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Yukti Vault

Algebra

Quadratic Equation: JEE Main & Advanced PYQs

292 previous year questions on Quadratic Equation from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Topics covered: Common Roots, Equations Reducible to Quadratic, Equations with Modulus and Greatest Integer, Higher Degree Polynomial Equations, Location of Roots, Miscellaneous, Nature of Roots and Discriminant, Powers of Roots (αⁿ ± βⁿ Recurrence), Relation Between Roots and Coefficients, Sign, Range and Inequalities of Quadratic Expressions.

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Questions 1-50 of 292

  1. Let $\alpha,\beta$ be the roots of the equation $x^{2}-3x+r=0$, and $\frac{\alpha}{2},2\beta$ be the roots of the equation $x^{2}+3x+r=0$. If the roots of the equation …
  2. Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c)$, $(2,b)$ and $(a,b)$ be $\left(\frac{10}{3},\frac{7}{3}\right)$. If …
  3. The integer '$k$', for which the inequality $x^{2}-2(3k-1)x+8k^{2}-7>0$ is valid for every $x$ in $\mathbf{R}$, is :
  4. 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 :
  5. Let $\alpha,\beta$ be the roots of the equation $x^{2}-x+p=0$ and $\gamma,\delta$ be the roots of the equation $x^{2}-4x+q=0$; $p,q\in\mathbf{Z}$. If …
  6. If the quadratic equation $(\lambda+2)x^{2}-3\lambda x+4\lambda=0$, $\lambda\neq-2$, has two positive roots, then the number of possible integral values of $\lambda$ is:
  7. If $\alpha=1$ and $\beta=1+i\sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation $x^{3}+ax^{2}+bx+c=0$, $a,b,c\in\mathbb{R}$, then …
  8. Consider the quadratic equation $(n^{2}-2n+2)x^{2}-3x+(n^{2}-2n+2)^{2}=0$, $n\in\mathbf{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be …
  9. If the equation, $x^2+bx+45=0$ ($b\in\mathbf{R}$) has conjugate complex roots and they satisfy $|z+1|=2\sqrt{10}$, then :
  10. 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 :
  11. If $\alpha,\beta$ are the roots of the equation …
  12. The sum of the squares of the roots of $|x-2|^{2}+|x-2|-2=0$ and the squares of the roots of $x^{2}-2|x-3|-5=0$, is
  13. Let $\alpha=\max_{x\in\mathbf{R}}\left\{8^{2\sin3x}\cdot4^{4\cos3x}\right\}$ and $\beta=\min_{x\in\mathbf{R}}\left\{8^{2\sin3x}\cdot4^{4\cos3x}\right\}$. If …
  14. The number of real roots of the equation $e^{4x}+2e^{3x}-e^{x}-6=0$ is :
  15. The number of real solutions of $x^{7}+5x^{3}+3x+1=0$ is equal to
  16. Let $a$ be the sum of all coefficients in the expansion of $\left(1-2x+2x^{2}\right)^{2023}\left(3-4x^{2}+2x^{3}\right)^{2024}$ and …
  17. Let $P(x)=x^{2}+bx+c$ be a quadratic polynomial with real coefficients such that $\int_{0}^{1}P(x)\,dx=1$ and $P(x)$ leaves remainder $5$ when it is divided by $(x-2)$. …
  18. The number of integral solutions $x$ of $\log_{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2x-3}\right)^{2}\ge0$ is
  19. 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 :
  20. Let $\alpha,\beta,\gamma$ be the three roots of the equation $x^{3}+bx+c=0$. If $\beta\gamma=1=-\alpha$, then $b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3}$ is equal …
  21. Let the set of all values of $p\in\mathbb{R}$, for which both the roots of the equation $x^{2}-(p+2)x+(2p+9)=0$ are negative real numbers, be the interval …
  22. Let $\alpha$ and $\beta$ be the roots of $x^{2}-6x-2=0$. If $a_n=\alpha^{n}-\beta^{n}$ for $n\ge1$, then the value of $\frac{a_{10}-2a_{8}}{3a_{9}}$ is :
  23. Let $a,b\in\mathbf{R}$ be such that the equation $ax^{2}-2bx+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-2bx+21=0$ …
  24. If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3x^{2}+\lambda x-1=0$ is $15$, then $6(\alpha^{3}+\beta^{3})^{2}$ is …
  25. Let $S_1=\left\{x\in\mathbf{R}-\{1,2\}:\frac{(x+2)\left(x^{2}+3x+5\right)}{-2+3x-x^{2}}\ge0\right\}$ and …
  26. Let $\alpha$ and $\beta$ be the roots of the equation $px^{2}+qx-r=0$, where $p\neq0$. If $p,q$ and $r$ be the consecutive terms of a non constant G.P. and …
  27. Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k,\frac{k}{2}\right)$ from the line $3x+4y+5=0$ is
  28. The sum of all the real roots of the equation $\left(e^{2x}-4\right)\left(6e^{2x}-5e^{x}+1\right)=0$ is
  29. The set of all values of $k>-1$, for which the equation …
  30. The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^{x}+48=0$ is :
  31. Let $\alpha,\beta$ be the roots of the equation $x^{2}-\sqrt{2}x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to
  32. The minimum value of the sum of the squares of the roots of $x^2+(3-a)x+1=2a$ is:
  33. Let $S=\left\{x\in\mathbf{R}:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10\right\}$. Then the number of elements in $S$ is :
  34. The sum of the roots of the equation, $x+1-2\log_2(3+2^{x})+2\log_4\left(10-2^{-x}\right)=0$, is :
  35. For $0<c<b<a$, let $(a+b-2c)x^{2}+(b+c-2a)x+(c+a-2b)=0$ and $\alpha\neq1$ be one of its root. Then, among the two statements (I) If $\alpha\in(-1,0)$, then $b$ cannot be …
  36. Let $S$ be the set of positive integral values of $a$ for which $\frac{ax^{2}+2(a+1)x+9a+4}{x^{2}-8x+32}<0$, $\forall x\in\mathbb{R}$. Then, the number of elements in …
  37. Let $P_n=\alpha^{n}+\beta^{n}$, $n\in\mathbf{N}$. If $P_{10}=123$, $P_9=76$, $P_8=47$ and $P_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and …
  38. 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 $P_n=\alpha^{n}+\beta^{n}$ and …
  39. Let $\alpha,\beta$; $\alpha>\beta$, be the roots of the equation $x^{2}-\sqrt{2}x-\sqrt{3}=0$. Let $P_n=\alpha^{n}-\beta^{n}$, $n\in\mathbb{N}$. Then …
  40. The number of real solutions of the equation, $x^{2}-|x|-12=0$ is :
  41. The number of roots of the equation $\left(3+\sqrt{5}\right)^{|x^{2}-4|}+4\left(\frac{3-\sqrt{5}}{4}\right)^{|x^{2}-4|}=6$ is
  42. The set of all $a\in\mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is
  43. Let $S$ be the set of all integral values of $\alpha$ for which the sum of squares of two real roots of the quadratic equation $3x^{2}+(\alpha-6)x+(\alpha+3)=0$ is …
  44. The number of solutions, of the equation $e^{\sin x}-2e^{-\sin x}=2$, is :
  45. Let $\alpha,\beta$ be the distinct roots of the equation $x^{2}-\left(t^{2}-5t+6\right)x+1=0$, $t\in\mathbb{R}$ and $a_n=\alpha^{n}+\beta^{n}$. Then the minimum value of …
  46. The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is :
  47. Consider the equation $x^{2}+4x-n=0$, 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 …
  48. The number of real roots of the equation $e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^{x}+1=0$ is :
  49. Let $\alpha,\beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then …
  50. The number of distinct real roots of $x^{4}-4x+1=0$ is :

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