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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 201-250 of 292

  1. If $a,b,c$ are in G.P., then the equations $ax^{2}+2bx+c=0$ and $dx^{2}+2ex+f=0$ have a common root if $\dfrac da,\dfrac eb,\dfrac fc$ are in
  2. If the quadratic equations $x^{2}+ax+b=0$ and $x^{2}+bx+a=0$ $(a\neq b)$ have a common root, then the numerical value of $a+b$ is ______.
  3. Find the set of all $x$ for which $\dfrac{2x}{2x^{2}+5x+2}>\dfrac{1}{x+1}$.
  4. State TRUE or FALSE: If $P(x)=ax^{2}+bx+c$ and $Q(x)=-ax^{2}+dx+c$, where $ac\neq0$, then $P(x)Q(x)=0$ has at least two real roots.
  5. The equation $x-\dfrac{2}{x-1}=1-\dfrac{2}{x-1}$ has
  6. If $a^{2}+b^{2}+c^{2}=1$, then $ab+bc+ca$ lies in the interval
  7. Let $a,b,c$ be real numbers, $a\neq0$. If $\alpha$ is a root of $a^{2}x^{2}+bx+c=0$, $\beta$ is the root of $a^{2}x^{2}-bx-c=0$ and $0<\alpha<\beta$, then the equation …
  8. For $a\le0$, determine all real roots of the equation $x^{2}-2a|x-a|-3a^{2}=0$.
  9. The equation $x^{3/4(\log_2x)^2+\log_2x-5/4}=\sqrt2$ has
  10. State TRUE or FALSE: If $a<b<c<d$, then the roots of the equation $(x-a)(x-c)+2(x-b)(x-d)=0$ are real and distinct.
  11. The real numbers $x_1,x_2,x_3$ satisfying the equation $x^{3}-x^{2}+\beta x+\gamma=0$ are in A.P. Find the intervals in which $\beta$ and $\gamma$ lie.
  12. Let $p,q\in\{1,2,3,4\}$. The number of equations of the form $px^{2}+qx+1=0$ having real roots is
  13. The sum of all the real roots of the equation $|x-2|^{2}+|x-2|-2=0$ is ______.
  14. If $p,q,r$ are positive and are in A.P., the roots of the quadratic equation $px^{2}+qx+r=0$ are all real for
  15. Let $\alpha,\beta$ be the roots of the equation $(x-a)(x-b)=c,\ c\neq0$. Then the roots of the equation $(x-\alpha)(x-\beta)+c=0$ are
  16. The equation $(\cos p-1)x^{2}+(\cos p)x+\sin p=0$ in the variable $x$, has real roots. Then $p$ can take any value in the interval
  17. If $x<0$, $y<0$, $x+y+\dfrac{x}{y}=\dfrac12$ and $(x+y)\dfrac{x}{y}=-\dfrac12$, then $x=$ ______ and $y=$ ______.
  18. Let $p$ and $q$ be roots of the equation $x^{2}-2x+A=0$ and let $r$ and $s$ be the roots of the equation $x^{2}-18x+B=0$. If $p<q<r<s$ are in arithmetic progression …
  19. If $a,b,c,d$ are positive real numbers such that $a+b+c+d=2$, then $M=(a+b)(c+d)$ satisfies the relation
  20. Let $\alpha,\beta$ be the roots of $x^{2}-x+p=0$ and $\gamma,\delta$ be the roots of $x^{2}-4x+q=0$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then the integral …
  21. If $\alpha$ and $\beta$ $(\alpha<\beta)$ are the roots of the equation $x^{2}+bx+c=0$, where $c<0<b$, then
  22. The harmonic mean of the roots of the equation $\left(5+\sqrt2\right)x^{2}-\left(4+\sqrt5\right)x+8+2\sqrt5=0$ is
  23. The set of all real numbers $x$ for which $x^{2}-|x+2|+x>0$ is
  24. Let $a,b,c$ be real numbers with $a\neq0$ and let $\alpha,\beta$ be the roots of the equation $ax^{2}+bx+c=0$. Express the roots of $a^{3}x^{2}+abcx+c^{3}=0$ in terms of …
  25. If $b>a$, then the equation $(x-a)(x-b)-1=0$ has
  26. For all $x$, $x^{2}+2ax+10-3a>0$, then the interval in which $a$ lies is
  27. For the equation $3x^{2}+px+3=0$, $p>0$, if one of the roots is the square of the other, then $p$ is equal to
  28. If $x^2+(a-b)x+(1-a-b)=0$ where $a,b\in R$ then find the values of $a$ for which the equation has unequal real roots for all values of $b$.
  29. If the roots of the equation $x^{2}-2ax+a^{2}+a-3=0$ are real and less than $3$, then
  30. Let $f(x)=(1+b^{2})x^{2}+2bx+1$ and let $m(b)$ be the minimum value of $f(x)$. As $b$ varies, the range of $m(b)$ is
  31. If one root is square of the other root of the equation $x^{2}+px+q=0$, then the relation between $p$ and $q$ is
  32. Let $a,b,c$ be the sides of a triangle. No two of them are equal and $\lambda\in R$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real, then
  33. If roots of the equation $x^2-10cx-11d=0$ are $a,b$ and those of $x^2-10ax-11b=0$ are $c,d$, then the value of $a+b+c+d$ is ($a,b,c$ and $d$ are distinct numbers)
  34. In the quadratic equation $ax^{2}+bx+c=0$, $\Delta=b^{2}-4ac$ and $\alpha+\beta$, $\alpha^{2}+\beta^{2}$, $\alpha^{3}+\beta^{3}$ are in G.P., where $\alpha,\beta$ are …
  35. The sum of all the real solutions of the equation $\log_{(x+3)}\left(6x^2+28x+30\right)=5-2\log_{(6x+10)}\left(x^2+6x+9\right)$ is equal to :
  36. Let $\alpha,\beta$ be the roots of the quadratic equation $12x^2-20x+3\lambda=0$, $\lambda\in\mathbf{Z}$. If $\frac12\le|\beta-\alpha|\le\frac32$, then the sum of all …
  37. If the set of all solutions of $|x^{2}+x-9|=|x|+|x^{2}-9|$ is $[\alpha,\beta]\cup[\gamma,\infty)$, then $(\alpha^{2}+\beta^{2}+\gamma^{2})$ is equal to :
  38. Let one root of the quadratic equation in $x$: $(k^{2}-15k+27)x^{2}+9(k-1)x+18=0$ be twice the other. Then the length of the latus rectum of the parabola $y^{2}=6kx$ is …
  39. Let $a,b\in\mathbb{C}$. Let $\alpha,\beta$ be the roots of the equation $x^{2}+ax+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^{2}-\alpha^{2}=3i\sqrt{11}$, then …
  40. If $\alpha$ and $\beta$ ($\alpha<\beta$) are the roots of the equation $(-2+\sqrt3)\left(\left|\sqrt x-3\right|\right)+\left(x-6\sqrt x\right)+(9-2\sqrt3)=0$, $x\ge0$ …
  41. 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 …
  42. The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is :
  43. Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2ax+(3a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :
  44. The sum of all the roots of the equation $(x-1)^2-5|x-1|+6=0$, is :
  45. The number of the real solutions of the equation $x|x-3|+|x-1|-2=0$ is :
  46. The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to :
  47. Let $S=\{x^3+ax^2+bx+c:a,b,c\in\mathbf{N}\text{ and }a,b,c\le20\}$ be a set of polynomials. Then the number of polynomials in $S$ which are divisible by $x^2+2$ is :
  48. If $\alpha,\beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^2-(\lambda+3)x+3=0$ such that $\frac1\alpha-\frac1\beta=\frac13$, then the sum of all …
  49. Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ …
  50. Let $f(x)$ be a quadratic expression which is positive for all the real values of $x$. If $g(x)=f(x)+f''(x)$, then for any real $x$,

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