Write a value
Question: Write a value of $\int \mathrm{e}^{\mathrm{x}}(\sin \mathrm{x}+\cos \mathrm{x}) \mathrm{dx}$. Solution: we know $\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x \square=e^{x} f(x)+c$ Given, $\int e^{x}(\sin x+\cos x) d x$ Here $f(x)=\sin x$ and $f^{\prime}(x)=\cos x$ Therefore $\int e^{x}(\sin x+\cos x) d x=e^{x} \sin x+c$...
Read More →Find A–1 if and show that A-1 = (A2 – 3I)/ 2.
Question: Find A1if and show that A-1= (A2 3I)/ 2. $A=\left[\begin{array}{lll}0 1 1 \\ 1 0 1 \\ 1 1 0\end{array}\right]$ Solution: Given, $A=\left[\begin{array}{lll}0 1 1 \\ 1 0 1 \\ 1 1 0\end{array}\right]$ Co-factors are: $A_{11}=-1, A_{12}=1, A_{13}=1$ $A_{21}=1, A_{22}=-1, A_{23}=1$ $A_{31}=1, A_{31}=1, A_{32}=1 A_{33}=-1$ Now, $\operatorname{adj} A=\left[\begin{array}{ccc}-1 1 1 \\ 1 -1 1 \\ 1 1 -1\end{array}\right]^{T}=\left[\begin{array}{ccc}-1 1 1 \\ 1 -1 1 \\ 1 1 -1\end{array}\right]$ $...
Read More →Write a value
Question: Write a value of $\int \tan ^{3} x \sec ^{2} x d x$. Solution: let $\tan x=t$ Differentiating on both sides we get, $\sec ^{2} x d x=d t$ Substituting above equation in $\int \tan ^{3} x \sec ^{2} x d x$ we get, $=\int \mathrm{t}^{3} \mathrm{dt}$ $=\frac{t^{4}}{4}+c$ $=\frac{\tan ^{4} x}{4}+c$...
Read More →Solve this
Question: If $A=\left[\begin{array}{cccc}5 -2 6 1 \\ 7 0 8 -3 \\ \sqrt{2} \frac{3}{5} 4 3\end{array}\right]$ then write i. the number of rows in $\mathrm{A}$, ii. the number of columns in $A$, iii. the order of the matrix $\mathbf{A}$, iv. the number of all entries in A, v. the elements $a_{23}, a_{31}, a_{14}, a_{33}, a_{22}$ of $A$. Solution: (i) Number of rows $=3$ (ii) Number of columns $=4$ (iii) Order of matrix $=$ Number of rows $x$ Number of columns $=(3 \times 4)$ (iv) Number of entries...
Read More →Write a value
Question: Write a value of $\int x^{2} \sin x^{3} d x$. Solution: let $x^{3}=t$ Differentiating on both sides we get, $3 \mathrm{x}^{2} \mathrm{dx}=\mathrm{dt}$ $x^{2} d x=\frac{1}{3} d t$ substituting above equation in $\int x^{2} \sin x^{3} d x$ we get, $=\int \frac{1}{3} \sin t d t$ $=-\frac{1}{3} \cos t+c$ $=-\frac{1}{3} \cos x^{3}+c$...
Read More →Show that the DABC is an isosceles
Question: Show that the DABC is an isoscelestriangle if the determinant $\Delta=\left[\begin{array}{ccc}1 1 1 \\ 1+\cos A 1+\cos B 1+\cos C \\ \cos ^{2} A+\cos A \cos ^{2} B+\cos B \cos ^{2} C+\cos C\end{array}\right]=0$ Solution: Given, $\quad \Delta=\left|\begin{array}{ccc}1 1 1 \\ 1+\cos A 1+\cos B 1+\cos C \\ \cos ^{2} A+\cos A \cos ^{2} B+\cos B \cos ^{2} C+\cos C\end{array}\right|=0$ [Applying $C_{1} \rightarrow C_{1}-C_{3}$ and $C_{2} \rightarrow C_{2}-C_{3}$ ] $\Rightarrow\left|\begin{ar...
Read More →Write a value
Question: Write a value of $\int e^{3 \log x} x^{4} d x$. Solution: Consider $\int e^{3 \log x} x^{4}$ $e^{3 \log x}=e^{\log x^{3}}$ $=x^{3}$ $\int e^{3 \log x} x^{4}=\int x^{3} x^{4} d x$ $=\int x^{7} d x$ $=\frac{x^{8}}{8}+c$...
Read More →Write a value
Question: Write a value of $\int \frac{1+\cot \mathrm{x}}{\mathrm{x}+\log \sin \mathrm{x}} \mathrm{dx}$. Solution: let $x+\log \sin x=t$ Differentiating it on both sides we get, $(1+\cot x) d x=d t-i$ Given that $\int \frac{1+\cot x}{x+\log \sin x} d x$ Substituting $i$ in above equation we get, $=\int \frac{d t}{t}$ $=\log t+c$ $=\log (x+\log \sin x)+c$...
Read More →Show that the points (a + 5, a – 4), (a – 2, a + 3)
Question: Show that the points (a+ 5,a 4), (a 2,a+ 3) and (a,a) do not lie on a straight line for any value ofa. Solution: Given points are (a+ 5,a 4), (a 2,a+ 3) and (a,a). Now, we have to prove that these points do not lie on a straight line. So, if we prove that these points form a triangle then it cant line on a straight line. Area, $\Delta=\frac{1}{2}\left|\begin{array}{ccc}a+5 a-4 1 \\ a-2 a+3 1 \\ a a 1\end{array}\right|$ [Applying $R_{1} \rightarrow R_{1}-R_{3}$ and $R_{2} \rightarrow R_...
Read More →If a1, a2, a3, …, ar are in G.P.,
Question: Ifa1,a2,a3, ,arare in G.P., then prove that the determinant $\left|\begin{array}{lll}a_{r+1} a_{r+5} a_{r+9} \\ a_{r+7} a_{r+11} a_{r+15} \\ a_{r+11} a_{r+17} a_{r+21}\end{array}\right|$ is independent ofr. Solution: We know that, $a_{r+1}=A R^{(r+1)-1}=A R^{r}$ where $a_{r}=r$ th term of G.P., $A=$ First term of G.P. and $R=$ Common ratio of G.P. Now, $\left|\begin{array}{lll}a_{r+1} a_{r+5} a_{r+9} \\ a_{r+7} a_{r+11} a_{r+15} \\ a_{r+11} a_{r+17} a_{r+21}\end{array}\right|=\left|\be...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\log x^{x}}{x} d x$ Solution: given $\int \frac{\log x^{x}}{x} d x$ $=\int \frac{x \log x}{x} d x$ $=\int \log x$ $=x \log x-x+c$...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \sec ^{2}(7-4 x) d x$ Solution: let $7-4 x=t$ Differentiating on both sides we get, $-4 d x=d t$ $d x=-\frac{1}{4} d t$ substituting it in $\int \sec ^{2}(7-4 x) d x$ we get, $=\int-\frac{1}{4} \sec ^{2} t d t$ $=\tan t+c$ $=\tan (7-4 x)+c$...
Read More →If , then find values of x.
Question: If , then find values ofx. $\left[\begin{array}{ccc}4-x 4+x 4+x \\ 4+x 4-x 4+x \\ 4+x 4+x 4-x\end{array}\right]$ Solution: Given, $\quad\left|\begin{array}{lll}4-x 4+x 4+x \\ 4+x 4-x 4+x \\ 4+x 4+x 4-x\end{array}\right|=0$ [Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ ], we have $\Rightarrow \quad\left|\begin{array}{ccc}12+x 12+x 12+x \\ 4+x 4-x 4+x \\ 4+x 4+x 4-x\end{array}\right|=0$ Now, $\left[\right.$ Taking $(12+x)$ common from $\left.R_{1}\right]$ $\Rightarrow \quad(12+x)\left|...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{(1+\log x)^{2}}{x} d x$. Solution: let $1+\log x=t$ Differentiating on both sides we get, $\frac{1}{x} d x=d t$ Substituting it in $\int \frac{(1+\log x)^{2}}{x}$ we get, $=\int t^{2} d t$ $=\frac{t^{3}}{3}+c$ $=\frac{(1+\log x)^{3}}{3}+c$...
Read More →Find the value of q satisfying
Question: Find the value of q satisfying $\left[\begin{array}{ccc}1 1 \sin 3 \theta \\ -4 3 \cos 2 \theta \\ 7 -7 -2\end{array}\right]=0$ Solution: Given, $\left|\begin{array}{ccc}1 1 \sin 3 \theta \\ -4 3 \cos 2 \theta \\ 7 -7 -2\end{array}\right|=0$ On expanding along $\mathrm{C}_{3}$, we have $\sin 3 \theta \times(28-21)-\cos 2 \theta \times(-7-7)-2(3+4)=0$ $7 \sin 3 \theta+14 \cos 2 \theta-14=0$ $\sin 3 \theta+2 \cos 2 \theta-2=0$ $\left(3 \sin \theta-4 \sin ^{3} \theta\right)+2\left(1-2 \si...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$. Solution: let $\sqrt{x}=t$ Differentiating on both sides we get, $\frac{1}{2 \sqrt{x}} d x=d t$ $\frac{1}{\sqrt{x}} d x=2 \mathrm{dt}$ substituting it in $\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$ we get, $=\int 2 \cos t d t$ $=2 \sin t+c$ $=2 \sin \sqrt{x}+c$...
Read More →If the co-ordinates of the vertices of
Question: If the co-ordinates of the vertices ofan equilateral triangle with sides of length a are (x1,y1), $\left|\begin{array}{lll}x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1\end{array}\right|^{2}=\frac{3 a^{4}}{4} .$ (x2,y2), (x3,y3), then Solution: We know that, the area of a triangle with vertices $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ is given by $\Delta=\frac{1}{2}\left|\begin{array}{lll}x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \\ x_{3} y_{3} 1\end...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\sin \sqrt{x}}{\sqrt{x}} \mathrm{dx}$ Solution: let $\sqrt{x}=t$ Differentiating on both sides we get, $\frac{1}{2 \sqrt{x}} d x=d t$ $\frac{1}{\sqrt{x}} d x=2 d t$ substituting it in $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$ we get, $=\int 2 \sin t d t$ $=-2 \cos t+c$ $=-2 \cos \sqrt{x}+c$...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\sec ^{2} \sqrt{x}}{\sqrt{x}} \mathrm{dx}$ Solution: let $\sqrt{x}=t$ Differentiating on both sides we get, $\frac{1}{2 \sqrt{x}} d x=d t$ $\frac{1}{\sqrt{x}} d x=2 d t$ substituting it in $\int \frac{\sec ^{2} \sqrt{x}}{\sqrt{x}} d x$ we get, $=\int 2 \sec ^{2} t d t$ $=2 \tan t+c$ $=2 \tan \sqrt{x}+c$...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: Domain of $\sec ^{-1} x$ is A. $[-1,1]$ B. $R-\{0\}$ C. $R-[-1,1]$ D. $R-\{-1,1\}$ Solution: To Find: The Domain of $\sec ^{-1}(x)$ Here,the inverse function is given by $y=\mathrm{f}^{-1}(x)$ The graph of the function $y=\sec ^{-1}(x)$ can be obtained from the graph of $Y=\sec x$ by interchanging $x$ and $y$ axes.i.e, if $(a, b)$ is a point on $Y=\sec x$ then $(b, a)$ is the point on the function $y=\sec ^{-1}(x)$ Below is the...
Read More →If A + B + C = 0, then prove that
Question: If A + B + C = 0, then prove that $\left|\begin{array}{ccc}1 \cos C \cos B \\ \cos C 1 \cos A \\ \cos B \cos A 1\end{array}\right|=0$ Solution: Given, $\left|\begin{array}{ccc}1 \cos C \cos B \\ \cos C 1 \cos A \\ \cos B \cos A 1\end{array}\right|$ On finding the determinant, we have $=1\left(1-\cos ^{2} A\right)-\cos C(\cos C-\cos A \cdot \cos B)+\cos B(\cos C \cdot \cos A-\cos B)$ $=\sin ^{2} A-\cos ^{2} C+\cos A \cdot \cos B \cdot \cos C+\cos A \cdot \cos B \cdot \cos C-\cos ^{2} B$...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\mathrm{x}^{2}+4 \mathrm{x}}{\mathrm{x}^{3}+6 \mathrm{x}^{2}+5} \mathrm{dx}$ Solution: let $x^{3}+6 x^{2}+5=t$ Differentiating on both sides we get, $\left(3 x^{2}+12 x\right) d x=d t$ $3\left(x^{2}+4 x\right) d x=d t$ $\left(x^{2}+4 x\right) d x=\frac{1}{3} d t$ Substituting it in $\int \frac{x^{2}+4 x}{x^{3}+6 x^{2}+5} d x$ we get, $=\int \frac{1}{3 t} d t$ $=\frac{1}{3 \log \left(x^{3}+6 x^{2}+5\right)}+c$...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: Domain of $\cos -1 \times$ is A. $[0,1]$ B. $[-1,1]$ C. $[-1,0]$ D. None of these Solution: To Find: The Domain of $\cos ^{-1}(x)$ Here,the inverse function of $\cos$ is given by $\mathrm{y}=\mathrm{f}^{-1}(x)$ The graph of the function $y=\cos ^{-1}(x)$ can be obtained from the graph of $Y=\cos x$ by interchanging $x$ and $y$ axes.i.e, if $(a, b)$ is a point on $Y=\cos x$ then $(b, a)$ is the point on the function $y=\cos ^{-1...
Read More →Evaluate the following integrals:
Question: Evaluate: $\int \frac{\mathrm{x}^{2}}{1+\mathrm{x}^{3}}$ Solution: let $1+x^{3}=t$ Differentiating on both sides we get, $3 x^{2} d x=d t$ $x^{2} d x=\frac{1}{3} d t$ substituting it in $\int \frac{x^{2}}{1+x^{3}} d x$ we get, $=\int \frac{1}{3 t} d t$ $=\frac{1}{3} \log t+c$ $=\frac{1}{3} \log \left(1+x^{3}\right)+c$...
Read More →Prove the equation
Question: $\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|=(a-1)^{3}$ Solution: Given, $\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|$ [Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{2} \rightarrow R_{2}-R_{3}$ ] $=\left|\begin{array}{ccc}a^{2}-1 a-1 0 \\ 2 a-2 a-1 0 \\ 3 3 1\end{array}\right|$ Now, [Taking $(a-1)$ common from $R_{1}$ and $R_{2}$ ] $(a-1)^{2}\left|\begin{array}{ccc}a+1 1 0 \\ 2 1 0 \\ 3 3 1\end{array}\ri...
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