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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 251-292 of 292

  1. If $a$, $b$ and $c$ are distinct positive numbers, then the expression $(b+c-a)(c+a-b)(a+b-c)-abc$ is
  2. A value of $b$ for which the equations $$x^2+bx-1=0$$ $$x^2+x+b=0,$$ have one root in common is
  3. The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))=0$$ has
  4. For $x\in\mathbb{R}$, the number of real roots of the equation $$3x^2-4|x^2-1|+x-1=0$$ is ___ .
  5. The smallest value of $k$, for which both the roots of the equation $$x^2-8kx+16(k^2-k+1)=0$$ are real, distinct and have values at least 4, is
  6. Let p and q be real numbers such that $p\neq0$, $p^3\neq q$ and $p^3\neq -q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha+\beta=-p$ and …
  7. Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$, with $\alpha>\beta$. If $a_n=\alpha^n-\beta^n$ for $n\ge1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is
  8. The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is
  9. Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha=0$ has two distinct real roots $x_1$ and $x_2$ satisfying …
  10. Let $-\frac{\pi}{6} \beta_1$ and $\alpha_2>\beta_2$, then $\alpha_1+\beta_2$ equals
  11. 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 …
  12. For a polynomial $g(x)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real …
  13. The sum of squares of all the roots of the equation $x^{2}+|2x-3|-4=0$ is
  14. The real number $s$ lies in the interval
  15. If $a_4=28$, then $p+2q=$
  16. Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$, then …
  17. The quadratic equations $x^{2}-6x+a=0$ and $x^{2}-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$. …
  18. Let the set $C=\left\{(x,y)\mid x^{2}-2^{y}=2023,\ x,y\in\mathbb N\right\}$. Then $\sum_{(x,y)\in C}(x+y)$ is equal to ______.
  19. Solve for $x$: $\sqrt{x+1}-\sqrt{x-1}=1$.
  20. If $x,y$ and $z$ are real and different and $u=x^{2}+4y^{2}+9z^{2}-6yz-3zx-2xy$, then $u$ is always
  21. Solve for $x$: $4^{x}-3^{x-\frac12}=3^{x+\frac12}-2^{2x-1}$.
  22. Let $f(x)=\frac{x^2-6x+5}{x^2-5x+6}$. Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the …
  23. If one root of the quadratic equation $ax^{2}+bx+c=0$ is equal to the $n$-th power of the other, then show that …
  24. Let $a,b,c$ be real. If $ax^{2}+bx+c=0$ has two real roots $\alpha$ and $\beta$, where $\alpha 1$, then show that $1+\dfrac{c}{a}+\left|\dfrac{b}{a}\right|<0$.
  25. If $\alpha,\beta$ are the roots of $ax^{2}+bx+c=0$, $(a\neq0)$ and $\alpha+\delta,\ \beta+\delta$ are the roots of $Ax^{2}+Bx+C=0$, $(A\neq0)$ for some constant …
  26. Show that the square of $\dfrac{\sqrt{26-15\sqrt3}}{5\sqrt2-\sqrt{38+5\sqrt3}}$ is a rational number.
  27. The integral value of $a$, for which the equation, $\left(x^{2}+x+2\right)^{2}-(a-3)\left(x^{2}+x+2\right)\left(x^{2}+x+1\right)+(a-4)\left(x^{2}+x+1\right)^{2}=0$ has …
  28. The number of elements in the set $\{x\in\mathbf{R}:(|x|-3)\,|x+4|=6\}$ is equal to :
  29. If $2$ and $6$ are the roots of the equation $ax^{2}+bx+1=0$, then the quadratic equation, whose roots are $\dfrac{1}{2a+b}$ and $\dfrac{1}{6a+b}$, is :
  30. Let $S=\left\{\sin^{2}2\theta:\left(\sin^{4}\theta+\cos^{4}\theta\right)x^{2}+(\sin2\theta)x+\left(\sin^{6}\theta+\cos^{6}\theta\right)=0\text{ has real roots}\right\}$. …
  31. The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is $\_\_\_\_\_\_\_\_$ .
  32. The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is $\_\_\_\_\_\_\_\_$ .
  33. If $\alpha$ and $\beta$ are the roots of the equation, $7x^{2}-3x-2=0$, then the value of $\dfrac{\alpha}{1-\alpha^{2}}+\dfrac{\beta}{1-\beta^{2}}$ is equal to :
  34. If $\alpha$ and $\beta$ be two roots of the equation $x^{2}-64x+256=0$. Then the value of …
  35. The product of all solutions of the equation $e^{5(\log_e x)^{2}+3}=x^{8}$, $x>0$, is :
  36. Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^{2}+(\cos\theta)x-1=0$, $\theta\in(0,2\pi)$. If $m$ and $M$ are the minimum and the maximum values of …
  37. If the equation $a(b-c)x^{2}+b(c-a)x+c(a-b)=0$ has equal roots, where $a+c=15$ and $b=\dfrac{36}{5}$, then $a^{2}+c^{2}$ is equal to $\_\_\_\_\_\_\_\_$ .
  38. The number of real solution(s) of the equation $x^{2}+3x+2=\min\{|x-3|,|x+2|\}$ is :
  39. If $2a+3b+6c=0$ $(a,b,c\in\mathbb{R})$ then the quadratic equation $ax^{2}+bx+c=0$ has
  40. If $2a+3b+6c=0$, then at least one root of the equation $ax^{2}+bx+c=0$ lies in the interval
  41. The least integral value $\alpha$ of $x$ such that $\frac{x-5}{x^{2}+5x-14}>0$, satisfies :
  42. The number of solutions of the equation $\log_{(x+1)}(2x^2+7x+5)+\log_{(2x+5)}(x+1)^2-4=0$, $x>0$, is :

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