Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
If $a_1, a_2, a_3, ...,$ ar are in G.P., then prove that the determinant $\begin{bmatrix}\text{a}_{\text{r}+1}&\text{a}_{\text{r}+5}&\text{a}_{\text{r}+9}\\\text{a}_{\text{r}+7}&\text{a}_{\text{r}+11}&\text{a}_{\text{r}+15}\\\text{a}_{\text{r}+11}&\text{a}_{\text{r}+17}&\text{a} _{\text{r}+21}\end{bmatrix}$ is independent of r.
✓

Answer

We know that, $\text{a}_{\text{r}+1}=\text{AR}^{(\text{r}+1)}=\text{AR}^\text{r}$ where $r = r^{th}$ term of a GA, A = First term of a GP and R = Common ratio of GP We have $\begin{vmatrix}\text{a}_{\text{r}+1}&\text{a}_{\text{r}+5}&\text{a}_{\text{r}+9}\\\text{a}_{\text{r}+7}&\text{a}_{\text{r}+11}&\text{a}_{\text{r}+15}\\\text{a}_{\text{r}+11}&\text{a}_{\text{r}+17}&\text{a} _{\text{r}+21}\end{vmatrix}$ $=\text{AR}^{\text{r}}\text{AR}^{\text{r}+6}\text{AR}^{\text{r}+10}\begin{vmatrix}1&\text{AR}^4&\text{AR}^8\\1&\text{AR}^4&\text{AR}^8\\1&\text{AR}^6&\text{AR}^{10}\end{vmatrix}$[Taking $\text{AR}^\text{r}.\text{AR}^{\text{r}+6}.\text{AR}^{\text{r}+10}$ common from $\text{R}_1,\text{R}_2\text{ and R}_2$ respectively]
$=0\ [\text{Since, R}_1\text{ and R}_2\text{ are identicals}]$

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 "3 Marks Question" 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

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$
→
Verify the following:
$\int\frac{2\text{x}-1}{2\text{x}+3}\text{dx}=\text{x}-\log\big|(2\text{x}+3)^3\big|+\text{c}$
→
Write the value of $\sin^{-1}(\sin(-600^\circ))\sin(-600^\circ).$
→
Differentiate w.r.t. $x$ the function in $x^x + x^a + a^x + a^a ,$ for some fixed $a > 0$ and $x > 0.$
→
Integrate the function in Exercise:
$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$
→
Prove the following results:
$\sin^{-1}\frac{63}{65}=\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}$
→
Show that the minimum of Z occurs at more than two points.
Maximize Z = x + y, subject to $x - y \leq - 1, - x + y \leq 0, \ x, \ y \geq 0$.
→
Evaluate the following integrals:$\int2\text{x}^3\text{e}^{\text{x}^{2}}\text{dx}$
→
Write the value of b which $\text{f(x)}=\begin{cases}5\text{x}-4,&0<\text{x}\leq1\\4\text{x}^2+3\text{bx},&1<\text{x}<2\end{cases}$ is continuous at x = 1.
→
Write the following in the simplest form:
$\tan^{-1}\Big\{\text{x}+\sqrt{1+\text{x}^2}\Big\},\text{x}\in\text{R}$
→