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Let \( \alpha,\beta \) be roots of \( x^{2}+\sqrt{2}\,x-8=0 \). If \( U_{n}=\alpha_{n}+\beta^{n} \), then \( \frac{U_{10}+\sqrt{2}\,U_{9}}{2U_{8}} \) is equal to ______.

Let \( S \) be the set of positive integral values of \( a \) for which \( \dfrac{a x^{2}+2(a+1)x+9a+4}{x^{2}-8x+32}<0 \, \forall x \in \mathbb{R} \). Then, the number of elements in \( S \) is:

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \( \dfrac{\left(\sqrt{\alpha - 1} + \sqrt{\beta - 1}\right)(\lambda
+ 35)}{|\alpha - \beta|} \) is equal to: ______.

Let \( S = \{ x \in R : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10 \} \). Then the number of elements in \( S \) is:

Let \( \alpha \) and \( \beta \) be the roots of the equation \( p x^{2} + qx - r = 0 \) , where \( p \neq 0 \). If \( p \), \( q \) and \( r \) be the consecutive terms of a non constant G.P. and \( \dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{3}{4} \) , then the value of \( (\alpha - \beta)^{2} \) is:

Let the sum of the maximum and the minimum values of the function \( f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8} \) be \( \dfrac{m}{n} \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to:

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:

The sum of all the solutions of the equation \( (8)^{2x} - 16\cdot(8)^{x} + 48 = 0 \) 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:

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JEE Main 2024 1st Feb Morning Shift

Let \( S = \{ x \in R : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10 \} \). Then the number of elements in \( S \) is: