1. Consider three Airports A, B, and C. Two friends Ananya and Madhuri want to meet at Airport C. Ananya Boarded Flight 1 from Point A to C which is 1200 km, due to bad weather, Flight 1 slowed down, and the average speed was reduced by 200 km/h and the time increased by 30 minutes. Madhuri boarded Flight 2 from Point B to C which is 1800 km, the average speed of Flight 2 is 720 km/h. What is the waiting time, and who will be waiting at the airport? (Given Ananya and Madhuri boarded at the same time)
Who waits: Madhuri arrives later, so Ananya waits.
Final Answer:
\boxed{Waiting Time is 30 min and Ananya is waiting.}
Multiple Choice Questions (MSQ)
Question 1
If the slope of the parabola $y = a x^{2} + b x + c$ at $(2,3)$ is 5 and the $X$-coordinate of the vertex of the parabola is 3, then which of the following is/are true?
Options:
(a) $y = -\frac{5}{2}x^{2} + 15x - 17$
(b) $y = \frac{5}{2}x^{2} + 15x - 17$
(c) $y = -\frac{5}{2}x^{2} + 15x + 17$
(d) $6y = -15x^{2} + 90x - 102$
Solution:
Option (a): Slope at $x=2$ is $-5(2)+15=5$. Vertex at $x=3$ (matches).
Option (b): Slope at $x=2$ is $5(2)+15=25$ (does not match).
Option (c): $y(2) = 27 \neq 3$ (does not pass through $(2,3)$).
Option (d): Equivalent to option (a).
Correct options: (a), (d)
Question 2
Two parabolas $y = x^{2} + 3x + 2$ and $y = -x^{2} - 5x - 4$ are intersecting at two points $A$ (not on the $X$-axis) and $B$. Suppose a straight line $\ell_{1}$ passes through $A$ with slope equal to the slope of the parabola $y = -x^{2} - 5x - 4$ at point $A$ and two straight lines $\ell_{2}$ and $\ell_{3}$ pass through $B$ with slopes equal to the slopes of the parabolas $y = x^{2} + 3x + 2$ and $y = -x^{2} - 5x - 4$ at point $B$, respectively. Which of the following is/are true?
Options:
(a) $\ell_{1}$ and $\ell_{2}$ are parallel.
(b) $\ell_{1}$ and $\ell_{3}$ are parallel.
(c) $\ell_{1}$ and $\ell_{3}$ are intersecting at point $(-2,3)$.
(d) $\ell_{2}$ and $\ell_{3}$ are intersecting at point $(-1,0)$.
Solution:
Intersection points: $(-3,2)$ and $(-1,0)$.
Slopes:
At $A(-3,2)$: slope of $y = -x^{2} - 5x - 4$ is $1$.
At $B(-1,0)$: slope of $y = x^{2} + 3x + 2$ is $1$; slope of $y = -x^{2} - 5x - 4$ is $-3$.
Equations:
$\ell_{1}$: $y = x + 5$
$\ell_{2}$: $y = x + 1$
$\ell_{3}$: $y = -3x - 3$
Analysis:
$\ell_{1}$ and $\ell_{2}$ are parallel (both have slope $1$).
$\ell_{1}$ and $\ell_{3}$ intersect at $(-2,3)$.
$\ell_{2}$ and $\ell_{3}$ intersect at $(-1,0)$.
Correct options: (a), (c), (d)
Numerical Answer Type (NAT)
Question 3
Suppose $f(x) = a x^{2} + b x + c$, where $a, b, c \in \mathbb{Z} \setminus {0}$. The sum and product of roots of $f(x)$ are $\frac{-7}{4}$ and $\frac{-1}{2}$ respectively. Find the value of $2a + 4b + 2c$.
Question 4
A class of 140 students are arranged in rows such that the number of students in a row is one less than thrice the number of rows. Find the number of students in each row.
Solution:
Let rows = $p$: $p(3p - 1) = 140$
Solve: $p=7$
Students per row: $3(7) - 1 = 20$
Answer: $20$
Comprehension Type Questions
The daily production cost (in lakh ₹) of manufacturing an electric device is $p(x) = 7400 - 60x + 15x^{2}$, where $x$ is the number of electric devices produced per day. The daily transportation cost (in lakh ₹) of $x$ devices is given by the slope of $p(x)$ at point $x$.
Question 5
How many electric devices should be produced per day to yield minimum production cost?
Solution:
Minimum at $x = 2$Answer: $2$
Question 6
If the transportation cost on a day is 30 (in lakh ₹), find the number of devices transported.
Solution:
Slope: $30x - 60 = 30 \implies x = 3$
Answer: $3$
Question 7
If the production cost on a day is 7475 (in lakh ₹), find the number of devices produced.
Solution:
Solve: $x=5$ (only positive solution)
Answer: $5$
Question 8
If the slope of parabola $y = a x^{2} + b x + c$ at points $(3,2)$ and $(2,3)$ are 16 and 12 respectively, find $a$.
Solution:
Solve: $6a + b = 16$, $4a + b = 12$
Subtract: $2a = 4 \implies a = 2$
Answer: $2$
Question 9
The product of two consecutive odd natural numbers is 143. Find the largest number among them.
Solution:
Numbers: $11, 13$
Answer: $13$
Question 10
The slope of a parabola $y = 3x^{2} - 11x + 10$ at a point $P$ is 7. Find the $y$-coordinate of $P$.
Solution:
Slope: $6x - 11 = 7 \implies x = 3$
$y(3) = 4$Answer: $4$
Additional Questions (from later sections of the PDF)
Question 11
Find out the points where the curve $y = 4x^{2} + x$ and the straight line $y = 2x - 3$ intersect with each other.
Solution:
No real intersection points.Answer: The curve and the straight line do not intersect.
Question 12
Let $a$ and $b$ be two consecutive positive odd natural numbers such that $a^{2} + b^{2} = 394$. Find the value of $a + b$.
Solution:
Numbers: $13, 15$
Sum: $28$
Answer: $28$
Question 13
A class of 352 students are arranged in rows such that the number of students in a row is one less than thrice the number of rows. Find the number of students in each row.
Solution:
Let rows = $x$: $x(3x - 1) = 352$
Solve: $x=11$
Students per row: $3(11) - 1 = 32$
Answer: $32$
Question 14
In order to cover a fixed distance of 48 km, two vehicles start from the same place. The faster one takes 2 hrs less and has a speed 4 km/hr more than the slower one.
(a) What is the speed (in km/hr) of the slower vehicle?
Solution:
Speed: $x = 8$ km/hr
Answer: $8$
(b) What is the time (in hrs) taken by the faster one?
Solution:
Time: $48 / 12 = 4$ hrs
Answer: $4$
Question 15
The maximum value of a quadratic function $f$ is $-3$, its axis of symmetry is $x=2$, and the value at $x=0$ is $-9$. What will be the coefficient of $x^{2}$ in the expression of $f$?
Solution:
Coefficient: $-1.5$
Answer: $-1.5$
Question 16
A ball is thrown from 3 m off the ground and reaches a maximum height of 5 m. It returns to 3 m after 2 seconds. Let $h(t) = a t^{2} + b t + c$ be the height function. What is the value of $a$?
Solution:
Using points: $(0,3)$, $(1,5)$, $(2,3)$
Solve: $a = -2$
Answer: $-2$
Question 17
The product of two consecutive odd natural numbers is 255. Find the largest number among them.
Solution:
Numbers: $15, 17$
Answer: $17$
Question 18
The slope of a parabola $y = 3x^{2} - 11x + 10$ at a point $P$ is 1. Find the $y$-coordinate of $P$.
Solution:
Slope: $6x - 11 = 1 \implies x = 2$
$y(2) = 0$Answer: $0$
Question 19
A ball is thrown. The height function is $h(t) = -0.5 t^{2} + 4 t + 1$. What is the time taken to reach maximum height?
Solution:
Time: $t = 4$
Answer: $4$
Question 20
What is the maximum height attained?
Solution:
Height: $h(4) = 9$
Answer: $9$
Summary Table
Q
Description
Solution/Answer
1
Parabola slope/vertex
(a), (d)
2
Parabola intersections, lines
(a), (c), (d)
3
Quadratic roots, sum/product
32
4
Students in rows
20
5
Minimum production cost
2
6
Transportation cost
3
7
Production cost
5
8
Parabola slope at points
2
9
Product of odd numbers
13
10
Parabola slope, y-coord
4
11
Curve-line intersection
No intersection
12
Sum of consecutive odds
28
13
Students in rows (352)
32
14a
Slower vehicle speed
8 km/hr
14b
Faster vehicle time
4 hrs
15
Quadratic coefficient
-1.5
16
Ball height, a value
-2
17
Product of odd numbers (255)
17
18
Parabola slope, y-coord
0
19
Ball max height time
4
20
Ball max height
9
This covers all questions and solutions from the Week-3.pdf file1.
