JEE Maths DPP – Common Roots
SEO Keywords: Common Roots Condition, Quadratic Equations, Elimination Method, JEE Advanced Problems, Parameter Based Equations
DPP Reference Key: QE-COM-22-005
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ have a common root ($a \neq b$), then the value of $a+b$ is:
Answer:
- (a) 1
- (b) -1
- (c) 0
- (d) 2
Question 2: If the equations $x^2 – px + q = 0$ and $x^2 – qx + p = 0$ have a common root, then which of the following is true?
Answer:
- (a) $p+q = 1$
- (b) $p+q = -1$
- (c) $p-q = 1$
- (d) $p^2 + q^2 = 1$
Question 3: If the equation $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0$ ($a, b, c \in \mathbb{R}$) have at least one common root, then $a:b:c$ is:
Answer:
- (a) $1:2:3$
- (b) $3:2:1$
- (c) $1:3:2$
- (d) $2:1:3$
Question 4: The value of $k$ for which the equations $x^2 – kx – 21 = 0$ and $x^2 – 3kx + 35 = 0$ have a common root is:
Answer:
- (a) $\pm 1$
- (b) $\pm 4$
- (c) $\pm 3$
- (d) $\pm 2$
Question 5: If $x^2 – ax + b = 0$ and $x^2 – ex + f = 0$ have a common root and the second equation has equal roots, then:
Answer:
- (a) $ae = b+f$
- (b) $be = a+f$
- (c) $fe = a+b$
- (d) $ae = 2(b+f)$
Question 6: If $a, b, c \in \mathbb{R}$ and the equations $ax^2 + bx + c = 0$ and $x^2 + 2x + 9 = 0$ have a common root, then:
Answer:
- (a) $a+b+c=0$
- (b) $b^2 – 4ac < 0$
- (c) $a/1 = b/2 = c/9$
- (d) Both (b) and (c)
Question 7: The condition that $ax^2 + bx + c = 0$ and $a’x^2 + b’x + c’ = 0$ have both roots common is:
Answer:
- (a) $ab’ = a’b$
- (b) $bc’ = b’c$
- (c) $a/a’ = b/b’ = c/c’$
- (d) $ac’ = a’c$
Question 8: If the equations $x^2 + bx + ca = 0$ and $x^2 + cx + ab = 0$ have a common root, their other roots satisfy the equation:
Answer:
- (a) $x^2 + ax + bc = 0$
- (b) $x^2 + bx + ac = 0$
- (c) $x^2 – ax + bc = 0$
- (d) $x^2 + (b+c)x + a^2 = 0$
Question 9: If $x^2 + 3x + 5 = 0$ and $ax^2 + bx + c = 0$ have a common root, then the value of $(a+c)/b$ is:
Answer:
- (a) 2
- (b) 1
- (c) 3
- (d) 6
Question 10: If $x^2 – 11x + a = 0$ and $x^2 – 14x + 2a = 0$ have a common root, then the non-zero value of $a$ is:
Answer:
- (a) 24
- (b) 32
- (c) 48
- (d) 12
Question 11: If $\alpha$ is a common root of $x^2 + px + q = 0$ and $x^2 + qx + p = 0$, then the value of $\alpha^2 + \alpha + 1$ is:
Answer:
- (a) $p+q+1$
- (b) 0
- (c) 1
- (d) Depends on $p, q$
Question 12: If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a+b+c \neq 0$, then:
Answer:
- (a) $a=b=c$
- (b) $a^2=bc$
- (c) $a+b=c$
- (d) $a^2+b^2+c^2 = ab+bc+ca$
Question 13: If the equation $x^2 + px + q = 0$ and $x^2 + qx + p = 0$ have a common root, the sum of their other roots is:
Answer:
- (a) $-(p+q)$
- (b) $1-(p+q)$
- (c) $p+q$
- (d) $-1$
Part II: Subjective Questions (Q14–Q15)
Question 14: Find the value of $a$ so that the equations $x^2 – x – 12 = 0$ and $ax^2 + 10x + 3 = 0$ have a common root.
Answer:
[Enter solution here]
Question 15: If the equations $x^2 + px + q = 0$ and $x^2 + p’x + q’ = 0$ have a common root, show that it must be equal to $\frac{pq’ – p’q}{q – q’}$ or $\frac{q – q’}{p’ – p}$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of values of $k$ for which the equations $x^2 – x – k = 0$ and $x^2 + kx – 1 = 0$ have a common root.
Answer:
Question 17: If the equations $x^2 + ax + 1 = 0$ and $x^2 – x – a = 0$ have a common root, find the value of $a$. (Integer only).
Answer:
Question 18: If $x^2 – 7x + 10 = 0$ and $x^2 – 10x + k = 0$ have a common root, find the sum of possible values of $k$.
Answer:
Question 19: If $a, b \in \{1, 2, 3, 4\}$, the number of pairs $(a, b)$ for which $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ have a common root is:
Answer:
Question 20: If the quadratic equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ have a common root, find the value of $(a+b)^2$ if the common root is not 1.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): If $x^2 + x + 1 = 0$ and $ax^2 + bx + c = 0$ have a common root, then $a=b=c$.
Reason (R): If one root of a quadratic with real coefficients is imaginary, the other root must be its conjugate.
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): The equations $x^2 – 3x + 2 = 0$ and $2x^2 – 6x + 4 = 0$ have only one common root.
Reason (R): Two quadratic equations have both roots common if their coefficients are proportional.
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 false but R is true.
- (d) A is true but R is false.
Answer Key
| Question | Answer | Question | Answer |
|---|---|---|---|
| Q1 | B | Q2 | B |
| Q3 | A | Q4 | B |
| Q5 | D | Q6 | D |
| Q7 | C | Q8 | A |
| Q9 | A | Q10 | A |
| Q11 | B | Q12 | D |
| Q13 | D | Q14 | See Sol |
| Q15 | — | Q16 | 2 |
| Q17 | -2 | Q18 | 41 |
| Q19 | 4 | Q20 | 1 |
| Q21 | A | Q22 | C |