The sum of all the solutions of the equation \( (8)^{2x} - 16\cdot(8)^{x} + 48 = 0 \) is:
\( \log_{8}(4) \)
\( 1 + \log_{8}(6) \)
\( \log_{8}(6) \)
\( 1 + \log_{8}(8) \)
Let the set \( C = \{(x,y)\mid x^2 - 2^y = 2023,\, x,y \in N\} \). Then \(\displaystyle \sum_{(x,y)\in C} (x+y) \) is equal to _______________.
The number of real solutions of the equation \( x\big(x^{2}+3|x|+5|x-1|+6|x-2|\big)=0 \) is __________.
If \( 2 \) and \( 6 \) are the roots of the equation \( ax^2+bx+1=0 \), then the quadratic equation, whose roots are \( \dfrac{1}{2a+b} \) and \( \dfrac{1}{6a+b} \), is:
The number of distinct real roots of the equation \( |x+1|\,|x+3|-4|x+2|+5=0 \), is __________.
The number of solutions, of the equation \( e^{\sin x}-2e^{-\sin x}=2 \), is:
Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \( 4x^{4} + 8x^{3} - 17x^{2} - 12x + 9 = 0 \) and \( \left(4 + x_{1}^{2}\right) \left(4 + x_{2}^{2}\right) \left(4 + x_{3}^{2}\right) \left(4 + x_{4}^{2}\right) = \dfrac{125}{16}\, m \). Then the value of \( m \) is:
If \( \alpha,\beta \) are the roots of the equation, \( x^{2}-x-1=0 \) and \( S_{n}=2023\,\alpha^{n}+2024\,\beta^{n} \), then:
Let \( \alpha, \beta \) be the distinct roots of the equation \( x^2 - (t^2 - 5t + 6)x + 1 = 0, \; t \in \mathbb{R} \) and \( a_n = \alpha^n + \beta^n \). Then the minimum value of \( \dfrac{a_{2023} + a_{2025}}{a_{2024}} \) is:
Let \( \alpha,\beta;\alpha>\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x-\sqrt{3}=0 \). Let \( P_{n}=\alpha^{n}-\beta^{n},\, n\in\mathbb{N} \). Then \( \left(11\sqrt{3}-10\sqrt{2}\right)P_{10}+\left(11\sqrt{2}+10\right)P_{11}-11P_{12} \) is equal to
Let \( \alpha, \beta \) be the roots of the equation \( x^2 + 2\sqrt{2}\,x - 1 = 0 \). The quadratic equation, whose roots are \( \alpha^4 + \beta^4 \) and \( \dfrac{1}{10}(\alpha^6 + \beta^6) \), is:
