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Question: $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ Solution: $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ $\therefore \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)=\sin ^{-1}(\sin 2 \theta)=2 \theta$\ $\Rightarrow \int \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x=\int 2 \theta \cdot \sec ^{2} \theta d \theta=2 \int \theta \cdot \sec ^{2} \theta d \theta$ Integrating by parts, we obtain $2\left[\theta \cdot \int \sec ^{2} \theta d...
Read More →Find the value of k for which each of the following system of equations have no solution :
Question: Find the value ofkfor which each of the following system of equations have no solution : $x+2 y=0$ $2 x+k y=5$ Solution: GIVEN: $x+2 y=0$ $2 x+k y=5$ To find: To determine for what value ofkthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{1}{2}=\frac{2}{k} \neq \frac{0}{5}$ $\frac{1}{2}=\frac{2}{k}$ $k=4$ Hence for $k...
Read More →Find the value of k for which each of the following system of equations have no solution :
Question: Find the value ofkfor which each of the following system of equations have no solution : $k x-5 y=2$ $6 x+2 y=7$ Solution: GIVEN: $k x-5 y=2$ $6 x+2 y=7$ To find: To determine for what value ofkthe system of equation has no solution We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ Here, $\frac{k}{6}=\frac{-5}{2} \neq \frac{2}{7}$ $\frac{k}{6}=\frac{-5}{2}$ $2 k=-30$ $k...
Read More →Plot the following points on the graph paper:
Question: Plot the following points on the graph paper: (i) (2, 5) (ii) (4,- 3) (iii) (- 5,- 7) (iv) (7, - 4) (v) (-3, 2) (vi) (7, 0) (vii) (- 4, 0) (viii) (0, 7) (ix) (0, - 4) (x) (0, 0) Solution: The given points are, A (2, 5), B (4, -3), C (-5, -7), D (7, - 4), E (- 3, 2), F (7, 0), G (- 4, 0), H (0, 7), I (0, - 4), J (0, 0) Let X 'OX and Y ' OY be the coordinate axes. (i) Here for the given point the abscissa is 2 units and ordinate is 5 units. The point is in the first quadrant. So it will ...
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Question: $e^{2 x} \sin x$ Solution: Let $I=\int e^{2 x} \sin x d x$ ...(1) Integrating by parts, we obtain $I=\sin x \int e^{2 x} d x-\int\left\{\left(\frac{d}{d x} \sin x\right) \int e^{2 x} d x\right\} d x$ $\Rightarrow I=\sin x \cdot \frac{e^{2 x}}{2}-\int \cos x \cdot \frac{e^{2 x}}{2} d x$ $\Rightarrow I=\frac{e^{2 x} \sin x}{2}-\frac{1}{2} \int e^{2 x} \cos x d x$ Again integrating by parts, we obtain $I=\frac{e^{2 x} \cdot \sin x}{2}-\frac{1}{2}\left[\cos x \int e^{2 x} d x-\int\left\{\l...
Read More →Find the domain of each of the following real valued functions of real variable:
Question: Find the domain of each of the following real valued functions of real variable: (i) $f(x)=\frac{1}{x}$ (ii) $f(x)=\frac{1}{x-7}$ (iii) $f(x)=\frac{3 x-2}{x+1}$ (iv) $f(x)=\frac{2 x+1}{x^{2}-9}$ (v) $f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$ Solution: (i) Given: $f(x)=\frac{1}{x}$ Domain off: We observe thatf(x) is defined for allxexcept atx= 0. At $x=0, f(x)$ takes the intermediate form $\frac{1}{0}$. Hence, domain (f) =R{ 0 } (ii) Given: $f(x)=\frac{1}{(x-7)}$ Domain off: Clearly,f(x) i...
Read More →Find the value of k for which each of the following system of equations have infinitely many solutions :
Question: Find the value ofkfor which each of the following system of equations have infinitely many solutions : $2 x+3 y=k$ $(k-1) x+(k+2) y=3 k$ Solution: GIVEN: $2 x+3 y=k$ $(k-1) x+(k+2) y=3 k$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution Here, $\frac{2}{(k-1)}=\frac{3}{(k+2)}=\frac{k}{3 k}$ Consider the following to find outk $\fra...
Read More →Find the domain of each of the following real valued functions of real variable:
Question: Find the domain of each of the following real valued functions of real variable: (i) $f(x)=\frac{1}{x}$ (ii) $f(x)=\frac{1}{x-7}$ (iii) $f(x)=\frac{3 x-2}{x+1}$ (iv) $f(x)=\frac{2 x+1}{x^{2}-9}$ (v) $f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$ Solution: (i) Given: $f(x)=\frac{1}{x}$ Domain off: We observe thatf(x) is defined for allxexcept atx= 0. At $x=0, f(x)$ takes the intermediate form $\frac{1}{0}$. Hence, domain (f) =R{ 0 } (ii) Given: $f(x)=\frac{1}{(x-7)}$ Domain off: Clearly,f(x) i...
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Question: $\frac{(x-3) e^{x}}{(x-1)^{3}}$ Solution: $\int e^{x}\left\{\frac{x-3}{(x-1)^{3}}\right\} d x=\int e^{x}\left\{\frac{x-1-2}{(x-1)^{3}}\right\} d x$ $=\int e^{x}\left\{\frac{1}{(x-1)^{2}}-\frac{2}{(x-1)^{3}}\right\} d x$ Let $f(x)=\frac{1}{(x-1)^{2}} \Rightarrow f^{\prime}(x)=\frac{-2}{(x-1)^{3}}$ It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$ $\therefore \int e^{x}\left\{\frac{(x-3)}{(x-1)^{2}}\right\} d x=\frac{e^{x}}{(x-1)^{2}}+C$...
Read More →Find the value of k for which each of the following system of equations have infinitely many solutions :
Question: Find the value of k for which each of the following system of equations have infinitely many solutions : $2 x+3 y=7$ $(k+1) x+(2 k-1) y=4 k+1$ Solution: GIVEN: $2 x+3 y=7$ $(k+1) x+(2 k-1) y=4 k+1$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here $\frac{2}{(k+1)}=\...
Read More →If for non-zero x,
Question: If for non-zero $x, \operatorname{af}(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5$, where $a \neq b$, then find $f(x)$ Solution: Given: $a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5$ ...(i) $\Rightarrow a f\left(\frac{1}{x}\right)+b f(x)=\frac{1}{\frac{1}{x}}-5$ $\Rightarrow a f\left(\frac{1}{x}\right)+b f(x)=x-5$ ....(2) On adding equations (i) and (ii), we get: $a f(x)+b f(x)+b f\left(\frac{1}{x}\right)+a f\left(\frac{1}{x}\right)=\frac{1}{x}-5+x-5$ $\Rightarrow(a+b) f(x)+(a+b) ...
Read More →Plot the following points on the graph paper:
Question: Plot the following points on the graph paper: (i) (2, 5) (ii) (4,- 3) (iii) (- 5,- 7) (iv) (7, - 4) (v) (-3, 2) (vi) (7, 0) (vii) (- 4, 0) (viii) (0, 7) (ix) (0, - 4) (x) (0, 0) Solution: The given points are, A (2, 5), B (4, -3), C (-5, -7), D (7, - 4), E (- 3, 2), F (7, 0), G (- 4, 0), H (0, 7), I (0, - 4), J (0, 0) Let X 'OX and Y ' OY be the coordinate axes. (i) Here for the given point the abscissa is 2 units and ordinate is 5 units. The point is in the first quadrant. So it will ...
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Question: $e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$ Solution: Let $I=\int e^{x}\left[\frac{1}{x}-\frac{1}{x^{2}}\right] d x$ Also, let $\frac{1}{x}=f(x) \Rightarrow f^{\prime}(x)=\frac{-1}{x^{2}}$ It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$ $\therefore I=\frac{e^{x}}{x}+\mathrm{C}$...
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Question: $e^{x}\left(\frac{1+\sin x}{1+\cos x}\right)$ Solution: $e^{x}\left(\frac{1+\sin x}{1+\cos x}\right)$ $=e^{x}\left(\frac{\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}+2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}\right)$ $=\frac{e^{x}\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}}{2 \cos ^{2} \frac{x}{2}}$ $=\frac{1}{2} e^{x} \cdot\left(\frac{\sin \frac{x}{2}+\cos \frac{x}{2}}{\cos \frac{x}{2}}\right)^{2}$ $=\frac{1}{2} e^{x}\left[\tan \frac{x}{2}+1\right]^{2}$ $=\fra...
Read More →If f(x) =
Question: Iff(x) = (axn)1/n,a 0 andn N, then prove thatf(f(x)) =xfor allx. Solution: Given: f(x) = (axn)1/n,a 0 Now, f{f(x)} =f(axn)1/n $=\left[a-\left\{\left(a-x^{n}\right)^{1 / m}\right\}^{n}\right]^{1 / n}$ $=\left[a-\left(a-x^{n}\right)\right]^{1 / n}$ $\left.=\left[a-a+x^{n}\right)\right]^{1 / n}=\left(x^{n}\right)^{1 / n}=x^{(n \times 1 / n)}=x$ Thus,f(f(x)) =x. Hence proved....
Read More →If f(x) =
Question: Iff(x) = (axn)1/n,a 0 andn N, then prove thatf(f(x)) =xfor allx. Solution: Given: f(x) = (axn)1/n,a 0 Now, f{f(x)} =f(axn)1/n $=\left[a-\left\{\left(a-x^{n}\right)^{1 / m}\right\}^{n}\right]^{1 / n}$ $=\left[a-\left(a-x^{n}\right)\right]^{1 / n}$ $\left.=\left[a-a+x^{n}\right)\right]^{1 / n}=\left(x^{n}\right)^{1 / n}=x^{(n \times 1 / n)}=x$ Thus,f(f(x)) =x. Hence proved....
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Question: $\frac{x e^{x}}{(1+x)^{2}}$ Solution: Let $I=\int \frac{x e^{x}}{(1+x)^{2}} d x=\int e^{x}\left\{\frac{x}{(1+x)^{2}}\right\} d x$ $=\int e^{x}\left\{\frac{1+x-1}{(1+x)^{2}}\right\} d x$ $=\int e^{x}\left\{\frac{1}{1+x}-\frac{1}{(1+x)^{2}}\right\} d x$ Let $f(x)=\frac{1}{1+x} \Rightarrow f^{\prime}(x)=\frac{-1}{(1+x)^{2}}$ $\Rightarrow \int \frac{x e^{x}}{(1+x)^{2}} d x=\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x$ It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^...
Read More →If f(x)=
Question: If $f(x)=\frac{x-1}{x+1}$, then show that (i) $f\left(\frac{1}{x}\right)=-f(x)$ (ii) $f\left(-\frac{1}{x}\right)=-\frac{1}{f(x)}$ Solution: Given: $f(x)=\frac{x-1}{x+1}$ ...(1) (i) Replacing $x$ by $\frac{1}{x}$ in $(1)$, we get $f\left(\frac{1}{x}\right)=\frac{\frac{1}{x}-1}{\frac{1}{x}+1}$ $=\frac{1-x}{1+x}$ $=-\frac{x-1}{x+1}$ $=-f(x)$ (ii) Replacing $x$ by $-\frac{1}{x}$ in (1), we get $f\left(-\frac{1}{x}\right)=\frac{-\frac{1}{x}-1}{-\frac{1}{x}+1}$ $=\frac{-1-x}{-1+x}$ $=-\frac{...
Read More →Find the value of $k$ for which each of the following system of equations have infinitely many solutions:
Question: Find the value of $k$ for which each of the following system of equations have infinitely many solutions: $2 x+(k-2) y=k$ $6 x+(2 k-1) y=2 k+5$ Solution: GIVEN: $2 x+(k-2) y=k$ $6 x+(2 k-1) y=2 k+5$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{1}}=\frac{c_{1}}{c_{2}}$ Here, $\frac{2}{6}=\fr...
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Question: $e^{x}(\sin x+\cos x)$ Solution: Let $I=\int e^{x}(\sin x+\cos x) d x$ Let $f(x)=\sin x$ $\Rightarrow f^{\prime}(x)=\cos x$ $\therefore I=\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x$ It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$ $\therefore I=e^{x} \sin x+\mathrm{C}$...
Read More →If f(x)=
Question: If $f(x)=\frac{2 x}{1+x^{2}}$, show that $f(\tan \theta)=\sin 2 \theta$ Solution: Given: $f(x)=\frac{2 x}{1+x^{2}}$ Thus, $f(\tan \theta)=\frac{2(\tan \theta)}{1+\tan ^{2} \theta}$ $=\frac{2 \times \frac{\sin \theta}{\cos \theta}}{1+\left(\frac{\sin ^{2} \theta}{\cos ^{2} \theta}\right)}$ $=\frac{2 \sin \theta}{\cos \theta} \times \frac{\cos ^{2} \theta}{\cos ^{2} \theta+\sin ^{2} \theta}$ $=\frac{2 \sin \theta \cos \theta}{1} \quad\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\rig...
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Question: $\left(x^{2}+1\right) \log x$ Solution: Let $I=\int\left(x^{2}+1\right) \log x d x=\int x^{2} \log x d x+\int \log x d x$ Let $I=I_{1}+I_{2} \ldots$ ...(1) Where, $I_{1}=\int x^{2} \log x d x$ and $I_{2}=\int \log x d x$ $I_{1}=\int x^{2} \log x d x$ Taking $\log x$ as first function and $x^{2}$ as second function and integrating by parts, we obtain $I_{1}=\log x-\int x^{2} d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x^{2} d x\right\} d x$ $=\log x \cdot \frac{x^{3}}{3}-\int ...
Read More →Fill in the blanks to make the following statements true.
Question: Fill in the blanks to make the following statements true. (i) In a right triangle the hypotenuse is the ___ side. (ii) The sum of three altitudes of a triangle is ___ than its perimeter. (iii) The sum of any two sides of a triangle is ___ than the third side. (iv) If two angles of a triangle are unequal, then the smaller angle has the ___ side opposite to it. (v) Difference of any two sides of a triangle is ___ than the third side. (vi) If two sides of a triangle are unequal, then the ...
Read More →Find the value of k for which each of the following system of equations have infinitely many solutions :
Question: Find the value ofkfor which each of the following system of equations have infinitely many solutions : $k x+3 y=2 k+1$ $2(k+1) x+9 y=7 k+1$ Solution: GIVEN: $k x+3 y=2 k+1$ $2(k+1) x+9 y=7 k+1$ To find: To determine for what value ofkthe system of equation has infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here, $\frac{k}{2(k+1)}=\fr...
Read More →If f(x) =
Question: If $f(x)=x^{3}-\frac{1}{x^{3}}$, show that $f(x)+f\left(\frac{1}{x}\right)=0$ Solution: Given: $f(x)=x^{3}-\frac{1}{x^{3}} \quad \ldots(\mathrm{i})$ Thus, $f\left(\frac{1}{x}\right)=\left(\frac{1}{x}\right)^{3}-\frac{1}{\left(\frac{1}{x}\right)^{3}}$ $=\frac{1}{x^{3}}-\frac{1}{\frac{1}{x^{3}}}$ $\therefore f\left(\frac{1}{x}\right)=\frac{1}{x^{3}}-x^{3} \ldots$ (ii) $f(x)+f\left(\frac{1}{x}\right)=\left(x^{3}-\frac{1}{x^{3}}\right)+\left(\frac{1}{x^{3}}-x^{3}\right)$ $=x^{3}-\frac{1}{x...
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