JEE Maths DPP – Nature of Roots
SEO Keywords: Discriminant, Real and Distinct roots, Rational roots, Imaginary roots, JEE Advanced Quadratic Equations, Parameter based questions
DPP Reference Key: QE-NATURE-22-004
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If the roots of the equation $(b-c)x^2 + (c-a)x + (a-b) = 0$ are equal, then $a, b, c$ are in:
Answer:
- (a) A.P.
- (b) G.P.
- (c) H.P.
- (d) None of these
Question 2: For the equation $3x^2 + px + 3 = 0, p > 0$, if one root is the square of the other, then $p$ is equal to:
Answer:
- (a) 1/3
- (b) 1
- (c) 3
- (d) 6
Question 3: If $a, b, c$ are rational and $a+b+c=0$, then the roots of the equation $(b+c-a)x^2 + (c+a-b)x + (a+b-c) = 0$ are:
Answer:
- (a) Real and distinct
- (b) Rational
- (c) Imaginary
- (d) Equal
Question 4: The number of real roots of the equation $e^{x^2} + e^{-x^2} = 2 \cos x$ is:
Answer:
- (a) 1
- (b) 2
- (c) 0
- (d) Infinitely many
Question 5: If the roots of $x^2 – 2cx + ab = 0$ are real and unequal, then the roots of $x^2 – 2(a+b)x + a^2 + b^2 + 2c^2 = 0$ are:
Answer:
- (a) Real
- (b) Rational
- (c) Imaginary
- (d) Equal
Question 6: If the roots of the equation $x^2 + ax + b = 0$ are of opposite signs, then:
Answer:
- (a) $a > 0$
- (b) $b < 0$
- (c) $b > 0$
- (d) $a < 0$
Question 7: The values of $k$ for which the equation $x^2 – 2(1+3k)x + 7(3+2k) = 0$ has real and equal roots are:
Answer:
- (a) $2, -10/9$
- (b) $2, 10/9$
- (c) $0, 2$
- (d) $1, -10/9$
Question 8: If $a, b, c \in \mathbb{R}$ and $ac < 0$, then the roots of $ax^2 + bx + c = 0$ are:
Answer:
- (a) Real and distinct
- (b) Real and equal
- (c) Purely imaginary
- (d) Rational
Question 9: If the equation $x^2 + 2(k+1)x + 9k – 5 = 0$ has only negative roots, then:
Answer:
- (a) $k \geq 6$
- (b) $k \leq 1$
- (c) $k \in [6, \infty) \cup \{1\}$
- (d) $k \geq 5/9$
Question 10: If the roots of $ax^2 + bx + c = 0$ are irrational, and $a, b, c$ are rational, then:
Answer:
- (a) The roots occur in conjugate pairs
- (b) One root is rational, other is irrational
- (c) Both roots are equal
- (d) None of these
Question 11: If $p$ and $q$ are odd integers, then the roots of $x^2 + px + q = 0$ are:
Answer:
- (a) Rational
- (b) Non-real
- (c) Real and equal
- (d) Not rational
Question 12: If $a < b < c < d$, then the roots of $(x-a)(x-c) + 2(x-b)(x-d) = 0$ are:
Answer:
- (a) Real and distinct
- (b) Real and equal
- (c) Imaginary
- (d) Rational
Question 13: The condition that roots of $ax^2 + bx + c = 0$ are of the same sign is:
Answer:
- (a) $ac > 0$
- (b) $ac < 0$
- (c) $ab > 0$
- (d) $bc > 0$
Part II: Subjective (Q14–Q15)
Question 14: Find the set of all real values of $m$ for which the equation $(m-2)x^2 – 2(m+1)x + m = 0$ has roots of opposite signs.
Answer:
[Enter solution here]
Question 15: Let $a, b, c$ be distinct real numbers. Prove that the roots of the equation $(a-b)x^2 + (b-c)x + (c-a) = 0$ are rational and find them.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of integral values of $k$ for which the equation $x^2 – 2(4k-1)x + 15k^2 – 2k – 7 = 0$ has real roots.
Answer:
Question 17: If the discriminant of $x^2 – px + 4 = 0$ is zero, and $p > 0$, find $p$.
Answer:
Question 18: If the roots of $x^2 – kx + 1 = 0$ are imaginary, find the number of integers in the range of $k$.
Answer:
Question 19: If roots of $ax^2 + x + 1 = 0$ are real and $a > 0$, find the maximum integer value of $a$.
Answer:
Question 20: Find the number of real roots of $x^2 + |x| – 6 = 0$.
Answer:
Part IV: Assertion Reason (Q21–Q22)
Question 21:
Assertion (A): If $D < 0$ for $ax^2 + bx + c = 0$, the roots are always of the form $p \pm iq$.
Reason (R): Complex roots of a quadratic equation with real coefficients always occur in conjugate pairs.
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 (A): If $a, b, c$ are odd integers, the equation $ax^2 + bx + c = 0$ cannot have rational roots.
Reason (R): For rational roots, the discriminant $D$ must be a perfect square.
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 | A | Q2 | C |
| Q3 | B | Q4 | A |
| Q5 | C | Q6 | B |
| Q7 | A | Q8 | A |
| Q9 | C | Q10 | A |
| Q11 | D | Q12 | A |
| Q13 | A | Q14 | (0, 2) |
| Q15 | $1, \frac{c-a}{a-b}$ | Q16 | $\infty$ |
| Q17 | 4 | Q18 | 3 |
| Q19 | 0 | Q20 | 2 |
| Q21 | A | Q22 | A |