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If the sum of all the roots of equation \( e^{2x}-11e^{x}-45e^{-x}+\dfrac{81}{2}=0 \) is \( \log_{e} p \), then \( p \) is equal to \( \_\_\_\_\_ \).

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-4\lambda x+5=0 \) and \( \alpha,\gamma \) be the roots of the equation \( x^{2}-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda\sqrt{3}=0 \), \( \lambda>0 \). If \( \beta+\gamma=3\sqrt{2} \), then \( (\alpha+2\beta+\gamma)^{2} \) is equal to _______.

Let \( f(x) \) be a quadratic polynomial with leading coefficient \( 1 \) such that \( f(0) = p, \; p \neq 0, \) and \( f(1) = \dfrac{1}{3} \). If the equations \( f(x) = 0 \) and \( fof of of(x) = 0 \) have a common real root, then \( f(-3) \) is equal to \( \_ \_ \_ \_ \_ \_ \).

If for some \( p, q, r \in \mathbb{R} \), not all have same sign, one of the roots of the equation \( (p^{2}+q^{2})x^{2}-2q(p+r)x+q^{2}+r^{2}=0 \) is also a root of the equation \( x^{2}+2x-8=0 \), then \( \dfrac{q^{2}+r^{2}}{p^{2}} \) is equal to __________.

The \( \alpha,\beta \) are the roots of the equation
\( x^{2}-\left(5+3^{\sqrt{\log_{3}5}-5\sqrt{\log_{5}3}}\right)+3\left(3^{(\log_{3}5)^{\dfrac{1}{3}}}-5^{(\log_{5}3)^{\dfrac{2}{3}}} -1\right)=0 \), then the equation, whose roots are \( \alpha+\dfrac{1}{\beta} \) and \( \beta+\dfrac{1}{\alpha} \), is:

\( \alpha=\sin 36^{\circ} \) is a root of which of the following equation?

Let \( f(x)=ax^{2}+bx+c \) be such that \( f(1)=3 \), \( f(-2)=\lambda \) and \( f(3)=4 \). If \( f(0)+f(1)+f(-2)+f(3)=14 \), then \( \lambda \) is equal to:

The number of real solution of the equation \( e^{4x} + 4e^{3x} - 58e^{2x} + 4e^{x} + 1 = 0 \) is ______.

If the sum of the squares of the reciprocals of the roots \( \alpha \) and \( \beta \) of the equation \( 3x^{2} + \lambda x - 1 = 0 \) is \( 15 \), then \( 6(\alpha^{3} + \beta^{3})^{2} \) is equal to:

The sum of all real values of \( x \) for which \( \dfrac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \dfrac{5x^2 - 7x + 19}{3x^2 + 5x + 12} \) is equal to __________.

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+\sqrt{6}=0 \) and \( \dfrac{1}{\alpha^{2}}+1,\, \dfrac{1}{\beta^{2}}+1 \) be the roots of the equation \( x^{2}+ax+b=0 \). Then the roots of the equation \( x^{2}-(a+b-2)x+(a+b+2)=0 \) are:

The sum of the cubes of all the roots of the equation \( x^{4} - 3x^{3} - 2x^{2} + 3x + 1 = 0 \) is __________ .

Let \( a, b \in R \) be such that the equation \( a x^{2} - 2 b x + 15 = 0 \) has a repeated root \( \alpha \). If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2} - 2 b x + 21 = 0 \), then \( \alpha^{2} + \beta^{2} \) is equal to:

The sum of all the real roots of the equation \( (e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0 \) is:

Let \( p \) and \( q \) be two real numbers such that \( p+q=3 \) and \( p^{4}+q^{4}=369 \). Then \( \left(\frac{1}{p}+\frac{1}{q}\right)^{-2} \) is equal to ______.

The minimum value of the sum of the squares of the roots of \( x^2 + (3-a)x + 1 = 2a \) is:

Let \( f(x) \) and \( g(x) \) be two real polynomials of degree \( 2 \) and \( 1 \) respectively. If \( f(g(x)) = 8x^2 - 2x \), and \( g(f(x)) = 4x^2 + 6x + 1 \), then the value of \( f(2) + g(2) \) is ________.

Let \( \alpha, \beta (\alpha > \beta) \) be the roots of the quadratic equation \( x^{2}-x-4=0 \). If \( P_{n}=\alpha^{n}-\beta^{n}, \ n \in \mathbb{N} \), then \( \dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^{2}+P_{14}P_{15}}{P_{13}P_{14}} \) is equal to __________.

Let \( f(x) \) be a quadratic polynomial such that \( f(-2) + f(3) = 0 \). If one of the roots of \( f(x) = 0 \) is \( -1 \), then the sum of the roots of \( f(x) = 0 \) is equal to:

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JEE Main 2022 24th June Evening Shift

The sum of all the real roots of the equation \( (e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0 \) is: