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Quadratic Equation and Inequalities

24 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Quadratic Equation and Inequalities

Quadratic Equation and Inequalities

24 questions · Mathematics · JEE Advanced

Let $R$ denote the set of all real numbers. Let $a_{i}, b_{i} \in R$ for $i \in {1, 2, 3}$ . Define the functions $f : R \to R$ , $g : R \to R$ , and $h : R \to R$ by $f (x) = a_{1} + 10 x + a_{2} x^{2} + a_{3} x^{3} + x^{4} g (x) = b_{1} + 3 x + b_{2} x^{2} + b_{3} x^{3} + x^{4} h (x) = f (x + 1) - g (x + 2)$ …2025 · Shift 1 · Q17 · MCQ
Let $a = 32$ and $b = \frac{1}{5 ^{1/6} 6}$ . If $x, y \in R$ are such that $3 x + 2 y = lo g_{a} (18)^{\frac{5}{4}} and 2 x - y = lo g_{b} (1080),$ then $4 x + 5 y$ …2024 · Shift 1 · Q25 · Numerical
The product of all positive real values of $x$ satisfying the equation $x^{(16 (l o g_{5} x)^{3} - 68 l o g_{5} x)} = 5^{- 16}$ is .2022 · Shift 2 · Q22 · Numerical
For x $\in$ R, the number of real roots of the equation $3 x^{2} - 4 x^{2} - 1 + x - 1 = 0$ is .2021 · Shift 1 · Q36 · Numerical
Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x $-$ 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2 $-$ 20x + 2020. Then the value of ac(a $-$ c) + ad(a $-$ d) + bc(b $-$ …2020 · Shift 1 · Q19 · MCQ
Let $α$ and $β$ be the roots of $x^{2} - x - 1 = 0$ , with $α$ > $β$ . For all positive integers n, define $a_{n} = \frac{α ^{n} - β ^{n}}{α - β}, n \geq 1 b_{1} = 1 an d b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2$ …2019 · Shift 1 · Q26 · Multiple correct
Let a, b, c three non-zero real numbers such that the equation $3 a cos x + 2 b sin x = c, x \in [- \frac{π}{2}, \frac{π}{2}]$ , has two distinct real roots $α$ and $β$ with $\alpha + \beta = {\pi \over…2018 · Shift 1 · Q31 · Numerical
Let p, q be integers and let $α$ , $β$ be the roots of the equation, x2 $-$ x $-$ 1 = 0 where $α e β$ . For n = 0, 1, 2, ........, let an = p $α$ n + q $β$ n. FACT : If a and b are rational numbers and a + b $5$ …2017 · Shift 2 · Q35 · MCQ
Let p, q be integers and let $α$ , $β$ be the roots of the equation, x2 $-$ x $-$ 1 = 0 where $α e β$ . For n = 0, 1, 2, ........, let an = p $α$ n + q $β$ n. FACT : If a and b are rational numbers and a + b $5$ …2017 · Shift 2 · Q36 · MCQ
Let $- \frac{π}{6} < θ < - \frac{π}{12} .$ Suppose $α_{1}$ and $β_{1}$ are the roots of the equation $x^{2} - 2 x sec θ + 1 = 0$ and $α_{2}$ and $β_{2}$ are the roots of the equation $x^{2} + 2 x tan θ - 1 = 0. I f α_{1} > β_{1}$ …2016 · Shift 1 · Q31 · MCQ
Let $S$ be the set of all non-zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $∣ x_{1} - x_{2} ∣ < 1.$ …2015 · Shift 2 · Q21 · Multiple correct
The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has2014 · Shift 2 · Q28 · MCQ
If $3^{x} = 4^{x - 1},$ then $x =$ 2013 · Shift 2 · Q35 · Multiple correct
The value of $6 + lo g_{3/2} (\frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} ...)$ is .2012 · Shift 1 · Q40 · Numerical
Let $α$ (a) and $β$ (a) be the roots of the equation $(3 1 + a - 1) x^{2} + (1 + a - 1) x + (6 1 + a - 1) = 0$ where $a > - 1$ . Then $a \to 0^{+} lim α (a)$ and $a \to 0^{+} lim β (a)$ …2012 · Shift 2 · Q37 · MCQ
The minimum value of the sum of real numbers $a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}$ and $a^{10}$ where $a > 0$ is2011 · Shift 1 · Q26 · Numerical
Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0,$ with $α > β .$ If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ then the value of $\frac{a _{10} - 2 a _{8}}{2 a _{9}}$ is2011 · Shift 1 · Q27 · MCQ
Let $(x_{0}, y_{0})$ be the solution of the following equations $(2 x)^{ℓ n 2} = (3 y)^{ℓ n 3} 3^{ℓ n x} = 2^{ℓ n y}$ Then $x_{0}$ …2011 · Shift 1 · Q28 · MCQ
The number of distinct real roots of $x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0$ 2011 · Shift 2 · Q21 · Numerical
A value of $b$ for which the equations $x^{2} + b x - 1 = 0 x^{2} + x + b = 0$ have one root in common is2011 · Shift 2 · Q23 · MCQ
Let $p$ and $q$ be real numbers such that $p e 0, p^{3} e q$ and $p^{3} e - q .$ If $p^{3} e - q .$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q,$ then a quadratic…2010 · Shift 1 · Q33 · MCQ
The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is2009 · Shift 2 · Q30 · Numerical
Let $a, b, c$ , $α, β$ be real numbers. Suppose $α, β$ are the roots of the equation $x^{2} + 2 p x + q = 0$ and $α, \frac{1}{β}$ are the roots of the equation $a x^{2} + 2 b x + c = 0,$ where $β^{2} \in {- 1, 0, 1}$ …2008 · Shift 2 · Q40 · MCQ
Let $x^{2} - p x + r = 0$ be the roots of the equation $x^{2} - p x + r = 0$ and $\frac{α}{2}, 2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of r is2007 · Shift 1 · Q23 · MCQ