JEE Maths DPP – Relation Between Roots and Coefficients
SEO Keywords: Sum of roots, Product of roots, Symmetric expressions, Equation transformation, Reciprocal roots, JEE Advanced Maths
DPP Reference Key: QE-REL-22-003
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If $\alpha$ and $\beta$ are the roots of $x^2 – px + q = 0$, then the value of $\alpha^2 + \beta^2$ is:
Answer:
- (a) $p^2 – 2q$
- (b) $p^2 + 2q$
- (c) $q^2 – 2p$
- (d) $p^2 – q$
Question 2: If the roots of $x^2 – bx + c = 0$ are two consecutive integers, then $b^2 – 4c$ is equal to:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) 3
Question 3: If $\alpha, \beta$ are roots of $ax^2 + bx + c = 0$, the equation whose roots are $1/\alpha$ and $1/\beta$ is:
Answer:
- (a) $ax^2 – bx + c = 0$
- (b) $cx^2 + bx + a = 0$
- (c) $cx^2 – bx + a = 0$
- (d) $x^2 + bx + ac = 0$
Question 4: If the ratio of the roots of $x^2 + px + q = 0$ is equal to the ratio of the roots of $x^2 + lx + m = 0$, then:
Answer:
- (a) $p^2m = l^2q$
- (b) $pm^2 = lq^2$
- (c) $p^2l^2 = qm$
- (d) $p^2q = l^2m$
Question 5: If $\alpha, \beta$ are roots of $x^2 – 2x + 3 = 0$, then the equation whose roots are $\alpha^3$ and $\beta^3$ is:
Answer:
- (a) $x^2 + 10x + 27 = 0$
- (b) $x^2 – 10x + 27 = 0$
- (c) $x^2 + 10x – 27 = 0$
- (d) $x^2 – 10x – 27 = 0$
Question 6: Let $\alpha, \beta$ be roots of $x^2 + x + 1 = 0$. The value of $\alpha^{2026} + \beta^{2026}$ is:
Answer:
- (a) 1
- (b) -1
- (c) 2
- (d) -2
Question 7: If $\alpha, \beta$ are roots of $x^2 – px + q = 0$, then $(\alpha+\beta)x^2 – (\alpha^2+\beta^2)x + \alpha^2\beta + \alpha\beta^2$ is equal to:
Answer:
- (a) $px^2 – (p^2-2q)x + pq$
- (b) $px^2 – (p^2+2q)x + pq$
- (c) $qx^2 – (p^2-2q)x + pq$
- (d) $px^2 – (p^2-2q)x + p+q$
Question 8: If one root of $ax^2 + bx + c = 0$ is square of the other, then $ca(a+c+3b)$ is:
Answer:
- (a) $b^3$
- (b) $-b^3$
- (c) $abc$
- (d) 0
Question 9: If the sum of the roots of $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then $bc^2, ca^2, ab^2$ are in:
Answer:
- (a) A.P.
- (b) G.P.
- (c) H.P.
- (d) None of these
Question 10: If $\alpha, \beta$ are the roots of $4x^2 + 3x + 7 = 0$, then the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is:
Answer:
- (a) $3/7$
- (b) $-3/7$
- (c) $3/4$
- (d) $-7/3$
Question 11: The condition that the roots of $ax^2 + bx + c = 0$ may be in the ratio $m:n$ is:
Answer:
- (a) $mna^2 = (m+n)^2 c$
- (b) $mnb^2 = (m+n)^2 ac$
- (c) $mnc^2 = (m+n)^2 b$
- (d) $(m+n)b^2 = mnac$
Question 12: If $\alpha, \beta$ are roots of $x^2 – px + q = 0$, the value of $\alpha^4 + \alpha^2\beta^2 + \beta^4$ is:
Answer:
- (a) $(p^2-q)(p^2-3q)$
- (b) $(p^2-q)(p^2+3q)$
- (c) $p^4 – 4p^2q + q^2$
- (d) $p^4 – 4p^2q + 3q^2$
Question 13: If $\alpha, \beta$ are roots of $x^2 – x + 1 = 0$, then $\alpha^2 + \beta^2$ and $\alpha^3 + \beta^3$ are roots of:
Answer:
- (a) $x^2 + 3x + 2 = 0$
- (b) $x^2 + 3x – 2 = 0$
- (c) $x^2 – 3x + 2 = 0$
- (d) $x^2 + 2x + 3 = 0$
Part II: Subjective Questions (Q14–Q15)
Question 14: Let $\alpha, \beta$ be the roots of the equation $x^2 – 6x – 2 = 0$. If $a_n = \alpha^n – \beta^n$ for $n \geq 1$, find the numerical value of $\frac{a_{10} – 2a_8}{2a_9}$.
Answer:
[Enter solution here]
Question 15: Find the quadratic equation whose roots are the arithmetic mean and the geometric mean of the roots of the equation $x^2 – 18x + 64 = 0$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: If $\alpha, \beta$ are roots of $x^2 – 4x + 1 = 0$, find the value of $\alpha^3 + \beta^3$.
Answer:
Question 17: If the product of the roots of $(k+1)x^2 + 2kx + 3 = 0$ is 1, find $k$.
Answer:
Question 18: If the sum of roots of $ax^2 + bx + c = 0$ is 5 and sum of their squares is 13, find the value of $c/a$.
Answer:
Question 19: If $\alpha, \beta$ are roots of $x^2 – 5x + k = 0$ such that $\alpha – \beta = 1$, find $k$.
Answer:
Question 20: Find the number of real roots of the equation formed by $x = \sqrt{6+\sqrt{6+\sqrt{6+\dots}}}$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): If $\alpha, \beta$ are roots of $x^2 + x + 1 = 0$, then $\alpha^2 = \beta$ and $\beta^2 = \alpha$.
Reason (R): For $x^2 + x + 1 = 0$, the roots are $\omega$ and $\omega^2$, where $\omega^3 = 1$.
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 equation whose roots are reciprocals of the roots of $ax^2 + bx + c = 0$ is $cx^2 + bx + a = 0$.
Reason (R): If $f(x) = 0$ has roots $\alpha, \beta$, then $f(1/x) = 0$ has roots $1/\alpha, 1/\beta$.
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 | B |
| Q3 | B | Q4 | A |
| Q5 | A | Q6 | B |
| Q7 | A | Q8 | B |
| Q9 | A | Q10 | B |
| Q11 | B | Q12 | A |
| Q13 | A | Q14 | 3 |
| Q15 | $x^2-17x+72=0$ | Q16 | 52 |
| Q17 | 2 | Q18 | 6 |
| Q19 | 6 | Q20 | 1 |
| Q21 | A | Q22 | A |