Write a value
Question: Write a value of $\int \tan x \sec ^{3} x d x$. Solution: given $\int \tan x \sec ^{3} x d x$ $=\int(\tan x \sec x) \sec ^{2} x d x$ Let $\sec x=t$ Differentiating on both sides we get, $\tan x \sec x d x=d t$ Substituting above equation in $\int \tan x \sec ^{3} x d x$ we get, $=\int t^{2} d t$ $=\frac{t^{3}}{3}+c$ $=\frac{\sec ^{3} x}{3}+c$...
Read More →If x + y + z = 0, prove that
Question: Ifx+y+z= 0, prove that Solution: $\begin{aligned}\text { Taking, } \\\text { L.H.S. }\end{aligned}=\left|\begin{array}{lll}x a y b z c \\ y c z a x b \\ z b x c y a\end{array}\right|$ [Expanding] $=x a\left(a^{2} y z-x^{2} b c\right)-y b\left(y^{2} a c=b^{2} x z\right)+z c\left(c^{2} x y-z^{2} a b\right)$ $=x y z a^{3}-x^{3} a b c-y^{3} a b c+b^{3} x y z+c^{3} x y z-z^{3} a b c$ $=x y z\left(a^{3}+b^{3}+c^{3}\right)-a b c\left(x^{3}+y^{3}+z^{3}\right)$ $=x y z\left(a^{3}+b^{3}+c^{3}\ri...
Read More →Write a value
Question: Write a value of $\int \cos ^{4} x \sin x d x$ Solution: let $\cos x=t$ Differentiating on both sides we get, $-\sin x d x=d t$ Substituting above equation in $\int \cos ^{4} x \sin x d x$ we get, $=\int-t^{4} d t$ $=-\frac{t^{5}}{5}+c$ $=-\frac{\cos ^{5} x}{5}+c$...
Read More →Prove that is divisible by
Question: Prove that is divisible bya+b+cand find the quotient. $\left|\begin{array}{lll}b c-a^{2} c a-b^{2} a b-c^{2} \\ c a-b^{2} a b-c^{2} b c-a^{2} \\ a b-c^{2} b c-a^{2} c a-b^{2}\end{array}\right|$ Solution: $\Delta=\left|\begin{array}{ccc}b c-a^{2} c a-b^{2} a b-c^{2} \\ c a-b^{2} a b-c^{2} b c-a^{2} \\ a b-c^{2} b c-a^{2} c a-b^{2}\end{array}\right|$ Now, [Applying $C_{1} \rightarrow C_{1}-C_{2}$ and $\left.C_{2} \rightarrow C_{2}-C_{3}\right]$$\Delta=\left|\begin{array}{lll}b c-a^{2}-c ...
Read More →Solve this
Question: If $A=\left[\begin{array}{cc}3 5 \\ -2 0 \\ 6 -1\end{array}\right], B=\left[\begin{array}{cc}-1 -3 \\ 4 2 \\ -2 3\end{array}\right]$ and $C=\left[\begin{array}{cc}0 2 \\ 3 -4 \\ 1 6\end{array}\right]$, verify that $(A+B)+C=A+(B+C)$ Solution: $(A+B)+C=\left(\left[\begin{array}{cc}3 5 \\ -2 0 \\ 6 -1\end{array}\right]+\left[\begin{array}{cc}-1 -3 \\ 4 2 \\ -2 3\end{array}\right]\right)+\left[\begin{array}{cc}0 2 \\ 3 -4 \\ 1 6\end{array}\right]$ $=\left(\left[\begin{array}{ll}2 2 \\ 2 2 ...
Read More →Write a value
Question: Write a value of $\int \sin ^{3} x \cos x d x$. Solution: let $\sin x=t$ Differentiating on both sides we get, $\cos x d x=d t$ Substituting above equation in $\int \sin ^{3} x \cos x d x$ we get, $=\int t^{3} d t$ $=\frac{t^{4}}{4}+c$ $=\frac{\sin ^{4} x}{4}+c$...
Read More →Solve this
Question: If $A=\left[\begin{array}{ccc}2 -3 5 \\ -1 0 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}3 2 -2 \\ 4 -3 1\end{array}\right]$, verify that $(A+B)=(B+A)$ Solution: $A+B=\left[\begin{array}{ccc}2 -3 5 \\ -1 0 3\end{array}\right]+\left[\begin{array}{ccc}3 2 -2 \\ 4 -3 1\end{array}\right]$ $=\left[\begin{array}{lll}5 -1 3 \\ 3 -3 4\end{array}\right]$ $B+A=\left[\begin{array}{ccc}3 2 -2 \\ 4 -3 1\end{array}\right]+\left[\begin{array}{ccc}2 -3 5 \\ -1 0 3\end{array}\right]$ $=\left[\b...
Read More →Write a value
Question: Write a value of $\int e^{\log \sin x} \cos x d x$. Solution: given $\int e^{\log \sin x} \cos x d x$ $=\int \sin x \cos x d x\left(\because e^{\log x}=x\right)$ Let $\sin x=t$ Differentiating on both sides we get, $\cos x d x=d t$ Substituting above equations in given equation we get, $=\int \mathrm{t} \mathrm{dt}$ $=\frac{t^{2}}{2}+c$ $=\frac{\sin ^{2} x}{2}+c$...
Read More →If a + b + c ¹ 0 and then prove that a = b = c
Question: Ifa+b+c 0 and then prove thata=b=c $\left|\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right|=0$ Solution: Let $\Delta=\left|\begin{array}{lll}a b c \\ b c a \\ c a b\end{array}\right|$ $\left[\right.$ Applying $\left.R_{1} \rightarrow R_{1}+R_{2}+R_{3}\right]$ $\Delta=\left|\begin{array}{ccc}a+b+c a+b+c a+b+c \\ b c a \\ \cdot c a b\end{array}\right|=(a+b+c)\left|\begin{array}{ccc}1 1 1 \\ b c a \\ c a b\end{array}\right|$ Now, [Applying $C_{1} \rightarrow C_{1}-C_{3}$ and $C_...
Read More →Construct a 3 × 4 matrix whose elements are given by
Question: Construct a $3 \times 4$ matrix whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$. Solution: It is a (3 x 4) matrix. So, it has 3 rows and 4 columns. Given $a_{i j}=\frac{|-3 i+j|}{2}$ So, $a_{11}=1, a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$, $a_{21}=\frac{5}{2}, a_{22}=2, a_{23}=\frac{3}{2}, a_{13}=1$ $a_{31}=4, a_{32}=\frac{7}{2}, a_{33}=3, a_{13}=\frac{5}{2}$ So, the matrix $=\left[\begin{array}{cccc}1 \frac{1}{2} 0 \frac{1}{2} \\ \frac{5}{2} 2 \frac{3}{2} 1 \\ 4 \fr...
Read More →Write a value
Question: Write a value of $\int \frac{(\log x)^{n}}{x} d x$. Solution: let $\log x=t$ Differentiating on both sides we get, $\frac{1}{x} d x=d t$ Substituting above equations in $\int \frac{(\log x)^{n}}{x} d x$ we get, $\int t^{n} d t$ $=\frac{t^{n+1}}{n+1}+c$ $=\frac{(\log x)^{n+1}}{n+1}+c$...
Read More →Write a value
Question: Write a value of $\int \frac{\log \mathrm{x}^{\mathrm{n}}}{\mathrm{x}} \mathrm{dx}$. Solution: let $\log x^{n}=t$ Differentiating on both sides we get, $\frac{1}{x^{n}} n x^{n-1} d x=d t$ $\frac{n}{x} d x=d t$ $\frac{1}{x} d x=\frac{1}{n} d t$ Substituting above equations in $\int \frac{\log x^{n}}{x} d x$ we get, $\int \frac{1}{n} t d t$ $=\frac{1}{n} \frac{t^{2}}{2}+c$ $=\frac{\left(\log x^{n}\right)^{2}}{2 n}+c$...
Read More →Write a value
Question: Write a value of $\int e^{x} \sec x(1+\tan x) d x$. Solution: given, $\int e^{x} \sec x(1+\tan x) d x=\int e^{x}(\sec x+\sec x \tan x) d x$ $=e^{x} \sec x+c$ $\because \int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+c$...
Read More →Given find BA and use this to
Question: Given find BA and use this to solve the system of equationsy+ 2z= 7,xy= 3, 2x+ 3y+ 4z= 17. Solution: Given, $A=\left[\begin{array}{ccc}2 2 -4 \\ -4 2 -4 \\ 2 -1 5\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 -1 0 \\ 2 3 4 \\ 0 1 2\end{array}\right]$ Now, $B A=\left[\begin{array}{ccc}1 -1 0 \\ 2 3 4 \\ 0 1 2\end{array}\right]\left[\begin{array}{ccc}2 2 -4 \\ -4 2 -4 \\ 2 -1 5\end{array}\right]=\left[\begin{array}{lll}6 0 0 \\ 0 6 0 \\ 0 0 6\end{array}\right]=6 I$ Thus, $B^{-1}=\f...
Read More →Write a value
Question: Write a value of $\int \frac{\cos \mathrm{x}}{3+2 \sin \mathrm{x}} \mathrm{dx}$. Solution: let $3+2 \sin x=t$ Differentiating on both sides we get, $2 \cos x d x=d t$ $\cos x d x=\frac{1}{2} d t$ Substituting above equation in $\int \frac{\cos x}{3+2 \sin x} d x$ we get, $\int \frac{1}{2 t} d t$ $=\frac{1}{2} \log t+c$ $=\frac{1}{2} \log (3+2 \sin x)+c$...
Read More →Using matrix method, solve the system of equations
Question: Using matrix method, solve the system of equations 3x+ 2y 2z= 3,x+ 2y+ 3z= 6, 2xy+z= 2. Solution: Given system of equations are: 3x+ 2y 2z= 3 x+ 2y+ 3z= 6 and 2xy+z= 2 Or, AX = B So, $\left[\begin{array}{ccc}3 2 -2 \\ 1 2 3 \\ 2 -1 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}3 \\ 6 \\ 2\end{array}\right]$ Hence, $\quad X=A^{-1} B$ Now, for $\mathrm{A}^{-1}$ the co-factors are $A_{11}=5, A_{12}=5, A_{13}=-5$ $A_{21}=0, A_{22}=7, A_{23}=7$...
Read More →Write a value
Question: Write a value of $\int \tan ^{6} x \sec ^{2} x d x$ 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, $=\int t^{6} d t$ $=\frac{t^{7}}{7}+c$ $=\frac{\tan ^{7} x}{7}+c$...
Read More →Construct a 2 × 3 matrix whose elements are
Question: Construct a $2 \times 3$ matrix whose elements are $a_{i j}=\frac{(i-2 j)^{2}}{2}$. Solution: It is a (2 x 3) matrix. So, it has 2 rows and 3 columns. Given $a_{i j}=\frac{(i-2 j)^{2}}{2}$ So, $a_{11}=\frac{1}{2}, a_{12}=\frac{9}{2}, a_{13}=\frac{25}{3}$ $a_{21}=0 \cdot a_{22}=2 \cdot a_{23}=8$ So, the matrix $=\left[\begin{array}{lll}\frac{1}{2} \frac{9}{2} \frac{25}{2} \\ 0 2 8\end{array}\right]$ Conclusion: Therefore, Matrix is $\left[\begin{array}{lll}\frac{1}{2} \frac{9}{2} \frac{...
Read More →Construct a 2 × 2 matrix whose elements are
Question: Construct a $2 \times 2$ matrix whose elements are $\mathrm{a}_{\mathrm{ij}}=\frac{(\mathrm{i}+2 \mathrm{j})^{2}}{2}$ Solution: It is a $(2 \times 2)$ matrix. So, it has 2 rows and 2 columns. Given $a_{i j}=\frac{((t+2))^{2}}{2}$ So, $a_{11}=\frac{9}{2}, a_{12}=\frac{25}{2}$, $a_{21}=8, a_{22}=18$ So, the matrix $=\left[\begin{array}{cc}\frac{9}{2} \frac{25}{2} \\ 8 18\end{array}\right]$ Conclusion: Therefore, Matrix is $=\left[\begin{array}{cc}\frac{9}{2} \frac{25}{2} \\ 8 18\end{arra...
Read More →Solve that equation
Question: If A = , find A-1. Using A1, solve the system of linear equations x 2y= 10 , 2xyz= 8 , 2y+z= 7. Solution: Given, $\quad A=\left[\begin{array}{ccc}1 2 0 \\ -2 -1 -2 \\ 0 -1 1\end{array}\right]$ Co-factors are: $A_{11}=-3, A_{12}=2, A_{13}=2$ $A_{21}=-2, A_{22}=1, A_{23}=1$ $A_{31}=-4, A_{32}=2, A_{33}=3$ Now, $\operatorname{adj} A=\left[\begin{array}{ccc}-3 2 2 \\ -2 1 1 \\ -4 2 3\end{array}\right]^{T}=\left[\begin{array}{ccc}-3 -2 -4 \\ 2 1 2 \\ 2 1 3\end{array}\right]$ $|A|=1(-3)-2(-2...
Read More →Construct a 4 × 3 matrix whose elements are given
Question: Construct a $4 \times 3$ matrix whose elements are given by $a_{i j}=\frac{i}{j}$. Solution: It is (4 x 3) matrix. So it has 4 rows and 3 columns Given $a_{i j}=\frac{i}{j}$ So, $a_{11}=1, a_{12}=\frac{1}{2}, a_{13}=\frac{1}{3}$ $a_{21}=2, a_{22}=1, a_{23}=\frac{2}{3}$ $a_{31}=3 \cdot a_{32}=\frac{3}{2}, a_{33}=1$ $a_{41}=4 \cdot a_{42}=2 \cdot a_{43}=\frac{4}{3}$ So, the matrix $=\left[\begin{array}{ccc}1 \frac{1}{2} \frac{1}{3} \\ 2 1 \frac{2}{3} \\ 3 \frac{3}{2} 1 \\ 4 2 \frac{4}{3}...
Read More →Construct a 3 × 2 matrix whose elements are given
Question: Construct a $3 \times 2$ matrix whose elements are given by $a_{i j}=(2 i-j)$. Solution: Given: $a_{i j}=(2 i-j)$ Now, $a_{11}=(2 \times 1-1)=2-1=1$ $a_{12}=2 \times 1-2=2-2=0$ $a_{21}=2 \times 2-1=4-1=3$ $a_{22}=2 \times 2-2=4-2=2$ $a_{31}=2 \times 3-1=6-1=5$ $a_{32}=2 \times 3-2=6-2=4$ Therefore, $A=\left[\begin{array}{ll}1 0 \\ 3 2 \\ 5 4\end{array}\right]$...
Read More →Find all possible orders of matrices having 7 elements.
Question: Find all possible orders of matrices having 7 elements. Solution: Number of entries $=$ (Number of rows) $\times$ (Number of columns) $=7$ If order is $(a x b)$ then, Number of entries $=a x b$ So now $\mathrm{a} \times \mathrm{b}=7$ (in this case) Possible cases are $(1 \times 7),(7 \times 1)$ Conclusion: If a matrix has 18 elements, then possible orders are $(1 \times 7),(7 \times 1)$...
Read More →If a matrix has 18 elements, what are the possible orders it can have?
Question: If a matrix has 18 elements, what are the possible orders it can have? Solution: Number of entries $=$ (Number of rows) $\times$ (Number of columns) $=18$ If order is $(a \times b)$ then, Number of entries $=a \times b$ So now $\mathrm{a} \times \mathrm{b}=18$ (in this case) Possible cases are $(1 \times 18),(2 \times 9),(3 \times 6),(6 \times 3),(9 \times 2),(18 \times 1)$ Conclusion: If a matrix has 18 elements, then possible orders are $(1 \times 18),(2 \times 9),(3 \times 6),(6 \ti...
Read More →Write the order of each of the following matrices:
Question: Write the order of each of the following matrices: i. $A=\left[\begin{array}{cccc}3 5 4 -2 \\ 0 \sqrt{3} -1 \frac{4}{9}\end{array}\right]$ ii. $B=\left[\begin{array}{cc}6 -5 \\ \frac{1}{2} \frac{3}{4} \\ -2 -1\end{array}\right]$ iii. $\mathrm{C}=\left[\begin{array}{lll}7-\sqrt{2} 5 0\end{array}\right]$ iv. $D=[8-3]$ v. $E=\left[\begin{array}{c}-2 \\ 3 \\ 0\end{array}\right]$ vi, $F=[6]$ Solution: i. $A=\left[\begin{array}{cccc}3 5 4 -2 \\ 0 \sqrt{3} -1 \frac{4}{9}\end{array}\right]$ Or...
Read More →