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JEE Mains | PYQ

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 ______.

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: