MCQ
Solve the equations $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0, a \neq 0$
- ✓$x=-\frac{a}{3}$
- B$x=-\frac{a}{6}$
- C$x=-\frac{a}{2}$
- D$x=-\frac{a}{4}$
Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3},$ we get:
$\left|\begin{array}{ccc}3 x+a & 3 x+a & 3 x+a \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0$
$\Rightarrow(3 x+a)\left|\begin{array}{ccc}1 & 1 & 1 \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0$
Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1},$ we have:
$\Rightarrow(3 x+a)\left|\begin{array}{lll}1 & 1 & 1 \\ x & a & x \\ x & x & a\end{array}\right|=0$
Expanding along $R_{1},$ we have:
$(3 x+a)\left[1 \mathrm{x} a^{2}\right]=0$
$\Rightarrow a^{2}(3 x+a)=0$
But $a \neq 0$
Therefore, we have:
$3 x+a=0$
$\Rightarrow x=-\frac{a}{3}$
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| Column $I$ | Column $II$ |
| $(A)$ Circle | $(p)$ The locus of the point $(h, k)$ for which the line $h x+k y=1$ touches the circle $x^2+y^2=4$ |
| $(B)$ Parabola | $(q)$ Points $z$ in the complex plane satisfying $|z+2|-|z-2|= \pm 3$ |
| $(C)$ Ellipse | $(r)$ Points of the conic have parametric representation $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$ |
| $(D)$ Hyperbola | $(s)$ The eccentricity of the conic lies in the interval $1 \leq x<\infty$ |
| $(t)$ Points $z$ in the complex plane satisfying $\operatorname{Re}(z+1)^2=|z|^2+1$ |