{\prime}(c)=0$....
Read More →Consider the quantities pressure, power, energy,
Question: Consider the quantities pressure, power, energy, impulse gravitational potential, electric charge, temperature, area. Out of these, the only vector quantities are (a) impulse, pressure, and area (b) impulse and area (c) area and gravitational potential (d) impulse and pressure Solution: The correct answer is b) impulse and area...
Read More →Prove that
Question: Prove that $\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}=\tan 4 x$ Solution: $\frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}$ $=\frac{2 \sin \frac{5 x+3 x}{2} \cos \frac{5 x-3 x}{2}}{2 \cos \frac{5 x^{2}+3 x}{2} \cos \frac{5 x-3 x}{2}}$ $=\frac{2 \sin 4 x \cos x}{2 \cos 4 x \cos x}$ $=\tan 4 x$ Using the formula, $\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$ $\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$...
Read More →If the solve the problem
Question: If $f(x)=A x^{2}+B x+C$ is such that $f(a)=f(b)$, then write the value of $c$ in Rolle's theorem. Solution: We have $f(x)=A x^{2}+B x+C$ Differentiating the given function with respect tox,we get $f^{\prime}(x)=2 A x+B$ $\Rightarrow f^{\prime}(c)=2 A c+B$ $\therefore f^{\prime}(c)=0 \Rightarrow 2 A c+B=0 \Rightarrow c=\frac{-B}{2 A}$ .....(1) $\because f(a)=f(b)$ $\therefore A a^{2}+B a+C=A b^{2}+b B+C$ $\Rightarrow A a^{2}+B a=A b^{2}+b B$ $\Rightarrow A\left(a^{2}-b^{2}\right)+B(a-b)...
Read More →The horizontal range of a projectile fired
Question: The horizontal range of a projectile fired at an angle of 15ois 50 m. If it is fired with the same speed at an angle of 45o, its range will be (a) 60 m (b) 71 m (c) 100 m (d) 141 m Solution: The correct answer is (c) 100 m...
Read More →If the solve the problem
Question: If $f(x)=A x^{2}+B x+C$ is such that $f(a)=f(b)$, then write the value of $c$ in Rolle's theorem. Solution: We have $f(x)=A x^{2}+B x+C$ Differentiating the given function with respect tox,we get $f^{\prime}(x)=2 A x+B$ $\Rightarrow f^{\prime}(c)=2 A c+B$ $\therefore f^{\prime}(c)=0 \Rightarrow 2 A c+B=0 \Rightarrow c=\frac{-B}{2 A}$ .....(1) $\because f(a)=f(b)$ $\therefore A a^{2}+B a+C=A b^{2}+b B+C$ $\Rightarrow A a^{2}+B a=A b^{2}+b B$ $\Rightarrow A\left(a^{2}-b^{2}\right)+B(a-b)...
Read More →Prove that
Question: Prove that $\frac{\sin 7 x-\sin 5 x}{\cos 7 x+\cos 5 x}=\tan x$ Solution: $\frac{\sin 7 x-\sin 5 x}{\cos 7 x+\cos 5 x}$ $=\frac{2 \cos \frac{7 x+5 x}{2} \sin \frac{7 x_{-} 5 x}{2}}{2 \cos \frac{7 x+5 x}{2} \cos \frac{7 x-5 x}{2}}$ $=\frac{2 \cos 6 x \sin x}{2 \cos 6 x \cos x}$ $=\frac{\sin x}{\cos x}$ $=\tan x$ Using the formula, $\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}$ $\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$...
Read More →The component of a vector r along
Question: The component of a vector r along X-axis will have maximum value if (a) r is along positive Y-axis (b) r is along positive X-axis (c) r makes an angle of 45owith the X-axis (d) r is along negative Y-axis Solution: The correct answer is (b) r is along positive X-axis...
Read More →Which one of the following statements is true?
Question: Which one of the following statements is true? (a) a scalar quantity is the one that is conserved in a process (b) a scalar quantity is the one that can never take negative values (c) a scalar quantity is the one that does not vary from one point to another in space (d) a scalar quantity has the same value for observers with different orientations of the axes Solution: The correct answer is d) a scalar quantity has the same value for observers with different orientations of the axes...
Read More →Prove that
Question: Prove that $\frac{\sin x+\sin 3 x}{\cos x-\cos 3 x}=\cot x$ Solution: $\frac{\sin x+\sin 3 x}{\cos x-\cos 3 x}$ $=\frac{2 \sin \frac{3 x+x}{2} \cos \frac{3 x-x}{2}}{-2 \sin \frac{x+3 x}{2} \sin \frac{x-3 x}{2}}$ $=\frac{2 \sin \frac{4 x}{2} \cos \frac{2 x}{2}}{2 \sin \frac{4 x}{2} \sin \frac{2 x}{2}}$ $=\frac{\cos x}{\sin x}$ $=\cot x$ Using the formula, $\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$ $\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}$...
Read More →Express each of the following as an algebraic sum of sines or cosines :
Question: Express each of the following as an algebraic sum of sines or cosines : (i) $2 \sin 6 x \cos 4 x$ (ii) $2 \cos 5 x \operatorname{din} 3 x$ (iii) $2 \cos 7 x \cos 3 x$ (iv) $2 \sin 8 x \sin 2 x$ Solution: (i) $2 \sin 6 x \cos 4 x=\sin (6 x+4 x)+\sin (6 x-4 x)$ $=\sin 10 x+\sin 2 x$ Using, $2 \sin A \cos B=\sin (A+B)+\sin (A-B)$ (ii) $2 \cos 5 x \sin 3 x=\sin (5 x+3 x)-\sin (5 x-3 x)$ $=\sin 8 x-\sin 2 x$ Using, $2 \cos A \sin B=\sin (A+B)-\sin (A-B)$ (iii) $2 \cos 7 x \cos 3 x=\cos (7 x...
Read More →The value of c in Rolle's theorem for the function
Question: The value of $c$ in Rolle's theorem for the function $f(x)=x^{3}-3 x$ in the interval $[0, \sqrt{3}]$ is Solution: The given function is $f(x)=x^{3}-3 x$. f(x) is a polynomial function. We know that a polynomial function is everywhere continuous and differentiable. So, $f(x)$ is continuous on $[0, \sqrt{3}]$ and differentiable on $(0, \sqrt{3})$. Also, $f(0)=0$ and $f(\sqrt{3})=(\sqrt{3})^{3}-3 \sqrt{3}=3 \sqrt{3}-3 \sqrt{3}=0$ $\therefore f(0)=f(\sqrt{3})$ Thus, all the conditions of ...
Read More →The value of c in Rolle's theorem for the function
Question: The value of $c$ in Rolle's theorem for the function $f(x)=x^{3}-3 x$ in the interval $[0, \sqrt{3}]$ is Solution: The given function is $f(x)=x^{3}-3 x$. f(x) is a polynomial function. We know that a polynomial function is everywhere continuous and differentiable. So, $f(x)$ is continuous on $[0, \sqrt{3}]$ and differentiable on $(0, \sqrt{3})$. Also, $f(0)=0$ and $f(\sqrt{3})=(\sqrt{3})^{3}-3 \sqrt{3}=3 \sqrt{3}-3 \sqrt{3}=0$ $\therefore f(0)=f(\sqrt{3})$ Thus, all the conditions of ...
Read More →Express each of the following as a product.
Question: Express each of the following as a product. 1. $\sin 10 x+\sin 6 x$ 2. $\sin 7 x-\sin 3 x$ 3. $\cos 7 x+\cos 5 x$ 4. $\cos 2 x-\cos 4 x$ Solution: 1. $\sin 10 x+\sin 6 x=2 \sin \frac{10 x+6 x}{2} \cos \frac{10 x-\sin x}{2}$ $=2 \sin \frac{18 x}{2} \cos \frac{4 x}{2}$ $=2 \sin 9 x \cos 2 x$ Using, $\sin (A+B)=\sin A \cos B+\cos A \sin B$ 2. $\sin 7 x-\sin 3 x=2 \cos \frac{7 x+3 x}{2} \sin \frac{7 x-3 x}{2}$ $=2 \cos \frac{10 x}{2} \sin \frac{4 x}{2}$ $=2 \cos 5 x \sin 2 x$ Using, $\sin ...
Read More →For the function
Question: For the function $f(x)=\log _{e} x, x \in[1,2]$, the value of $c$ for the lagrange's mean value theorem is____________ Solution: The given function is $f(x)=\log _{e} x$. Now, $f(x)=\log _{e} x$ is differentiable and so continuous for all $x0 .$ So, $f(x)$ is continuous on $[1,2]$ and differentiable on $(1,2)$. Thus, both the conditions of $L$ agrange's mean value theorem are satisfied. So, there must exist at least one real numberc (1, 2) such that $f^{\prime}(c)=\frac{f(2)-f(1)}{2-1}...
Read More →The angle between
Question: The angle between $A=\hat{i}+\hat{j} \quad$ and $B=\hat{i}-\hat{j}$ is (a) 40o (b) 90o (c) -45o (d) 180o Solution: The correct answer is (b) 90o...
Read More →A man is standing on top of a building 100 m high.
Question: A man is standing on top of a building 100 m high. He throws two balls vertically, one at t = 0 and other after a time interval. The later ball is thrown at a velocity of half the first. The vertical gap between first and second ball is +15m at t = 2s. The gap is found to remain constant. Calculate the velocity with which the balls were thrown and the exact time interval between their throw. Solution: Let the speed of ball 1 = u1= 2u m/s Then the speed of ball 2 = u2= u m/s The height ...
Read More →Prove that
Question: Prove that $\tan \left(\frac{\pi}{4}-x\right)=\frac{1-\tan x}{1+\tan x}$ Solution: In this question the following formulas will be used: $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$ $\tan \left(\frac{\pi}{4}-x\right)=\frac{\tan \frac{\pi}{4}-\tan x}{1+\tan \frac{\pi}{4} \tan x}$ $=\frac{1-\tan x}{1+\tan x} \because \tan \frac{\pi}{4}=1$...
Read More →If the function
Question: If the function $f(x)=x^{3}-6 x^{2}+a x+b$ defined on $[1,3]$ satisfies Roll's theorem for $c=2+\frac{1}{\sqrt{3}}$, thena= ___________,b = __________. Solution: The given function is $f(x)=x^{3}-6 x^{2}+a x+b$. It is given that $f(x)$ defined on $[1,3]$ satisfies Rolle's theorem for $c=2+\frac{1}{\sqrt{3}}$. $\therefore f(1)=f(3)$ and $f^{\prime}(c)=0$ Now, $f(1)=f(3)$ $\Rightarrow 1-6+a+b=27-54+3 a+b$ $\Rightarrow-5+a=-27+3 a$ $\Rightarrow 2 a=22$ $\Rightarrow a=11$ Also, $f(x)=x^{3}...
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