JEE Maths DPP – Graphical Representation
SEO Keywords: Parabola Vertex, Quadratic Graphs, Range of Quadratic, Maximum Minimum Value, Sign of Quadratic Expressions, JEE Advanced Problems
DPP Reference Key: QE-GRAPH-22-007
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If $a < 0$ and $b^2 - 4ac < 0$, then the graph of $f(x) = ax^2 + bx + c$ always:
Answer:
- (a) Lies above the x-axis
- (b) Lies below the x-axis
- (c) Touches the x-axis
- (d) Crosses the x-axis at two points
Question 2: The coordinates of the vertex of the parabola $y = 2x^2 – 4x + 5$ are:
Answer:
- (a) $(1, 3)$
- (b) $(2, 5)$
- (c) $(1, 5)$
- (d) $(-1, 11)$
Question 3: The maximum value of the expression $-3x^2 + 6x + 7$ is:
Answer:
- (a) 7
- (b) 10
- (c) 13
- (d) 4
Question 4: For the quadratic expression $f(x) = ax^2 + bx + c$, if $a > 0$ and $f(1) = f(5)$, then the x-coordinate of the vertex is:
Answer:
- (a) 2
- (b) 3
- (c) 4
- (d) 6
Question 5: If the graph of $y = x^2 + (2k+1)x + k^2$ touches the x-axis, then the value of $k$ is:
Answer:
- (a) $-1/4$
- (b) $1/4$
- (c) $-1/2$
- (d) $1/2$
Question 6: The range of the function $f(x) = \frac{x^2 – x + 1}{x^2 + x + 1}$ for $x \in \mathbb{R}$ is:
Answer:
- (a) $[1/3, 3]$
- (b) $(1/3, 3)$
- (c) $[0, \infty)$
- (d) $(-\infty, \infty)$
Question 7: If $f(x) = x^2 + 2bx + c$ and the minimum value of $f(x)$ is greater than the maximum value of $g(x) = -x^2 + 2ax + a$, then:
Answer:
- (a) $c – b^2 > a^2 + a$
- (b) $c – b^2 < a^2 + a$
- (c) $c + a^2 > b^2 + a$
- (d) $b^2 – c > a^2 + a$
Question 8: The set of values of $k$ for which $x^2 + kx + 1 > 0$ for all $x \in \mathbb{R}$ is:
Answer:
- (a) $(-2, 2)$
- (b) $(-\infty, -2) \cup (2, \infty)$
- (c) $[-2, 2]$
- (d) $(0, 2)$
Question 9: If the vertex of the parabola $y = x^2 – 8x + c$ lies on the x-axis, then $c$ is:
Answer:
- (a) 4
- (b) 8
- (c) 16
- (d) 64
Question 10: Let $a, b, c$ be real numbers, $a \neq 0$. If the graph of $y = ax^2 + bx + c$ does not intersect the x-axis and $a + b + c < 0$, then:
Answer:
- (a) $c > 0$
- (b) $c < 0$
- (c) $b^2 – 4ac > 0$
- (d) $a > 0$
Question 11: The axis of symmetry of the parabola $f(x) = ax^2 + bx + c$ is $x = 2$. If $f(0) = 3$, then $f(4)$ is:
Answer:
- (a) 3
- (b) 6
- (c) 0
- (d) Insufficient data
Question 12: The range of $y = x^2 – 4x + 3$ for $x \in [0, 3]$ is:
Answer:
- (a) $[-1, 3]$
- (b) $[-1, 0]$
- (c) $[0, 3]$
- (d) $[-1, \infty)$
Question 13: If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$, where $ac \neq 0$, then $P(x)Q(x) = 0$ has at least:
Answer:
- (a) Four real roots
- (b) Two real roots
- (c) No real roots
- (d) Three real roots
Part II: Subjective Questions (Q14–Q15)
Question 14: Find the range of $y = \frac{x+2}{2x^2 + 3x + 6}$ for all real values of $x$.
Answer:
[Enter solution here]
Question 15: A parabola $y = ax^2 + bx + c$ passes through $(1, 2)$ and its vertex is $(2, 3)$. Find the value of $a + b + c$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the minimum value of $y = |x^2 – 4x + 3|$.
Answer:
Question 17: If the expression $x^2 + 2(a-1)x + a+5$ is always positive, find the number of possible integral values of $a$.
Answer:
Question 18: The vertex of $y = x^2 + kx + 4$ lies on the line $y = x$. Find the sum of all possible real values of $k$.
Answer:
Question 19: If the range of $f(x) = x^2 – 2x + c$ is $[5, \infty)$, find the value of $c$.
Answer:
Question 20: If $f(x) = x^2 + bx + c$ satisfies $f(2+t) = f(2-t)$ for all $t \in \mathbb{R}$, find the value of $b$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): The expression $x^2 + x + 1$ is positive for all real $x$.
Reason (R): For $ax^2 + bx + c$, if $a > 0$ and $D < 0$, the expression is always positive.
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 maximum value of $y = -x^2 + 4x + 5$ occurs at $x = 2$.
Reason (R): For $y = ax^2 + bx + c$ with $a < 0$, the maximum occurs at the vertex $x = -b/2a$.
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 | B | Q2 | A |
| Q3 | B | Q4 | B |
| Q5 | A | Q6 | A |
| Q7 | A | Q8 | A |
| Q9 | C | Q10 | B |
| Q11 | A | Q12 | A |
| Q13 | B | Q14 | — |
| Q15 | 2 | Q16 | 0 |
| Q17 | 5 | Q18 | -1 |
| Q19 | 6 | Q20 | -4 |
| Q21 | A | Q22 | A |