JEE Maths DPP – Quadratic Equations
SEO Keywords: Quadratic Formula, Discriminant, Equation Formation, Parameter Based Problems, JEE Advanced, JEE Mains
DPP Reference Key: QE-STDF-22-002
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If the discriminant of the equation $ax^2 + bx + c = 0$ is a perfect square of a non-zero integer (where $a, b, c \in \mathbb{Q}$), then the roots are always:
Answer:
- (a) Irrational and distinct
- (b) Rational and distinct
- (c) Rational and equal
- (d) Imaginary
Question 2: If $\alpha, \beta$ are the roots of $x^2 – 5x + 3 = 0$, then the equation whose roots are $\frac{1}{2\alpha-5}$ and $\frac{1}{2\beta-5}$ is:
Answer:
- (a) $3x^2 + 5x + 1 = 0$
- (b) $12x^2 + 10x + 1 = 0$
- (c) $12x^2 – 10x + 1 = 0$
- (d) $3x^2 – 5x + 1 = 0$
Question 3: The value of $k$ for which the equation $(k-2)x^2 + 2(2k-3)x + (5k-6) = 0$ has equal roots is:
Answer:
- (a) $1, 3$
- (b) $2, 3$
- (c) $3$
- (d) $1$
Question 4: If the roots of $ax^2 + bx + c = 0$ are $\sin \theta$ and $\cos \theta$, then:
Answer:
- (a) $a^2 + b^2 = 2ac$
- (b) $b^2 – a^2 = 2ac$
- (c) $a^2 – b^2 = 2ac$
- (d) $a^2 + b^2 = ac$
Question 5: The number of real solutions of the equation $2^{2x} – 3 \cdot 2^{x+2} + 32 = 0$ is:
Answer:
- (a) 1
- (b) 2
- (c) 0
- (d) 3
Question 6: If one root of $x^2 – px + q = 0$ is the square of the other, then $p^3 – q(3p+1)$ is equal to:
Answer:
- (a) $q$
- (b) $q^2$
- (c) $-q^2$
- (d) $0$
Question 7: For what values of $m$ will the equation $x^2 – 2(1+3m)x + 7(3+2m) = 0$ have equal roots?
Answer:
- (a) $2, -10/9$
- (b) $2, 10/9$
- (c) $-2, 10/9$
- (d) $0, 2$
Question 8: If $a, b, c \in \mathbb{R}$ and $a+b+c=0$, then the roots of the equation $3ax^2 + 2bx + c = 0$ are:
Answer:
- (a) Imaginary
- (b) Real
- (c) Equal
- (d) None of these
Question 9: The equation formed by decreasing the roots of $ax^2 + bx + c = 0$ by $h$ is $a(x+h)^2 + b(x+h) + c = 0$. If the roots of $x^2 – 4x + 3 = 0$ are increased by $2$, the new equation is:
Answer:
- (a) $x^2 – 8x + 15 = 0$
- (b) $x^2 + 8x + 15 = 0$
- (c) $x^2 – 4x + 15 = 0$
- (d) $x^2 – 8x – 15 = 0$
Question 10: If roots of $x^2 – bx + c = 0$ are two consecutive integers, then $b^2 – 4c$ is:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Question 11: If $\alpha, \beta$ are roots of $x^2 – 2x + 4 = 0$, then $\alpha^n + \beta^n$ is equal to:
Answer:
- (a) $2^{n+1} \cos(n\pi/3)$
- (b) $2^{n+1} \sin(n\pi/3)$
- (c) $2^n \cos(n\pi/3)$
- (d) $2^n \sin(n\pi/3)$
Question 12: The values of $p$ for which the equation $(p-3)x^2 – 2px + 5p = 0$ has real roots is:
Answer:
- (a) $p \in [0, 15/4]$
- (b) $p \in (3, 15/4]$
- (c) $p \in [0, 3) \cup (3, 15/4]$
- (d) $p \in [0, 15/4] \setminus \{3\}$
Question 13: If $x^2 + px + 1$ is a factor of $ax^3 + bx + c$, then:
Answer:
- (a) $a^2 + c^2 = -ab$
- (b) $a^2 – c^2 = -ab$
- (c) $a^2 – c^2 = ab$
- (d) $a^2 + c^2 = ab$
Part II: Subjective (Q14–Q15)
Question 14: Find all the values of the parameter $a$ for which the roots of the equation $(a-1)x^2 – 2(a+1)x + a = 0$ are real and distinct.
Answer:
[Enter student solution here]
Question 15: If $\alpha, \beta$ are the roots of $ax^2 + bx + c = 0$, find the equation whose roots are $\alpha^2 + \beta^2$ and $\alpha^{-2} + \beta^{-2}$.
Answer:
[Enter student solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of integral values of $m$ for which the equation $(m-5)x^2 + 2(m-2)x + (m-1) = 0$ has real roots.
Answer:
Question 17: If the equation $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ have a common root, find the value of $|a+b|$ (given $a \neq b$).
Answer:
Question 18: If the sum of the roots of the equation $kx^2 + 2x + 3k = 0$ is equal to their product, find the value of $3k+1$.
Answer:
Question 19: Find the value of $k$ such that the ratio of the roots of $x^2 + kx + 12 = 0$ is $3:1$. (Take positive $k$).
Answer:
Question 20: If $D_1$ and $D_2$ are discriminants of $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$, and $D_1 + D_2 = 0$, find the number of equations that MUST have real roots if $a_i, b_i, c_i \in \mathbb{R}$.
Answer:
Part IV: Assertion Reason (Q21–Q22)
Question 21:
Assertion: The equation $x^2 + 2x + 5 = 0$ has no real roots.
Reason: For any quadratic equation $ax^2 + bx + c = 0$, if $b^2 – 4ac < 0$, the roots are imaginary.
Answer:
- (a) Both A and R are true and R is the correct explanation of A.
- (b) Both A and R are true but R is not the correct explanation of A.
- (c) A is true but R is false.
- (d) A is false but R is true.
Question 22:
Assertion: If $a+b+c=0$ and $a, b, c$ are rational, then the roots of $ax^2 + bx + c = 0$ are rational.
Reason: The discriminant of $ax^2 + bx + c = 0$ becomes $(b+2a)^2$ when $a+b+c=0$.
Answer:
- (a) Both A and R are true and R is the correct explanation of A.
- (b) Both A and R are true but R is not the correct explanation of A.
- (c) A is true but R is false.
- (d) A is false but R is true.
Answer Key
| Question | Answer | Question | Answer |
|---|---|---|---|
| Q1 | B | Q2 | C |
| Q3 | C | Q4 | B |
| Q5 | B | Q6 | B |
| Q7 | A | Q8 | B |
| Q9 | A | Q10 | B |
| Q11 | A | Q12 | D |
| Q13 | B | Q14 | — |
| Q15 | — | Q16 | 5 |
| Q17 | 1 | Q18 | -1 |
| Q19 | 8 | Q20 | 1 |
| Q21 | A | Q22 | C |