Special Determinants

Q.

Determinant of skew symmetric matrix is always 0.

1. True

2. False

Q. If $\omega$ is an imaginary cube root of unity, then the value of the determinant

$\left|\begin{array}{ccc}1+\omega & {\omega }^2& -\omega \\ 1+{\omega }^2& \omega & -{\omega }^2\\ \omega +{\omega }^2& \omega & -{\omega }^2\end{array}\right|$ is

1. $-2\omega$
2. $-3{\omega }^2$
3. $-1$
4. $0$ (Zero)

Q. Express the following matrix as the sum of a symmetric and a skew symmetric matrix:
$\left(i\right)$ $\left[\begin{array}{cc}3& 5\\ 1& -1\end{array}\right]$
$\left(ii\right)$ $\left[\begin{array}{ccc}6& -2& 2\\ -2& 3& -1\\ 2& -1& 3\end{array}\right]$
$\left(iii\right)$ $\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$
$\left(iv\right)$ $\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]$

Q. If $A$ is a skew symmetric matrix, then which of the following is/are true?

1. $(A^7{)}^T=-A^7$
2. $(A^4{)}^T=-A^4$
3. $(A^8{)}^T=A^8$
4. $(A^{11}{)}^T=A^{11}$

Q. Let $M$ denote the maximum value of $f\left(x\right)={\sin }^{2}x-4\sin x+10$ and $m$ denote the minimum value of $g\left(x\right)=4{\sec }^{2}x+36{\text{cosec}}^{2}x-14.$ Then the value of $(m-M)$ is

Q. If matrix $A=\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]=B+C$, where $B$ is symmetric matrix and $C$ is skew-symmetric matrix.
Then the matrices $B$ and $C$ are:

1. $B=\left[\begin{array}{ccc}0& \frac{5}{2}& 1\\ \frac{5}{2}& -3& \frac{1}{2}\\ 1& \frac{1}{2}& 4\end{array}\right]$
2. $B=\left[\begin{array}{ccc}0& -\frac{5}{2}& -1\\ \frac{5}{2}& 3& \frac{1}{2}\\ 1& -\frac{1}{2}& 4\end{array}\right]$
3. $C=\left[\begin{array}{ccc}0& -\frac{3}{2}& -2\\ \frac{3}{2}& 0& \frac{7}{2}\\ 2& -\frac{7}{2}& 0\end{array}\right]$
4. $C=\left[\begin{array}{ccc}0& -\frac{3}{2}& 2\\ -\frac{3}{2}& 0& -\frac{7}{2}\\ 2& -\frac{7}{2}& 0\end{array}\right]$

Q.

Suppose the cubic $x^3-px+q$ has $3$ distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

1. The cubic has minima at $-\sqrt{\frac{p}{3}}$ and maxima at $\sqrt{\frac{p}{3}}$

2. The cubic has minima at both $\sqrt{\frac{p}{3}}$ and $-\sqrt{\frac{p}{3}}$

3. The cubic has maxima at both $\sqrt{\frac{p}{3}}$ and $-\sqrt{\frac{p}{3}}$

4. The cubic has minima at both $\sqrt{\frac{p}{3}}$ and maxima at $-\sqrt{\frac{p}{3}}$

Q. If $\left|\begin{array}{ccc}2x^2+x& 3x-5& 4x+3\\ x+5& -x^2& 6x-6\\ 2x-3& 5x+6& -3x\end{array}\right|=a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f\forall x\in R,$ then the value of $f$ is

1. $14$
2. $5$
3. $0$
4. $-8$

Q. $\int {\left|x\right|}^{3}dx$ is equal to

(a) $\frac{-x^4}{4}+C$

(b) $\frac{{\left|x\right|}^4}{4}+C$

(c) $\frac{x^4}{4}+C$

(d) none of these

Q. $\left|\begin{array}{ccc}0& a& -b\\ -a& 0& c\\ b& -c& 0\end{array}\right|=$

1. -2abc
2. abc
3. \N
4. $a^2+b^2+c^2$

Q.

1. R_1→R₂-R₁

2. R_2→R₂+R₁

3. R_1→R₁+R₂

4. R_1→R₁-R₂

Q.

Let \( a , b , c \) be positive and not all equal, the value of the determinant \( \left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right| i \)

Q.

Let A be a square matrix of order n and B be its adjoint, then for a scalar K, $|AB+K{I}_n|$ is ___

1. $(|A|+K{)}^{n-1}$

2. $(|A|+K{)}^{n-2}$

3. $(|A|+K{)}^n$

4. None of these

Q. A b c + b a c = d a d c .find the value of a b c d

Q.

1. 0

2. -1

3. 2

4. 1

Q. Let $P=\left[a_{ij}\right]$ be a 3 $\times$ 3 matrix and let $Q=\left[b_{ij}\right]$, where $b_{ij}=2^{i+j}{a}_{ij}$ for $1\le i,j\le 3$. If the determinant of P is 2, then the determinant of the matrix Q is

1. $2^{10}$
2. $2^{11}$
3. $2^{12}$
4. $2^{13}$

Q. Let $A=\left[\begin{array}{ccc}4& -1& 4\\ 4& 0& 2\\ 3& -2& 4\end{array}\right]$
Find $A^{-1}$ by using elementery row transformations.

Q.

$\text{Given f (x)=}\left|\begin{array}{ccc}1& x& x^2\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|\text{Then, is (x - a) a factor to this determinant}$

1. True

2. False

Q.

If 2P(A) = P(B) = 5/13 and P(A|B) = 2/5, then P(A U B) = ...

1. 

2. 

3. 

4. 

Q. Let $A=\{1,3,5,7\},B=\{2,4,6,8\}$ and $f:A\to B,$ then number of function(s) $f$ that can be defined from $A$ to $B$ is

1. $24$ if $i+f\left(i\right)<10,\forall i=\{1,3,5,7\}$
2. $12$ if $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$
3. $12$ if $i+f\left(i\right)<10,\forall i=\{1,3,5,7\}$
4. $0$ if $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$

Q.

$\text{Given f (x)=}\left|\begin{array}{ccc}1& x& x^2\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|\text{Then, is (x - a) a factor to this determinant}$

1. True

2. False

Q.

a, b, c are different real numbers. factor of the determinant

$\left|\begin{array}{ccc}0& x^2-a& x^3-b\\ x^2+a& 0& x^2+c\\ x^4+b& x-c& 0\end{array}\right|$ is

1. x
2. x-a
3. x-b
4. x-c

Q. If $\frac{x^4+1}{(x-1)(x-2)}=A{x}^2+Bx+C-\frac{2}{x-1}+\frac{17}{x-2}$, then $C=$

1. $1$
2. $3$
3. $5$
4. $7$

Q. Let
$P_1=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|,$$P_2=\left|\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right|,$$P_3=\left|\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right|,$$P_4=\left|\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right|,$$P_5=\left|\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right|,$$P_6=\left|\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right|,$ and $X=\sum _{k=1}^6{P}_k\left[\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 3& 2& 1\end{array}\right]P_k^T,$ where $P_k^T$ denotes the transpose of the matrix $P_k.$ Then which of the following options is/are correct?

1. $X$ is a symmetric matrix
2. $\text{If}X\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\alpha \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\text{then}\alpha =30$
3. $X-30I$ is an invertible matrix
4. The sum of diagonal entries of $X$ is $18$

Q. $\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=$

1. $a^2+b^2+c^2$
2. (a-b) (b-c) (c-a)
3. (a+b) (b+c) (c+a)
4. None of these

Q.

If value of $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ is A then value of $\left|\begin{array}{ccc}d& e& f\\ a& b& c\\ g& h& i\end{array}\right|$ is also equal to A.

1. True

2. False

Q.

$\text{Express the det}A=\left|\begin{array}{ccc}a& bc& abc\\ b& ca& abc\\ c& ab& abc\end{array}\right|\text{as product of factors}$

1. $abc(a+b)(b+c)(c+a)$

2. $abc(a-b)(b-c)(c-a)$

3. $ab(a-b)(b-c)$

4. $abc(a+b)(b-c)(c-a)$

Q.

1. 2

2. 3

3. 4

4. 1

Q.

1. (a)²

2. (a - b)²

3. (a+b)²

4. 1

Q.

If p + q + r = a + b + c = 0, then the determinant
$\Delta =\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

1. 0

2. 1

3. pa + qb + rc

4. none of these