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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 51-100 of 292

  1. If $S_1=\{x\in\mathbf{R}:x^{2}+|x|-2=0\}$ and $S_2=\{x\in\mathbf{R}:x^{2}+x-2=0\}$, then
  2. 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 …
  3. The number of real roots of the equation $x|x|-5|x+2|+6=0$, is
  4. Let $\alpha,\beta$ be two roots of the equation $x^2+(20)^{\frac14}x+(5)^{\frac12}=0$. Then $\alpha^8+\beta^8$ is equal to :
  5. If the equations $2x^{2}+kx-5=0$ and $x^{2}-3x-4=0$ have one root in common, then a value of '$k$' is
  6. The number of distinct real roots of the equation $x^{7}-7x-2=0$ is
  7. The sum of all the roots of the equation $\left|x^{2}-8x+15\right|-2x+7=0$ is :
  8. Let $\alpha,\beta$ be the roots of the equation $x^2-\sqrt2x+\sqrt6=0$ and $\frac1{\alpha^2}+1,\frac1{\beta^2}+1$ be the roots of the equation $x^2+ax+b=0$. Then the …
  9. The number of real solutions of the equation $x\left(x^{2}+3|x|+5|x-1|+6|x-2|\right)=0$ is $\_\_\_\_\_\_$ .
  10. Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha\beta=-1$. Let $p_n=(\alpha)^{n}+(\beta)^{n}$, $p_{n-1}=11$ and $p_{n+1}=29$ for some …
  11. If $\alpha$ and $\beta$ are the roots of the equation $2x(2x+1)=1$, then $\beta$ is equal to :
  12. If $\alpha,\beta$ are roots of the equation $x^{2}+5\sqrt{2}\,x+10=0$, $\alpha>\beta$ and $P_n=\alpha^{n}-\beta^{n}$ for each positive integer $n$, then the value of …
  13. The number of distinct real roots of the equation $3x^{4}+4x^{3}-12x^{2}+4=0$ is $\_\_\_\_\_\_$ .
  14. If for some $p,q,r\in\mathbf{R}$, not all have same sign, one of the roots of the equation $(p^2+q^2)x^2-2q(p+r)x+q^2+r^2=0$ is also a root of the equation $x^2+2x-8=0$ …
  15. If $a+b+c=1$, $ab+bc+ca=2$ and $abc=3$, then the value of $a^{4}+b^{4}+c^{4}$ is equal to $\_\_\_\_\_\_$ .
  16. The sum of squares of all the real solutions of the equation $\log_{(x+1)}(2x^{2}+5x+3)=4-\log_{(2x+3)}(x^{2}+2x+1)$ is equal to $\_\_\_\_\_\_$ .
  17. For $p,q\in\mathbf{R}$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf{R}$ and $q>0$. Let $a_1,a_2,a_3$ and $a_4$ be in an arithmetic progression with …
  18. The set of all real values of $\lambda$ for which the quadratic equations, $(\lambda^2+1)x^2-4\lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is :
  19. Let $\lambda\neq0$ be in $\mathbf{R}$. 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 …
  20. Let $f(x)$ be a quadratic polynomial with leading coefficient $1$ such that $f(0)=p$, $p\ne0$, and $f(1)=\frac13$. If the equations $f(x)=0$ and …
  21. The least positive value of `a' for which the equation, $2x^2+(a-10)x+\frac{33}2=2a$ has real roots is ________.
  22. The sum of all real values of $x$ for which $\frac{3x^2-9x+17}{x^2+3x+10}=\frac{5x^2-7x+19}{3x^2+5x+12}$ is equal to ________.
  23. Let $\lambda\ne0$ be in $\mathbf{R}$. 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 …
  24. The product of the roots of the equation $9x^2-18|x|+5=0$, is :
  25. Let $a,b\in\mathbf{R}$, $a\ne0$ be such that the equation, $ax^2-2bx+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^2-2bx-10=0$. If $\beta$ …
  26. The sum of all integral values of $k$ ($k\neq0$) for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is $\_\_\_\_\_\_$ .
  27. 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 :
  28. The number of real roots of the equation $(x+1)^{2}+|x-5|=\frac{27}{4}$ is $\_\_\_\_\_\_$ .
  29. Let $\alpha$ and $\beta$ be the roots of the equation, $5x^2+6x-2=0$. If $S_n=\alpha^n+\beta^n$, $n=1,2,3,\ldots$, then :
  30. The number of distinct real roots of the equation $x^5(x^3-x^2-x+1)+x(3x^3-4x^2-2x+4)-1=0$ is ________.
  31. Let $S$ be the set of all real roots of the equation, $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. Then $S$ :
  32. Let $f(x)=2x^2-x-1$ and $S=\{n\in\mathbb{Z}:|f(n)|\le800\}$. Then, the value of $\sum_{n\in S}f(n)$ is equal to ________.
  33. The number of elements in the set $\{n\in\mathbb{Z}:|n^{2}-10n+19|<6\}$ is $\_\_\_\_\_\_$ .
  34. The number of elements in the set $\{x\in\mathbb{R}:(|x|-2)\cdot|2x+3|=1\}$ is $\_\_\_\_\_\_$ .
  35. The number of real solutions of the equation $e^{4x}+4e^{3x}-58e^{2x}+4e^{x}+1=0$ is $\_\_\_\_\_\_$ .
  36. The sum of the cubes of all the roots of the equation $x^{4}-3x^{3}-2x^{2}+3x+1=0$ is $\_\_\_\_\_\_$ .
  37. Let $a$ be an integer such that all the real roots of the polynomial $2x^{5}+5x^{4}+10x^{3}+10x^{2}+10x+10$ lie in the interval $(a,a+1)$. Then, $|a|$ is equal to …
  38. Let $x_1,x_2,x_3,x_4$ be the solution of the equation $4x^{4}+8x^{3}-17x^{2}-12x+9=0$ and …
  39. The sum of modulus of all non-real roots of the equation $(x^{2}+4x+6)(2x^{2}+8x+14)=12$ is equal to $\_\_\_\_\_\_$ .
  40. Let $a,b,c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^{2}+b^{2}\right)x^{2}-2b(a+c)x+\left(b^{2}+c^{2}\right)=0$. If the set of all …
  41. Let $\alpha,\beta\in\mathbf{N}$ be roots of the equation $x^{2}-70x+\lambda=0$, where $\frac{\lambda}{2},\frac{\lambda}{3}\notin\mathbf{N}$. If $\lambda$ assumes the …
  42. If $a$ and $b$ are the roots of the equation $x^{2}-7x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to $\_\_\_\_\_\_$ .
  43. Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to $\_\_\_\_\_\_$ .
  44. Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^{2}-12x+[x]+31=0$ and $x^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest …
  45. Let $a,b,c$ be three distinct positive real numbers such that $(2a)^{\log_ea}=(bc)^{\log_eb}$ and $b^{\log_e2}=a^{\log_ec}$. Then $6a+5bc$ is equal to $\_\_\_\_\_\_$ .
  46. The number of solutions of the equation $\log_4(x-1)=\log_2(x-3)$ is $\_\_\_\_\_\_$ .
  47. The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is $\_\_\_\_\_\_$ .
  48. If $x^{2}-x+1=0$, then the value of $\sum_{n=1}^{5}\left(x^{n}+\frac{1}{x^{n}}\right)^{2}$ is $\_\_\_\_\_\_$ .
  49. Let $\alpha,\beta$ be roots of $x^{2}+\sqrt{2}x-8=0$. If $U_n=\alpha^{n}+\beta^{n}$, then $\frac{U_{10}+\sqrt{2}U_9}{2U_8}$ is equal to $\_\_\_\_\_\_$ .
  50. Let $\alpha,\beta$ be the roots of the equation $x^{2}-4\lambda x+5=0$ and $\alpha,\gamma$ be the roots of the equation …

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