RD Sharma Solutions Class 12 Mathematics Chapter 5 FBQ

Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with

Plan, Prepare & Make the Best Career Choices

# RD Sharma Solutions Class 12 Mathematics Chapter 5 FBQ

Edited By Kuldeep Maurya | Updated on Jan 25, 2022 06:54 PM IST

The RD Sharma class 12th exercise FBQ solution tops the list of NCERT solutions as per the opinion of students. Since CBSE schools require students to practice and master their NCERT book, they need to answer the given questions correctly and learn the concepts clearly. RD Sharma solutions Mathematics is a rather complex subject that can only be mastered with self-practice. Therefore, the RD Sharma class 12th exercise FBQ will be a good guide in your endeavor to score high in exams and ace the paper.

## Determinants Excercise:FBQ

Determinants exercise Fill in the blanks question 1

Answer: $\left | A \right |=6$
Hint: Here, we use basic concept of diagonal and determinant of matrix
Given: $\begin{bmatrix} 1 &0 &0 \\ 0&2 &0 \\ 0&0 & 3 \end{bmatrix}$
Solution:
$det(A)=\left | A \right |=\begin{bmatrix} 1 &0 &0 \\ 0&2 &0 \\ 0&0 & 3 \end{bmatrix}$
$=1\begin{bmatrix} 2 & 0\\ 0&3 \end{bmatrix}-0\begin{bmatrix} 0 &0 \\ 0&3 \end{bmatrix}+0\begin{bmatrix} 0 &0 \\ 2&0 \end{bmatrix}$
$=1(6-0)-0(0-0)+0(0-0)$
$=1(6)-0+0$
$\left | A \right |=6$

Determinants exercise Fill in the blanks question 2

Answer:$\left | A \right |=0$
Hint: Here, we use basic concept of skew symmetric matrix and determinant of matrix
Given: A is $3\times 3$ order skew symmetric matrix
Solution: Here, $A_{\left [ 3\times 3 \right ]}$ and $A=-A^{T}$ ( Skew Symmetric )
So, $A=-A^{T}$
$\left | A \right |=-\left | A^{T} \right |$ [Taking determinant both side]
$\left | A \right |=-\left | A \right |\; \; \; \; \; \; \; \;$ $\left [ \left | A^{T} \right |=\left | A \right | \right ]$
$\left | A \right |+\left | A \right |=0$
$2\left | A \right |=0$
$\left | A \right |=0$
So, here determinant of 3 $3\times 3$ skew symmetric matrix always be zero.

Determinants exercise Fill in the blanks question 3

Hint: Here, we use basic concept of singular matrix
Given:$A=\begin{bmatrix} 1 &3 &x+2 \\ 2& 4 & 8\\ 3& 5& 10 \end{bmatrix}$ is singular
Solution: Here, A is regular
So, $\left | A \right |$ is zero
$\left | A \right |=0$
So,
$\begin{bmatrix} 1 &3 &x+2 \\ 2& 4 & 8\\ 3& 5& 10 \end{bmatrix}=0$
$1\begin{bmatrix} 4 & 8\\ 5& 10 \end{bmatrix}-3\begin{bmatrix} 2 & 8\\ 3& 10 \end{bmatrix}+(x+2)\begin{bmatrix} 2 & 4\\ 3& 5 \end{bmatrix}=0$
$1(40-40)-3(20-24)+(x+2)(10-12)=0$
$0-3(-4)+(x+2)(-2)=0$
$12+\left [ -2x-4 \right ]=0$
$8-2x=0$
$8=2x$
$x=\frac{8}{2}=4$

Determinants exercise Fill in the blanks question 4

Answer: $K^{2}$
Hint: Here, we use basic concept of singular matrix
Given: A and B non singular square matrix of order ‘n’
Such that A = KB
Solution: Here, A and B are non singular
And both are n order of singular matrix
Here
$A=KB$
$\left | A \right |=\left | K \right |\left | B \right |$ (Let's take both side determinate)
$\left | A \right |=K^{n}\left | B \right |$
$\frac{\left | A \right |}{\left |B \right |}=K^{n}$ [A and B are $n\times n$ matrix]
So, $\frac{\left | A \right |}{\left |B \right |}=K^{n}$

Determinants exercise Fill in the blanks question 6

Answer:$\left | 5A \right |=125$
Hint: Here, we use basic concept $\left | KA \right |=K^{n}\left | A \right |$
Given: Here, A is $2\times 2$ matrix
So, n = 2
$\left | A \right |=5$
Solution: Here, $n=2$ and $\left | A \right |=5$
So, $\left |5 A \right |=?$
So, here 5 is just like k
$\left |K A \right |=K^{n}\left | A \right |$
$\left |5 A \right |=5^{2}\times 5\; \; \; \; \; \; \; \; \left [ K=5,\left | A \right |=5,n=2 \right ]$
$\left |5 A \right |=125$
$\left |5 A \right |$ is equal to 125.

Determinants exercise Fill in the blanks question 5

Answer: $R-\left | 1 \right |$
Hint: Here, we use basic concept of non-singular matrix
Given:$A=\begin{bmatrix} a & 2\\ 2&4 \end{bmatrix}$ is non-singular
Solution: Here, A is non-singular
So, $\left | A \right |\neq 0$
$\begin{bmatrix} a & 2\\ 2&4 \end{bmatrix}\neq 0$
$(a(4))-(2)(2)\neq 0$
$4a-4\neq 0$
$4a\neq 4$
$a\neq 1$
So, $a$ should be in $R-\left | 1 \right |$

Determinants exercise Fill in the blanks question 7

Answer: $\left | 3AB \right |=-81$
Hint: Here, we use basic concept of $\left | KA \right |=K^{n}\left | A \right |$
Given: A and B are square matrix of order 3. So, n = 3
$\left | A \right |=(-1), \; \; \left | B \right |=3$
Solution: Here, $\left | 3AB \right |$
Here 3 is just like K. So,
$\left | KAB \right |=K^{n}\left | A \right |\left | B \right |$
$\left | 3AB \right |=3^{3}\times -1(3)\; \; \; \; \; \; \; \left [ n=3,\left | A \right | =-1,\left | B \right |=3\right ]$
$\left | 3AB \right |=-81$
So, here $\left | 3AB \right |=-81$

Determinants exercise Fill in the blanks question 8

Answer: $\partial =\pm 3$
Hint: Here, we use basic concept of determinant of matrix
Given: $\left | A \right |^{3}=125$, $\inline A=\begin{bmatrix} \partial & 2\\ 2& \partial \end{bmatrix}$
Solution: Here, $\inline A=\begin{bmatrix} \partial & 2\\ 2& \partial \end{bmatrix}$
So, $\inline \left | A \right |=\partial \times \partial -2\times 2$
$\inline \left | A \right |=\partial^{2} -4$ ........$\inline (1)$
And,$\inline \left | A \right |^{3}=125$
$\inline \left ( \left | A \right | \right )^{3}=125$
$\inline \left ( \left | A \right | \right )^{3}=(5)^{3}$
$\inline \left | A \right |=5$ ...........$\inline (2)$
From equation (1) and equation (2)
$\inline \partial ^{2}-4=5$
$\inline \partial ^{2}=4+5$
$\inline \partial ^{2}=9$
$\inline \partial=\pm 3$

Determinants exercise Fill in the blanks question 9

Answer: $x=e^{2}$
Hint: Here, we use basic concept of determinant of matrix
Given: $A=\left[\begin{array}{cc} \log x & -1 \\ -\log x & 2 \end{array}\right]$
$det (A) = 2$
Solution: Here, $A=\left[\begin{array}{cc} \log x & -1 \\ -\log x & 2 \end{array}\right]$
\begin{aligned} &|A|=\log x \times 2-(-1 \times-\log x) \\ &=2 \log x-\log x \\ &|A|=\log x \end{aligned}
\begin{aligned} &\text { And }|A|=2 \text { (Given) }\\ &\text { So, }|A|=\log x=2 \end{aligned}
\begin{aligned} &e^{\log x}=e^{2} \\ &x=e^{2}\left[e^{\log x}=x\right] \end{aligned}

Determinants exercise Fill in the blanks question 10

Hint: Here, we use basic concept of identity matrix and determinant of matrix
Given: Here I is order of 10 and identity matrix
Solution: Here, I is identity matrix and it’s order is 10.
Any order of identity matrix is determinant of matrix be 1
Let I is $2\times 2$ matrix
\begin{aligned} &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ &|I|=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=1-0=1 \\ &S o,\left|I_{10}\right|=1 \end{aligned}

Determinants exercise Fill in the blanks question 11

Answer: $8\left | A \right |$
Hint: Here, we use basic concept of determinant of matrix $\left | KA \right |=K^{n}\left | A \right |$
Given: A is $3\times 3$ matrix. So, n = 3
Solution: Here A is $3\times 3$ matrix. So, n = 3
$\left | -2A \right |=-2A$ is constant. So,$K=(-2)$
Let’s put value of K in below, Formula,
\begin{aligned} &|K A|=K^{n}|A| \\ &|-2 A|=(-2)^{n}|A| \end{aligned} \quad[K=-2]
\begin{aligned} &|-2 A|=(-2)^{3}|A| \quad[n=3] \\ &|-2 A|=-8|A| \end{aligned}
So, here$8\left | A \right |$ is our answer.

Determinants exercise Fill in the blanks question 12

Hint: Here, we use basic concept of determinant of matrix
Given: A is $3\times 3$ matrix. So, n = 3
$|A|=5 \; \; C_{i j}=\operatorname{cofactor} \text { of } a_{i j} . \text { So, let } C_{i j}=A_{i j}, C=A$
Solution: A is $\left | a_{ij} \right |$ be a square matrix
So, $A=\left(\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) 1 \leq i \leq 3$
\begin{aligned} &|A|=5 \\ &|A|=a_{11}\left[\begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right]-a_{12}\left[\begin{array}{cc} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array}\right]+a_{13}\left[\begin{array}{cc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right] \\ &|A|=a_{11}\left[\begin{array}{cc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right]+a_{12}\left[\begin{array}{cc} a_{23} & a_{21} \\ a_{33} & a_{31} \end{array}\right]+a_{13}\left[\begin{array}{ll} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right] \end{aligned}
[Interchange column of a12]
\begin{aligned} &\Rightarrow|A|=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \\ &5=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \\ &5=a_{11} C_{11}+a_{12} C_{12}+a_{13} C_{13} \quad[A=C] \end{aligned}

Determinants exercise Fill in the blanks question 13

Hint: Here we use basic concept of determinant of matrix
Given: A is $3\times 3$ matrix. So, n = 3
\begin{aligned} &|A|=5\\ &C_{i y} \text { is cofactor of } C_{11 j} \end{aligned}
Solution:
\begin{aligned} &A_{i j}=\left(\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) \\ &\text { and cofactor }\left(A_{i j}\right)=(-1)^{i+j}|\operatorname{mij}| \end{aligned}
So, here
\begin{aligned} &a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{23} \\ &A_{21}=(-1)^{2+1}\left[a_{12} a_{33}-a_{13} a_{32}\right] \\ &A_{22}=(-1)^{2+2}\left[a_{11} a_{33}-a_{13} a_{31}\right] \\ &A_{23}=(-1)^{2+3}\left[a_{11} a_{32}-a_{12} a_{31}\right] \end{aligned}
\begin{aligned} &=a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{23} \\ &=\left(-a_{11} a_{12} a_{33}\right)+a_{11} a_{13} a_{32}+a_{11} a_{12} a_{33}-a_{12} a_{13} a_{31}-a_{12} a_{13} a_{32}+a_{12} a_{13} a_{31} \\ &=\left[a_{11} a_{12} a_{33}-a_{11} a_{12} a_{33}\right]+\left[a_{11} a_{13} a_{32}-a_{11} a_{13} a_{32}\right]+\left[a_{12} a_{13} a_{31}-a_{12} a_{13} a_{31}\right] \\ &\text { So } 0+0+0 \\ &=0 \end{aligned}

Determinants exercise Fill in the blanks question 14

Hint: Here, we use basic concept of diagonal matrix and determinant of matrix
Given: $A= diag(2,34)$
Solution: $diag(2,34)$
$\text { So, } A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]$
Let’s find determinant of A
\begin{aligned} &|A|=2\left[\begin{array}{ll} 3 & 0 \\ 0 & 4 \end{array}\right]-0\left[\begin{array}{ll} 0 & 4 \\ 0 & 0 \end{array}\right]+0\left[\begin{array}{ll} 0 & 3 \\ 0 & 0 \end{array}\right] \\ &=2(12-0)-0(0-0)+0(0-0) \\ &=2(12)-0+0 \\ &|A|=24 \end{aligned}
So,
\begin{aligned} &\left|A^{2}\right|=|A|^{2}=(24)^{2} \\ &\left|A^{2}\right|=576 \end{aligned}

Determinants exercise Fill in the blanks question 15

Answer: $|C|=|A|^{2}=2^{2}=4$
Hint: Here, we use basic concept of diagonal matrix and determinant of matrix
Given: \begin{aligned} &A=\left[c_{i j}\right]_{3 \times 3}|A|=2 \text { and } c_{i j} \text { be cofactor of } a_{i j}\\ &C=\left[c_{i j}\right] \end{aligned}
Solution: c is cofactor matrix
\begin{aligned} &A d j A=(C)^{T}\\ &\text { But Adj } A=\text { Det. of } C \end{aligned}
Because both are transpose of each other,
$|\operatorname{adj} A|=|C|=|A|^{n-1}$
Here n = 3 because A is $3\times 3$ matrix,
\begin{aligned} &|C|=|A|^{3-1} \\ &|C|=|A|^{2} \end{aligned}
Here,
\begin{aligned} &|A|=2 \\ &|C|=(2)^{2}=4 \end{aligned}

Determinants exercise Fill in the blanks question 16

Hint: Here, we use basic concept of determinant of matrix.
Given: $\Delta=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 x & 6 x & 9 x \end{array}\right]$
Solution:
$\Delta=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 x & 6 x & 9 x \end{array}\right]$
\begin{aligned} &=1\left[\begin{array}{cc} 5 & 6 \\ 6 x & 9 x \end{array}\right]-2\left[\begin{array}{cc} 4 & 6 \\ 3 x & 9 x \end{array}\right]+3\left[\begin{array}{cc} 4 & 5 \\ 3 x & 6 x \end{array}\right] \\ &=1(45 x-36 x)-2(36 x-18 x)+3(24 x-15 x) \\ &=9 x-36 x+27 x \\ &=0 \end{aligned}

Determinants exercise Fill in the blanks question 17

Hint: Here, we use basic concept of determinant of matrix
Given: $\Delta=\left[\begin{array}{ccc} \sec ^{2} \theta & \tan ^{2} \theta & 1 \\ \tan ^{2} \theta & \sec ^{2} \theta & -1 \\ 12 & 10 & 2 \end{array}\right]$
Solution: We know that
$\sec ^{2} \theta-\tan ^{2} \theta=1$ ..............(1)
Applying $C_{2}\rightarrow C_{2}-C_{1}$
\begin{aligned} \Delta &=\left[\begin{array}{ccc} \sec ^{2} \theta & \sec ^{2} \theta-\tan ^{2} \theta & 1 \\ \tan ^{2} \theta & \tan ^{2} \theta-\sec ^{2} \theta & -1 \\ 12 & 12-10 & 2 \end{array}\right] \\ \Delta &=\left[\begin{array}{ccc} \sec ^{2} \theta & 1 & 1 \\ \tan ^{2} \theta & -1 & -1 \\ 12 & 2 & 2 \end{array}\right] \end{aligned}[From equation (1)]
$\Delta =0$ because according to the property of determinant, if any 2 columns are same then $\Delta =0$

Determinants exercise Fill in the blanks question 18

Hint: Here, we use basic concept of determinant of matrix
Given: $\Delta=\left[\begin{array}{ccc} 0 & x-y & y-z \\ y-x & 0 & z-x \\ z-y & x-z & 0 \end{array}\right]$
Solution: Here $\Delta$ is skew symmetric matrix.
Here, clearly we can see that $\Delta$ is skew symmetric and its diagonal are 0
So, $\Delta=0$

Determinants exercise Fill in the blanks question 19

Hint: Here, we use basic concept of determinant of matrix
Given:
\begin{aligned} &b_{i 1}=2 a_{i 1} \\ &b_{i 2}=3 a_{i 2} \\ &b_{i 3}=4 a_{i 3} i, i=1,2,3 \end{aligned}
Let $\left | A \right |=\left [ a_{ij} \right ]$ and $B=b_{ij}$ be a square matrix of order 3 and $\left | A \right |=5$
Solution:
\begin{aligned} &|B|=\left(\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array}\right) \\ &|B|=\left(\begin{array}{lll} 2 a_{11} & 3 a_{12} & 4 a_{13} \\ 2 a_{21} & 3 a_{22} & 4 a_{23} \\ 2 a_{31} & 3 a_{32} & 4 a_{33} \end{array}\right) \end{aligned}
Let’s transfer $\left | B \right |=\left | B^{T} \right |$ So,
\begin{aligned} &|B|=\left(\begin{array}{ccc} 2 a_{11} & 2 a_{12} & 2 a_{13} \\ 3 a_{21} & 3 a_{22} & 3 a_{23} \\ 4 a_{31} & 4 a_{32} & 4 a_{33} \end{array}\right) \\ &|B|=2 \times 3 \times 4\left(\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) \end{aligned}
\begin{aligned} &=24 \times\left|A^{T}\right| \\ &=24 \times 5 \quad\left[|A|=\left|A^{T}\right|=5\right] \\ &=120 \end{aligned}

Determinants exercise Fill in the blanks question 20

Hint: Here, we use basic concept of determinant
$|K A B|=K^{n}|A||B|$
Given:$n = 3$
$\left | A \right |=-2$
$\left | B \right |=4$
Solution: A and B are $3\times 3$ order of matrix
So, n = 3
Here, $\left | 2AB \right |\; \; \; \; \; \; \; So,K=2$
So, now
$|K A B|=K^{n}|A||B|$
$|2 A B|=2^{3} \times-2 \times 4 \; \; \; \; \; \; \; \; \; \quad[n=3,|A|=-2,|B|=4]$
\begin{aligned} &|2 A B|=8 \times-2 \times 4 \\ &|2 A B|=8 \times-8 \\ &|2 A B|=-64 \end{aligned}

Determinants exercise Fill in the blanks question 21

Hint: Here, we use basic concept of determinant of matrix
Given: $\left[\begin{array}{ccc} \cos (x+y) & -\sin (x+y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ -\cos x & \sin x & \sin y \end{array}\right]$
Solution: Formula expanding,
\begin{aligned} &{\left[\begin{array}{ccc} \cos x \cos y-\sin x \sin y & -\sin x \cos y+\cos x \sin y & \cos ^{2} y-\sin ^{2} y \\ \sin x & \cos x & \sin y \\ -\cos x & \sin x & \sin y \end{array}\right]} \\ &R_{1} \rightarrow R_{1}+R_{2}(\sin y)+R_{3}(\cos y) & \end{aligned}
$\left[\begin{array}{ccc} \cos x \cos y-\sin x \sin y+\sin y \sin x-\cos y \cos x & -\sin x \cos y-\cos x \sin y+\sin y \cos x+\cos y \sin x & \cos ^{2} y-\sin ^{2} y+\sin ^{2} y-\cos ^{2} y \\ \sin x & \cos x & \sin y \\ -\cos x & \sin x & -\sin y \end{array}\right]$$\text { So }\left[\begin{array}{ccc} 0 & 0 & 0 \\ \sin x & \cos x & \sin x \\ -\cos x & \sin x & -\cos x \end{array}\right]=0$
Here, we get answer is zero which is obtained by multiplying 2nd row with sin y and 3rd row with cosy.
So, Value of determinant depends on y.

Determinants exercise Fill in the blanks question 22

Hint: Here, we use basic concept of determinant of matrix
Given: $\left[\begin{array}{lll} \left(2^{x}+2^{-x}\right)^{2} & \left(2^{x}+2^{-x}\right)^{2} & 1 \\ \left(3^{x}+3^{-x}\right)^{2} & \left(3^{x}+3^{-x}\right)^{2} & 1 \\ \left(4^{x}+4^{-x}\right)^{2} & \left(4^{x}+4^{-x}\right)^{2} & 1 \end{array}\right]$
Solution: Here, clearly 2 column has same value
According to property of determinant, if two column has same value then determinant of matrix should be zero.

Determinants exercise Fill in the blanks question 23

Hint: Here, we use basic concept of determinant.
Given: $\left[\begin{array}{lll} (1+x)^{17} & (1+x)^{19} & (1+x)^{23} \\ (1+x)^{23} & (1+x)^{24} & (1+x)^{34} \\ (1+x)^{41} & (1+x)^{43} & (1+x)^{47} \end{array}\right]$
$A+B x+c x^{2}+\ldots \ldots \ldots \text { then find } A=?$
Solution: Let’s take something common from 3 rows
$(1+x)^{17}\left(1+x^{23}\right)\left(1+x^{41}\right)\left[\begin{array}{ccc} 1 & (1+x)^{2} & (1+x)^{6} \\ 1 & (1+x)^{6} & (1+x)^{11} \\ 1 & (1+x)^{2} & (1+x)^{6} \end{array}\right]$
Here, we clearly see that 1st and 3rd row has same value
So, f(x) = 0
So, here x = 0 and
\begin{aligned} &A+B x+C x^{2} \ldots \ldots \ldots .=0 \\ &S o, A=0 \end{aligned}

Determinants exercise Fill in the blanks question 24

Answer:$\frac{1}{2}$
Hint: Here, we use basic concept of determinant of matrix.
Given: $\cos 2 \theta=0$
$\left[\begin{array}{ccc} 0 & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & 0 \\ \sin \theta & 0 & \cos \theta \end{array}\right]^{2}=?$
Solution:
$\cos 2 \theta=0$
\begin{aligned} &\cos 2 \theta=\cos 90^{\circ} \quad\left(\cos 90^{\circ}=0\right) \\ &2 \theta=90^{\circ} \\ &\theta=45^{\circ} \end{aligned}
Let’s put $\theta=45^{\circ}$ in matrix
$\left[\begin{array}{ccc} 0 & \cos 45^{\circ} & \sin 45^{\circ} \\ \cos 45^{\circ} & \sin 45^{\circ} & 0 \\ \sin 45^{\circ} & 0 & \cos 45^{\circ} \end{array}\right]^{2}$
$\left[\begin{array}{ccc} 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{array}\right]^{2} \quad\left[\sin 45^{\circ}=\frac{1}{\sqrt{2}}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}\right]$
Let’s take $\frac{1}{\sqrt{2}}$ in common
$=\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$
\begin{aligned} &=\left[\frac{1}{2 \sqrt{2}}[-1 \times(1-0)+1(0-1)]\right]^{2} \\ &=\frac{1}{8} \times[-1-1]^{2} \\ &=\frac{1}{8} \times(-2)^{2} \\ &=\frac{1}{2} \end{aligned}

Determinants exercise Fill in the blanks question 25

Hint: Here, we use basic concept of minors in matrix
Given: Here A is $3\times 3$ matrix
Solution: In any matrix minors are equal to total number of elements
So, here total number of elements of $A=3\times 3=9$
So, total minors in A is 9

Determinants exercise Fill in the blanks question 26

Hint: Here, we use basic concept of determinant of matrix
Given:$x=-y$ and $\left[\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]=0$
Solution:
$\left[\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right]$
\begin{aligned} &=x\left[\begin{array}{ll} x & 2 \\ 6 & x \end{array}\right]-3\left[\begin{array}{cc} 2 & 2 \\ 7 & x \end{array}\right]+7\left[\begin{array}{cc} 2 & x \\ 7 & 6 \end{array}\right] \\ &=x\left(x^{2}-12\right)-3(2 x-14)+7(12-7 x) \\ &f(x)=x^{3}-12 x-6 x+42+84-49 x=0 \\ &f(x)=x^{3}-67 x+126=0 \end{aligned}
Let put x = 2 and x = 7
\begin{aligned} &f(2)=(2)^{3}-67(2)+126 \\ &=8-134+126 \\ &=0 \\ &f(7)=(7)^{3}-67(7)+126 \\ &=49 \times 7-67+126 \\ &=0 \end{aligned}
So, 2 and 7 are other roots.

Determinants exercise Fill in the blanks question 27

Answer: $(y-z)(z-x)[y-x+x y z]$
Hint: Here, we use basic concept of determinant of matrix
Given: $\left[\begin{array}{ccc} 0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0 \end{array}\right]$
Solution:
$0\left[\begin{array}{cc} 0 & y-z \\ z-y & 0 \end{array}\right]-x y z\left[\begin{array}{cc} y-x & y-z \\ z-x & 0 \end{array}\right]+x-z\left[\begin{array}{cc} y-x & 0 \\ z-x & z-y \end{array}\right]$
\begin{aligned} &=0(0-(y-z)) \times(z-y)-x y z(0-(y-z)(z-x))+(x-z)((y-x)(z-y)-0) \\ &=x y z(y-z)(z-x)+(z-x)(y-z)(y-z) \\ &=(y-z)(z-x)[x y z+(y-x)] \\ &=(y-z)(z-x)[y-x+x y z] \end{aligned}

Determinants exercise Fill in the blanks question 28

Hint: Here, we use basic concept of determinant of matrix $|K A B|=K^{n}|A||B|$
Given: A is $3\times 3$ matrix. So, n = 3
$|A|=5,|B|=5$
Solution: $|3 A B|$
Here 3 is constant K. So,
\begin{aligned} &|K A B|=K^{n}|A||B| \\ &|3 A B|=3^{n}|A||B| \quad(K=3) \\ &|3 A B|=3^{3} \times 5 \times 5[n=3,|A|=5,|B|=5] \\ &=27 \times 25 \\ &|3 A B|=675 \end{aligned}

Determinants exercise Fill in the blanks question 29

Hint: Here, we use basic concept of determinant of matrix.
Given:$\left[\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{array}\right]=0$
Solution:
So, Let’s
\begin{aligned} &R_{2} \rightarrow R_{2}-R_{1} \\ &R_{3} \rightarrow R_{3}-R_{1} \\ &{\left[\begin{array}{ccc} x+1 & x+2 & x+a \\ 1 & 1 & b-a \\ 2 & 2 & c-a \end{array}\right]=0} \end{aligned}
Let’s perform differentiate
\begin{aligned} &=x+a[0+0]-1[(0-a)-2(b-a)]+(x+a) 0=0 \\ &c+a-2 b=0 \\ &2 b=a+c \end{aligned}
So a,b,c are in A.P

Determinants exercise Fill in the blanks question 30

Hint: Here, we use basic concept of determinant of matrix.
Given:$\left[\begin{array}{ccc} \sin A & \cos A & \sin A+\cos B \\ \sin B & \cos A & \sin B+\cos B \\ \sin C & \cos A & \sin C+\cos B \end{array}\right]=0$
Solution: Let’s take sin A, sin B and sin C common from respectively 1st , 2nd and 3rd row
$\sin A \sin B \sin C\left[\begin{array}{lll} 1 & \cos A & 1+\cos B \\ 1 & \cos A & 1+\cos B \\ 1 & \cos A & 1+\cos B \end{array}\right]$
Here, all rows are same
According to properties of determinants, if there are 2 or more rows are same then, value of determinants is zero.

Determinants exercise Fill in the blanks question 31

Hint: Here, we use basic concept of determinant of matrix.
Given: $\left[\begin{array}{lll} x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h \end{array}\right] K=?$
Solution: Here, we use the property of determinant if elements of row and column of determinant are expressed as sum of two (or more) terms, then it can be expressed as sum of two ( or more) determinant.
Let’s take $\Delta$ of determinant
$\Delta=\left[\begin{array}{ccc} x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h \end{array}\right]$
Let’s use property for C1
\begin{aligned} &\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right] \\ &\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{lll} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right] \end{aligned}
Property using for C2
$\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]$$+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]$
Property using for C3
$\Delta=\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]$
$+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} x & p+u & l+f \\ y & q+v & m+g \\ z & r+w & n+h \end{array}\right]$
$+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]+\left[\begin{array}{ccc} a & p+u & l+f \\ b & q+v & m+g \\ c & r+w & n+h \end{array}\right]$
So, here we split in 8 parts.
So, k = 8

Determinants exercise Fill in the blanks question 32

Answer: $\frac{1}{2}$
Hint: Here we use basic concept of determinant of matrix.
Given: $\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cot \theta \end{array}\right]$
Solution:
$\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cot \theta \end{array}\right]$
\begin{aligned} &C_{2} \rightarrow C_{2}-C_{1}, C_{3} \rightarrow C_{3}-1 \\ &\Delta=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & \sin \theta & 0 \\ 1 & 0 & \cot \theta \end{array}\right] \\ &1\left[\begin{array}{cc} \sin \theta & 0 \\ 0 & \cot \theta \end{array}\right]-0\left[\begin{array}{cc} 1 & 0 \\ 1 & \cot \theta \end{array}\right]+0\left[\begin{array}{cc} 1 & \sin \theta \\ 1 & 0 \end{array}\right] \end{aligned}
\begin{aligned} &1(\sin \theta \times \cot \theta)-0(0)+0(0) \\ &\sin \theta \times \cot \theta-0+0 \\ &\sin \theta \times \frac{\cos \theta}{\sin \theta}=\cos \theta \\ &\Delta=\cos \theta \end{aligned}
Let's$\Delta =0$
$\cos \; \theta =0=\frac{\pi }{2} \left [ So,maximum\; value\; is\; \frac{1}{2}\; of \; \cos \theta \right ]$

Determinants exercise Fill in the blanks question 34

Hint: Here, we use basic concept of determinant of matrix
Given:$D=\left[\begin{array}{ccc} \sqrt{23}+\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{46} & 5 & \sqrt{10} \\ 3+\sqrt{115} & \sqrt{15} & 5 \end{array}\right]$
Solution: Let’s use separation property of determinant
\begin{aligned} &{\left[\begin{array}{ccc} \sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}\right]+\left[\begin{array}{ccc} \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}\right]} \\ &{\left[\begin{array}{lll} \sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{2 \times 23} & \sqrt{5 \times 5} & \sqrt{2 \times 5} \\ \sqrt{5 \times 23} & \sqrt{5 \times 3} & \sqrt{5 \times 5} \end{array}\right]+\left[\begin{array}{ccc} \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{3 \times 5} & \sqrt{5 \times 5} & \sqrt{2 \times 5} \\ \sqrt{3 \times 3} & \sqrt{5 \times 3} & \sqrt{5 \times 5} \end{array}\right]} \end{aligned}
\begin{aligned} C_{1} & \rightarrow \frac{C_{1}}{\sqrt{3}} \\ C_{2} & \rightarrow \frac{C_{2}}{\sqrt{5}} \text { and } C_{3} \rightarrow \frac{C_{3}}{\sqrt{5}} \end{aligned}
$\sqrt{23} \sqrt{5} \sqrt{5}\left[\begin{array}{ccc} 1 & 1 & 1 \\ \sqrt{2} & \sqrt{5} & \sqrt{2} \\ \sqrt{5} & \sqrt{3} & \sqrt{5} \end{array}\right]+\sqrt{3} \sqrt{5} \sqrt{5}\left[\begin{array}{ccc} 1 & 1 & 1 \\ \sqrt{5} & \sqrt{5} & \sqrt{2} \\ \sqrt{3} & \sqrt{3} & \sqrt{5} \end{array}\right]$
$C_{1} \rightarrow C_{1}-C_{3} \text { and } C_{1} \rightarrow C_{1}-C_{2}$
$5 \sqrt{23}\left[\begin{array}{ccc} 0 & 1 & 1 \\ 0 & \sqrt{5} & \sqrt{2} \\ 0 & \sqrt{3} & \sqrt{5} \end{array}\right]+5 \sqrt{23}\left[\begin{array}{ccc} 0 & 1 & 1 \\ 0 & \sqrt{5} & \sqrt{2} \\ 0 & \sqrt{3} & \sqrt{5} \end{array}\right]$
Here, both column has 0 value of each column
So, determinant value is zero.

Determinants exercise Fill in the blanks question 35

Hint: Here, we use basic concept of determinant of matrix
Given: $\left[\begin{array}{ccc} \sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 67^{\circ} & -\sin ^{2} 23^{\circ} & \cos ^{2} 180^{\circ} \\ -1 & \sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ} \end{array}\right]$
Solution:
$\left[\begin{array}{ccc} \sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 67^{\circ} & -\sin ^{2} 23^{\circ} & \cos ^{2} 180^{\circ} \\ -1 & \sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ} \end{array}\right]$
\begin{aligned} &{\left[\begin{array}{ccc} \sin ^{2} 23^{\circ} & \sin ^{2}\left(90^{\circ}-23^{\circ}\right) & -1 \\ -\sin ^{2}\left(90^{\circ}-23^{\circ}\right) & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \cos ^{2}\left(90^{\circ}-23^{\circ}\right) \end{array}\right]} \\ &{\left[\begin{array}{ccc} \sin ^{2} 23^{\circ} & \cos ^{2} 23 & -1 \\ -\cos ^{2} 23 & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} \end{array}\right]} \end{aligned}
Let's apply $C_{1} \rightarrow C_{1}+C_{2}$
\begin{aligned} &{\left[\begin{array}{ccc} \sin ^{2} 23^{\circ}+\cos ^{2} 23^{\circ} & \cos ^{2} 23 & -1 \\ -\cos ^{2} 23-\sin ^{2} 23^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1+\sin ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} \end{array}\right]} \\ &{\left[\begin{array}{ccc} 1 & 1 & -1 \\ -1 & -\sin ^{2} 23^{\circ} & 1 \\ -\cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} \end{array}\right]} \\ &(-1)\left[\begin{array}{ccc} 1 & 1 & -1 \\ -1 & -\sin ^{2} 23^{\circ} & 1 \\ -\cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} \end{array}\right] \end{aligned}
So, here two columns become same. So, determinant value becomes zero.

Determinants exercise Fill in the blanks question 36

Hint: Here we use basic determinant
Given: $\left[\begin{array}{ccc} 2-x & 2+x & 2+x \\ 2+x & 2-x & 2+x \\ 2+x & 2+x & 2-x \end{array}\right]=0$
Solution: Here, firstly
$2-x\left[\begin{array}{ll} 2-x & 2+x \\ 2+x & 2-x \end{array}\right]-2+x\left[\begin{array}{cc} 2+x & 2+x \\ 2+x & 2-x \end{array}\right]+2+x\left[\begin{array}{cc} 2+x & 2-x \\ 2+x & 2+x \end{array}\right]$
\begin{aligned} &0=(2-x)\left[(2-x)^{2}-(2+x)^{2}\right]-[2+x]\left[2^{2}-x^{2}[2+x]^{2}\right]+ &(2+x)\left[(2+x)^{2}-\left[\left(2^{2}-x^{2}\right)\right]\right. \end{aligned}
\begin{aligned} &0=8 x^{2}+16 x+8 x+8 x^{2}+8 x^{2}+4 x^{3}\end{aligned}
\begin{aligned} &0=8 x^{2}+8 x^{2}+8 x^{2}+4 x^{3} \end{aligned}
\begin{aligned} &0=24 x^{2}+4 x^{3} \end{aligned}
Let’s take 4x2 common
\begin{aligned} &4 x^{2}(6+x)=0 \\ &4 x^{2}=0 \text { and } 6+x=0 \\ &x^{2}=0 \text { and } x=-6 \end{aligned}

Determinants exercise Fill in the blanks question 37

Hint: Here, we use basic component of determinant
Given: A is square matrix of order 3
So, n = 3
$\left | A \right |=2$
Solution:
Here, $\left |-A A^{T} \right |$
So, $\left | -A \right |\times \left |A^{T} \right |$
Here, $\left |A^{T} \right |=\left | A \right |$ because both are transpose to each other
So, $\left | -A \right |\times \left | A \right |$
Let’s take -1 common
\begin{aligned} &-1([A]) \times|A| \\ &-1(2) \times 2[|A|=2] \\ &S o,\left|-A A^{T}\right|=-4 \end{aligned}

Determinants exercise Fill in the blanks question 33

Hint: Here, we use basic concept of determinant of matrix
Given: $D=\left[\begin{array}{ccc} \sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1 \\ \sin ^{2} C & \cot C & 1 \end{array}\right]$
Solution: Let’s perform some row operations
\begin{aligned} &R_{1} \rightarrow R_{1}-R_{3} \\ &R_{2} \rightarrow R_{2}-R_{3} \\ &{\left[\begin{array}{ccc} \sin ^{2} A-\sin ^{2} C & \cot A-\cot c & 0 \\ \sin ^{2} B-\sin ^{2} C & \cot B-\cot c & 0 \\ \sin ^{2} C & \cot C & 1 \end{array}\right]} \end{aligned}
$=\sin (A-C) \times \sin (B-C)\left[\begin{array}{ccc} \sin B & \frac{1}{\sin A \sin C} & 0 \\ \sin A & \frac{1}{\sin B \sin C} & 0 \\ \sin ^{2} C & \cot c & 1 \end{array}\right]$
\begin{aligned} &=\sin (A-C) \sin (B-C)\left[\frac{1}{\sin C}-\frac{1}{\sin C}\right] \\ &=\sin (A-C) \sin (B-C) \times 0 \\ &=0 \end{aligned}

The class 12 RD Sharma chapter 5 exercise FBQ solution is highly trusted by students and teachers in the entire country. The answers provided in the RD Sharma class 12th exercise FBQ book are expert-created, making them accurate and easy to understand. You will also find some exceptional tips in the book that might not be taught in class.

The class 12 RD Sharma chapter 5 exercise FBQ solution includes the chapter Determinants, which explores linear equations that need to be solved using determinants. The FBQ section is specifically essential because it contains questions from the entire chapter like symmetric matrix, skew-symmetric matrix, singular matrix, order of a matrix, minor, cofactor, and determinant value. Thus, there are 38 questions that students must answer.

The RD Sharma class 12 solutions Determinants ex FBQ can be an excellent guide for students due to the following reasons:

• The RD Sharma Class 12 Chapter 5 Exercise FBQ solutions are trusted by thousands of students in India.

• You might find common questions in your board paper if you practice the RD Sharma solutions thoroughly and diligently.

• RD Sharma class 12 solutions Determinants ex FBQ has the latest and updated syllabus, which is created to correspond to the NCERT textbooks used by teachers.

• Teachers will often use the RD Sharma class 12 solutions chapter 5 ex FBQ to give homework and test students' progress. This means using the RD Sharma solutions will help you to solve your homework no matter how difficult it is.

• RD Sharma class 12 solutions Determinants ex FBQ can be used by students for self-practice. They can use these solutions to compare answers and mark themselves at home.

• RD Sharma class 12th exercise FBQ Solutions can be downloaded from the Career360 website, for all subjects and classes.

• RD Sharma class 12 solutions Determinants ex FBQ book can be downloaded free of cost from Career360. So you won't need to pay anything to access or download the pdf.

RD Sharma Chapter wise Solutions

1. Where can I download the class 12 RD Sharma chapter 5 exercise FBQ solution?

2. Can I use RD Sharma class 12th exercise FBQ for JEE Mains preparations?

Yes, students can use RD Sharma class 12th exercise FBQ for their JEE mains preparations. The answers will help them test their knowledge and enhance their skills.

3. Does RD Sharma class 12 solutions chapter 5 ex FBQ has the latest syllabus?

RD Sharma class 12 solutions chapter 5 ex FBQ is regularly updated to correspond to the syllabus of NCERT Books. You can download the latest version of the pdf by entering your school year.

4. Can I use RD Sharma class 12 chapter 5 exercise FBQ for solving homework?

Teachers like to test students' progress by assigning homework from RD Sharma class 12 chapter 5 exercise FBQ. Therefore,  you can use these solutions to solve your homework.

5. How can I use class 12 RD Sharma chapter 5 exercise FBQ solution for self-practice?

You can solve questions from your NCERT textbooks and compare answers with class 12 RD Sharma chapter 5 exercise FBQ solution to mark yourself.

## Upcoming School Exams

#### All India Sainik Schools Entrance Examination

Application Date:06 November,2023 - 15 December,2023

#### National Institute of Open Schooling 12th Examination

Application Date:20 November,2023 - 19 December,2023

#### National Institute of Open Schooling 10th examination

Application Date:20 November,2023 - 19 December,2023

#### West Bengal Board 12th Examination

Exam Date:30 November,2023 - 14 December,2023

#### National Means Cum-Merit Scholarship

Application Date:03 December,2023 - 18 December,2023

Get answers from students and experts
##### Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

4 Jobs Available

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available
##### GIS Expert

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

3 Jobs Available
##### Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi

3 Jobs Available
##### Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available
##### Geotechnical engineer

The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.

3 Jobs Available
##### Finance Executive

A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

3 Jobs Available
##### Investment Banker

An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

3 Jobs Available
##### Bank Branch Manager

Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

3 Jobs Available
##### Treasurer

Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

3 Jobs Available
##### Bank Probationary Officer (PO)

A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as  Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

3 Jobs Available
##### Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available
##### Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available
##### Conservation Architect

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available
##### Architect

Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.

2 Jobs Available
##### Landscape Architect

Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.

2 Jobs Available
##### Plumber

An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.

2 Jobs Available
##### Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available
##### Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available
##### Veterinary Doctor

A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

5 Jobs Available
##### Gynaecologist

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

4 Jobs Available
##### Oncologist

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available
##### Surgical Technologist

When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

3 Jobs Available
##### Critical Care Specialist

A career as Critical Care Specialist is responsible for providing the best possible prompt medical care to patients. He or she writes progress notes of patients in records. A Critical Care Specialist also liaises with admitting consultants and proceeds with the follow-up treatments.

2 Jobs Available
##### Ophthalmologist

Individuals in the ophthalmologist career in India are trained medically to care for all eye problems and conditions. Some optometric physicians further specialize in a particular area of the eye and are known as sub-specialists who are responsible for taking care of each and every aspect of a patient's eye problem. An ophthalmologist's job description includes performing a variety of tasks such as diagnosing the problem, prescribing medicines, performing eye surgery, recommending eyeglasses, or looking after post-surgery treatment.

2 Jobs Available
##### Actor

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

4 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Acrobat

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
##### Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available
##### Talent Agent

The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

3 Jobs Available
##### Videographer

Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Visual Communication Designer

Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available

Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available
##### Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
##### Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available
##### Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available
##### Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available
##### Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### .NET Developer

.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### DevOps Architect

A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.

2 Jobs Available
##### Cloud Solution Architect

Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

2 Jobs Available