Determinants Test

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Determinants Test - 55

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Question 1
1 / -0

In a triangle ABC, with usual notations, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix}=0,$$ then $$4sin^2A+24sin^2B+36sin^2C$$ is equal to

A
$$48$$

B
$$50$$

C
$$44$$

D
$$34$$

Solution

$$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix} = 0$$
using $$R_2 \rightarrow R_2 - R_1$$ and $$R_3 \rightarrow R_3 - R_1$$ property.
$$\Rightarrow \begin{vmatrix} 1 & a & b \\ 0 & c-a & a-b \\ 0 & b-a & c-b \end{vmatrix} = 0$$
Expanding along first column.
$$\Rightarrow 1[(c -a ) (c - b) - (b - a)(a - b)] + 0 = 0$$
$$\Rightarrow c^2 - ac - bc + ab - (ab - a^2 - b^2 + ab) = 0$$
$$\Rightarrow c^2 - ac - bc + ab - ab + a^2 + b^2 - ab = 0$$
$$\Rightarrow a^2 + b^2 + c^2 - ab - bc - ca = 0$$
using identify
$$a^2 + b^2 + c^2 - ab - bc - ca = \dfrac{1}{2}\left [(a - b)^2 + (b - c)^2 + (c- a)^2\right]$$
$$\Rightarrow \dfrac{1}{2} [(a - b)^2 + (b - c)^2 + (c - a)^2] = 0$$
It us only possible if $$a + b = c$$
$$\Rightarrow $$ Triangle is equilateral
$$\Rightarrow \angle A = \angle B = \angle C = 60^{\circ}$$
$$\Rightarrow 4 \sin^2 A + 24 \sin^2 B + 36 \sin^2 c$$
$$= 4 \sin^2 60^{\circ} + 24 \sin^2 60^{\circ} + 36 \sin^2 60^{\circ}$$
$$= 64 \sin^2 60^{\circ} = 64 \times \left(\dfrac{\sqrt{3}}{2} \right)^2$$
$$= 64 \times \dfrac{3}{4}$$
$$= 48$$
Question 2
1 / -0

If adj $$A=\begin{bmatrix}20 & -20 \\ 10 & 10 \end{bmatrix}$$ , then $$|A|=$$.....

A
400

B
200

C
$$\pm 20$$

D
0

Solution
Question 3
1 / -0

Sum of the real roots of the equation $$\begin{vmatrix} 1 & 4 & 20\\ 1 & -2 & 5 \\ 1 & 2x & 5{x}^{2} \end{vmatrix}=0$$ is

A
$$-2$$

B
$$-1$$

C
$$0$$

D
$$1$$

Solution
Question 4
1 / -0

Let $$\omega =-\dfrac {1}{2}+i\dfrac {\sqrt {3}}{2}$$. Then the value of the determinant.
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -1-{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{vmatrix}$$ is

A
$$-3\omega $$

B
$$3\omega (\omega -1)$$

C
$$3\omega^{2}$$

D
$$3\omega (1-\omega)$$

Solution

$$\left| \begin{matrix} 1 & 1 & 1 \\ 1 & -1-{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \\ $$
$$\Rightarrow R_1=R_1-R_3$$
$$\Rightarrow \left| \begin{matrix} 0 & 1-{ \omega }^{ 2 } & 1-{ \omega }^{ 2 } \\ 1 & -1-{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \\ $$
$$\Rightarrow C_2=C_2-C_3$$
$$\Rightarrow \left| \begin{matrix} 0 & 0 & 1-{ \omega }^{ 2 } \\ 1 & -1-2{ \omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & 0 & { \omega }^{ 2 } \end{matrix} \right| \\ $$
$$Det=(1-\omega^2)\times[1\times0+(1+2\omega^2)]\\ \quad=(1-\omega^2)[(1+\omega^2)+\omega^2]\\ \quad=(1-\omega^4)+\omega^2(1-\omega^2)\\ \quad=1-\omega+\omega^2-\omega^4\\ \quad=1-\omega+\omega^2-\omega\\ \quad=1+\omega^2-2\omega\\ \quad=-3\omega$$
Question 5
1 / -0

The determinant $$\begin{bmatrix} b_{1}+c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\ b_{2}+c_{2} & c_{2}+a_{2} & a_{2}+b_{2} \\ b_{3}+c_{3} & c_{3}+a_{3} & a_{3}+b_{3}\end{bmatrix}=$$_____

A
$$\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$

B
$$2\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$

C
$$3\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$

D
$$4\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{bmatrix}$$
Question 6
1 / -0

Find the values of $$x$$ if, $$\left| \begin{matrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^{ 2 } \end{matrix} \right| =0$$

A
$$-1, 2$$

B
$$-1, -2$$

C
$$1, -2$$

D
$$1, 2$$

Solution
Question 7
1 / -0

29 If $$z=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ where 0, I are 2x2 null and identity matrix then det $$\left( \left[ z \right] \right) $$ is _______________.

A
1

B
-1

C
0

D
None of these

Solution
Question 8
1 / -0

The cofactor of the element $$4$$ in the determinant $$\begin{vmatrix} 1 & 3 & 5 & 1\\ 2 & 3 & 4 & 2\\ 8 & 0 & 1 & 1\\ 0 & 2 & 1 & 1\end{vmatrix}$$ is?

A
$$4$$

B
$$10$$

C
$$-10$$

D
$$-4$$
Question 9
1 / -0

If $$\Delta =\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}$$ for $$x \neq 0, y \neq 0$$, then $$\Delta $$ is

A
divisible by neither $$x$$ nor $$y$$.

B
divisible by both $$x$$ and $$y$$

C
divisible by $$x$$ but not $$y$$

D
divisible by $$y$$ but not $$x$$.

Solution

ANSWER IS B
Question 10
1 / -0

The number of distinct real roots of the equation,$$\left| \begin{matrix} cosx & sinx & sinx \\ sinx & cosx & sinx \\ sinx & sinx & cosx \end{matrix} \right| =0$$In t interval $$\left[ -\dfrac { \pi }{ 4 } \dfrac { \pi }{ 4 } \right]$$ is/are:

A
$$3$$

B
$$2$$

C
$$1$$

D
$$4$$

Solution