JEE Maths DPP – Descartes’ Rule of Signs
SEO Keywords: Descartes Rule of Signs, Positive Real Roots, Negative Real Roots, Polynomial Analysis, JEE Advanced Maths, Imaginary Roots Estimation
DPP Reference Key: QE-DRS-22-013
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: According to Descartes’ Rule of Signs, the maximum number of positive real roots of a polynomial $f(x)$ is equal to:
Answer:
- (a) The number of sign changes in $f(-x)$
- (b) The number of sign changes in $f(x)$
- (c) The degree of the polynomial
- (d) The number of terms in the polynomial
Question 2: The maximum number of negative real roots of $f(x) = x^4 + 3x^3 – x – 1$ is:
Answer:
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Question 3: For the polynomial $P(x) = x^5 – x^2 + 1$, the number of real roots is at most:
Answer:
- (a) 1
- (b) 2
- (c) 3
- (d) 5
Question 4: The equation $x^3 + 4x + 7 = 0$ has:
Answer:
- (a) Exactly one positive real root
- (b) At least two negative roots
- (c) Exactly one negative real root and two imaginary roots
- (d) Three real roots
Question 5: The minimum number of imaginary roots of $x^7 – 3x^4 + x^3 – 1 = 0$ is:
Answer:
- (a) 2
- (b) 4
- (c) 6
- (d) 0
Question 6: Let $f(x) = x^4 + x^2 + 1$. The number of sign changes in $f(x)$ and $f(-x)$ respectively are:
Answer:
- (a) 0, 0
- (b) 1, 1
- (c) 2, 2
- (d) 0, 2
Question 7: If a polynomial $f(x)$ has 3 sign changes, the possible number of positive real roots is:
Answer:
- (a) Exactly 3
- (b) 3 or 1
- (c) 3, 2, or 1
- (d) 1 only
Question 8: The equation $x^n – 1 = 0$ where $n$ is even, has:
Answer:
- (a) $n$ real roots
- (b) 2 real roots
- (c) 1 real root
- (d) No real roots
Question 9: The number of positive real roots of $x^{10} – 10x + 9 = 0$ is:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) 10
Question 10: For $f(x) = ax^3 + bx^2 + cx + d$, if all $a, b, c, d > 0$, then $f(x)$ has:
Answer:
- (a) No positive real roots
- (b) At least one positive real root
- (c) Exactly three negative roots
- (d) No negative real roots
Question 11: The number of real roots of $x^n + 1 = 0$ when $n$ is odd is:
Answer:
- (a) 0
- (b) 1
- (c) $n$
- (d) 2
Question 12: The equation $x^4 + 2x^2 + 3x – 1 = 0$ has:
Answer:
- (a) One positive and one negative root
- (b) Two positive roots
- (c) No real roots
- (d) Four real roots
Question 13: If $f(x) = x^5 + 2x^3 – x^2 + x – 5$, the maximum number of negative real roots is:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) 3
Part II: Subjective Questions (Q14–Q15)
Question 14: Discuss the nature of the roots of the equation $x^6 – 3x^2 – x + 1 = 0$ using Descartes’ Rule of Signs. Determine the possible combinations of positive, negative, and imaginary roots.
Answer:
[Enter solution here]
Question 15: Prove that the equation $x^5 + x^3 + x – 2 = 0$ has exactly one real root and that root lies between $x=0$ and $x=1$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the maximum number of real roots of $P(x) = x^6 + 4x^4 – 3x^2 – 1 = 0$.
Answer:
Question 17: If $f(x) = x^9 + x^5 + x + 1$, find the number of positive real roots.
Answer:
Question 18: How many negative real roots can $f(x) = x^4 – 5x^3 – 3x^2 + 2x + 1 = 0$ have at most?
Answer:
Question 19: Find the number of sign changes in $f(-x)$ for the polynomial $f(x) = 2x^5 – 3x^4 + x^3 + x^2 – x – 1$.
Answer:
Question 20: If $f(x) = x^8 + x^4 + 1$, find the total number of real roots.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): The equation $x^3 + x + 1 = 0$ has exactly one real root.
Reason (R): There are zero sign changes in $f(x)$ and one sign change in $f(-x)$.
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): Descartes’ Rule of Signs gives the exact number of real roots.
Reason (R): The rule only provides an upper bound on the number of positive and negative roots.
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 | C |
| Q3 | C | Q4 | C |
| Q5 | B | Q6 | A |
| Q7 | B | Q8 | B |
| Q9 | C | Q10 | A |
| Q11 | B | Q12 | A |
| Q13 | B | Q14 | — |
| Q15 | — | Q16 | 2 |
| Q17 | 0 | Q18 | 2 |
| Q19 | 2 | Q20 | 0 |
| Q21 | A | Q22 | C |