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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 101-150 of 292

  1. If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+2=0$ and $\frac1\alpha$ and $\frac1\beta$ are the roots of the equation $2x^2+2qx+1=0$, then …
  2. If the set of all $a\in\mathbf{R}-\{1\}$, for which the roots of the equation $(1-a)x^{2}+2(a-3)x+9=0$ are positive is $(-\infty,-\alpha]\cup[\beta,\gamma)$, then …
  3. The value of $a$ for which the sum of the squares of the roots of the equation $x^{2}-(a-2)x-a-1=0$ assumes the least value is
  4. If the roots of the quadratic equation $x^{2}+px+q=0$ are $\tan30^\circ$ and $\tan15^\circ$, respectively, then the value of $2+q-p$ is
  5. All the values of $m$ for which both roots of the equation $x^{2}-2mx+m^{2}-1=0$ are greater than $-2$ but less than $4$, lie in the interval
  6. If the roots of the equation $x^{2}-bx+c=0$ be two consecutive integers, then $b^{2}-4c$ equals
  7. If $x$ is real, the maximum value of $\frac{3x^{2}+9x+17}{3x^{2}+9x+7}$ is
  8. In a triangle $PQR$, $\angle R=\frac\pi2$. If $\tan\left(\frac P2\right)$ and $\tan\left(\frac Q2\right)$ are the roots of $ax^{2}+bx+c=0$, $a\neq0$ then
  9. If $(1-p)$ is a root of quadratic equation $x^{2}+px+(1-p)=0$ then its roots are
  10. Let two numbers have arithmetic mean $9$ and geometric mean $4$. Then these numbers are the roots of the quadratic equation
  11. If both the roots of the quadratic equation $x^{2}-2kx+k^{2}+k-5=0$ are less than $5$, then $k$ lies in the interval
  12. If $a,b,c$ are distinct $+ve$ real numbers and $a^{2}+b^{2}+c^{2}=1$ then $ab+bc+ca$ is
  13. If $x$ is a solution of the equation $\sqrt{2x+1}-\sqrt{2x-1}=1,\ \left(x\ge\frac12\right)$, then $\sqrt{4x^{2}-1}$ is equal to :
  14. If $p$ and $q$ are the roots of the equation $x^{2}+px+q=0$, then
  15. The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation $x^{2}+ax+b=0$, $\alpha^{2}-2$ is also a root of this equation, is :
  16. 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 :
  17. If the equations $ax^{2}+bx+c=0$, $(a,b,c\in R,\ a\neq0)$ and $2x^{2}+3x+4=0$ have a common root, then $a:b:c$ equals :
  18. Let $p,q\in Q$. If $2-\sqrt3$ is a root of the quadratic equation $x^{2}+px+q=0$, then
  19. The sum of all the real values of $x$ satisfying the equation $2^{(x-1)\left(x^{2}+5x-50\right)}=1$ is :
  20. The probability of selecting integers $a\in[-5,30]$ such that $x^{2}+2(a+4)x-5a+64>0$, for all $x\in R$, is :
  21. The equation $\sqrt{3x^{2}+x+5}=x-3$, where $x$ is real, has
  22. Let $S=\left\{x:x\in\mathbb R \text{ and } \left(\sqrt3+\sqrt2\right)^{x^2-4}+\left(\sqrt3-\sqrt2\right)^{x^2-4}=10\right\}$. Then $n(S)$ is equal to
  23. If $\alpha\neq\beta$ but $\alpha^{2}=5\alpha-3$ and $\beta^{2}=5\beta-3$ then the equation having $\alpha/\beta$ and $\beta/\alpha$ as its roots is
  24. The sum of all real values of $x$ satisfying the equation $\left(x^{2}-5x+5\right)^{x^{2}+4x-60}=1$ is
  25. If $\alpha$ and $\beta$ are the distinct roots of the equation $x^{2}+(3)^{\frac14}x+3^{\frac12}=0$, then the value of …
  26. The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x^2+(3-\lambda)x+2=\lambda$ has the least value is :
  27. If $\frac{1}{\sqrt{\alpha}},\frac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^{2}+bx+1=0$, $(a\neq0,\ a,b\in R)$, then the equation …
  28. The equation $x^{2}-4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:
  29. 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
  30. If one real root of the quadratic equation $81x^{2}+kx+256=0$ is cube of the other root, then a value of $k$ is :
  31. Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1)\tan^{2}x-\sqrt2\cdot\lambda\tan x=(1-k)$, where $k(\neq-1)$ and $\lambda$ are real numbers. If …
  32. The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation $6x^{2}-11x+\alpha=0$ are rational numbers is :
  33. Let $S=\left\{x\in R: x\ge0 \text{ and } 2\left|\sqrt x-3\right|+\sqrt x\left(\sqrt x-6\right)+6=0\right\}$. Then $S$:
  34. The number of real roots of the equation $5+\left|2^{x}-1\right|=2^{x}\left(2^{x}-2\right)$ is :
  35. If $\lambda\in\mathbb R$ is such that the sum of the cubes of the roots of the equation, $x^{2}+(2-\lambda)x+(10-\lambda)=0$ is minimum, then the magnitude of the …
  36. If $m$ is chosen in the quadratic equation $\left(m^{2}+1\right)x^{2}-3x+\left(m^{2}+1\right)^{2}=0$ such that the sum of its roots is greatest, then the absolute …
  37. If non-zero real numbers $b$ and $c$ are such that $\min f(x)>\max g(x)$, where $f(x)=x^{2}+2bx+2c^{2}$ and $g(x)=-x^{2}-2cx+b^{2}$, $(x\in R)$; then …
  38. The number of real solutions of the equation $x^{2}-3|x|+2=0$ is
  39. Let $p,q$ and $r$ be real numbers ($p\neq q$, $r\neq0$), such that the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac1r$ are equal in magnitude but opposite in …
  40. The value of $a$ for which one root of the quadratic equation $\left(a^{2}-5a+3\right)x^{2}+(3a-1)x+2=0$ is twice as large as the other is
  41. Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2}\sin\theta-x(\sin\theta\cos\theta+1)+\cos\theta=0$ $(0<\theta<45^\circ)$, and $\alpha<\beta$. Then …
  42. If the sum of the roots of the quadratic equation $ax^{2}+bx+c=0$ is equal to the sum of the squares of their reciprocals, then $\frac ac$, $\frac ba$ and $\frac cb$ are …
  43. Difference between the corresponding roots of $x^{2}+ax+b=0$ and $x^{2}+bx+a=0$ is same and $a\neq b$, then
  44. Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha)^k+(\beta)^k$, $k\ge1$, then which one of the following statements is not true?
  45. The number of integral values of $k$, for which one root of the equation $2x^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is …
  46. If the two roots of the equation $(a-1)\left(x^{4}+x^{2}+1\right)+(a+1)\left(x^{2}+x+1\right)^{2}=0$ are real and distinct, then the set of all values of $a$ is equal to
  47. If $5,\ 5r,\ 5r^{2}$ are the lengths of the sides of a triangle, then $r$ cannot be equal to :
  48. If the equations $x^{2}+bx-1=0$ and $x^{2}+x+b=0$ have a common root different from $-1$, then $|b|$ is equal to :
  49. If $\lambda$ be the ratio of the roots of the quadratic equation in $x$, $3m^{2}x^{2}+m(m-4)x+2=0$, then the least value of $m$ for which $\lambda+\frac1\lambda=1$, is :
  50. 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

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