Evaluate the following integrals:
Question: Evaluate $\int \frac{\log (\log x)}{x} d x$ Solution: Let, $\log x=t$ Differentiating both side with respect to $t$ $\frac{1}{x} \frac{d x}{d t}=1 \Rightarrow \frac{d x}{x}=d t$ Note:- Always use direct formula for $\int \log x d x$ $y=\int \log t d t$ $y=t \log t-t+c$ Again, put $t=\log x$ $y=(\log x) \log (\log x)-\log x+c$...
Read More →Compute A B and B A, which ever exists when
Question: Compute $A B$ and $B A$, which ever exists when Solution: Matrix A is of order $1 \times 4$ and Matrix $B$ is of order $4 \times 1$ To find : matrices $A B$ and $B A$ Formula used : Where $c_{i j}=a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+a_{i 3} b_{3 j}+\ldots \ldots \ldots \ldots \ldots . .+a_{i n} b_{n j}$ If $A$ is a matrix of order $a \times b$ and $B$ is a matrix of order $c \times d$, then matrix $A B$ exists and is of order $a \times d$, if and only if $b=$ $c$ If $A$ is a matrix of orde...
Read More →A swimming pool is to be drained for cleaning.
Question: A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pooltseconds after the pool has been plugged off to drain and L = 200 (10 t)2. How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds? Solution: Given, L = 200(10 t)2where L represents the number of liters of water in the pool. On differentiating both the sides w.r.t, t, we get dL/dt = 200 x 2(10 t) (-1)...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \log _{10} \mathrm{x} \mathrm{dx}$ Solution: Use the method of integration by parts $y=\int 1 \times \log _{10} x d x$ $y=\log _{10} x \int d x-\int \frac{d}{d x} \log _{10} x\left(\int d x\right) d x$ $y=x \log _{10} x-\int x \frac{1}{x \log _{e} 10} d x$ $y=x \log _{10} x-\frac{x}{\log _{e} 10}+c$ $y=x\left(\log _{e} x-1\right) \log _{10} e+c$...
Read More →A man, 2m tall, walks at the rate of m/s towards
Question: A man, 2m tall, walks at the rate of m/s towards a street light which is m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the $3 \frac{1}{3}$ shadow changing when he is m from the base of the light? Solution: Let AB is the height of street light post and CD is the height of the man such that AB = 5(1/3) = 16/3 m and CD = 2 m Let BC = x length (the distance of the man from the lamp post) And CE = y is the length of the shadow of the man at ...
Read More →Solve this following
Question: Compute $A B$ and $B A$, which ever exists when $A=\left[\begin{array}{rrr}0 1 -5 \\ 2 4 0\end{array}\right]$ and $B=\left[\begin{array}{cc}1 3 \\ -1 0 \\ 0 5\end{array}\right]$ Solution: Given : $A=\left[\begin{array}{rrr}0 1 -5 \\ 2 4 0\end{array}\right]$ and $B=\left[\begin{array}{cc}1 3 \\ -1 0 \\ 0 5\end{array}\right]$ Matrix A is of order $2 \times 3$ and Matrix $B$ is of order $3 \times 2$ To find : matrices $A B$ and $B A$ Formula used : Where $c_{i j}=a_{i 1} b_{1 j}+a_{i 2} b...
Read More →Evaluate the following integrals:
Question: Evaluate $\int\left(1+x^{2}\right) \cos 2 x d x$ Solution: $y=\int \cos 2 x+x^{2} \cos 2 x d x$ $A=\int \cos 2 x d x$ $A=\frac{\sin 2 x}{2}+c_{1}$ $B=\int x^{2} \cos 2 x d x$ Use the method of integration by parts $B=x^{2} \int \cos 2 x d x-\int \frac{d}{d x}\left(x^{2}\right)\left(\int \cos 2 x d x\right) d x$ $B=x^{2} \frac{\sin 2 x}{2}-\int x \sin 2 x d x$ $B=x^{2} \frac{\sin 2 x}{2}-\left(x \int \sin 2 x d x-\int \frac{d}{d x}(x)\left(\int \sin 2 x d x\right)\right.$ $B=x^{2} \frac...
Read More →Find the approximate volume of metal in a hollow spherical
Question: Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively. Solution: Given, The internal radius r = 3 cm And, external radius R = r + ∆r =3.0005 cm ∆r = 3.0005 3 = 0.0005 cm Let y = r3⇒ y + ∆y = (r + ∆r)3= R3= (3.0005)3 Differentiating both sides w.r.t., r, we get $\frac{d y}{d r}=3 r^{2}$ So, $\Delta y=\frac{d y}{d r} \times \Delta r=3 r^{2} \times 0.0005$ $=3 \times(3)^{2} \times 0.0005=27 \times 0.0005=0.0...
Read More →Compute A B and BA, which ever exists when
Question: Compute $A B$ and BA, which ever exists when $A=\left[\begin{array}{ll}-1 1 \\ -2 2 \\ -3 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}3 -2 1 \\ 0 1 2 \\ -3 4 -5\end{array}\right]$ Solution: Given : $A=\left[\begin{array}{cc}-1 1 \\ -2 2 \\ -3 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}3 -2 1 \\ 0 1 2 \\ -3 4 -5\end{array}\right]$ Matrix $A$ is of order $3 \times 2$, and Matrice $B$ is of order $3 \times 3$ To find: matrix $A B$ and $B A$ Formula used: Where $c_{i j}=a_...
Read More →Find the approximate value
Question: Find the approximate value of (1.999)5. Solution: (1.999)5= (2 0.001)5 Let x = 2 and ∆x = -0.001 Also, let y = x5 Differentiating both sides w.r.t, x, we get dy/dx = 5x4= 5(2)4= 80 Now, ∆y = (dy/dx). ∆x = 80. (-0.001) = -0.080 And, (1.999)5= y + ∆y = x5 0.080 = (2)5 0.080 = 32 0.080 = 31.92 Therefore, approximate value of (1.999)5is 31.92...
Read More →Find an angle q, 0 < q < /2,
Question: Find an angle q, 0 q /2, which increases twice as fast as its sine. Solution: According to the question, we have $\frac{d \theta}{d t}=2 \frac{d}{d t}(\sin \theta)$ $\Rightarrow \frac{d \theta}{d t}=2 \cos \theta \cdot \frac{d \theta}{d t} \Rightarrow 1=2 \cos \theta$ So, $\cos \theta=\frac{1}{2} \Rightarrow \cos \theta=\cos \frac{\pi}{3} \Rightarrow \theta=\frac{\pi}{3}$ Therefore, the required angle is /3....
Read More →Solve this following
Question: Compute $\mathrm{AB}$ and BA, which ever exists when $A=\left[\begin{array}{cc}2 -1 \\ 3 0 \\ -1 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 3 \\ 0 4\end{array}\right]$ Solution: Given : $A=\left[\begin{array}{cc}2 -1 \\ 3 0 \\ -1 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 3 \\ 0 4\end{array}\right]$ Matrix A is of order $3 \times 2$, and Matrix $B$ is of order $2 \times 2$ To find: matrix $\mathrm{AB}$ and $\mathrm{BA}$ Formula used: Where $c_{i j}=a_{i 1} b_{1 j}...
Read More →Two men A and B start with velocities v at the same time
Question: Two men A and B start with velocitiesvat the same time from the junction of two roads inclined at 45 to each other. If they travel by different roads, find the rate at which they are being separated. Solution: Lets consider P to be any point at which the two roads are inclined at an angle of 45o. Now, two men A and B are moving along the roads PA and PB respectively with same speed V. APB = 45oand they move with the same speed. So, ∆APB is an isosceles triangle. Now, draw PQ AB. We hav...
Read More →Evaluate the following integrals:
Question: Evaluate $\int(2 x+3) \sqrt{4 x^{2}+5 x+6} d x$ Solution: Make perfect square of quadratic equation $4 x^{2}+5 x+6=4\left[\left(x+\frac{5}{8}\right)^{2}+\frac{71}{64}\right]$ $y=2 \int(2 x+3) \sqrt{\left[\left(x+\frac{5}{8}\right)^{2}+\left(\frac{\sqrt{71}}{8}\right)^{2}\right]} d x$ Let, $x+\frac{5}{8}=t \Rightarrow x=t-\frac{5}{8}$ Differentiate both side with respect to $t$ $\frac{d x}{d t}=1 \Rightarrow d x=d t$ $y=2 \int\left(2 t+\frac{7}{4}\right) \sqrt{\left[t^{2}+\left(\frac{\s...
Read More →A kite is moving horizontally at a height
Question: A kite is moving horizontally at a height of $151.5$ meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m. Solution: Speed of the kite(V) = 10 m/s Let FD be the height of the kite and AB be the height of the kite and AB be the height of the boy. Now, let AF = x m So, BG = AF = x And, dx/dt = 10 m/s From the figure, its seen that GD = DF GF = DF AB = (151.5 1.5) m = 150 m...
Read More →If the area of a circle increases at a uniform rate,
Question: If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius. Solution: We know that the area of circle, A = r2, where r = radius of the circle And, perimeter = 2r According to the question, we have $\frac{d \mathrm{~A}}{d t}=\mathrm{K}$, where $\mathrm{K}=$ constant $\Rightarrow \frac{d}{d t}\left(\pi r^{2}\right)=\mathrm{K} \Rightarrow \pi \cdot 2 r \cdot \frac{d r}{d t}=\mathrm{K}$ So, $\frac{d r}{d t}=\frac{\mathrm{K}}{2 \pi r}$ Now,...
Read More →Solve this following
Question: If $\left[\begin{array}{cc}x-y 2 y \\ 2 y+z x+y\end{array}\right]=\left[\begin{array}{ll}1 4 \\ 9 5\end{array}\right]$ then write the value of $(x+y)$. Solution: If $\left[\begin{array}{ll}a b \\ c d\end{array}\right]=\left[\begin{array}{ll}e f \\ g h\end{array}\right]$ Then $a=e, b=f, c=g, d=h$ Given, $\left[\begin{array}{cc}x-y 2 y \\ 2 y+z x+y\end{array}\right]=\left[\begin{array}{cc}1 4 \\ 9 5\end{array}\right]$ So, $x-y=1, x+y=5,2 y=4$ and $2 y+z=9$ Therefore, $x+y=5$ Conclusion: ...
Read More →Solve this following
Question: Find the value of $(x+y)$ from the following equation : $2\left[\begin{array}{ll}1 3 \\ 0 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}\right]=\left[\begin{array}{ll}5 6 \\ 1 8\end{array}\right]$ Solution: Given $2\left[\begin{array}{ll}1 3 \\ 0 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}\right]=\left[\begin{array}{ll}5 6 \\ 1 8\end{array}\right]$ ${\left[\begin{array}{cc}2 6 \\ 0 2 x\end{array}\right]+\left[\begin{array}{ll}y 0 \\ 1 2\end{array}...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \mathrm{x} \sqrt{1+\mathrm{x}-\mathrm{x}^{2}} \mathrm{dx}$ Solution: Make perfect square of quadratic equation $1+x-x^{2}=\frac{5}{4}-\left(x^{2}-2\left(\frac{1}{2}\right)(x)+\left(\frac{1}{2}\right)^{2}\right)$ $=\left(\frac{\sqrt{5}}{2}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}$ $y=\int x \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}} d x$ Let, $x-\frac{1}{2}=t \Rightarrow x=t+\frac{1}{2}$ Differentiate both side with respect to $\mathrm{t}$ ...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \sqrt{1+2 \mathrm{x}-3 \mathrm{x}^{2}} \mathrm{dx}$ Solution: Make perfect square of quadratic equation $1+2 x-3 x^{2}=3\left[-\left(x^{2}-\frac{2}{3} x-\frac{1}{3}\right)\right]$ $=3\left[\frac{4}{9}-\left(x^{2}-2\left(\frac{1}{3}\right)(x)+\left(\frac{1}{3}\right)^{2}\right)\right]$ $=3\left[\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}\right]$ $y=\sqrt{3} \int\left[\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}\right] d x$ Using formula, $\i...
Read More →A spherical ball of salt is dissolving in water in such
Question: A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate. Solution: Given, a spherical ball of salt Then, the volume of ball V = 4/3 r3where r = radius of the ball Now, according to the question we have dV/dt S, where S = surface area of the ball $\frac{d}{d t}\left(\frac{4}{3} \pi r^{3}\right) \propto 4 \pi r^{2}$ $\left[\because S=4 \pi...
Read More →Evaluate the following integrals:
Question: Evaluate $\int \sqrt{3 \mathrm{x}^{2}+4 \mathrm{x}+1} \mathrm{dx}$ Solution: Make perfect square of quadratic equation $3 x^{2}+4 x+1=3\left(x^{2}+\frac{4}{3} x+\frac{1}{3}\right)$ $=3\left(x^{2}+2\left(\frac{2}{3}\right)(x)+\left(\frac{2}{3}\right)^{2}-\frac{1}{9}\right)$ $=3\left[\left(x+\frac{2}{3}\right)^{2}-\frac{1}{9}\right]$ $y=\int \sqrt{3\left[\left(x+\frac{2}{3}\right)^{2}-\frac{1}{9}\right]} d x$ $y=\sqrt{3} \int \sqrt{\left[\left(x+\frac{2}{3}\right)^{2}-\frac{1}{9}\right]}...
Read More →Statement :1 If three forces
Question: Statement :1 If three forces $\overrightarrow{\mathrm{F}}_{1}, \overrightarrow{\mathrm{F}}_{2}$ and $\overrightarrow{\mathrm{F}}_{3}$ are represented by three sides of a triangle and $\overrightarrow{\mathrm{F}}_{1}+\overrightarrow{\mathrm{F}}_{2}=-\overrightarrow{\mathrm{F}}_{3}$, then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement : II A triangle made up of three forces $\overrightarrow{\mathrm{F}}_{1}, \overrightarrow{\mathrm{F}}_{2}$ a...
Read More →The equivalent resistance
Question: The equivalent resistance of the given circuit between the terminals $\mathrm{A}$ and $\mathrm{B}$ is : $0 \Omega$$3 \Omega$$\frac{9}{2} \Omega$$1 \Omega$Correct Option: , 4 Solution: $R_{e q}=\frac{3 \times 3 / 2}{3+3 / 2}=\frac{9 / 2}{9 / 2}=1 \Omega$...
Read More →The magnetic field vector of an electromagnetic
Question: The magnetic field vector of an electromagnetic wave is given by $B=B_{0} \frac{\hat{i}+\hat{j}}{\sqrt{2}} \cos (k z-\omega t)$; where $\hat{i}, \hat{j}$ represents unit vector along $x$ and $y$-axis respectively. At $\mathrm{t}=0 \mathrm{~s}$, two electric charges $\mathrm{q}_{1}$ of $4 \pi$ coulomb and $\mathrm{q}_{2}$ of $2 \pi$ coulomb located at $\left(0,0, \frac{\pi}{\mathrm{k}}\right)$ and $\left(0,0, \frac{3 \pi}{\mathrm{k}}\right)$, respectively, have the same velocity of $0.5...
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