JEE Mains JEE Advance BITSAT
JEE Advance | PYQ

Year

All years 2026 2024 2021

Let \( \mathbb{R}^2 \) denote \( \mathbb{R} \times \mathbb{R} \). Let \( S = \{ (a, b, c) : a, b, c \in \mathbb{R} \} \) and \( ax^2 + 2bxy + cy^2 > 0 \) for all \( (x, y) \in \mathbb{R}^2 - \{(0, 0)\} \). Then which of the following statements is (are) TRUE?

Let \( f(x) = x^4 + ax^3 + bx^2 + c \) be a polynomial with real coefficients such that \( f(1) = -9 \). Suppose that \( i\sqrt{3} \) is a root of the equation \( 4x^3 + 3ax^2 + 2bx = 0 \), where \( i = \sqrt{-1} \). If \( \alpha_1, \alpha_2, \alpha_3 \) and \( \alpha_4 \) are all the roots of the equation \( f(x) = 0 \), then \( |\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2 \) is equal to ________.

Let \( a = 3\sqrt{2} \) and \( b = \dfrac{1}{5^{1/6}\sqrt{6}} \). If \( x, y \in \mathbb{R} \) are such that
\( 3x + 2y = \log_a (18)^{5/4} \) and \( 2x - y = \log_b \left(\sqrt{1080}\right) \) then \( 4x + 5y \) is equal to ________.

Let \( a, b, c \) be positive integers in arithmetic progression such that the equation \( ax^2 + bx + c = 0 \) has only integer solutions.
Then which of the following statements is (are) TRUE?

Match each entry in List-I to the correct entry in List-II and choose the correct option.

For \( x \in \mathbb{R} \), the number of real roots of the equation \( 3x^2 - 4\lvert x^2 - 1 \rvert + x - 1 = 0 \) is ______.

← PreviousQuestion 3 of 6Next →

JEE Advanced 2024 Paper - 1

Let \( a = 3\sqrt{2} \) and \( b = \dfrac{1}{5^{1/6}\sqrt{6}} \). If \( x, y \in \mathbb{R} \) are such that
\( 3x + 2y = \log_a (18)^{5/4} \) and \( 2x - y = \log_b \left(\sqrt{1080}\right) \) then \( 4x + 5y \) is equal to ________.