Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
The value of $\begin{vmatrix}1&1&1\\^\text{n}\text{C}_1&^{\text{n}+2}\text{C}_1&^{\text{n}+4}\text{C}_1\\^\text{n}\text{C}_2&^{\text{n}+2}\text{C}_2&^{\text{n}+4}\text{C}_2\end{vmatrix}$ is:
  • A
    $2$
  • B
    $4$
  • ✓
    $8$
  • D
    $n^2$
✓

Answer

Correct option: C.
$8$
$\begin{vmatrix}1&1&1\\^\text{n}\text{C}_1&^{\text{n}+2}\text{C}_1&^{\text{n}+4}\text{C}_1\\^\text{n}\text{C}_2&^{\text{n}+2}\text{C}_2&^{\text{n}+4}\text{C}_2\end{vmatrix}$
$=\begin{vmatrix}1&1&1\\\text{n}&\text{n}+2&\text{n}+3\\\frac{\text{n}(\text{n}-1)}{2}&\frac{(\text{n}+2)(\text{n}+1)}{2}&\frac{(\text{n}+4)(\text{n}+3)}{2}\end{vmatrix}$
$=\begin{vmatrix}1&0&0\\\text{n}&2&4\\\frac{\text{n}(\text{n}+1)}{2}&\frac{4\text{n}+2}{2}&\frac{8\text{n}+12}{2}\end{vmatrix} [$Applying $C_2 \rightarrow C_2 - C_1$ and $C_3 \rightarrow C_3 - C_1]$
$=\begin{vmatrix}1&0&0\\\text{n}&2&4\\\frac{\text{n}(\text{n}+1)}{2}&(2\text{n}+1)&(4\text{n}+6)\end{vmatrix}$
$=8\text{n}+12-8\text{n}-4$
$=8$
Hence, the correct option is $(c)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Determinants→Determinants — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\int_{}^{} {\frac{1}{{{x^2}}}{{(2x + 1)}^3}} dx = $
→
The general solution of the differential equation $\frac{{dy}}{{dx}} = \cot x\cot y$ is
→
${d \over {dx}}\,\,\left[ {{{\tan }^{ - 1}}\left( {{{\sqrt x (3 - x)} \over {1 - 3x}}} \right)} \right] =$
→
The area bounded by the circle ${x^2} + {y^2} = 4,$ line $x = \sqrt 3 y$ and $x - $ axis lying in the first quadrant, is
→
If $A = \left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ - \sin \alpha }\\{\sin \alpha }&{\cos \alpha }\end{array}} \right]$and $B = \left[ {\begin{array}{*{20}{c}}{\cos \beta }&{ - \sin \beta }\\{\sin \beta }&{\cos \beta }\end{array}} \right]$, then the correct relation is
→
The equation of tangent to the curve $y\left(1+x^2\right)=2-x, w$ here it crosses $x-$axis is:
→
The number of commutative binary operation that can be defined on a set of $2$ elements is:
→
If the following system of linear equations

$2 x+y+z=5$

$x-y+z=3$

$x+y+a z=b$

has no solution, then :

→
The existence of the unique solution of the system of equations:
$x + y + z = \lambda$
$5x − y + µz = 10$
$2x + 3y − z = 6$
depends on
→
The function $f(x)=\left|x^{2}-2 x-3\right| \cdot e^{\left|9 x^{2}-12 x+4\right|}$ is not differentiable at exactly :
→