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If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a-5)x + 15 = 3a \) has no real root, is the interval \( (\alpha,\beta) \), and \( X = \{ x \in \mathbb{Z} : \alpha < x < \beta \} \), then \( \sum\limits_{x \in X} x^2 \) is equal to:

Let \( \alpha \) and \( \beta \) be the roots of \( x^2+\sqrt{3}x-16=0 \), and \( \gamma \) and \( \delta \) be the roots of \( x^2+3x-1=0 \). If \( P_n=\alpha^n+\beta^n \) and \( Q_n=\gamma^n+\delta^n \), then \( \dfrac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}} \) is equal to:

Let the set of all values of $p \in \mathbb{R}$, for which both the roots of the equation $x^2 - (p + 2)x + (2p + 9) = 0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta - 2\alpha$ is equal to:

If the set of all \( a \in \mathbb{R} - \{1\} \), for which the roots of the equation \( (1 - a)x^{2} + 2(a - 3)x + 9 = 0 \) are positive is \( (-\infty, -\alpha] \cup [\beta, \gamma) \), then \( 2\alpha + \beta + \gamma \) is equal to ________________.

Let \( P_n=\alpha^n+\beta^n,\, n\in \mathbb{N} \). If \( P_{10}=123,\, P_9=76,\, P_8=47 \) and \( P_1=1 \), then the quadratic equation having roots \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:

The sum of the squares of the roots of \( |x-2|^{2}+|x-2|-2=0 \) and the squares of the roots of \( x^{2}-2|x-3|-5=0 \), is:

If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to ________.

If the set of all values of \( a \), for which the equation \( 5x^{3} - 15x - a = 0 \) has three distinct real roots, is the interval \( (\alpha, \beta) \), then \( \beta - 2\alpha \) is equal to ________.

The roots of the quadratic equation \( 3x^2 - px + q = 0 \) are \( 10^{\text{th}} \) and \( 11^{\text{th}} \) terms of an arithmetic progression with common difference \( \dfrac{3}{2} \). If the sum of the first \( 11 \) terms of this arithmetic progression is \( 88 \), then \( q - 2p \) is equal to _________.

The number of solutions of the equation \( \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right)\left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is:

The number of real roots of the equation \( x|x-2|+3|x-3|+1=0 \) is:

Let the equation \( x(x+2)(12-k)=2 \) have equal roots. Then the distance of the point \( \left(k,\dfrac{k}{2}\right) \) from the line \( 3x+4y+5=0 \) is:

Consider the equation \( x^{2} + 4x - n = 0 \), where \( n \in [20,100] \) is a natural number. Then the number of all distinct values of \( n \), for which the given equation has integral roots, is equal to:

The sum, of the squares of all the roots of the equation \( x^{2}+|2x-3|-4=0 \), is:

The product of all the rational roots of the equation \( (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 \), is equal to:

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JEE MAIN 2025 4th April Morning Shift

Consider the equation \( x^{2} + 4x - n = 0 \), where \( n \in [20,100] \) is a natural number. Then the number of all distinct values of \( n \), for which the given equation has integral roots, is equal to: