Quadratic Equations
105 JEE Maths previous year questions on Quadratic Equations — options free on every question; 10 include the answer & explanation free, the rest unlock with PYQ Pass.
If the set of all \(\mathrm{a} \in \mathbf{R}\), for which the equation \(2 x^2+(a-5) x+15=3a\) has no real root, is the interval \((\alpha, \beta)\) and \(\mathrm{X}=\{x \in Z: \alpha [JEE Main 2025, 29 Jan (Shift 2)]
2139
Given quadratic equation does not have real roots.
\(2 x^2+(a-5) x+15=3\)
so \(D<0\)
\((a-5)^2-8(15-3 a)<0\)
\(a^2+14 a+25-120<0\)
\(a^2+14 a-95<0\)
\((a+19)(a-5)<0\)
\(a \in(-19,5)\)
\(\therefore-19 \( \therefore \sum_{x \in X} x^2= \left(1^2+2^2+\ldots+4^2\right) +\left(1^2+2^2+\ldots+18^2\right)\) \(=\frac{4 \times 5 \times 9}{6}+\frac{18 \times 19 \times 37}{6}\) \(=2139\)
Let \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}},\mathrm{n}\in \mathrm{N}\). If \({\mathrm{P}}_{10}=123,{\mathrm{P}}_{9}=76\), \({\mathrm{P}}_{8}=47\) and \({\mathrm{P}}_{1}=1\), then the quadratic equation having roots \(\frac{1}{\alpha }\) and \(\frac{1}{\beta }\) is :
[JEE Main 2025, 2 Apr (Shift 1)]
\({x}^{2}+x-1=0\)
\(S=\alpha+\beta\)
\(p=\alpha\beta\)
\(P_1=\alpha+\beta=1\)
\(S=1\)
\(P_n=sP_{n-1}-pP_{n-2}\)
\(P_{10}=123,\quad P_9=76,\quad P_8=47\)
\(123=P_9-pP_8\)
\(123=76-p(47)\)
\(47p=-47\)
\(p=-1\)
\(\alpha+\beta=1\)
\(\alpha\beta=-1\).
Therefore \(\alpha\) and \(\beta\) are roots of
\(x^2-x-1=0\)
Now the roots required are \(\dfrac1\alpha\) and \(\dfrac1\beta\)
Their sum is \(\dfrac{\alpha+\beta}{\alpha\beta}=\dfrac{1}{-1}=-1\)
Their product is
\(\dfrac1{\alpha\beta}=-1\)
Hence the required quadratic equation is
\(x^2-(\text{sum})x+(\text{product})=0\)
\(x^2+x-1=0\)
For \(0
(II) If \(\alpha \in(0,1)\), then \(b\) may be the geometric mean of \(a\) and \(c\)
[JEE Main 2024, 31 Jan (Shift 1)]
Both (I) and (II) are true
Given
\((a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0\)
Observe that
\((a+b-2c)+(b+c-2a)+(c+a-2b)=0\)
Hence \(x=1\) is always a root.
Since \(\alpha\ne1\) is the other root,
\(\alpha=\dfrac{c+a-2b}{a+b-2c}\).
Let
\(p=a-b>0,\quad q=b-c>0\).
Then
\(a=b+p,\quad c=b-q\).
Substituting,
\(\alpha=\dfrac{(b-q)+(b+p)-2b}{(b+p)+b-2(b-q)}=\dfrac{p-q}{p+2q}\).
Since \(p,q>0\),
\(-1<\dfrac{p-q}{p+2q}<1\).
Thus \(\alpha\in(-1,1)\).
Now let \(b\) be the geometric mean of \(a\) and \(c\):
\(b^2=ac=(b+p)(b-q)\).
This gives
\(b(p-q)=pq\).
Since \(b>0\),
\(p-q=\dfrac{pq}{b}>0\).
Hence
\(\alpha=\dfrac{p-q}{p+2q}>0\).
Therefore, if \(b\) is the geometric mean of \(a\) and \(c\), then necessarily
\(\alpha\in(0,1)\).
So:
(I) If \(\alpha\in(-1,0)\), then \(b\) cannot be the geometric mean of \(a\) and \(c\). True.
(II) If \(\alpha\in(0,1)\), then \(b\) may be the geometric mean of \(a\) and \(c\). True.
For example, take \(a=4,\ b=2,\ c=1\) \((b=\sqrt{ac})\).
Then
\(\alpha=\dfrac{4+1-4}{4+2-2}=\dfrac14\in(0,1)\).
Hence both statements are true
If the quadratic equation \((\lambda +2){\mathrm{x}}^{2}-3\mathrm{λx}+4\lambda =0,\lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:
[JEE Main 2026, 4 Apr (Shift 2)]
\(2\)
\( f(x)=(\lambda+2) x^2-3 \lambda x+4 \lambda\)
Case - 1: \(\frac{c}{a}>0\) or \(a c>0\)
\( (\lambda+2) 4 \lambda>0\)
\( \Rightarrow \lambda<-2 \text { or } \lambda>0\)
Case - 2: \(-\frac{\mathrm{b}}{2 \mathrm{a}}>0 \)
\( \frac{3 \lambda}{2(\lambda+2)}>0 \)
\( \Rightarrow \lambda<-2 \text { or } \lambda>0\)
Case - 3: \(\mathrm{D} \geq 0\)
\( (-3 \lambda)^2-4(\lambda+2) \times 4 \lambda \geq 0 \)
\( \lambda(7 \lambda+32) \leq 0 \)
\( \Rightarrow \lambda \in\left[\frac{-32}{7}, 0\right]\)
Intersection of Case-1, Case-2 and Case-3
\( \Rightarrow \lambda \in\left[\frac{-32}{7},-2\right)\)
\( \Rightarrow \lambda \in[-4.57,-2)\)
\( \Rightarrow \lambda=-4,-3\)
Number of values of \(\lambda=2\)
Consider the equation \({x}^{2}+4x-n=0\), where \(\mathrm{n}\in [20,100]\) is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
\(6\)
x² + 4x + 4 = n + 4
(x + 2)² = n + 4
x = -2 ± √n + 4
∵ 20 ≤ n ≤ 100
√24 ≤ √n + 4 ≤ √104
⇒ √n + 4 ∈ {5,6,7,8,9,10}
∴ '6' integral values of 'n' are possible
If \(\alpha, \beta\), where \(\alpha<\beta\),are the roots of the equation \(\lambda {x}^{2}−\left(\lambda +3\right)x+3=0\) such that \(\frac{1}{\alpha }−\frac{1}{\beta }=\frac{1}{3}\), then the sum of all possible values of λ is
[JEE Main 2026, 28 Jan (Shift 1)]
6
Since \(\alpha, \beta\) be the roots
of the quadratic equation \(\lambda x^2-(\lambda+3) x+3=0\)
\(\alpha +\beta =\frac{\lambda +3}{\lambda },\alpha \beta =\frac{3}{\lambda }\)
Now, \(\frac{\beta −\alpha }{\alpha \beta }=\frac{1}{3}\)
\(\Rightarrow \beta −\alpha =\frac{\alpha \beta }{3}=\frac{1}{\lambda }\)
on squaring
\({\alpha }^{2}+{\beta }^{2}−2\alpha \beta =\frac{1}{{\lambda }^{2}}...\left(1\right)\)
\({\alpha }^{2}+{\beta }^{2}+2\alpha \beta =\frac{{(\lambda +3)}^{2}}{{\lambda }^{2}}...\left(2\right)\)
eq(2) –eq(1)
\(4\alpha \beta =\frac{{(\lambda +3)}^{2}−1}{{\lambda }^{2}}\)
\(\Rightarrow \frac{12}{\lambda }=\frac{{\lambda }^{2}+6\lambda +8}{{\lambda }^{2}}\)
\(\Rightarrow {\lambda }^{2}−6\lambda +8=0\)
\(\Rightarrow \lambda =2,4\)
Sum of possible values of \(\lambda\) is \(=6\)
If the exhaustive values of \(a\) for which the equation \(2{x}^{2}+(a-5)x+15=3a\) has no real roots is \((\alpha ,\beta )\) then \(|4(\alpha +\beta )|\) is equal to
56
Given. equation has no real roots. i.e. \(D<0\)
Now, \({(a-5)}^{2}-4.2.(15-3a)<0\\ {a}^{2}+14a-95<0\\ (a+19)(a-5)<0\\ a\in (-19,5)\\ \alpha =-19,\beta =5\\ and|4(\alpha +\beta )|=|4(-19+5)|=56\)
if a function \(f(x)= 5x³-15x-a\) has three distinct and real roots for \(a € ( \alpha,\beta)\) then find \(\beta-2(\alpha)\)
30
\(\begin{aligned}f(x) & =5 x^3-15 x-a \\f^{\prime}(x) & =15 x^2-15 \\f^{\prime}(x) & =15\left(x^2-1\right) \\& =15(x-1)(x+1)\end{aligned}\)
\((-1) \rightarrow\) point of maxima \(1 \rightarrow\) point of minima
\(\begin{aligned}& f(1)=-10-9 \\& f(-1)=10-9\end{aligned}\)
For 3 distinet meakerw)
\(\begin{array}{lll}-10-a<0 \Rightarrow & a>-10 \\a \text { ard } \quad 10 a>0 & a<10\end{array}\)
\(\begin{aligned}& =1 \quad a \in(-10,10) \\& \alpha=-10, \quad \beta=10 \\& \Rightarrow \beta-2 \alpha=10-2 \times(-10) \\& =10+20 \\& =30\end{aligned}\)
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 :
[JEE Main 2024, 31 Jan (Shift 1)]
0
Given \(\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0 \quad \forall x \in \mathbb{R}\)
In \(x^2-8 x+32\), we have \(D=64-128<0\)
\( \therefore x^2-8 x+32>0\) \(\forall x \in \mathbb{R} \)
\( \Rightarrow a x^2+2(a+1) x+9 a+4<0\) \(\forall x \in R \)
\( \Rightarrow a<0\) and \(D<0\)
Since Question has asked for positive integral values of \(a\)
\( \therefore|S|=0\)
but we need positive integral value of a.
So, No solution
Let \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}},\mathrm{n}\in N\). If \({\mathrm{P}}_{10}=123,{\mathrm{P}}_{9}=76\), \({\mathrm{P}}_{8}=47\) and \({\mathrm{P}}_{1}=1\), then the quadratic equation having roots \(\frac{1}{\alpha }\) and \(\frac{1}{\beta }\) is :
[JEE Main 2025, 2 Apr (Shift 1)]
\({x}^{2}+x-1=0\)
\({P}_{10}={P}_{8}+{P}_{9}\\ \Rightarrow {x}^{2}=x+1\mathrm{has}\mathrm{roots}\alpha \mathrm{and}\beta \\ \Rightarrow \mathrm{Required}\mathrm{equation}\mathrm{is}\\ {\mathrm{x}}^{2}+\mathrm{x}-1=0\)
Consider the equation \({x}^{2}+4x-n=0\), where \(\mathrm{n}\in [20,100]\) is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha\) and \(\beta (\alpha <\beta )\) are the roots of the equation \((-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x⩾0,\)
then \(\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }\) is equal to:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha_\theta\) and \(\beta_\theta\) be the distinct roots of \(2 \mathrm{x}^2+(\cos \theta) \mathrm{x}-1=0, \theta \in(0,2 \pi)\). If m and M are the minimum and the maximum values of \(\alpha_\theta^4+\beta_\theta^4\), then \(16(M+m)\) equals :
[JEE Main 2025, 22 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of \({x}^{2}+\sqrt{3}x-16=0\), and \(\gamma\) and \(\delta\) be the roots of \({x}^{2}+3x-1=0\). If \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}}\) and \({\mathrm{Q}}_{\mathrm{n}}={\gamma }^{\mathrm{n}}+{\delta }^{\mathrm{n}}\), then \(\frac{{\mathrm{P}}_{25}+\sqrt{3}{\mathrm{P}}_{24}}{2{\mathrm{P}}_{23}}+\frac{{\mathrm{Q}}_{25}-{\mathrm{Q}}_{23}}{{\mathrm{Q}}_{24}}\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The product of all the rational roots of the equation \({\left({x}^{2}-9x+11\right)}^{2}-(x-4)(x-5)=3,\) is equal to
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the exhaustive values of \(a\) for which the equation \(2{x}^{2}+(a-5)x+15=3a\) has no real roots is \((\alpha ,\beta )\) then \(|4(\alpha +\beta )|\) is equal to
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of the equation \(p{x}^{2}+qx−r=0\), where \(p\neq 0\). If p, q and r be the consecutive terms of a non constant G.P. and \(\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}\), then the value of \({(\alpha −\beta )}^{2}\) is:EndFragment
[JEE Main 2024, 01 Feb (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the set of all values of \(p\in \mathrm{ℝ}\), for which both the roots of the equation \({x}^{2}-(p+2)x+(2p+9)=0\) are negative real numbers, be the interval \((\alpha ,\beta ]\). Then \(\beta -2\alpha\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The product of all the rational roots of the equation \(\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3\), is equal to
[JEE Main 2025, 24 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha, \beta\), where \(\alpha<\beta\),are the roots of the equation \(\lambda {x}^{2}−\left(\lambda +3\right)x+3=0\) such that \(\frac{1}{\alpha }−\frac{1}{\beta }=\frac{1}{3}\), then the sum of all possible values of λ is
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the set of all \(\mathrm{a} \in \mathbf{R}\), for which the equation \(2 x^2+(a-5) x+15=3a\) has no real root, is the interval \((\alpha, \beta)\) and \(\mathrm{X}=\{x \in Z: \alpha
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha_\theta\) and \(\beta_\theta\) be the distinct roots of \(2 \mathrm{x}^2+(\cos \theta) \mathrm{x}-1=0, \theta \in(0,2 \pi)\). If \(m\) and \(M\) are the minimum and the maximum values of \(\alpha_\theta^4+\beta_\theta^4\), then \(16(M+m)\) equals:
[JEE Main 2025, 22 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha ,\beta ;\alpha >\beta\), be the roots of the equation \({x}^{2}-\sqrt{2}x-\sqrt{3}=0\). Let \({P}_{n}={\alpha }^{n}-{\beta }^{n},n\in N\). Then \((11\sqrt{3}-10\sqrt{2}){P}_{10}+(11\sqrt{2}+10){P}_{11}-11{P}_{12}\) is equal to
[JEE Main 2024, 9 Apr (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha, \beta\) are the roots of the equation, \(x^{2}-x-1=0\) and \(S_{n}=2023 \alpha^{n}+2024 \beta^{n}\), then :
[JEE Main 2024, 27 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If 2 and 6 are the roots of the equation \(a{x}^{2}+bx+1=0\), then the quadratic equation, whose roots are \(\frac{1}{2a+b}\) and \(\frac{1}{6a+b}\), is
[JEE Main 2024, 4 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta\) be the distinct roots of the equation \(x^2-\left(t^2-5 t+6\right) x+1=0, t \in R\) and \(a_n=\alpha^n+\beta^n\). Then the minimum value of \(\frac{a_{2023}+a_{2025}}{a_{2024}}\) is
[JEE Main 2024, 6 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of solutions of the equation \(\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0\) is:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}\left(\alpha^6+\beta^6\right)\), is:
[JEE Main 2024, 09 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}\left(\alpha^6+\beta^6\right)\), is:
[JEE Main 2024, 09 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}, a>2\). If \(\alpha\) is such that \(a, 4, \alpha, b\) are in A.P., then the equation \(\alpha x^2-a x+2(\alpha-2 b)=0\) has:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Roots of the equation \(x^2+b x-c=0(b, c>0)\) are:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha ,\beta\) be the roots of the quadratic equation \(12{x}^{2}−20x+3\lambda =0,\) \(\lambda \in Z.\) If \(\frac{1}{2}\leq \left|\beta −\alpha \right|\leq \frac{3}{2},\) then the sum of all possible values of \(\lambda\) is:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\mathbb{R}\) denote the set of all real numbers. Let \({a}_{i},{b}_{i}\in \mathrm{ℝ}\) for \(i\in {1,2,3}\).
Define the functions \(f:\mathrm{ℝ}\to \mathrm{ℝ},g:\mathrm{ℝ}\to \mathrm{ℝ}\), and \(h:\mathrm{ℝ}\to \mathrm{ℝ}\) by
\(f(x)={a}_{1}+10x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+{x}^{4}\\ g(x)={b}_{1}+3x+{b}_{2}{x}^{2}+{b}_{3}{x}^{3}+{x}^{4}\\ h(x)=f(x+1)-g(x+2)\)
If \(f(x) \neq g(x)\) for every \(x \in \mathbb{R}\), then the coefficient of \(x^3\) in \(h(x)\) is
[JEE Advanced 2025]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(S\) be the set of postive 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 :
[JEE Main 2024, 31 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha\) and \(\beta (\alpha <\beta )\) are the roots of the equation \((-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x⩾0,\)
then \(\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }\) is equal to:
[JEE Main 2026, 23 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The sum, of the squares of all the roots of the equation \({x}^{2}+|2x-3|-4=0,\) is:
[JEE Main 2025, 28 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Roots of the equation \(x^2+b x-c=0(b, c>0)\) are:
Options are free to see. Unlock the correct answer and full explanation with Pass.
A building construction work can be completed by two masons \(A\) and \(B\) together in \(22.5\) days. Mason \(A\) alone can complete the construction work in \(24\) days less than mason \(B\) alone. Then mason \(A\) alone will complete the construction work in:
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of \({x}^{2}+\sqrt{3}x-16=0\), and \(\gamma\) and \(\delta\) be the roots of \({x}^{2}+3x-1=0\). If \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}}\) and \({\mathrm{Q}}_{\mathrm{n}}={\gamma }^{\mathrm{n}}+{\delta }^{\mathrm{n}}\), then \(\frac{{\mathrm{P}}_{25}+\sqrt{3}{\mathrm{P}}_{24}}{2{\mathrm{P}}_{23}}+\frac{{\mathrm{Q}}_{25}-{\mathrm{Q}}_{23}}{{\mathrm{Q}}_{24}}\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha, \beta\) are the roots of the equation, \(x^{2}-x-1=0\) and \(S_{n}=2023 \alpha^{n}+2024 \beta^{n}\), then :
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the set of all values of \(p\in \mathrm{ℝ}\), for which both the roots of the equation \({x}^{2}-(p+2)x+(2p+9)=0\) are negative real numbers, be the interval \((\alpha ,\beta ]\) Then \(\beta -2\alpha\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
A building construction work can be completed by two masons \(A\) and \(B\) together in \(22.5\) days. Mason \(A\) alone can complete the construction work in \(24\) days less than mason \(B\) alone. Then mason \(A\) alone will complete the construction work in:
[JEE Main 2026, 23 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of \({x}^{2}+\sqrt{3}x-16=0\), and \(\gamma\) and \(\delta\) be the roots of \({x}^{2}+3x-1=0\). If \({\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}+{\beta }^{\mathrm{n}}\) and \({\mathrm{Q}}_{\mathrm{n}}={\gamma }^{\mathrm{n}}+{\delta }^{\mathrm{n}}\), then \(\frac{{\mathrm{P}}_{25}+\sqrt{3}{\mathrm{P}}_{24}}{2{\mathrm{P}}_{23}}+\frac{{\mathrm{Q}}_{25}-{\mathrm{Q}}_{23}}{{\mathrm{Q}}_{24}}\) is equal to
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the set of all values of \(p\in \mathrm{ℝ}\), for which both the roots of the equation \({x}^{2}-(p+2)x+(2p+9)=0\) are negative real numbers, be the interval \((\alpha ,\beta ]\) Then \(\beta -2\alpha\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of solutions of the equation \(\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0\) is:
[JEE Main 2025, 29 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
if a function \(f(x)= 5x³-15x-a\) has three distinct and real roots for \(a € ( \alpha,\beta)\) then find \(\beta-2(\alpha)\)
Options are free to see. Unlock the correct answer and full explanation with Pass.
The sum, of the squares of all the roots of the equation \({x}^{2}+|2x-3|-4=0,\) is:
[JEE Main 2025, 28 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let the arithemetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}, a>2\). If \(\alpha\) is such that \(a, 4, \alpha, b\) are in A.P., then the equation \(\alpha x^2-a x+2(\alpha-2 b)=0\) has:
[JEE Main 2026, 28 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha ,\beta\) be the roots of the quadratic equation \(12{x}^{2}−20x+3\lambda =0,\) \(\lambda \in Z.\) If \(\frac{1}{2}\leq \left|\beta −\alpha \right|\leq \frac{3}{2},\) then the sum of all possible values of \(\lambda\) is:
[JEE Main 2026, 22 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The product of the roots of the equation \(9 x^2-18|x|+5=0\), is :
[JEE Main 2020, 5 Sep (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(a, b \in R, a \neq 0\) be such that the equation, \(a x^2-2 b x+5=0\) has a repeated root \(\alpha\), which is also a root of the equation, \(x^2-2 b x-10=0\). If \(\beta\) is the other root of this equation, then \(\alpha^2+\beta^2\) is equal to:
[JEE Main 2020, 9 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \( \alpha \) and \( \beta \) are the roots of the equation, \( 7 x^{2}-3 x-2=0 \), then the value of \( \frac{\alpha}{1-\alpha^{2}}+\frac{\beta}{1-\beta^{2}} \) is equal to
[JEE Main 2020, 5 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of the equation, \(5 x^2+6 x-2=0\). If \(S_n=\alpha^n+\beta^n, n=1,2,3\) , then
[JEE Main 2020, 2 Sep (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(p\) and \(q\) be two positive numbers such that \(p+q=2\) and \(p^4+q^4=272\). Then \(p\) and \(q\) are roots of the equation:
[JEE Main 2021, 24 Feb (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha=\max _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}\) and \(\beta=\min _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}\). If \(8 x^2+b x+c=0\) is a quadratic equation whose roots are \(\alpha^{1 / 5}\) and \(\beta^{1 / 5}\), then the value of \(c-b\) is equal to:
[JEE Main 2021, 27 Jul (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The equation \(e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R\) has:
[JEE Main 2023, 31 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\lambda \neq 0\) be a real number. Let \(\alpha ,\beta\) be the roots of the equation \(14{\mathrm{x}}^{2}−31\mathrm{x}+3\lambda =0\) and \(\alpha ,\gamma\) be the roots of the equation 35x2 – 53x + 4λ =0. Then \(\frac{3\alpha }{\beta }\) and \(\frac{4\alpha }{\gamma }\) are the roots of the equation
[JEE Main 2023, 29 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If one root of the equation \({x}^{2}+px+12=0\), is 4 while the equation \({x}^{2}+px+q=0\) has equal roots, then the value of \('q'\) is
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2}+p x+2 \) \( =0 \) and \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) are the roots of the equation \( 2 x^{2}+ \) \( 2 q x+1=0 \), then \( \left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right) \) is equal to
[JEE Main 2020, 3 Sep (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(S=\left\{x: x \in R\right.\) and \(\left.(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\right\}\)
Then \(n(S)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The value of \(4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \infty }}}}\)
[JEE Main 2021, 17 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^2+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real solutions of the equation \(3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0\)
Options are free to see. Unlock the correct answer and full explanation with Pass.
If (α, β), (β, γ ), (γ, α) are respectively the roots of x2 – 2px + 2 = 0; x2 – 2qx + 3 = 0; x2 – 2rx + 6 = 0 where α, β, γ are all positive then the value of p + q + r is
Options are free to see. Unlock the correct answer and full explanation with Pass.
The set of all values of \(k>-1\), for which the equation \(\left(3 x^2+4 x+3\right)^2-(k+1)\left(3 x^2+4 x+3\right)\left(3 x^2+4 x+2\right)+\) \(k\left(3 x^2+4 x+2\right)^2=0\) has real roots is:
[JEE Main 2021, 27 Aug (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\lambda \neq 0\) be a real number. Let \(\alpha, \beta\) be the roots of the equation \(14 x^2-31 x+3 \lambda=0\) and \(\alpha\), \(\gamma\) be the roots of the equation \(35 x^2-53 x+4 \lambda=0\). Then \(\frac{3 \alpha}{\beta}\) and \(\frac{4 \alpha}{\gamma}\) are the roots of the equation:
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real solutions of the equation \(3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0\),
[JEE Main 2023, 24 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The value of \(4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \ldots . . \infty}}}}\)
[JEE Main 2021, 17 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The value of \( 3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty}}}} \) is equal to
[JEE Main 2021, 18 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \alpha \) and \( \beta \) be the roots of \( x^{2}-6 x-2=0 \). If \( a_{n}=\alpha^{n}-\beta^{n} \) for \( n \geq 1 \), then the value of \( \frac{a_{10}-2 a_{8}}{3 a_{9}} \)
[JEE Main 2021, 25 Feb (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The sum of all the roots of the equation \(\left|x^2-8 x+15\right|-2 x\) \(+7=0\) is:
[JEE Main 2023, 6 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta, \gamma\) be the real roots of the equation, \(x^3+a x^2+b x\) \(+c=0,(a, b, c \in R\) and \(a, b \neq 0)\). If the system of equations (in \(u, v, w\) ) given by \(\alpha u+\beta v+\gamma w=0 ; \beta u+\gamma v+\alpha w=0\); \(\gamma u+\alpha v+\beta w=0\) has non-trivial solution, then the value of \(\frac{a^2}{b}\) is:
[JEE Main 2021, 18 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real roots of the equation \(\sqrt{x^2-4 x+3}+\sqrt{x^2-9}=\sqrt{4 x^2-14 x+6}\), is:
[JEE Main 2023, 31 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Suppose \(a,b\) denote the distinct real roots of the quadratic polynomial \({x}^{2}+20x-2020\) and suppose \(c,d\) denote the distinct complex roots of the quadratic polynomial \({x}^{2}-20x+2020.\) Then the value of \(ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)\)\(+bd\left(b-d\right)\) is
[JEE Advanced 2020]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real solutions of the equation
\({(x-1)}^{2}+{(x-2)}^{2}+{(x-3)}^{2}=0\) is
Options are free to see. Unlock the correct answer and full explanation with Pass.
The set of all real values of \( \lambda \) for which the quadratic equations, \( \left(\lambda^{2}+1\right) x^{2}-4 \lambda x+2=0 \) always have exactly one root in the interval \( (0,1) \) is
[JEE Main 2020, 3 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real solutions of the equation, \(x^2-|x|-12\) \(=0\) is:
[JEE Main 2021, 25 Jul (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The roots of the given equation \((\mathbf{p}-\mathbf{q}) \mathbf{x}^2+(\mathbf{q}-\mathbf{r}) \mathbf{x}+(\mathbf{r}-\mathbf{p})=0\) are
Options are free to see. Unlock the correct answer and full explanation with Pass.
If x, y and z are real numbers, then
\({x}^{2}+4{y}^{2}+9{z}^{2}−6yz−3zx−2xy\) is always
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \alpha, \beta \) be two roots of the equation \( x^{2}+(20)^{\frac{1} { 4}} x+(5)^{\frac{1} {2}}=0 \). Then \( \alpha^{8}+\beta^{8} \) is equal to
[JEE Main 2021, 27 Jul (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If the functions \(f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}\) and \(g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b\) have a common extreme point, then \(a+2 b+7\) is equal to
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\lambda \neq 0\) be a real number. Let \(\alpha, \beta\) be the roots of the equation \(14 x^2-31 x+3 \lambda=0\) and \(\alpha\), \(\gamma\) be the roots of the equation \(35 x^2-53 x+4 \lambda=0\). Then \(\frac{3 \alpha}{\beta}\) and \(\frac{4 \alpha}{\gamma}\) are the roots of the equation:
[JEE Main 2023, 29 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
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 ' \(q\) ' is
Options are free to see. Unlock the correct answer and full explanation with Pass.
The equation \(x^2-4 x+[x]+3=x[x]\), where \([x]\) denotes the greatest integer function, has:
[JEE Main 2023, 24 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta, \gamma\) be the three roots of the equation \(x^3+b x+c=0\). If \(\beta \gamma=1=-\alpha\), then \(b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3\) is equal to
[JEE Main 2023, 8 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of pairs \((a, b)\) of real numbers, such that whenever \(\alpha\) is a root of the equation \(x^2+a x+b=0\), \(\alpha^2-2\) is also a root of this equation, is:
[JEE Main 2021, 1 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
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:
[JEE Main 2022, 25 Jun (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Find the value of a such that the sum of the squares of the roots of the equation \(\ x^2-(a-2) x-(a+1)=0 \) is least.
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \( \alpha \) and \( \beta \) be two roots of the equation \( x^{2}-64 x+256=0 \). Then the value of \( \left(\frac{\alpha^{3}}{\beta^{5}}\right)^{1 / 8}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{1 / 8} \) is :
[JEE Main 2020, 6 Sep (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The value of \(3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots }}}}\) is equal to:
[JEE Main 2021, 18 Mar (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(x^2+9 y^2-4 x+3=0, x, y \in R\), then \(x\) and \(y\) respectively lie in the intervals:
[JEE Main 2021, 27 Aug (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha\) and \(\beta\) be the roots of the equation, \(5 x^2+6 x-2=0\). If \(S_n=\alpha^n+\beta^n, n=1,2,3 \ldots\). , then
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha ,\beta\) are the roots of \(a{x}^{2}+bx+c=0\) and \(c\text{ }\neq \text{ }0,\) then the value of \(\frac{1}{{\left(a\alpha +b\right)}^{2}}+\frac{1}{{\left(a\beta +b\right)}^{2}}\) in terms of \(a,b,c\) is
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \lambda \neq 0 \) be in \( \mathrm{R} \). If \( \alpha \) and \( \beta \) are the roots of the equation, \( x^{2}-x+2 \lambda=0 \) and \( \alpha \) and \( \gamma \) are the roots of the equation, \( 3 x^{2}-10 x+27 \lambda=0 \), then \( \frac{\beta \gamma}{\lambda} \) is equal to:
[JEE Main 2020, 4 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^2+\sqrt{6} x+3=0\). Then \(\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \(\alpha\) and \(\beta\) are the roots of the equation \(2 x(2 x+1)=1\), then \(\beta\) is equal to:
[JEE Main 2020, 6 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
The integer \( k \) for which the inequality \( x^{2}-2(3 k-1) x+ \) \( 8 k^{2}-7>0 \) is valid for every \( x \) in \( R \) is:
[JEE Main 2021, 25 Feb (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \alpha=\max _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\} \) and \( \beta=\min _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\} \). If \( 8 x^{2}+b x+c=0 \) is a quadratic equation whose roots are \( a^{1 / 5} \) and \( \beta^{1 / 5} \), then the value of \( c-b \) is equal to
[JEE Main 2021, 27 Jul (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2}-x-1=0 \). If \( p_{k}= \) \( (\alpha)^{k}+(\beta)^{k}, \mathrm{k} \geq 1 \), then which one of the following statements is not true?
[JEE Main 2020, 7 Jan (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
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 :
[JEE Main 2020, 2 Sep (Shift 2)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \(S=\left\{x:x\in R\text{and }(\sqrt{3}+\sqrt{2}{)}^{{x}^{2}-4}+(\sqrt{3}-\sqrt{2}{)}^{{x}^{2}-4}=10\right\}\). Then \(n(S)\) is equal to
[JEE Main 2023, 1 Feb (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
If \( \alpha \) and \( \beta \) are the distinct roots of the equation \( x^{2}+3^{1 / 4} x \) \( +3^{1 / 2}=0 \) then the value of \( \alpha^{96}\left(\alpha^{12}-1\right)+\beta^{96}\left(\beta^{12}-1\right) \) is equal to
[JEE Main 2021, 20 Jul (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
Let \( \alpha \) and \( \beta \) two real roots of the equation \( (k+1) \tan ^{2} x \) \( -\sqrt{2} \cdot \lambda \tan x=(1-k) \), where \( k(\neq-1) \) and \( \lambda \) are real numbers. If \( \tan ^{2}(\alpha+\beta)=50 \), then a value of \( \lambda \) is
[JEE Main 2020, 7 Jan (Shift 1)]
Options are free to see. Unlock the correct answer and full explanation with Pass.
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 – c) + bd(b – d) is
Options are free to see. Unlock the correct answer and full explanation with Pass.
The number of real solutions of the equation \(3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0\),
Options are free to see. Unlock the correct answer and full explanation with Pass.
Practice more JEE Maths PYQs
Browse every Maths chapter, or explore the full JEE question bank.
All Maths chapters →