If $\alpha $ and $\beta $ are the roots of the quadratic equation, $x^2 + x\, sin\,\theta -2sin\,\theta = 0$, $\theta \in \left( {0,\frac{\pi }{2}} \right)$ then $\frac{{{\alpha ^{12}} + {\beta ^{12}}}}{{\left( {{\alpha ^{ - 12}} + {\beta ^{ - 12}}} \right){{\left( {\alpha - \beta } \right)}^{24}}}}$ is equal to
$\frac{{{2^{12}}}}{{{{\left( {\sin \,\theta + 8} \right)}^{12}}}}$
$\frac{{{2^{12}}}}{{{{\left( {\sin \,\theta - 4} \right)}^{12}}}}$
$\frac{{{2^{12}}}}{{{{\left( {\sin \,\theta - 8} \right)}^{6}}}}$
$\frac{{{2^{6}}}}{{{{\left( {\sin \,\theta + 8} \right)}^{12}}}}$
Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :
Consider a three-digit number with the following properties:
$I$. If its digits in units place and tens place are interchanged, the number increases by $36$ ;
$II.$ If its digits in units place and hundreds place are interchanged, the number decreases by $198 .$
Now, suppose that the digits in tens place and hundreds place are interchanged. Then, the number
Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?
If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$ then
If $a,b,c$ are distinct real numbers and $a^3 + b^3 + c^3 = 3abc$ , then the equation $ax^2 + bx + c = 0$ has two roots, out of which one root is