Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}})^2&(2^{\text{x}}-2^{-\text{x}})^2&1\\(3^{\text{x}}+3^{-\text{x}})^2&(3^{\text{x}}-3^{-\text{x}})^2&1\\(4^{\text{x}}+4^{-\text{x}})^2&(4^{\text{x}}-4^{-\text{x}})^2&1\end{vmatrix}$
✓

Answer

$\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}})^2&(2^{\text{x}}-2^{-\text{x}})^2&1\\(3^{\text{x}}+3^{-\text{x}})^2&(3^{\text{x}}-3^{-\text{x}})^2&1\\(4^{\text{x}}+4^{-\text{x}})^2&(4^{\text{x}}-4^{-\text{x}})^2&1\end{vmatrix}$
$=\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}}+2)&(2^{\text{x}}-2^{-\text{x}}-2)&1\\(3^{\text{x}}+3^{-\text{x}}+2)&(3^{\text{x}}-3^{-\text{x}}-2)&1\\(4^{\text{x}}+4^{-\text{x}}+2)&(4^{\text{x}}-4^{-\text{x}}-2)&1\end{vmatrix}$
$=\begin{vmatrix}4&(2^{\text{x}}+2^{-\text{x}}-2)&1\\4&(3^{\text{x}}+3^{-\text{x}}-2)&1\\4&(4^{\text{x}}+4^{-\text{x}}-2)&1\end{vmatrix}$ [Applying $C_1 → C_1 - C_2$]
$=4\begin{vmatrix}1&(2^{\text{x}}+2^{-\text{x}}-2)&1\\1&(3^{\text{x}}+3^{-\text{x}}-2)&1\\1&(4^{\text{x}}+4^{-\text{x}}-2)&1\end{vmatrix}$ $=0$

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 "5 Marks Questions" 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

A library has to accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weigh 1kg and $1\frac{1}{2}$ kg each respectively. The shelf is 96 cm long and at most can support a weight of 21kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LPP and solve it graphically.
→
Make a sketch of the recion ${(x, y) : 0 < y < x^2 + 3 : 0 < y < 2x + 3 : 0 < x < 3}$ and find using interation.
→
Show that $\tan\Big(\frac{1}{2}\sin^{-1}\frac{3}{4}\Big)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?
→
If $x^m + y^n = 1$, Prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$
→
If $\text{x}=\text{a}(\cos\theta+\theta\sin\theta),\text{y}=\text{a}(\sin\theta-\theta\cos\theta)$ prove that $\frac{\text{d}^2\text{x}}{\text{d}\theta^2}=\text{a}(\cos\theta-\theta\sin\theta),\frac{\text{d}^2}{\text{d}\theta^2}$ $=\text{a}(\sin\theta-\theta\cos\theta)\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\sec^3\theta}{\text{a}\theta}$
→
Prove the following results:
$\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}=\pi$
→
Evaluate the following integrals as limit of sum:
$\int\limits^3_{2}\big(2\text{x}^2+1\big)\text{ dx}$
→
If $\text{u}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ and $\text{v}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big),$ where -1 < x < 1 then write the value of $\frac{\text{dy}}{\text{dx}}.$
→
Evaluate the following integrals:
$\int\limits^\pi_0\sin^{100}\text{x}\cos^{101}\text{x dx}$
→
Using integration find the area of the region bounded by the curves $\text{y} = \sqrt{4 - \text{x}^{2}}, \text{x}^{2} + \text{y}^{2} \text{4x} = 0$ and the x-axis.
→