JEE Maths DPP – Introduction to Equations
Chapter: Quadratic Equations | Key: QE-INTRO-22-001
SEO Keywords: Quadratic Equations, Polynomial Degree, Roots of Equations, JEE Advanced, JEE Mains, Fundamental Theorem of Algebra
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: The number of real roots of the equation $x^2 + 5|x| + 6 = 0$ is:
Answer:
- (a) 0
- (b) 2
- (c) 4
- (d) 1
Question 2: If the equation $(k-2)x^2 + (k^2-4)x + (k^2-5k+6) = 0$ is an identity in $x$, then the value of $k$ is:
Answer:
- (a) 2
- (b) 3
- (c) 2 and 3
- (d) None of these
Question 3: A polynomial $P(x)$ of degree 3 leaves a remainder 4 when divided by $(x-1)$, $(x-2)$, and $(x-3)$. If $P(0) = -2$, then $P(4)$ is:
Answer:
- (a) 10
- (b) 12
- (c) 14
- (d) 16
Question 4: The degree of the equation $\sqrt{x+1} – \sqrt{x-1} = 1$, after rationalization to polynomial form, is:
Answer:
- (a) 1
- (b) 2
- (c) 1/2
- (d) 4
Question 5: If $\alpha$ is a root of $x^7 – 1 = 0$ and $\alpha \neq 1$, then the value of $\sum_{k=0}^{6} \alpha^k$ is:
Answer:
- (a) 7
- (b) 1
- (c) 0
- (d) -1
Question 6: The equation $x^4 + 2x^2 + 1 = 0$ has:
Answer:
- (a) Four real roots
- (b) Two real roots
- (c) No real roots
- (d) One real root
Question 7: If the equation $ax^2 + bx + c = 0$ has more than two roots, then:
Answer:
- (a) $a=b=c=0$
- (b) $a=b, c=0$
- (c) $b^2 – 4ac > 0$
- (d) $a+b+c=1$
Question 8: The number of solutions of the equation $2^{x^2-1} + 2^{1-x^2} = 2$ is:
Answer:
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Question 9: If $P(x)$ is a polynomial of degree $n$, then the equation $P(x) = P'(x)$ has:
Answer:
- (a) At most $n$ real roots
- (b) Exactly $n$ real roots
- (c) No real roots
- (d) Exactly one real root
Question 10: The number of ways the expression $x^2 + 2x + 2$ can be zero for real $x$ is:
Answer:
- (a) 1
- (b) 2
- (c) Infinitely many
- (d) None
Question 11: If $f(x) = 0$ is a polynomial equation of degree 5 with real coefficients, then it must have:
Answer:
- (a) At least one real root
- (b) At least one imaginary root
- (c) All real roots
- (d) No real roots
Question 12: The equation $|x-1|^2 – 3|x-1| + 2 = 0$ has:
Answer:
- (a) 2 real roots
- (b) 4 real roots
- (c) 1 real root
- (d) No real roots
Question 13: Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$. If the roots are $i, -i, 2, 3$, then $d$ is:
Answer:
- (a) 6
- (b) -6
- (c) 5
- (d) -5
Part II: Subjective (Q14–Q15)
Question 14: Find all real values of $x$ that satisfy the equation $x^4 – 4x^3 – 1 = 0$ given that it has a root of the form $2 + \sqrt{a + \sqrt{b}}$.
Answer:
[Enter Solution Here]
Question 15: A polynomial $f(x)$ satisfies $f(x) \cdot f(1/x) = f(x) + f(1/x)$ for all $x \neq 0$. If $f(3) = 28$, find the value of $f(4)$.
Answer:
[Enter Solution Here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of real roots of $x^4 – 4x – 1 = 0$.
Answer:
Question 17: If the equation $x^2 – px + q = 0$ has roots $r_1$ and $r_2$, find the value of $p$ if $r_1, r_2$ are consecutive integers and $q=6$.
Answer:
Question 18: Find the number of points of intersection of the curves $y = x^2$ and $y = \cos x$.
Answer:
Question 19: If $(x-1)^2$ is a factor of $ax^3 + bx^2 + 1$, find the value of $a + b$.
Answer:
Question 20: If a polynomial $P(x)$ of degree 6 has $P(k) = \frac{1}{k}$ for $k=1, 2, \dots, 7$, find the value of $7! \cdot P(8)$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): The equation $ax^2 + bx + c = 0$ cannot have more than two roots if $a, b, c$ are not all zero.
Reason (R): A polynomial of degree $n$ has exactly $n$ roots in the complex number system.
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): Every polynomial equation of even degree has at least one real root.
Reason (R): Complex roots of a polynomial 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 false but R is true.
- (d) A is true but R is false.
Answer Key
| Question | Answer | Question | Answer |
|---|---|---|---|
| Q1 | A | Q2 | A |
| Q3 | A | Q4 | B |
| Q5 | C | Q6 | C |
| Q7 | A | Q8 | B |
| Q9 | A | Q10 | D |
| Q11 | A | Q12 | B |
| Q13 | A | Q14 | See Sol |
| Q15 | 65 | Q16 | 2 |
| Q17 | 5 | Q18 | 2 |
| Q19 | -1 | Q20 | 0 |
| Q21 | A | Q22 | C |