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

Solve the following determinant equations:
$\begin{vmatrix}\text{x}+1&3&5\\2&\text{x}+2&5\\2&3&\text{x}+4\end{vmatrix}=0$
→
Solve the following initial value problems:
$\Big\{\text{x}\sin^2\Big(\frac{\text{y}}{\text{x}}\Big)-\text{y}\Big\}\text{dx + x dy}=0,\text{y}(1)=\frac{\pi}4$
→
Find the maximum and the minimum values, if any, without using derivaives of the following functions:$f(x) = 2x^3+ 5$ on R.
→
If $\text{y}=\text{x}^3\log\text{x},$ Prove that $\frac{\text{d}^4\text{y}}{\text{dx}^4}=\frac{6}{\text{x}}$
→
Let f be a function defined on $[a, b]$ such that $f ′(x) > 0,$ for all x $\in (a, b).$ Then prove that f is an increasing function on $(a, b).$
→
A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?
→
Differentiate the following functions with respect to x:
$\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$
→
Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:
$2\text{x}+4\text{y}\leq8$
$3\text{x}+\text{y}\leq6$
$\text{x}+\text{y}\leq4$
$\text{x}\geq0,\text{y}\geq0$
→
Solve the following systems of linear equations by cramer's rule:
5x - 7y + z = 11,
6x - 8y - z = 15,
3x + 2y - 6z = 7
→
Verify the Rolle’s theorem for each of the functions:
$\text{f(x)}=\log(\text{x}^2+2)-\log3\text{ in }[-1,1].$
→