JEE Maths DPP – Symmetric Functions of Roots
SEO Keywords: Symmetric Functions, Newton’s Theorem, Sum of Powers, Quadratic Roots, JEE Advanced Maths, Vieta’s Relations
DPP Reference Key: QE-SYM-22-010
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If $\alpha, \beta$ 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 $\alpha, \beta$ are the roots of $ax^2 + bx + c = 0$, the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$ is:
Answer:
- (a) $\frac{b^2 – 2ac}{c^2}$
- (b) $\frac{b^2 + 2ac}{c^2}$
- (c) $\frac{b^2 – 2ac}{a^2}$
- (d) $\frac{b^2 – 4ac}{c^2}$
Question 3: If $\alpha, \beta$ are the roots of $x^2 – 5x + 3 = 0$, then the value of $\alpha^3 + \beta^3$ is:
Answer:
- (a) 125
- (b) 80
- (c) 170
- (d) 45
Question 4: If $\alpha, \beta$ are the roots of $x^2 – px + q = 0$, then $(\alpha – \beta)^2$ is:
Answer:
- (a) $p^2 – 4q$
- (b) $p^2 + 4q$
- (c) $q^2 – 4p$
- (d) $p^2 – 2q$
Question 5: Let $S_n = \alpha^n + \beta^n$. If $\alpha, \beta$ are roots of $x^2 – 6x – 2 = 0$, then the value of $\frac{S_{10} – 2S_8}{2S_9}$ is:
Answer:
- (a) 3
- (b) 1
- (c) 6
- (d) 2
Question 6: If $\alpha, \beta$ are roots of $x^2 + x + 1 = 0$, then $\alpha^{2026} + \beta^{2026}$ is:
Answer:
- (a) -1
- (b) 2
- (c) 1
- (d) 0
Question 7: If $\alpha, \beta$ are roots of $x^2 – x – 1 = 0$, then the value of $\alpha^4 + \beta^4$ is:
Answer:
- (a) 7
- (b) 5
- (c) 9
- (d) 11
Question 8: If $\alpha, \beta$ are roots of $x^2 – 4x + 1 = 0$, then the value of $\alpha^2\beta + \beta^2\alpha$ is:
Answer:
- (a) 4
- (b) 1
- (c) -4
- (d) 16
Question 9: The value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ for the equation $3x^2 – 5x + 1 = 0$ is:
Answer:
- (a) 19/3
- (b) 25/3
- (c) 22/3
- (d) 7/3
Question 10: If $\alpha, \beta$ are roots of $x^2 + px + 1 = 0$ and $\gamma, \delta$ are roots of $x^2 + qx + 1 = 0$, then $(\alpha – \gamma)(\beta – \gamma)(\alpha + \delta)(\beta + \delta)$ is:
Answer:
- (a) $q^2 – p^2$
- (b) $p^2 – q^2$
- (c) $pq$
- (d) 0
Question 11: If $\alpha, \beta$ are roots of $x^2 – 7x + 1 = 0$, the value of $\sqrt{\alpha} + \sqrt{\beta}$ is:
Answer:
- (a) 3
- (b) $\sqrt{7}$
- (c) $\sqrt{5}$
- (d) 9
Question 12: Let $\alpha, \beta$ be roots of $x^2 + bx + c = 0$. The equation whose roots are $\alpha^2, \beta^2$ is:
Answer:
- (a) $x^2 – (b^2 – 2c)x + c^2 = 0$
- (b) $x^2 + (b^2 – 2c)x + c^2 = 0$
- (c) $x^2 – (b^2 + 2c)x + c^2 = 0$
- (d) $x^2 – (b^2 – 4c)x + c^2 = 0$
Question 13: If $\alpha, \beta$ are roots of $x^2 – x + 1 = 0$, then $\alpha^n + \beta^n = 2 \cos(n\pi/3)$. For $n=6$, the value is:
Answer:
- (a) 2
- (b) -2
- (c) 1
- (d) 0
Part II: Subjective Questions (Q14–Q15)
Question 14: Let $\alpha, \beta$ be the roots of $x^2 – x – 1 = 0$. Let $a_n = \alpha^n + \beta^n$. Prove that $a_n = a_{n-1} + a_{n-2}$ for $n \geq 2$. Also, find $a_5$.
Answer:
[Enter solution here]
Question 15: If $\alpha, \beta$ are roots of $x^2 – 6x + a = 0$, find the value of $a$ such that $3\alpha + 2\beta = 20$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: If $\alpha, \beta$ are roots of $x^2 – px + q = 0$, and $\alpha^2 + \beta^2 = 10, \alpha^3 + \beta^3 = 26$, find the value of $p$.
Answer:
Question 17: Let $f(n) = \alpha^n + \beta^n$. If $\alpha, \beta$ are roots of $x^2 – 5x + 3 = 0$, find the value of $f(4) – 5f(3) + 3f(2)$.
Answer:
Question 18: If $\alpha, \beta$ are roots of $x^2 – 2x + 4 = 0$, find the value of $\frac{\alpha^3 + \beta^3}{4}$.
Answer:
Question 19: Find the value of $k$ if the sum of squares of the roots of $x^2 – (k-2)x – (k+1) = 0$ is minimum.
Answer:
Question 20: If $\alpha, \beta$ are roots of $x^2 – 6x + 2 = 0$, find the value of $\alpha^4 + \beta^4$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): If $\alpha, \beta$ are roots of $x^2 – 3x + 1 = 0$, then $\alpha^n + \beta^n$ is always an integer for $n \in \mathbb{N}$.
Reason (R): Newton’s Sum states $S_n = 3S_{n-1} – S_{n-2}$, and since $S_1, S_2$ are integers, all subsequent terms are integers.
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): $\alpha^2 + \beta^2$ is a symmetric function of roots $\alpha, \beta$.
Reason (R): An expression is symmetric if it remains unchanged when $\alpha$ and $\beta$ are interchanged.
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 | A |
| Q3 | B | Q4 | A |
| Q5 | A | Q6 | A |
| Q7 | A | Q8 | A |
| Q9 | A | Q10 | B |
| Q11 | A | Q12 | A |
| Q13 | A | Q14 | 11 |
| Q15 | -16 | Q16 | 4 |
| Q17 | 0 | Q18 | -4 |
| Q19 | 1 | Q20 | 1084 |
| Q21 | A | Q22 | A |