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Week 3 - Graded Assignment 3

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Week 3 - Graded Assignment 3

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Course: Jan 2026 - Mathematics I

Topic: Real Roots and Intersection (Discriminant) | Marks: 3

Question 1

Find out the points where the curve

y = 4x^2+x

and the straight line

y = 2x-3

intersect with each other.

$(\frac{3}{2},0)$ and $(\frac{3}{2},\frac{21}{2})$ .
Only at the origin.
The curve and the straight line do not intersect.
$(1, -1)$ and $(1, 5)$ .

MCQ

3 Unit Assessment

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Accepted Answer: The curve and the straight line do not intersect. Abstract Solution (Strategy):

Set Equations Equal: Equate $y_{curve}$ and $y_{line}$ to find intersection points.
Form Quadratic: Rearrange into standard form $ax^2 + bx + c = 0$ .
Discriminant Check: Calculate $D = b^2 - 4ac$ . If $D < 0$ , there are no real intersection points. ^maths-w3-q1-strategy

Procedure:

$4x^2 + x = 2x - 3 \implies 4x^2 - x + 3 = 0$ .
$a = 4, b = -1, c = 3$ .
$D = (-1)^2 - 4(4)(3) = 1 - 48 = -47$ .
Since $D < 0$ , the roots are imaginary, so the curve and line do not intersect in the real plane.

Topic: Consecutive Number Theory (Quadratic Modeling) | Marks: 3

Question 2

Let

a

and

b

two consecutive positive odd natural numbers such that

a^2+b^2=394

. Then find the value of

a+b

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 28 Abstract Solution (Strategy):

Variable Definition: Let consecutive odd numbers be $a$ and $a+2$ .
Equation Setup: $a^2 + (a+2)^2 = 394$ .
Solve for $a$ : Find the positive odd root. ^maths-w3-q2-strategy

Procedure:

$2a^2 + 4a + 4 = 394 \implies a^2 + 2a - 195 = 0$ .
Roots: $a=13$ or $a=-15$ .
Positive integers: $a=13, b=15$ .
Sum: $13+15=28$ .

Topic: Axis of Symmetry and Vertex Form | Marks: 3

Question 3

The maximum value of a quadratic function

f

-3

, its axis of symmetry is

x=2

and the value of the quadratic function at

x=0

-9

. What will be the coefficient of

x^2

in the expression of

f

$-1$
1
$-1.5$
$-0.5$

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: -1.5 Abstract Solution (Strategy):

Vertex Form: $f(x) = a(x-2)^2 - 3$ .
Solve for $a$ : Use $f(0) = -9$ . ^maths-w3-q3-strategy

Procedure:

$-9 = a(0-2)^2 - 3 \implies 4a = -6 \implies a = -1.5$ .

Topic: Parabolic Motion (Vertex and Symmetry) | Marks: 4

Question 4

A ball is thrown from

3

m off the ground and reaches a maximum height of

5

m. Assume that the ball was released from the point

(0,3)

in the

xy

-plane as shown in the Figure M1W3GA-3. The ball returns to a height of

3

m after

2

seconds. Let

h(t)=at^2+bt+c

be the quadratic function which represents the height of the ball after

t

seconds. What is the value of

a

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: -2 Abstract Solution (Strategy):

Symmetry: With points $(0,3)$ and $(2,3)$ , the vertex is at $t=1$ . Max height is $5$ .
Vertex Form: $h(t) = a(t-1)^2 + 5$ . ^maths-w3-q4-strategy

Procedure:

Using $(0,3)$ : $3 = a(0-1)^2 + 5 \implies a = -2$ .

Topic: Economic Modeling (Minimization and Slope) | Marks: 4

Question 5

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 and the daily transportation cost (in lakh ₹) of

x

number of electric devices is given by the slope of the function

p(x)

at point

x

. If the transportation cost of the electric devices on a particular day is 30 (in lakh ₹), then find the number of transported electric devices.

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 3 Abstract Solution (Strategy):

Derivative: Slope $p'(x) = 30x - 60$ .
Solve: $30x - 60 = 30$ . ^maths-w3-q5-strategy

Procedure:

$30x = 90 \implies x = 3$ .

Topic: Arch Modeling (Parabola with Axis Symmetry) | Marks: 4

Question 6

An Architect is designing an arch which is in the shape of figure M1W3-PARABOLA,
The details of the design are as follows:
Height of arch is 20m, width of arch at height of 8m is 4m.
Consider the axis of symmetry as

x=0

, find the equation of the arch.

fig:M1W3-PARABOLA

$y = -3x^2 +20$
$y = \frac{-3}{4} x^2 +20$
$y = -x^2+x+20$
$y = -2x^2-2x+20$

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer:

y = -3x^2 + 20

Abstract Solution (Strategy):

Model: Axis of symmetry $x=0 \implies y = ax^2 + k$ .
Vertex: Given height 20m $\implies$ vertex is $(0, 20) \implies k = 20$ .
Use Point: Width is 4m at height 8m $\implies$ points $(-2, 8)$ and $(2, 8)$ are on the parabola. Plug $(2, 8)$ to solve for $a$ . ^maths-w3-q6-strategy

Procedure:

$y = ax^2 + 20$ .
$8 = a(2)^2 + 20 \implies 4a = -12 \implies a = -3$ .
Equation: $y = -3x^2 + 20$ .

Topic: Quadratic Maximization (Fountain Path) | Marks: 3

Question 7

A water fountain is designed to shoot a stream of water in the shape of a parabolic arc. The equation of the parabola is given by

h(t) = -0.5t^2 + 4t + 1

, where

h(t)

represents the height of the water stream in meters and t represents the time in seconds since the water was shot.
Determine the maximum height reached (in meters).

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 9 Abstract Solution (Strategy):

Find Vertex Time: $t_{vertex} = -b / 2a$ .
Calculate Height: $h(t_{vertex})$ . ^maths-w3-q7-strategy

Procedure:

$a = -0.5, b = 4$ .
$t = -4 / (2 \cdot -0.5) = 4$ .
$h(4) = -0.5(16) + 4(4) + 1 = -8 + 16 + 1 = 9$ .

Topic: Quadratic Properties and Definitions | Marks: 2

Question 8

Which of the following is/are correct

$x$ intercepts of the quadratic function $f(x)$ are known as the real roots of the quadratic equation $f(x)=0$ .
If discriminant for two quadratic equations are same then they must be the same quadratic equations.
The slope at the vertex of the quadratic function is zero.
Every Quadratic function has axis of symmetry.
Suppose $P(x)$ is a Quadratic function and L be any straight line, then L must intersect the graph of $P(x)$ .

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: (Multiple True) Abstract Solution (Strategy):

Roots mapping: $f(x)=0$ solutions are indeed the x-intercepts.
Discriminant limit: $D$ same does not imply same equation (e.g., $x^2-4=0$ vs $x^2+4=0$ , or different $a$ ).
Vertex Slope: At any local extremum (vertex of parabola), the derivative (slope) is 0.
Axis of Symmetry: Universal property of quadratic parabolas. ^maths-w3-q8-strategy

Topic: Variation in Speed and Time (PIE/Quadratic) | Marks: 5

Question 9

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)

Waiting Time is 1 hr and Ananya is waiting.
Waiting Time is 1 hr and Madhuri is waiting.
Waiting Time is 30 min and Ananya is waiting.
Waiting Time is 30 min and Madhuri is waiting.

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: Waiting Time is 30 min and Ananya is waiting. Abstract Solution (Strategy):

Madhuri's Time: $t = \text{Distance} / \text{Speed} = 1800 / 720 = 2.5$ hours.
Ananya's Original: Let speed be $v$ , time be $1200 / v$ .
Ananya's Delayed: $1200 / (v-200) = (1200/v) + 0.5$ .
Solve for $v$ : Then compare total times. ^maths-w3-q9-strategy

Procedure:

Ananya's equation: $1200v = 1200(v-200) + 0.5v(v-200)$
$1200v = 1200v - 240000 + 0.5v^2 - 100v$
$0.5v^2 - 100v - 240000 = 0 \implies v^2 - 200v - 480000 = 0$ .
$(v-800)(v+600) = 0 \implies v = 800$ .
Ananya's final time: $1200 / (800-200) = 2$ hours.
Madhuri's time: $2.5$ hours.
Ananya arrives first. She waits $2.5 - 2 = 0.5$ hours (30 min).

Topic: Slope and Tangency at a Point | Marks: 3

Question 10

Which of the following options is/are true?

The point at which the slope of the equation $x^2 +2x-5$ equals 10 is (4,17)
$x=2$ is the axis of symmetry of the quadratic function $f(x)= x^2+4x+5$
If two different quadratic equations have the same discriminant then the roots of both equations can be the same.
The point at which the slope of the equation $x^2 +2x-5$ equals 10 is (4,19)

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: (Multiple True) Abstract Solution (Strategy):

Same Roots/Discriminant: If two equations have same roots, they are proportional. Proportional quadratics have the same discriminant only if the scaling factor is 1 or -1. So "can be the same" is technically true or common for specific cases.
Slope Check: For $f(x) = x^2 + 2x - 5$ , find $x$ where $f'(x) = 10$ . ^maths-w3-q10-strategy

Procedure:

$f'(x) = 2x + 2 = 10 \implies 2x = 8 \implies x = 4$ .
$f(4) = 16 + 8 - 5 = 19$ .
Point: $(4, 19)$ .

Topic: Calculating Model Parameters from Slope | Marks: 3

Question 11

If the slope of parabola

y=Ax^2 + Bx+ C

, where

A, B, C \in \mathbb{R}

at points (3, 2) and (2, 3) are 16 and 12 respectively.

Calculate the value of

A

Your Answer: 2

Status: Yes, the answer is correct.

Accepted Answers:

(Type: Numeric) 2

Topic: Root Restoration (Leading Coefficient 1) | Marks: 3

Question 12

Ram and Shyam want to solve a quadratic equation. Ram made a mistake in writing down the constant term and ended up in getting roots as 3 and 4. Shyam made a mistake in writing down the coefficient of

x

and got the roots as 2 and 3. Consider the leading coefficient to be 1 in all cases. The correct roots of the quadratic equation are:

1 and 5
2 and 6
1 and 6
2 and 5

Status: Yes, the answer is correct.

Accepted Answers:

1 and 6

Question 13

Consider a quadratic function

q(x) = ax^2+ 20x+ 15

, where

a \in \mathbb{R}\setminus\{0\}

. If the slope of

q(x)

x=2

is equal to the slope of the line

y=40x+5

Then which of the following options is true?

$a= 5$
$a= 8$
$q(x)$ has a unique root.
$q(x)$ has the minimum value at $x=2$ .

Status: Yes, the answer is correct.

Accepted Answers:

a= 5

Question 14

Find out the maximum height(in meters) attained by the missile.

Your Answer: 72

Status: Yes, the answer is correct.

Accepted Answers:

(Type: Numeric) 72

Question 15

Find out the time (in seconds) when the missile hits the tank.

Your Answer: 5

Status: Yes, the answer is correct.

Accepted Answers:

(Type: Numeric) 5

Question 16

Suppose an air defense system is present at the origin, and it follows the straight line path

h(t) = 10t

, find the height from the ground at which the air defense missile will destroy the ballistic missile in the air.

40 m
12.5 m
4 m
1.25 m

Status: Yes, the answer is correct.

Accepted Answers:

40 m

Question 17

The polynomial

p(x) = a (x − 4) (x − 6) (x − 8) (x − 10)

passes through the vertex of the quadratic function

q(x) = − (x − 7)^2 − 9

. Calculate the value of

a

Your Answer: -1

Status: Yes, the answer is correct.

Accepted Answers:

(Type: Numeric) -1

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