Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x$ Solution: re-writing the given equation as $\int \frac{1+\frac{1}{x^{2}}}{x^{2}+1+\frac{1}{x^{2}}} d x$ $\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+3} d x$ Let $\mathrm{x}-\frac{1}{\mathrm{x}}$ as $\mathrm{t}$ $\left(1+\frac{1}{x^{2}}\right)=d t$ $\int \frac{1}{t^{2}+3} d t$ Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$ $\frac{1}{\sqrt{3}} \arctan \left(\frac{t}{\sqrt{3}}\right)+c$ Sub...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{\left(x^{2}+1\right)\left(x^{2}+4\right)}{\left(x^{2}+3\right)\left(x^{2}-5\right)} d x$ Solution: $\frac{\left(x^{2}+1\right)\left(x^{2}+4\right)}{\left(x^{2}+3\right)\left(x^{2}-5\right)}=\frac{x^{4}+5 x^{2}+4}{x^{4}-2 x^{2}-15}$ $=\frac{\left(x^{4}-2 x^{2}-15\right)+7 x^{2}+19}{x^{4}-2 x^{2}-15}$ $=1+\frac{7 x^{2}+19}{x^{4}-2 x^{2}-15}$ Now, $\frac{7 x^{2}+19}{x^{4}-2 x^{2}-15}=\frac{7 x^{2}+19}{\left(x^{2}+3\right)\left(x^{2}-5\right)}$ ...
Read More →Find the matrix A satisfying
Question: Find the matrix A satisfying the matrix equation: $\left[\begin{array}{ll}2 1 \\ 3 2\end{array}\right] \mathrm{A}\left[\begin{array}{cc}-3 2 \\ 5 -3\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ Solution: Given, $\left[\begin{array}{ll}2 1 \\ 3 2\end{array}\right] \mathrm{A}\left[\begin{array}{cc}-3 2 \\ 5 -3\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ Let $\mathrm{P}=\left[\begin{array}{ll}2 1 \\ 3 2\end{array}\right]_{\mathrm{Q}}=\l...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: The principal value of $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ is A. $\frac{-\pi}{4}$ B. $\frac{\pi}{4}$ C. $\frac{3 \pi}{4}$ D. $\frac{5 \pi}{4}$ Solution: To Find: The Principle value of $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ Let the principle value be given by $x$ Now, let $x=\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ $\Rightarrow \cos x=\frac{-1}{\sqrt{2}}$ $\Rightarrow \cos x=-\cos \left(\frac{\pi}{4}\right)\le...
Read More →Show that satisfies the equation
Question: Show that satisfies the equation A2 3A 7I = 0 and hence find A-1. Solution: Given, $A=\left[\begin{array}{cc}5 3 \\ -1 -2\end{array}\right]$ So, $A^{2}=A \cdot A=\left[\begin{array}{cc}5 3 \\ -1 -2\end{array}\right] \cdot\left[\begin{array}{cc}5 3 \\ -1 -2\end{array}\right]=\left[\begin{array}{cc}25-3 15-6 \\ -5+2 -3+4\end{array}\right]=\left[\begin{array}{cc}22 9 \\ -3 1\end{array}\right]$ $3 A=3\left[\begin{array}{cc}5 3 \\ -1 -2\end{array}\right]=\left[\begin{array}{cc}15 9 \\ -3 -6...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: The principal value of $\operatorname{cosec}^{-1}(2)$ is A. $\frac{\pi}{3}$ B. $\frac{\pi}{6}$ C. $\frac{2 \pi}{3}$ D. $\frac{5 \pi}{6}$ Solution: To Find: The Principle value of $\operatorname{cosec}^{-1}(2)$ Let the principle value be given by $x$ Now, let $x=\operatorname{cosec}^{-1}(2)$ $\Rightarrow \operatorname{cosec} x=2$ $\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(\frac{\pi}{6}\right)\left(\because \co...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: Solution: Let, I $=\int \frac{x^{2}}{x^{4}+x^{2}-2} d x$ Let, $\frac{\mathrm{x}^{2}}{\mathrm{x}^{4}+\mathrm{x}^{2}-2}=\frac{\mathrm{A}}{\mathrm{x}+1}+\frac{\mathrm{B}}{\mathrm{x}-1}+\frac{\mathrm{C}}{\mathrm{x}^{2}+2}$ $\Rightarrow x^{2}=A(x-1)\left(x^{2}+2\right)+B(x+1)\left(x^{2}+2\right)+C\left(x^{2}-1\right)$ For, $x=1, A=\frac{1}{6}$ For, $x=-1, B=-\frac{1}{6}$ For, $x=0, C=-\frac{2}{3}$ $\therefore \mathrm{I}=\frac{1}{6} \int \frac{\mathrm{dx}}{\m...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: The principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is A. $\frac{\pi}{6}$ B. $\frac{5 \pi}{6}$ C. $\frac{7 \pi}{6}$ D. none of these Solution: To Find:The Principle value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ Let the principle value be given by $\mathrm{x}$ Now, let $x=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ $\Rightarrow \cos x=\frac{\sqrt{3}}{2}$ $\Rightarrow \cos x=\cos \left(\frac{\pi}{6}\right)\le...
Read More →Find the value of x if
Question: Find the value of x if $\left[\begin{array}{lll}1 x 1\end{array}\right]\left[\begin{array}{ccc}1 3 2 \\ 2 5 1 \\ 15 3 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=\mathrm{O}$ Solution: Given, $\left[\begin{array}{lll}1 x 1\end{array}\right]\left[\begin{array}{ccc}1 3 2 \\ 2 5 1 \\ 15 3 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=O$ $\Rightarrow\left[\begin{array}{lll}1+2 x+15 3+5 x+3 2+x+2\end{array}\right]\left[\begin{array}{l}1 \\ ...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}}{1-x^{4}} d x$ Solution: Let, $\mathrm{I}=\int \frac{\mathrm{x}^{2}}{1-\mathrm{x}^{4}} \mathrm{dx}$ Let, $\frac{\mathrm{x}^{2}}{1-\mathrm{x}^{4}}=\frac{\mathrm{A}}{1-\mathrm{x}}+\frac{\mathrm{B}}{1+\mathrm{x}}+\frac{\mathrm{C}}{1+\mathrm{x}^{2}}$ $\Rightarrow x^{2}=A(1+x)\left(x^{2}+1\right)+B(1-x)\left(x^{2}+1\right)+c(x+1)(1-x)$ For, $x=1, A=\frac{1}{4}$ For, $\mathrm{x}=-1, \mathrm{~B}=\frac{1}{4}$ For, $x=0, C=-\frac{1}{2}$ $\there...
Read More →If show that
Question: If show that(A + B) (A B) A2 B2 Solution: Given, $A=\left[\begin{array}{ll}0 1 \\ 1 1\end{array}\right]$ and $B=\left[\begin{array}{cc}0 -1 \\ 1 0\end{array}\right]$ So, $(A+B)=\left[\begin{array}{cc}0+0 1-1 \\ 1+1 1+0\end{array}\right]=\left[\begin{array}{ll}0 0 \\ 2 1\end{array}\right]$ And, $(A-B)=\left[\begin{array}{ll}0-0 1+1 \\ 1-1 1-0\end{array}\right]=\left[\begin{array}{ll}0 2 \\ 0 1\end{array}\right]$ $(A+B) \cdot(A-B)=\left[\begin{array}{ll}0 0 \\ 2 1\end{array}\right]\left[...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\operatorname{cosec}^{-1} x$ Solution: Principal value branch of $\operatorname{cosec}^{-1} \times$ is $\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]$...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}}{x^{4}-x^{2}-12} d x$ Solution: $\frac{x^{2}}{x^{4}-x^{2}-12}=\frac{x^{2}}{\left(x^{2}-4\right)\left(x^{2}+3\right)}$ Let, $\frac{\mathrm{x}^{2}}{\left(\mathrm{x}^{2}-4\right)\left(\mathrm{x}^{2}+3\right)}=\frac{\mathrm{A}}{\mathrm{x}-2}+\frac{\mathrm{B}}{\mathrm{x}+2}+\frac{\mathrm{C}}{\mathrm{x}^{2}+3}$ $\Rightarrow x^{2}=A(x+2)\left(x^{2}+3\right)+B(x-2)\left(x^{2}+3\right)+C(x-2)(x+2)$ For, $x=2, A=\frac{1}{7}$ For, $x=-2, B=-\frac...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\sec ^{-1} x$ Solution: Principal value branch of $\sec ^{-1} \times$ is $\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]$...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\cot ^{-1} x$ Solution: Principal value branch of $\cot ^{-1} \mathrm{x}$ is $(0, \pi)$...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{4}}{(x-1)\left(x^{2}+1\right)} d x$ Solution: $\frac{x^{4}}{(x-1)\left(x^{2}+1\right)}=\frac{x^{4}}{x^{3}-x^{2}+x-1}$ $=\frac{x\left(x^{3}-x^{2}+x-1\right)+1\left(x^{3}-x^{2}+x-1\right)+1}{x^{3}-x^{2}+x-1}$ $=x+1+\frac{1}{(x-1)\left(x^{2}+1\right)}$ Now, let $\frac{1}{(x-1)\left(x^{2}+1\right)}=\frac{A}{x-1}+\frac{B x+C}{x^{2}+1}$ $\Rightarrow 1=A\left(x^{2}+1\right)+(B x+C)(x-1)$ For, $\mathrm{x}=1, \mathrm{~A}=\frac{1}{2}$ For, $\mathrm...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\tan ^{-1} x$ Solution: Principal value branch of $\tan ^{-1} \mathrm{x}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$...
Read More →Find non-zero values of x satisfying
Question: Find non-zero values ofxsatisfying the matrix equation: $x\left[\begin{array}{cc}2 x 2 \\ 3 x\end{array}\right]+2\left[\begin{array}{ll}8 5 x \\ 4 4 x\end{array}\right]=2\left[\begin{array}{cc}\left(x^{2}+8\right) 24 \\ (10) 6 x\end{array}\right]$ Solution: Given, $x\left[\begin{array}{cc}2 x 2 \\ 3 x\end{array}\right]+2\left[\begin{array}{ll}8 5 x \\ 4 4 x\end{array}\right]=2\left[\begin{array}{cc}x^{2}+8 24 \\ 10 6 x\end{array}\right]$ $\Rightarrow$$\left[\begin{array}{cc}2 x^{2} 2 x...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\cos ^{-1} x$ Solution: Principal value branch of $\cos ^{-1} \mathrm{x}$ is $[0, \pi]$...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{4 x^{4}+3}{\left(x^{2}+2\right)\left(x^{2}+3\right)\left(x^{2}+4\right)} d x$ Solution: Let $\mathrm{I}=\int \frac{4 \mathrm{x}^{4}+3}{\left(\mathrm{x}^{2}+2\right)\left(\mathrm{x}^{2}+3\right)\left(\mathrm{x}^{2}+4\right)} \mathrm{dx}$ Let $x^{2}=y$ $\therefore \frac{4 \mathrm{x}^{4}+3}{\left(\mathrm{x}^{2}+2\right)\left(\mathrm{x}^{2}+3\right)\left(\mathrm{x}^{2}+4\right)}=\frac{4 \mathrm{y}^{2}+3}{(\mathrm{y}+2)(\mathrm{y}+3)(\mathrm{y}+4...
Read More →Write down the interval for the principal-value branch of each of the following functions and draw its graph:
Question: Write down the interval for the principal-value branch of each of the following functions and draw its graph: $\sin ^{-1} x$ Solution: Principal value branch of $\sin ^{-1} \mathrm{x}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$...
Read More →If (i) X + Y
Question: If (i) X + Y (ii) 2X 3Y (iii) A matrix Z such that X + Y + Z is a zero matrix. Solution: Given, $X=\left[\begin{array}{ccc}3 1 -1 \\ 5 -2 -3\end{array}\right]_{2 \times 3}$ and $Y=\left[\begin{array}{ccc}2 1 -1 \\ 7 2 4\end{array}\right]_{2 \times 3}$ (i) $X+Y=\left[\begin{array}{ccc}3+2 1+1 -1-1 \\ 5+7 -2+2 -3+4\end{array}\right]=\left[\begin{array}{ccc}5 2 -2 \\ 12 0 1\end{array}\right]$ (ii) $2 X-3 Y=2\left[\begin{array}{ccc}3 1 -1 \\ 5 -2 -3\end{array}\right]-3\left[\begin{array}{c...
Read More →Solve for x :
Question: Solve for x : $\sin ^{-1} x-\cos ^{-1} x=\frac{\pi}{6}$ Solution: Given: $\sin ^{-1} x-\cos ^{-1} x=\frac{\pi}{6}$ We know that $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$ So, $\sin ^{-1} \mathrm{x}=\frac{\pi}{2}-\cos ^{-1} \mathrm{X}$ Substituting in the given equation, $\frac{\pi}{2}-\cos ^{-1} x-\cos ^{-1} x=\frac{\pi}{6}$ Rearranging, $2 \cos ^{-1} x=\frac{\pi}{2}-\frac{\pi}{6}$ $2 \cos ^{-1} x=\frac{\pi}{3}$ $\cos ^{-1} x=\frac{\pi}{6}$ $x=\frac{\sqrt{3}}{2}$ Therefore, $x=\frac{\sq...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{\left(x^{2}+1\right)\left(x^{2}+2\right)}{\left(x^{2}+3\right)\left(x^{2}+4\right)} d x$ Solution: $\frac{\left(x^{2}+1\right)\left(x^{2}+2\right)}{\left(x^{2}+3\right)\left(x^{2}+4\right)}=\frac{x^{4}+3 x^{2}+2}{x^{4}+7 x^{2}+12}$ $=\frac{\left(x^{4}+7 x^{2}+12\right)-4 x^{2}-10}{x^{4}+7 x^{2}+12}$ $=1-\frac{4 x^{2}+10}{x^{4}+7 x^{2}+12}$ Now, $\frac{4 x^{2}+10}{x^{4}+7 x^{2}+12}=\frac{4 x^{2}+10}{\left(x^{2}+3\right)\left(x^{2}+4\right)}$ ...
Read More →If possible, find the sum of
Question: If possible, find the sum of the matrices A and B, where A = $\left[\begin{array}{cc}\sqrt{3} 1 \\ 2 3\end{array}\right]$ and $\mathrm{B}=$ $\left[\begin{array}{lll}x y z \\ a b 6\end{array}\right]$ Solution: The given two matrices A and B are of different orders. Two matrices can be added only if order of both the matrices is same. Thus, the sum of matrices A and B is not possible. $X=\left[\begin{array}{ccc}3 1 -1 \\ 5 -2 -3\end{array}\right]$ and $Y=\left[\begin{array}{ccc}2 1 -1 \\...
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