Properties of Determinants

Q.

Is the determinant of a transpose the same?

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

1. abc(a+b+c)
2. $3a^2{b}^2{c}^2$
3. \N
4. None of these

Q. Let a, b and c be such that b(a+c) $\ne$ 0.
If $\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n+2}a& (-1{)}^{n-1}b& (-1{)}^{n}c\end{array}\right|=0$, then the value of n is

1. zero

2. any even integer

3. any odd integer
4. any integer

Q.

What does determinant of $0$ mean?

Q. In a triangle ABC, the value of the determinant $\left|\begin{array}{ccc}sin\frac{A}{2}& sin\frac{B}{2}& sin\frac{c}{2}\\ sin(A+B+C)& sin\frac{B}{2}& cos\frac{A}{2}\\ cos\left(\frac{A+B+C}{2}\right)& tan(A+B+C)& sin\frac{C}{2}\end{array}\right|$ is less than or equal to

1. $\frac{1}{2}$
2. $\frac{1}{4}$
3. $\frac{1}{8}$
4. None of these

Q. If $\left|\begin{array}{ccc}y+z& x& y\\ z+x& z& x\\ x+y& y& z\end{array}\right|=k(x+y+z)(x-z{)}^2$, then k =

1. 2xyz
2. 1
3. xyz
4. $x^2{y}^2{z}^2$

Q. If $\lambda$ is a non real cube root of -2, then the value of $\left|\begin{array}{ccc}1& 2\lambda & 1\\ \lambda & 1& 3{\lambda }^2\\ 2& 2\lambda & 1\end{array}\right|$
is equal to

1. -11
2. -12
3. -13
4. \N

Q. Let $d\in R$, and
$A=\left[\begin{array}{ccc}-2& 4+d& \left(\sin \theta \right)-2\\ 1& \left(\sin \theta \right)+2& d\\ 5& \left(2\sin \theta \right)-d& (-\sin \theta )+2+2d\end{array}\right]$,
$\theta \in [0,2\pi ].$ If the minimum value of $det\left(A\right)$ is $8$, then a value of $d$ is:

1. $-5$
2. $2(\sqrt{2}+2)$
3. $-7$
4. $2(\sqrt{2}+1)$

Q.

What do determinants tell us?

Q. If $\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$ ; then for all

$\theta \in (\frac{3\pi }{4},\frac{5\pi }{4}),det\left(A\right)$ lies in the interval :

1. $(0,\frac{3}{2}]$
2. $(1,\frac{5}{2}]$
3. $(\frac{3}{2},3]$
4. $[\frac{5}{2},4)$

Q. Let a > 0, d > 0. Find the value of the determinant
$\left|\begin{array}{ccc}\frac{1}{a}& \frac{1}{a(a+d)}& \frac{1}{(a+d)(a+2d)}\\ \frac{1}{(a+d)}& \frac{1}{(a+d)(a+2d)}& \frac{1}{(a+2d)(a+3d)}\\ \frac{1}{(a+2d)}& \frac{1}{(a+2d)(a+3d)}& \frac{1}{(a+3d)(a+4d)}\end{array}\right|$ is

1. $\frac{4d}{a(a+d)(a+2d)(a+3d)(a+4d)}$
2. $\frac{4d}{a(a+d{)}^2(a+2d{)}^3(a+3d{)}^2(a+4d)}$
3. $\frac{4d^4}{a(a+d{)}^2(a+2d{)}^3(a+3d{)}^2(a+4d)}$
4. \N

Q.

Subtract $p-2q+r$ from the sum of $10p-r$and $5p+2q$

Q.

Subtract$-x^2+10x-5$ from $5x-10$

Q. The value of the determinant $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$ is

1. a+b+c
2. $(a+b+c{)}^2$
3. \N
4. 1+a+b+c

Q. $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|=$

1. $a^3+b^3+c^3-3abc$
2. $a^3+b^3+c^3+3abc$
3. $(a+b+c)(a-b)(b-c)(c-a)$
4. $Noneofthese$

Q.

If a² +b² + c² =-2 and

then f(x) is a polynomial of degree

1. 2

2. 3

3. 0

4. 1

Q. If $\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$

$=(a+b+c)(x+a+b+c{)}^2,x\ne 0$ and $a+b+c\ne 0$, then $x$ is equal to :

1. $abc$
2. $2(a+b+c)$
3. $-(a+b+c)$
4. $-2(a+b+c)$

Q. Let $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ be two $3\times 3$ real matrices such that $b_{ij}=(3{)}^{(i+j-2)}{a}_{ji}$, where $i,j=1,2,3$. If the determinant of $B$ is $81$, then the determinant of $A$ is :

1. $\frac{1}{9}$
2. $\frac{1}{81}$
3. $\frac{1}{3}$
4. $3$

Q. The total number of distinct $x\in R$ for which
$\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10\text{is}$

Q. Let $\omega$ be the complex number $cos\frac{2\pi }{3}+isin\frac{2\pi }{3}$. Then the number of distinct complex numbers z satisfying $\left|\begin{array}{ccc}z+1& \omega & {\omega }^2\\ \omega & z+{\omega }^2& 1\\ {\omega }^2& 1& z+\omega \end{array}\right|=0$ is equal to

1. 2
2. 1
3. \N
4. 3

Q. If a+b+c=0, then the solution of the equation $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$ is

1. $0$
2. $\pm \frac{3}{2}(a^2+b^2+c^2)$
3. $0,\pm \sqrt{\frac{3}{2}(a^2+b^2+c^2)}$
4. $0,\pm \sqrt{(a^2+b^2+c^2)}$

Q. If a, b, c are unequal what is the condition that the value of the following determinant is zero
$\Delta =\left|\begin{array}{ccc}a& a^2& a^3+1\\ b& b^2& b^3+1\\ c& c^2& c^3+1\end{array}\right|$

1. 1+abc = 0
2. a+b+c+1=0
3. (a-b) (b-c) (c-a) = 0
4. None of these

Q. If $a\ne b\ne c$, the value of x which satisfies the equation $\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$, is

1. x=0
2. x=a
3. x=b
4. x=c

Q. Find the value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$, where a, b and c are respectively the pth, qth and rth terms of a harmonic progression.

1. 1
2. pqr
3. \N
4. $\frac{1}{abc}$

Q. The roots of the equation $\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|=0$ are

1. -1, -2
2. -1, 2
3. 1, -2
4. 1, 2

Q. If n is positive integer, then $\left|\begin{array}{ccc}{n+2}_{C_n}& n+3_{C_{n+1}}& n+4_{C_{n+2}}\\ n+3_{C_{n+1}}& n+4_{C_{n+2}}& n+5_{C_{n+3}}\\ n+4_{C_{n+2}}& n+5_{C_{n+3}}& n+6_{C_{n+4}}\end{array}\right|$ is equal to

1. 3
2. -1
3. -5
4. -9

Q. Let $D_1=\left|\begin{array}{ccc}x& a& b\\ -1& 0& x\\ x& 2& 1\end{array}\right|$ and $D_2=\left|\begin{array}{ccc}c{x}^2& 2a& -b\\ x& 2& 1\\ -1& 0& x\end{array}\right|.$ If all the roots of the equation $(x^2-4x-7)(x^2-2x-3)=0$ satisfy the equation $D_1+D_2=0,$ then the value of $a+4b+c$ is

Q. For a fixed positive integer n, if
$D=\left|\begin{array}{ccc}n!& (n+1)!& (n+2)!\\ (n+1)!& (n+2)!& (n+3)!\\ (n+2)!& (n+3)!& (n+4)!\end{array}\right|$, then $[\frac{D}{(n!{)}^3}-4]$ is___

1. divisible by n.
2. divisible by n+1
3. divisible by n+2
4. divisible by n+3

Q. The roots of the equation $\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|=0$ are

1. -1, -2
2. -1, 2
3. 1, -2
4. 1, 2

Q. If a, b, c are respectively the $p^{th},q^{th},r^{th}$ terms of an A.P., the $\left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=$

1. 1
2. -1
3. \N
4. pqr