Evaluation of a Determinant

Q. The set of all values of $\lambda$ for which the system of linear equations $2x_1-2x_2+x_3=\lambda x_1,2x_1-3x_2+2x_3=\lambda x_2$ and $-x_1+2x_2=\lambda x_3$ has a non - tirvial solution.

1. is an empty set
2. is a singleton set

3. contains two elements

4. contains more than two elements

Q.

The determinant $\left|\begin{array}{ccc}1& ab& \frac{1}{a}+\frac{1}{b}\\ 1& bc& \frac{1}{b}+\frac{1}{c}\\ 1& ca& \frac{1}{c}+\frac{1}{a}\end{array}\right|$ is equal to

1. 0

2. bc + ca + ab

3. $a^3+b^3+c^3$

4. None of these

Q.

If$A,B,C$ are acute positive angles such that $A+B+C=\pi$ and $cotAcotBcotC=K$ then

1. $K\le \frac{1}{3\surd 3}$

2. $K\ge \frac{1}{3\surd 3}$

3. $K<\frac{1}{9}$

4. $k>\frac{1}{3}$

Q. The value of the determinant $\left|\begin{array}{ccc}4& -6& 1\\ -1& -1& 1\\ -4& 11& -1\end{array}\right|$ is

1. -75
2. 25
3. \N
4. -25

Q.

If a₁, a₂, a₃, . . . . . . , a_n . . . . . . .are in GP, then the value of the determinant

1. 0

2. 1

3. 2

4. -2

Q. Let $a_1,a_2,a_3,\cdots ,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\cdots ,10$ and $S$ be the set of pairs $(r,k),r,k\in N$ (the set of natural numbers) for which
$\left|\begin{array}{ccc}{\log }_e{a}_1^r{a}_2^k& {\log }_e{a}_2^r{a}_3^k& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{a}_4^r{a}_5^k& {\log }_e{a}_5^r{a}_6^k& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{a}_7^r{a}_8^k& {\log }_e{a}_8^r{a}_9^k& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|$

Then the number of elements in $S$, is :

1. $2$
2. $4$
3. $10$
4. infinitely many

Q.

The value of when m is equal to

1. 6

2. 5

3. 4

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

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

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

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

4. None of these

Q. $A=\left[\begin{array}{cc}1& \tan x\\ -\tan x& 1\end{array}\right]$ and $f\left(x\right)$ is defined as $f\left(x\right)=det.A^T{A}^{-1}$ then the value of $\frac{f\left(f\right(f......f\left(x\right)\left)\right))}{n\text{times}}(\text{for}n\ge 2)$ is

Q. If matrix $A$ is given by $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the determinant of $A^{2005}-6A^{2004}$ is

1. $2^{2006}$
2. $(-11)2^{2005}$
3. ${7.2}^{2006}$
4. $(-9)2^{2004}$

Q. A value of $\theta \in (0,\frac{\pi }{3})$, for which $\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & 1+{\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & {\sin }^2\theta & 1+4\cos 6\theta \end{array}\right|=0\text{is}:$

1. $\frac{\pi }{18}$
2. $\frac{\pi }{9}$
3. $\frac{7\pi }{24}$
4. $\frac{7\pi }{36}$

Q. If $p{\lambda }^4+q{\lambda }^3+r\lambda +5\lambda +t=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 2-\lambda & \lambda -4\\ \lambda -3& \lambda +4& 3\lambda \end{array}\right|$, the value of t is

1. 16
2. 18
3. 17
4. 19

Q.

If A is a square matrix of order n such that |adj (adj A)|$=|A{|}^9$, then the value of n can be

1. 4

2. 2

3. either 4 or 2

4. None of these

Q. If $a^3\left|\begin{array}{ccc}1& b& c\\ b^2& b^3+1& b^{2}c\\ c^2& b{c}^2& c^3+1\end{array}\right|+a^2\left|\begin{array}{ccc}\frac{1}{a^2}& 0& 0\\ b& b^3+1& b{c}^2\\ c& b^{2}c& c^3+1\end{array}\right|=18,$
where $a,b,c\ge 0,$ then the values of $(a,b,c)$ satisfying the given equation is/are

1. $(1,2,2)$
2. $(2,1,2)$
3. $(2,1,1)$
4. $(1,1,2)$

Q. The value of the determinant $\left(\begin{array}{ccc}1& 2& 4\\ -1& 3& 0\\ 4& 1& 0\end{array}\right|$ will be .

1. -13
2. -52
3. \N
4. -4

Q. Let $\omega \ne 1$ be a cube root of unity and S be the set of all non - singular matrices of the form $\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$, where each of a, b and c is either $\omega$ or ${\omega }^2$. Then, the number of distinct matrices in the set S is

1. 2
2. 6
3. 4
4. 8

Q. What will be the determinant of the following matrix
$\left[\begin{array}{cc}2& 3\\ a& c\end{array}\right]$

1. 3a - 2c
2. 2c - 3a
3. 6 - ac
4. ac - 6

Q. Let $\Delta =\left|\begin{array}{ccc}-bc& b^2+bc& c^2+bc\\ a^2+ac& -ac& c^2+ac\\ a^2+ab& b^2+ab& -ab\end{array}\right|$ and the equation $p{x}^2+q{x}^2+rx+s=0$ has roots, $a,b,c$ where $a,b,c\in R^{+}$

The value of $\Delta$ is

1. $\frac{r^2}{p^2}$
2. $\frac{r^3}{p^3}$
3. $-\frac{s}{p}$
4. None of these

Q. The set of all values of $\lambda$ for which the system of linear equations $2x_1-2x_2+x_3=\lambda x_1,2x_1-3x_2+2x_3=\lambda x_2$ and $-x_1+2x_2=\lambda x_3$ has a non - tirvial solution.

1. is an empty set
2. is a singleton set

3. contains two elements

4. contains more than two elements

Q. Let the system of linear equations
$4x+\lambda y+2z=0$
$2x-y+z=0$
$\mu +2y+3z=0,\lambda ,\mu \in R$
has a non-trivial solution. Then which of the following is true?

1. $\lambda =3,\mu \in R$
2. $\mu =6,\lambda \in R$
3. $\mu =-6,\lambda \in R$
4. $\lambda =2,\mu \in R$

Q. If $C_i,i=1,2,....9$ are perfect odd squares, then $\left|\begin{array}{ccc}{C}_1& C_2& C_3\\ C_4& C_5& C_6\\ C_7& C_8& C_9\end{array}\right|$ is always a multiple of

1. 14
2. 7
3. 16
4. 5

Q. If $A_1,B_1,C_1....$ are respectively the co-factors of the elements $a_1,b_1,c_1....$ of the determinant $\Delta =\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|,$ then $\left|\begin{array}{cc}{B}_2& C_2\\ B_3& C_3\end{array}\right|=$

1. $a_1\Delta$
2. $a_1{a}_3\Delta$
3. $(a_1+b_1)\Delta$
4. None of these

Q. If $\omega$ is a cube root of unity and $\Delta =\left|\begin{array}{cc}1& 2\omega \\ \omega & {\omega }^2\end{array}\right|$, then ${\Delta }^2$ is equal to

1. $-\omega$
2. ${\omega }^2$
3. 1
4. $\omega$

Q. If $\left[\right]$ denotes the greatest integer less than or equal to the real number under consideration, and $-1\le x<0,0\le y<1,1\le z<2,$ then the value of the determinant $\left|\begin{array}{ccc}\left[x\right]+1& \left[y\right]& \left[z\right]\\ \left[x\right]& \left[y\right]+1& \left[z\right]\\ \left[x\right]& \left[y\right]& \left[z\right]+1\end{array}\right|$ is

1. $\left[x\right]$
2. $\left[y\right]$
3. $\left[z\right]$
4. None of the above

Q.

The value of the determinant
$\Delta =\left|\begin{array}{ccc}logx& logy& logz\\ log2x& log2y& log2z\\ log3x& log3y& log3z\end{array}\right|$ is

1. 0

2. log (xyz)

3. log (6 xyz)

4. 6 log (xyz)

Q. Given $\Delta$$=\left|\begin{array}{ccc}P& 2-i& i+1\\ 2+i& q& 3+i\\ 1-i& 3-i& r\end{array}\right|$, $\Delta$ is

1. Real

2. Imaginary

3. Complex

4. Cannot be determined

Q. Area of triangle whose vertices are $(a,a^2),(b,b^2),(c,c^2)$ is 1, and area of another triangle whose vertices are $(p,p^2),(q,q^2)$ and $(r,r^2)$ is 4 then the value of $\left|\begin{array}{ccc}(1+ap{)}^2& (1+bp{)}^2& (1+cp{)}^2\\ (1+aq{)}^2& (1+bq{)}^2& (1+cq{)}^2\\ (1+ar{)}^2& (1+br{)}^2& (1+cr{)}^2\end{array}\right|$ is

1. 4
2. 8
3. 16
4. 32

Q.

The value of the determinant
$\left|\begin{array}{ccc}lo{g}_a\left(\frac{x}{y}\right)& lo{g}_a\left(\frac{y}{z}\right)& lo{g}_a\left(\frac{z}{x}\right)\\ lo{g}_b\left(\frac{y}{z}\right)& lo{g}_b\left(\frac{z}{x}\right)& lo{g}_b\left(\frac{x}{y}\right)\\ lo{g}_c\left(\frac{z}{x}\right)& lo{g}_c\left(\frac{x}{y}\right)& lo{g}_c\left(\frac{y}{z}\right)\end{array}\right|$ is

1. $lo{g}_a\left(xyz\right)+lo{g}_b\left(xyz\right)+lo{g}_c\left(xyz\right)$

2. $alo{g}_{e}x+blo{g}_{e}y+clo{g}_{e}z$

3. -1

4. None of these

Q. $\left|\begin{array}{ccc}1+i& 1-i& i\\ 1-i& i& 1+i\\ i& 1+i& 1-i\end{array}\right|=$

1. -4-7i
2. 4+7i
3. 3+7i
4. 7+4i