Quadratic Equation and Inequalities

182 questions · Mathematics · JEE Main

Quadratic Equation and Inequalities

182 questions · Mathematics · JEE Main

Let $P_{n} = α^{n} + β^{n}, n \in N$ . If $P_{10} = 123, P_{9} = 76, P_{8} = 47$ and $P_{1} = 1$ , then the quadratic equation having roots $\frac{1}{α}$ …2025 · 2 Apr · Shift 1 · Q37 · MCQ
If the set of all $a \in R - {1}$ , for which the roots of the equation $(1 - a) x^{2} + 2 (a - 3) x + 9 = 0$ are positive is $(- \infty, - α] \cup [β, γ)$ , then $2 α + β + γ$ is equal to .2025 · 2 Apr · Shift 2 · Q47 · Numerical
Let $α$ and $β$ be the roots of $x^{2} + 3 x - 16 = 0$ , and $γ$ and $δ$ be the roots of $x^{2} + 3 x - 1 = 0$ . If $P_{n} = α^{n} + β^{n}$ and $Q_{n} = γ^{n} + \overset{o}{^}^{n}$ , then $\frac{P _{25} + 3 P _{24}}{2 P _{23}} + \frac{Q _{25} - Q _{23}}{Q _{24}}$ …2025 · 3 Apr · Shift 1 · Q34 · MCQ
Let the equation $x (x + 2) (12 - k) = 2$ have equal roots. Then the distance of the point $(k, \frac{k}{2})$ from the line $3 x + 4 y + 5 = 0$ is2025 · 3 Apr · Shift 2 · Q35 · MCQ
Consider the equation $x^{2} + 4 x - n = 0$ , 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 to2025 · 4 Apr · Shift 1 · Q30 · MCQ
Let the set of all values of $p \in R$ , for which both the roots of the equation $x^{2} - (p + 2) x + (2 p + 9) = 0$ are negative real numbers, be the interval $(α, β]$ . Then $β - 2 α$ is equal to2025 · 7 Apr · Shift 1 · Q37 · MCQ
The number of real roots of the equation $x ∣ x - 2∣ + 3∣ x - 3∣ + 1 = 0$ is :2025 · 7 Apr · Shift 2 · Q32 · MCQ
The sum of the squares of the roots of $∣ x - 2 ∣^{2} + ∣ x - 2∣ - 2 = 0$ and the squares of the roots of $x^{2} - 2∣ x - 3∣ - 5 = 0$ , is2025 · 8 Apr · Shift 2 · Q33 · MCQ
Let $α_{θ}$ and $β_{θ}$ be the distinct roots of $2 x^{2} + (cos θ) x - 1 = 0, θ \in (0, 2 π)$ . If m and M are the minimum and the maximum values of $α_{θ}^{4} + β_{θ}^{4}$ , then $16 (M + m)$ equals :2025 · 22 Jan · Shift 2 · Q26 · MCQ
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 $a^{2} + c^{2}$ …2025 · 23 Jan · Shift 1 · Q48 · Numerical
The product of all the rational roots of the equation $(x^{2} - 9 x + 11)^{2} - (x - 4) (x - 5) = 3$ , is equal to2025 · 24 Jan · Shift 1 · Q28 · MCQ
The number of real solution(s) of the equation $x^{2} + 3 x + 2 = min {∣ x - 3∣, ∣ x + 2∣}$ is :2025 · 24 Jan · Shift 2 · Q43 · MCQ
The sum, of the squares of all the roots of the equation $x^{2} + ∣2 x - 3∣ - 4 = 0$ , is2025 · 28 Jan · Shift 1 · Q28 · MCQ
Let $f : R - {0} \to (- \infty, 1)$ be a polynomial of degree 2 , satisfying $f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x})$ . If $f (K) = - 2 K$ , then the sum of squares of all possible values…2025 · 28 Jan · Shift 2 · Q34 · MCQ
The number of solutions of the equation $(\frac{9}{x} - \frac{9}{x} + 2) (\frac{2}{x} - \frac{7}{x} + 3) = 0$ is :2025 · 29 Jan · Shift 1 · Q33 · MCQ
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval ( $α, β$ ), and $X = ∣ x \in Z; α < x < β ∣$ , then $x \in X \sum x^{2}$ is equal to:2025 · 29 Jan · Shift 2 · Q32 · MCQ
Let $S = {x \in R : (3 + 2)^{x} + (3 - 2)^{x} = 10}$ . Then the number of elements in $S$ is :2024 · 1 Feb · Shift 1 · Q39 · MCQ
Let $α$ and $β$ be the roots of the equation $p x^{2} + q x - r = 0$ , where $p e q 0$ . If $p, q$ and $r$ be the consecutive terms of a non constant G.P. and $\frac{1}{α} + \frac{1}{β} = \frac{3}{4}$ , then the value of $(α - β)^{2}$ …2024 · 1 Feb · Shift 2 · Q37 · MCQ
If 2 and 6 are the roots of the equation $a x^{2} + b x + 1 = 0$ , then the quadratic equation, whose roots are $\frac{1}{2 a + b}$ and $\frac{1}{6 a + b}$ , is :2024 · 4 Apr · Shift 1 · Q46 · MCQ
The number of distinct real roots of the equation $∣ x ∣∣ x + 2∣ - 5∣ x + 1∣ - 1 = 0$ is .2024 · 5 Apr · Shift 1 · Q60 · Numerical
The number of real solutions of the equation $x ∣ x + 5∣ + 2∣ x + 7∣ - 2 = 0$ is .2024 · 5 Apr · Shift 2 · Q58 · Numerical
Let $α, β$ be the distinct roots of the equation $x^{2} - (t^{2} - 5 t + 6) x + 1 = 0, t \in R$ and $a_{n} = α^{n} + β^{n}$ . Then the minimum value of $\frac{a _{2023} + a _{2025}}{a _{2024}}$ is2024 · 6 Apr · Shift 1 · Q41 · MCQ
Let $x_{1}, x_{2}, x_{3}, x_{4}$ be the solution of the equation $4 x^{4} + 8 x^{3} - 17 x^{2} - 12 x + 9 = 0$ and $(4 + x_{1}^{2}) (4 + x_{2}^{2}) (4 + x_{3}^{2}) (4 + x_{4}^{2}) = \frac{125}{16} m$ . Then the value of $m$ is …2024 · 6 Apr · Shift 1 · Q51 · Numerical
Let $α, β$ be roots of $x^{2} + 2 x - 8 = 0$ . If $U_{n} = α^{n} + β^{n}$ , then $\frac{U _{10} + 2 U _{9}}{2 U _{8}}$ is equal to .2024 · 6 Apr · Shift 2 · Q56 · Numerical
The sum of all the solutions of the equation $(8)^{2 x} - 16 \cdot (8)^{x} + 48 = 0$ is :2024 · 8 Apr · Shift 1 · Q44 · MCQ
The number of distinct real roots of the equation $∣ x + 1∣∣ x + 3∣ - 4∣ x + 2∣ + 5 = 0$ , is 2024 · 8 Apr · Shift 2 · Q52 · Numerical
Let $α, β$ be the roots of the equation $x^{2} + 22 x - 1 = 0$ . The quadratic equation, whose roots are $α^{4} + β^{4}$ and $\frac{1}{10} (α^{6} + β^{6})$ , is:2024 · 9 Apr · Shift 1 · Q42 · MCQ
Let $α, β; α > β$ , be the roots of the equation $x^{2} - 2 x - 3 = 0$ . Let $P_{n} = α^{n} - β^{n}, n \in N$ . Then $(113 - 102) P_{10} + (112 + 10) P_{11} - 11 P_{12}$ …2024 · 9 Apr · Shift 2 · Q37 · MCQ
If $α, β$ are the roots of the equation, $x^{2} - x - 1 = 0$ and $S_{n} = 2023 α^{n} + 2024 β^{n}$ , then :2024 · 27 Jan · Shift 2 · Q37 · MCQ
Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to .2024 · 29 Jan · Shift 2 · Q53 · Numerical
Let $α, β \in N$ be roots of the equation $x^{2} - 70 x + λ = 0$ , where $\frac{λ}{2}, \frac{λ}{3} o t in N$ . If $λ$ assumes the minimum possible value, then $\frac{( α - 1 + β - 1 ) ( λ + 35 )}{∣ α - β ∣}$ …2024 · 30 Jan · Shift 1 · Q54 · Numerical
The number of real solutions of the equation $x (x^{2} + 3∣ x ∣ + 5∣ x - 1∣ + 6∣ x - 2∣) = 0$ is .2024 · 30 Jan · Shift 2 · Q53 · Numerical
Let $S$ be the set of positive integral values of $a$ for which $\frac{a x ^{2} + 2 ( a + 1 ) x + 9 a + 4}{x ^{2} - 8 x + 32} < 0, \forall x \in R$ . Then, the number of elements in $S$ is :2024 · 31 Jan · Shift 1 · Q33 · MCQ
Let $a, b, c$ be the lengths of three sides of a triangle satistying the condition $(a^{2} + b^{2}) x^{2} - 2 b (a + c) x + (b^{2} + c^{2}) = 0$ . If the set of all possible values of $x$ is the interval $(α, β)$ , then $12 (α^{2} + β^{2})$ …2024 · 31 Jan · Shift 2 · Q55 · Numerical
Let $S = {x : x \in R and (3 + 2)^{x^{2} - 4} + (3 - 2)^{x^{2} - 4} = 10}$ . Then $n (S)$ is equal to2023 · 1 Feb · Shift 1 · Q27 · MCQ
The number of integral values of k, for which one root of the equation $2 x^{2} - 8 x + k = 0$ lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :2023 · 1 Feb · Shift 2 · Q32 · MCQ
Let $A = {x \in R : [x + 3] + [x + 4] \leq 3},$ $B = {x \in R : 3^{x} (r = 1 \sum \infty \frac{3}{10 ^{r}})^{x - 3} < 3^{- 3 x}},$ where [t] denotes greatest integer function. Then,2023 · 6 Apr · Shift 1 · Q23 · MCQ
The sum of all the roots of the equation $x^{2} - 8 x + 15 - 2 x + 7 = 0$ is :2023 · 6 Apr · Shift 1 · Q33 · MCQ
Let $α, β, γ$ be the three roots of the equation $x^{3} + b x + c = 0$ . If $β γ = 1 = - α$ , then $b^{3} + 2 c^{3} - 3 α^{3} - 6 β^{3} - 8 γ^{3}$ is equal to :2023 · 8 Apr · Shift 1 · Q33 · MCQ
Let m and $n$ be the numbers of real roots of the quadratic equations $x^{2} - 12 x + [x] + 31 = 0$ and $x^{2} - 5∣ x + 2∣ - 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^{2} + mn + n^{2}$ …2023 · 8 Apr · Shift 2 · Q37 · Numerical
If $a$ and $b$ are the roots of the equation $x^{2} - 7 x - 1 = 0$ , then the value of $\frac{a ^{21} + b ^{21} + a ^{17} + b ^{17}}{a ^{19} + b ^{19}}$ is equal to .2023 · 11 Apr · Shift 1 · Q47 · Numerical
The number of points, where the curve $f (x) = e^{8 x} - e^{6 x} - 3 e^{4 x} - e^{2 x} + 1, x \in R$ cuts $x$ -axis, is equal to .2023 · 11 Apr · Shift 2 · Q42 · Numerical
Let $α, β$ be the roots of the quadratic equation $x^{2} + 6 x + 3 = 0$ . Then $\frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ is equal to :2023 · 12 Apr · Shift 1 · Q30 · MCQ
The set of all $a \in R$ for which the equation $x ∣ x - 1∣ + ∣ x + 2∣ + a = 0$ has exactly one real root, is :2023 · 13 Apr · Shift 1 · Q29 · MCQ
Let $α, β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ . Then $α^{14} + β^{14}$ is equal to2023 · 13 Apr · Shift 2 · Q28 · MCQ
Let $[α]$ denote the greatest integer $\leq α$ . Then $[1] + [2] + [3] + \dots + [120]$ is equal to 2023 · 13 Apr · Shift 2 · Q42 · Numerical
The number of real roots of the equation $x ∣ x ∣ - 5∣ x + 2∣ + 6 = 0$ , is :2023 · 15 Apr · Shift 1 · Q33 · MCQ
The equation $x^{2} - 4 x + [x] + 3 = x [x]$ , where $[x]$ denotes the greatest integer function, has :2023 · 24 Jan · Shift 1 · Q32 · MCQ
Let $λ \in R$ and let the equation E be $∣ x ∣^{2} - 2∣ x ∣ + ∣ λ - 3∣ = 0$ . Then the largest element in the set S = { $x + λ : x$ is an integer solution of E} is 2023 · 24 Jan · Shift 1 · Q37 · Numerical
The number of real solutions of the equation $3 (x^{2} + \frac{1}{x ^{2}}) - 2 (x + \frac{1}{x}) + 5 = 0$ , is2023 · 24 Jan · Shift 2 · Q24 · MCQ
Let $α \in R$ and let $α, β$ be the roots of the equation $x^{2} + 6 0^{\frac{1}{4}} x + a = 0$ . If $α^{4} + β^{4} = - 30$ , then the product of all possible values of $a$ is .2023 · 25 Jan · Shift 2 · Q46 · Numerical
Let $λ e 0$ be a real number. Let $α, β$ be the roots of the equation $14 x^{2} - 31 x + 3 λ = 0$ and $α, γ$ be the roots of the equation $35 x^{2} - 53 x + 4 λ = 0$ . Then $\frac{3 α}{β}$ and $\frac{4 α}{γ}$ …2023 · 29 Jan · Shift 1 · Q22 · MCQ
Let $α_{1}, α_{2}, ...., α_{7}$ be the roots of the equation $x^{7} + 3 x^{5} - 13 x^{3} - 15 x = 0$ and $∣ α_{1} ∣ \geq ∣ α_{2} ∣ \geq ... \geq ∣ α_{7} ∣$ . Then $α_{1} α_{2} - α_{3} α_{4} + α_{5} α_{6}$ is…2023 · 29 Jan · Shift 2 · Q42 · Numerical
If the value of real number $a > 0$ for which $x^{2} - 5 a x + 1 = 0$ and $x^{2} - a x - 5 = 0$ have a common real root is $\frac{3}{2 β}$ then $β$ is equal to .2023 · 30 Jan · Shift 2 · Q39 · Numerical
The number of real roots of the equation $x^{2} - 4 x + 3 + x^{2} - 9 = 4 x^{2} - 14 x + 6$ , is :2023 · 31 Jan · Shift 1 · Q25 · MCQ
The equation $e^{4 x} + 8 e^{3 x} + 13 e^{2 x} - 8 e^{x} + 1 = 0, x \in R$ has :2023 · 31 Jan · Shift 2 · Q25 · MCQ
If the sum of the squares of the reciprocals of the roots $α$ and $β$ of the equation 3x2 + $λ$ x $-$ 1 = 0 is 15, then 6( $α$ 3 + $β$ 3)2 is equal to :2022 · 24 Jun · Shift 1 · Q27 · MCQ
The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0$ is2022 · 24 Jun · Shift 2 · Q23 · MCQ
The number of distinct real roots of the equation x7 $-$ 7x $-$ 2 = 0 is2022 · 24 Jun · Shift 2 · Q31 · MCQ
If $α, β, γ, δ$ are the roots of the equation $x^{4} + x^{3} + x^{2} + x + 1 = 0$ , then $α^{2021} + β^{2021} + γ^{2021} + δ^{2021}$ is equal to :2022 · 25 Jul · Shift 1 · Q22 · MCQ
Let $A = {x \in R : ∣ x + 1∣ < 2}$ and $B = {x \in R : ∣ x - 1∣ \geq 2}$ . Then which one of the following statements is NOT true?2022 · 25 Jun · Shift 2 · Q23 · MCQ
Let a, b $\in$ R be such that the equation $a x^{2} - 2 b x + 15 = 0$ has a repeated root $α$ . If $α$ and $β$ are the roots of the equation $x^{2} - 2 b x + 21 = 0$ , then $α^{2} + β^{2}$ is equal to :2022 · 25 Jun · Shift 2 · Q24 · MCQ
If for some $p, q, r \in R$ , not all have same sign, one of the roots of the equation $(p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0$ …2022 · 26 Jul · Shift 1 · Q39 · Numerical
The number of distinct real roots of the equation $x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is .2022 · 26 Jul · Shift 1 · Q41 · Numerical
The minimum value of the sum of the squares of the roots of $x^{2} + (3 - a) x + 1 = 2 a$ is:2022 · 26 Jul · Shift 2 · Q20 · MCQ
The sum of the cubes of all the roots of the equation $x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 0$ is .2022 · 26 Jun · Shift 1 · Q33 · Numerical
Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then $(\frac{1}{p} + \frac{1}{q})^{- 2}$ is equal to .2022 · 26 Jun · Shift 2 · Q40 · Numerical
If $α, β$ are the roots of the equation $x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{\frac{1}{3}}} - 5^{(l o g_{5} 3)^{\frac{2}{3}}} - 1) = 0$ , then the…2022 · 27 Jul · Shift 2 · Q24 · MCQ
The number of distinct real roots of x4 $-$ 4x + 1 = 0 is :2022 · 27 Jun · Shift 1 · Q24 · MCQ
If the sum of all the roots of the equation $e^{2 x} - 11 e^{x} - 45 e^{- x} + \frac{81}{2} = 0$ is $lo g_{e} p$ , then p is equal to .2022 · 27 Jun · Shift 1 · Q37 · Numerical
Let $α$ , $β$ be the roots of the equation $x^{2} - 4 λ x + 5 = 0$ and $α$ , $γ$ be the roots of the equation $x^{2} - (32 + 23) x + 7 + 3 λ 3 = 0$ , $λ$ > 0. If $β + γ = 32$ …2022 · 27 Jun · Shift 2 · Q38 · Numerical
The sum of all real values of $x$ for which $\frac{3 x ^{2} - 9 x + 17}{x ^{2} + 3 x + 10} = \frac{5 x ^{2} - 7 x + 19}{3 x ^{2} + 5 x + 12}$ is equal to .2022 · 28 Jul · Shift 1 · Q46 · Numerical
$Let S = {x \in [- 6, 3] - {- 2, 2} : \frac{∣ x + 3∣ - 1}{∣ x ∣ - 2} \geq 0} and T = {x \in Z : x^{2} - 7∣ x ∣ + 9 \leq 0} .$ Then the number of elements in $S \cap T$ is :2022 · 28 Jul · Shift 2 · Q21 · MCQ
Let $α$ , $β$ be the roots of the equation $x^{2} - 2 x + 6 = 0$ and $\frac{1}{α ^{2}} + 1, \frac{1}{β ^{2}} + 1$ be the roots of the equation $x^{2} + a x + b = 0$ . Then the roots of the equation $x^{2} - (a + b - 2) x + (a + b + 2) = 0$ …2022 · 28 Jul · Shift 2 · Q22 · MCQ
The number of real solutions of the equation $e^{4 x} + 4 e^{3 x} - 58 e^{2 x} + 4 e^{x} + 1 = 0$ is .2022 · 28 Jun · Shift 1 · Q36 · Numerical
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 :2022 · 28 Jun · Shift 2 · Q23 · MCQ
If $\frac{1}{( 20 - a ) ( 40 - a )} + \frac{1}{( 40 - a ) ( 60 - a )} + \dots + \frac{1}{( 180 - a ) ( 200 - a )} = \frac{1}{256}$ , then the maximum value of $a$ is :2022 · 29 Jul · Shift 1 · Q28 · MCQ
Let $α, β (α > β)$ be the roots of the quadratic equation $x^{2} - x - 4 = 0.$ If $P_{n} = α^{n} - β^{n}$ , $n \in N$ , then $\frac{P _{15} P _{16} - P _{14} P _{16} - P _{15}^{2} + P _{14} P _{15}}{P _{13} P _{14}}$ is equal…2022 · 29 Jul · Shift 2 · Q37 · Numerical
Let $α$ be a root of the equation 1 + x2 + x4 = 0. Then, the value of $α$ 1011 + $α$ 2022 $- α$ 3033 is equal to :2022 · 29 Jun · Shift 2 · Q24 · MCQ
Let $S_{1} = {x \in R - {1, 2} : \frac{( x + 2 ) ( x ^{2} + 3 x + 5 )}{- 2 + 3 x - x ^{2}} \geq 0}$ and $S_{2} = {x \in R : 3^{2 x} - 3^{x + 1} - 3^{x + 2} + 27 \leq 0}$ . Then, $S_{1} \cup S_{2}$ is…2022 · 30 Jun · Shift 1 · Q23 · MCQ
Let S be the set of all integral values of $α$ for which the sum of squares of two real roots of the quadratic equation $3 x^{2} + (α - 6) x + (α + 3) = 0$ is minimum. Then S :2022 · 30 Jun · Shift 1 · Q25 · MCQ
The numbers of pairs (a, b) of real numbers, such that whenever $α$ is a root of the equation x2 + ax + b = 0, $α$ 2 $-$ 2 is also a root of this equation, is :2021 · 1 Sep · Shift 2 · Q33 · MCQ
Let f(x) be a polynomial of degree 3 such that $f (k) = - \frac{2}{k}$ for k = 2, 3, 4, 5. Then the value of 52 $-$ 10f(10) is equal to :2021 · 1 Sep · Shift 2 · Q42 · Numerical
The value of $4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + ......\infty}}}}$ is :2021 · 17 Mar · Shift 1 · Q28 · MCQ
The value of $3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + ....\infty}}}}$ is equal to2021 · 18 Mar · Shift 1 · Q39 · MCQ
If $α$ and $β$ are the distinct roots of the equation $x^{2} + (3)^{1/4} x + 3^{1/2} = 0$ , then the value of $α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$ is equal to :2021 · 20 Jul · Shift 1 · Q25 · MCQ
Let [x] denote the greatest integer less than or equal to x. Then, the values of x $\in$ R satisfying the equation $[e^{x}]^{2} + [e^{x} + 1] - 3 = 0$ lie in the interval :2021 · 22 Jul · Shift 2 · Q32 · MCQ
Let p and q be two positive numbers such that p + q = 2 and p4+q4 = 272. Then p and q are roots of the equation :2021 · 24 Feb · Shift 1 · Q31 · MCQ
The number of the real roots of the equation $(x + 1)^{2} + ∣ x - 5∣ = \frac{27}{4}$ is .2021 · 24 Feb · Shift 2 · Q39 · Numerical
The integer 'k', for which the inequality x2 $-$ 2(3k $-$ 1)x + 8k2 $-$ 7 > 0 is valid for every x in R, is :2021 · 25 Feb · Shift 1 · Q27 · MCQ
Let $α$ and $β$ be the roots of x2 $-$ 6x $-$ 2 = 0. If an = $α$ n $- β$ n for n $\geq$ 1, then the value of $\frac{a _{10} - 2 a _{8}}{3 a _{9}}$ is :2021 · 25 Feb · Shift 2 · Q29 · MCQ
The number of real roots of the equation $e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0$ is :2021 · 25 Jul · Shift 1 · Q36 · MCQ
If $α$ , $β$ are roots of the equation $x^{2} + 5 (2) x + 10 = 0$ , $α$ > $β$ and $P_{n} = α^{n} - β^{n}$ for each positive integer n, then the value of $(\frac{P _{17} P _{20} + 5 2 P _{17} P _{19}}{P _{18} P _{19} + 5 2 P _{18}^{2}})$ …2021 · 25 Jul · Shift 1 · Q43 · Numerical
If [x] be the greatest integer less than or equal to x, then $n = 8 \sum 100 [\frac{( - 1 ) ^{n} n}{2}]$ is equal to :2021 · 25 Jul · Shift 2 · Q32 · MCQ
The number of real solutions of the equation, x2 $-$ |x| $-$ 12 = 0 is :2021 · 25 Jul · Shift 2 · Q35 · MCQ
If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to .2021 · 25 Jul · Shift 2 · Q44 · Numerical
The sum of all integral values of k (k $e$ 0) for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in x has no real roots, is .2021 · 26 Aug · Shift 1 · Q35 · Numerical
Let $λ e$ 0 be in R. If $α$ and $β$ are the roots of the equation x2 $-$ x + 2 $λ$ = 0, and $α$ and $γ$ are the roots of equation 3x2 $-$ 10x + 27 $λ$ = 0, then $\frac{β γ}{λ}$ is equal…2021 · 26 Aug · Shift 2 · Q43 · Numerical
The sum of 162th power of the roots of the equation x3 $-$ 2x2 + 2x $-$ 1 = 0 is .2021 · 26 Feb · Shift 1 · Q39 · Numerical
Let $α$ and $β$ be two real numbers such that $α$ + $β$ = 1 and $α β$ = $-$ 1. Let pn = ( $α$ )n + ( $β$ )n, pn $-$ 1 = 11 and pn+1 = 29 for some integer n $\geq$ 1. Then, the value of p $_{n}^{2}$ is …2021 · 26 Feb · Shift 2 · Q42 · Numerical
The set of all values of K > $-$ 1, for which the equation $(3 x^{2} + 4 x + 3)^{2} - (k + 1) (3 x^{2} + 4 x + 3) (3 x^{2} + 4 x + 2) + k (3 x^{2} + 4 x + 2)^{2} = 0$ has real roots, is :2021 · 27 Aug · Shift 2 · Q29 · MCQ
Let $α$ , $β$ be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then $α$ 8 + $β$ 8 is equal to2021 · 27 Jul · Shift 1 · Q36 · MCQ
Let $α = x \in R max {8^{2 s i n 3 x} . 4^{4 c o s 3 x}}$ and $β = x \in R min {8^{2 s i n 3 x} . 4^{4 c o s 3 x}}$ . If $8 x^{2} + b x + c = 0$ is a quadratic equation whose roots are $α$ …2021 · 27 Jul · Shift 2 · Q33 · MCQ
The number of real roots of the equation e4x $-$ e3x $-$ 4e2x $-$ ex + 1 = 0 is equal to .2021 · 27 Jul · Shift 2 · Q40 · Numerical
cosec18 $^{\circ}$ is a root of the equation :2021 · 31 Aug · Shift 1 · Q30 · MCQ
The sum of the roots of the equation $x + 1 - 2 lo g_{2} (3 + 2^{x}) + 2 lo g_{4} (10 - 2^{- x}) = 0$ , is :2021 · 31 Aug · Shift 2 · Q30 · MCQ
Let $α$ and $β$ be the roots of the equation 5x2 + 6x – 2 = 0. If Sn = $α$ n + $β$ n, n = 1, 2, 3...., then :2020 · 2 Sep · Shift 1 · Q34 · MCQ
Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :2020 · 2 Sep · Shift 2 · Q26 · MCQ
If $α$ and $β$ are the roots of the equation x2 + px + 2 = 0 and $\frac{1}{α}$ and $\frac{1}{β}$ are the roots of the equation 2x2 + 2qx + 1 = 0, then $(α - \frac{1}{α}) (β - \frac{1}{β}) (α + \frac{1}{β}) (β + \frac{1}{α})$ …2020 · 3 Sep · Shift 1 · Q23 · MCQ
The set of all real values of $λ$ for which the quadratic equations, ( $λ$ 2 + 1)x2 – 4 $λ$ x + 2 = 0 always have exactly one root in the interval (0, 1) is :2020 · 3 Sep · Shift 2 · Q40 · MCQ
Let $α$ and $β$ be the roots of x2 - 3x + p=0 and $γ$ and $δ$ be the roots of x2 - 6x + q = 0. If $α, β, γ, δ$ form a geometric progression.Then ratio (2q + p) : (2q - p) is:2020 · 4 Sep · Shift 1 · Q30 · MCQ
Let [t] denote the greatest integer $\leq$ t. Then the equation in x, [x]2 + 2[x+2] - 7 = 0 has :2020 · 4 Sep · Shift 1 · Q39 · MCQ
Let $λ e 0$ be in R. If $α$ and $β$ are the roots of the equation, x2 - x + 2 $λ$ = 0 and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is…2020 · 4 Sep · Shift 2 · Q38 · MCQ
The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :2020 · 5 Sep · Shift 1 · Q40 · MCQ
If $α$ and $β$ are the roots of the equation, 7x2 – 3x – 2 = 0, then the value of $\frac{α}{1 - α ^{2}} + \frac{β}{1 - β ^{2}}$ is equal to :2020 · 5 Sep · Shift 2 · Q32 · MCQ
If $α$ and $β$ be two roots of the equation x2 – 64x + 256 = 0. Then the value of $(\frac{α ^{3}}{β ^{5}})^{1/8} + (\frac{β ^{3}}{α ^{5}})^{1/8}$ is :2020 · 6 Sep · Shift 1 · Q30 · MCQ
If $α$ and $β$ are the roots of the equation 2x(2x + 1) = 1, then $β$ is equal to :2020 · 6 Sep · Shift 2 · Q32 · MCQ
Let $α$ and $β$ be two real roots of the equation (k + 1)tan2x - $2$ . $λ$ tanx = (1 - k), where k( $e$ - 1) and $λ$ are real numbers. if tan2 ( $α$ + $β$ ) = 50, then a value of $λ$ is:2020 · 7 Jan · Shift 1 · Q24 · MCQ
Let $α$ and $β$ be the roots of the equation x2 - x - 1 = 0. If pk = $(α)^{k} + (β)^{k}$ , k $\geq$ 1, then which one of the following statements is not true?2020 · 7 Jan · Shift 2 · Q38 · MCQ
The least positive value of 'a' for which the equation 2x2 + (a – 10)x + $\frac{33}{2}$ = 2a has real roots is2020 · 8 Jan · Shift 1 · Q22 · Numerical
Let $α = \frac{- 1 + i 3}{2}$ . If $a = (1 + α) k = 0 \sum 100 α^{2 k}$ and $b = k = 0 \sum 100 α^{3 k}$ , then a and b are the roots of the quadratic equation :2020 · 8 Jan · Shift 2 · Q22 · MCQ
Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :2020 · 8 Jan · Shift 2 · Q26 · MCQ
The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is :2020 · 9 Jan · Shift 1 · Q42 · MCQ
Let a, b $\in$ R, a $e$ 0 be such that the equation, ax2 – 2bx + 5 = 0 has a repeated root $α$ , which is also a root of the equation, x2 – 2bx – 10 = 0. If $β$ is the other root of this equation, then $α$ 2 + $β$ 2 is…2020 · 9 Jan · Shift 2 · Q35 · MCQ
The sum of the solutions of the equation $∣ x - 2 ∣ + x (x - 4) + 2 = 0$ (x > 0) is equal to:2019 · 8 Apr · Shift 1 · Q25 · MCQ
The number of integral values of m for which the equation (1 + m2 )x2 – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :2019 · 8 Apr · Shift 2 · Q30 · MCQ
Let p, q $\in$ R. If 2 - $3$ is a root of the quadratic equation, x2 + px + q = 0, then :2019 · 9 Apr · Shift 1 · Q24 · MCQ
If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-2019 · 9 Apr · Shift 2 · Q34 · MCQ
If both the roots of the quadratic equation x2 $-$ mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :2019 · 9 Jan · Shift 2 · Q37 · MCQ
The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, 6x2 $-$ 11x + $α$ = 0 are rational numbers is :2019 · 9 Jan · Shift 2 · Q46 · MCQ
If $α$ and $β$ are the roots of the quadratic equation, x2 + x sin $θ$ - 2 sin $θ$ = 0, $θ \in (0, \frac{π}{2})$ , then $\frac{α ^{12} + β ^{12}}{( α ^{- 12} + β ^{- 12} ) . ( α - β ) ^{24}}$ …2019 · 10 Apr · Shift 1 · Q28 · MCQ
All the pairs (x, y) that satisfy the inequality $2^{s i n^{2} x - 2 s i n x + 5} . \frac{1}{4 ^{s i n^{2} y}} \leq 1$ also satisfy the equation2019 · 10 Apr · Shift 1 · Q29 · MCQ
The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is2019 · 10 Apr · Shift 2 · Q25 · MCQ
Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c $e$ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the…2019 · 10 Jan · Shift 1 · Q41 · MCQ
The value of $λ$ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – $λ$ )x + 2 = $λ$ has the least value is -2019 · 10 Jan · Shift 2 · Q30 · MCQ
If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is2019 · 11 Jan · Shift 1 · Q26 · MCQ
Let $α$ and $β$ be the roots of the quadratic equation x2 sin $θ$ – x(sin $θ$ cos $θ$ + 1) + cos $θ$ = 0 (0 < $θ$ < 45o), and $α$ < $β$ . Then $n = 0 \sum \infty (α^{n} + \frac{( - 1 ) ^{n}}{β ^{n}})$ …2019 · 11 Jan · Shift 2 · Q31 · MCQ
If $α$ , $β$ and $γ$ are three consecutive terms of a non-constant G.P. such that the equations $α$ x 2 + 2 $β$ x + $γ$ = 0 and x2 + x – 1 = 0 have a common root, then $α$ ( $β$ + $γ$ ) is equal to :2019 · 12 Apr · Shift 2 · Q27 · MCQ
If $λ$ be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which $λ + \frac{1}{λ} = 1,$ is2019 · 12 Jan · Shift 1 · Q34 · MCQ
The number of integral values of m for which the quadratic expression, (1 + 2m)x2 – 2(1 + 3m)x + 4(1 + m), x $\in$ R, is always positive, is :2019 · 12 Jan · Shift 2 · Q32 · MCQ
If $λ \in$ R is such that the sum of the cubes of the roots of the equation, x2 + (2 $- λ$ ) x + (10 $- λ$ ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :2018 · 15 Apr · Shift 1 · Q31 · MCQ
If tanA and tanB are the roots of the quadratic equation, 3x2 $-$ 10x $-$ 25 = 0, then the value of 3 sin2(A + B) $-$ 10 sin(A + B).cos(A + B) $-$ 25 cos2(A + B) is :2018 · 15 Apr · Shift 1 · Q35 · MCQ
If f(x) is a quadratic expression such that f (1) + f (2) = 0, and $-$ 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :2018 · 15 Apr · Shift 2 · Q29 · MCQ
Let p, q and r be real numbers (p $e$ q, r $e$ 0), such that the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :2018 · 16 Apr · Shift 1 · Q37 · MCQ
If an angle A of a $Δ$ ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 = 0 are :2018 · 16 Apr · Shift 1 · Q40 · MCQ
Let S = { $x \in$ R : $x \geq$ 0 and $2 ∣ x - 3 ∣ + x (x - 6) + 6 = 0$ }. Then S2018 · Shift 0 · Q28 · MCQ
Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x $-$ 1 and it leaves remainder 6 when divided by x + 1; then :2017 · 8 Apr · Shift 1 · Q32 · MCQ
The sum of all the real values of x satisfying the equation 2(x $-$ 1)(x2 + 5x $-$ 50) = 1 is :2017 · 9 Apr · Shift 1 · Q29 · MCQ
If for a positive integer n, the quadratic equation $x (x + 1) + (x + 1) (x + 2) + .... + (x + \overline{n - 1}) (x + n)$ $= 10 n$ has two consecutive integral…2017 · Shift 0 · Q27 · MCQ
If the equations x2 + bx−1 = 0 and x2 + x + b = 0 have a common root different from −1, then $∣ b ∣$ is equal to :2016 · 9 Apr · Shift 1 · Q27 · MCQ
If x is a solution of the equation, $2 x + 1 - 2 x - 1 = 1,$ $(x \geq \frac{1}{2}),$ then $4 x^{2} - 1$ is equal to :2016 · 10 Apr · Shift 1 · Q26 · MCQ
The sum of all real values of $x$ satisfying the equation $(x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1$ is :2016 · Shift 0 · Q36 · MCQ
Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n},$ for $n \geq 1,$ then the value of $\frac{a _{10} - 2 a _{8}}{2 a _{9}}$ is equal to :2015 · Shift 0 · Q26 · MCQ
Let $α$ and $β$ be the roots of equation $p x^{2} + q x + r = 0, p e 0.$ If $p, q, r$ in A.P. and $\frac{1}{α} + \frac{1}{β} = 4,$ then the value of $∣ α - β ∣$ is :2014 · Shift 0 · Q40 · MCQ
If $a \in R$ and the equation $- 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0$ (where [ $x$ ] denotes the greater integer $\leq x$ ) has no integral solution, then all possible values of a…2014 · Shift 0 · Q41 · MCQ
If the equations $x^{2} + 2 x + 3 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R,$ have a common root, then $a : b : c$ is2013 · Shift 0 · Q41 · MCQ
The equation $e^{s i n x} - e^{- s i n x} - 4 = 0$ has:2012 · Shift 0 · Q42 · MCQ
If $α$ and $β$ are the roots of the equation $x^{2} - x + 1 = 0,$ then $α^{2009} + β^{2009} =$ 2010 · Shift 0 · Q46 · MCQ
If the roots of the equation $b x^{2} + c x + a = 0$ imaginary, then for all real values of $x$ , the expression $3 b^{2} x^{2} + 6 b c x + 2 c^{2}$ is :2009 · Shift 0 · Q44 · MCQ
The quadratic equations $x^{2} - 6 x + a = 0$ and $x^{2} - c x + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is2008 · Shift 0 · Q51 · MCQ
STATEMENT - 1 : For every natural number $n \geq 2, \frac{1}{1} + \frac{1}{2} + ........ + \frac{1}{n} > n .$ STATEMENT - 2 : For every natural number $n \geq 2,$ , $n (n + 1) < n + 1.$ 2008 · Shift 0 · Q52 · MCQ
If the difference between the roots of the equation $x^{2} + a x + 1 = 0$ is less than $5,$ then the set of possible values of $a$ is2007 · Shift 0 · Q60 · MCQ
If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $tan 3 0^{\circ}$ and $tan 1 5^{\circ}$ , respectively, then the value of $2 + q - p$ is2006 · Shift 0 · Q69 · MCQ
All the values of $m$ for which both roots of the equation $x^{2} - 2 m x + m^{2} - 1 = 0$ are greater than $- 2$ but less then 4, lie in the interval2006 · Shift 0 · Q70 · MCQ
If $x$ is real, the maximum value of $\frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7}$ is2006 · Shift 0 · Q71 · MCQ
The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is2005 · Shift 0 · Q109 · MCQ
If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals2005 · Shift 0 · Q111 · MCQ
If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals2005 · Shift 0 · Q68 · MCQ
The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is :2005 · Shift 0 · Q69 · MCQ
In a triangle $P QR, ∠ R = \frac{π}{2} . I f tan (\frac{P}{2})$ and $tan (\frac{Q}{2})$ are the roots of $a x^{2} + b x + c = 0, a e 0$ then2005 · Shift 0 · Q96 · MCQ
If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval2005 · Shift 0 · Q97 · MCQ
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation2004 · Shift 0 · Q100 · MCQ
If $(1 - p)$ is a root of quadratic equation $x^{2} + p x + (1 - p) = 0$ then its root are2004 · Shift 0 · Q101 · MCQ
If one root of the equation $x^{2} + p x + 12 = 0$ is 4, while the equation $x^{2} + p x + q = 0$ has equal roots, then the value of $^{'} q^{'}$ is2004 · Shift 0 · Q102 · MCQ
If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $\frac{a}{c}, \frac{b}{a}$ and $\frac{c}{b}$ are in2003 · Shift 0 · Q100 · MCQ
The value of ' $a$ ' for which one root of the quadratic equation $(a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0$ is twice as large as the other is2003 · Shift 0 · Q101 · MCQ
The number of real solutions of the equation $x^{2} - 3 ∣ x ∣ + 2 = 0$ is2003 · Shift 0 · Q102 · MCQ
If $α e β$ but $α^{2} = 5 α - 3$ and $β^{2} = 5 β - 3$ then the equation having $α / β$ and $β / α$ as its roots is2002 · Shift 0 · Q100 · MCQ
Product of real roots of equation $t^{2} x^{2} + ∣ x ∣ + 9 = 0$ 2002 · Shift 0 · Q101 · MCQ
Difference between the corresponding roots of $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ is same and $a e b,$ then2002 · Shift 0 · Q102 · MCQ
If $p$ and $q$ are the roots of the equation $x^{2} + p x + q = 0,$ then2002 · Shift 0 · Q103 · MCQ
If $a, b, c$ are distinct $+ v e$ real numbers and $a^{2} + b^{2} + c^{2} = 1$ then $ab + b c + c a$ is2002 · Shift 0 · Q104 · MCQ