JEE Maths DPP – Location of Roots
SEO Keywords: Location of roots, Quadratic inequalities, Interval of roots, Vertex of parabola, JEE Advanced Maths, Parameter range
DPP Reference Key: QE-LOC-22-006
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: The values of $a$ for which both roots of the equation $x^2 – ax + 1 = 0$ are less than 2 is:
Answer:
- (a) $(- \infty, 2]$
- (b) $[2, 5/2)$
- (c) $(-\infty, -2] \cup [2, 5/2)$
- (d) $(-2, 2)$
Question 2: If both roots of the equation $x^2 – 6ax + 2 – 2a + 9a^2 = 0$ are greater than 3, then $a$ lies in the interval:
Answer:
- (a) $(11/9, \infty)$
- (b) $(-\infty, 1)$
- (c) $(1, \infty)$
- (d) $(9/11, \infty)$
Question 3: If the roots of $x^2 – 2kx + k^2 + k – 5 = 0$ are less than 5, then $k$ lies in the interval:
Answer:
- (a) $(-\infty, 4)$
- (b) $(-\infty, 5]$
- (c) $(4, 5]$
- (d) $(-\infty, 2)$
Question 4: The value of $k$ for which one root of the equation $x^2 – (k+1)x + k^2 + k – 8 = 0$ is greater than 2 and the other is less than 2 is:
Answer:
- (a) $k \in (-2, 3)$
- (b) $k \in (-3, 2)$
- (c) $k \in (2, \infty)$
- (d) $k \in (-\infty, -2)$
Question 5: If $\alpha, \beta$ are the roots of $x^2 – 2ax + a^2 + a – 3 = 0$, then the values of $a$ for which $\alpha < 3 < \beta$ are:
Answer:
- (a) $a < 2$
- (b) $a > 3$
- (c) $2 < a < 3$
- (d) $a \in \emptyset$
Question 6: For the roots of $x^2 – (m-3)x + m = 0$ to be such that one root is smaller than 2 and the other root is greater than 2, $m$ must satisfy:
Answer:
- (a) $m > 10$
- (b) $m < 10$
- (c) $m = 10$
- (d) $m < 0$
Question 7: If both roots of the equation $x^2 + x + a = 0$ exceed $a$, then:
Answer:
- (a) $a > 0$
- (b) $a < -2$
- (c) $a = -2$
- (d) $a < -1$
Question 8: The set of values of $a$ for which both roots of $x^2 – 2ax + a^2 – 1 = 0$ lie in the interval $(-2, 4)$ is:
Answer:
- (a) $(-1, 3)$
- (b) $(1, 3)$
- (c) $(-3, 1)$
- (d) $(-2, 4)$
Question 9: If 2 and 3 lie between the roots of the equation $2x^2 – 2(m+n)x + mn = 0$, then:
Answer:
- (a) $mn < 0$
- (b) $m > 6, n > 6$
- (c) $m < 2, n > 3$
- (d) Information insufficient
Question 10: If the equation $x^2 – (k-1)x + (k+1) = 0$ has two roots between -1 and 3, then:
Answer:
- (a) $k \in [-\infty, 1)$
- (b) $k \in (-1, 15/4]$
- (c) $k \in (1, \infty)$
- (d) $k \in \emptyset$
Question 11: If $a < 0$, then both roots of the equation $x^2 - 2ax + a^2 - 1 = 0$ are:
Answer:
- (a) Greater than $a-1$
- (b) Less than $a+1$
- (c) Both (a) and (b)
- (d) None of these
Question 12: The values of $p$ for which the equation $x^2 – 2px + p^2 – 1 = 0$ has roots in the interval $(-1, 1)$ is:
Answer:
- (a) $p \in (0, 1)$
- (b) $p = 0$
- (c) $p \in (-1, 1)$
- (d) No such $p$ exists
Question 13: If the roots of the equation $x^2 – 2mx + m^2 – 1 = 0$ lie in the interval $(-2, 4)$, then $m$ belongs to:
Answer:
- (a) $(-1, 3)$
- (b) $(-3, 5)$
- (c) $(1, 3)$
- (d) $(-1, 1)$
Part II: Subjective (Q14–Q15)
Question 14: Find all the values of the parameter $a$ for which the quadratic equation $(a+1)x^2 – 3ax + 4a = 0$ has at least one root greater than 1.
Answer:
[Enter solution here]
Question 15: Determine the range of $k$ for which both roots of the equation $x^2 – 2kx + k^2 – 1 = 0$ lie between -2 and 4.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of integral values of $m$ for which exactly one root of $x^2 – (m+1)x + m^2 + m – 8 = 0$ lies in $(1, 2)$.
Answer:
Question 17: If both roots of $x^2 – kx + 4 = 0$ are real and distinct and lie in $(1, 5)$, find the number of possible integral values of $k$.
Answer:
Question 18: If the roots of $x^2 – 4x + a = 0$ are real and at least one root is greater than 3, find the maximum integer value of $a$.
Answer:
Question 19: If roots of $x^2 – 2mx + m^2 – 1 = 0$ are $\alpha, \beta$ and $-2 < \alpha < \beta < 4$, find the number of integers in the range of $m$.
Answer:
Question 20: If $f(x) = x^2 + 2(k-1)x + k+5 = 0$ has roots of opposite signs, find the largest negative integer value of $k$.
Answer:
Part IV: Assertion Reason (Q21–Q22)
Question 21:
Assertion: If $f(k_1) \cdot f(k_2) < 0$ for a quadratic $f(x)$, then exactly one root lies in $(k_1, k_2)$.
Reason: For any continuous function, if the signs at the endpoints of an interval are different, there is at least one root in that interval.
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: The roots of $x^2 – 2ax + a^2 = 0$ are always equal and equal to $a$.
Reason: If the discriminant of a quadratic equation is zero, the roots are $x = -b/2a$.
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 | C | Q2 | A |
| Q3 | A | Q4 | B |
| Q5 | D | Q6 | B |
| Q7 | B | Q8 | A |
| Q9 | D | Q10 | D |
| Q11 | C | Q12 | B |
| Q13 | A | Q14 | See Sol |
| Q15 | See Sol | Q16 | 0 |
| Q17 | 0 | Q18 | 3 |
| Q19 | 3 | Q20 | -6 |
| Q21 | A | Q22 | A |