ISC Class XI — Applied Mathematics (885)
Chapter: Quadratic Equations & Applications
NEP-Based HOTS Question Paper
Time: 3 Hours
Maximum Marks: 80
General Instructions
-
This question paper contains 4 Sections — A, B, C and D.
All questions are compulsory.
Use of calculator is allowed where necessary.
-
The paper is designed according to the NEP competency-based pattern.
-
Questions test:
Analytical thinking
Logical reasoning
Mathematical modelling
Real-life applications
Interpretation skills
Draw neat graphs wherever required.
Section A — Conceptual & Analytical MCQs
(1 mark each)
Q1.
A company models its monthly profit (P(x)) (in lakh rupees) by:
$P(x) = -2x^2 + 24x - 40$
where (x) represents the number of advertisements released.
What does the negative coefficient of ($x^2$) indicate?
A. Profit increases indefinitely
B. Profit eventually decreases after a
certain point
C. Profit is always negative
D. Advertisements have no
effect
Q2.
Two roots of a quadratic equation differ by 6 and their sum is 10.
The
quadratic equation is:
A. ($x^2 - 10x + 16 = 0$)
B. ($x^2 + 10x + 16 = 0$)
C. ($x^2 - 5x +
4 = 0$)
D. ($x^2 - 10x + 25 = 0$)
Q3.
A quadratic function has equal roots. Which of the following is true?
A. Graph cuts x-axis at two points
B. Discriminant is positive
C.
Vertex lies on x-axis
D. Function has no turning point
Q4.
A startup predicts customer growth using:
$C(t)= -5t^2 + 60t + 20$
What does the vertex represent?
A. Initial customers
B. Maximum customer count
C. Minimum customer
count
D. Time when company closes
Q5.
If one root of:
$x^2 - kx + 12 = 0$
is 3, then value of (k) is:
A. 4
B. 5
C. 7
D. 9
Q6.
A projectile follows path:
$h = -x^2 + 8x + 5$
What is the practical meaning of the roots?
A. Maximum height points
B. Time intervals
C. Points where
projectile touches ground level
D. Speed of projectile
Q7.
Which condition ensures that a quadratic equation has irrational roots?
A. (D = 0)
B. (D < 0)
C. (D > 0) but not a perfect square
D.
(D) is a perfect square
Q8.
A rectangular park has area 192 m². Its length exceeds breadth by 4 m.
Which
equation models the situation?
A. $x(x+4)=192$
B. $x(x-4)=192$
C. $2x+4=192$
D. $x^2+4=192$
Q9.
The graph of a quadratic function opens upward. This implies:
A. Minimum value exists
B. Maximum value exists
C. Roots are
imaginary
D. Vertex lies below x-axis
Q10.
If roots of a quadratic equation are reciprocal of each other, then:
A. Constant term = coefficient of (x)
B. Product of roots = 1
C. Sum
of roots = 1
D. Discriminant = 0
Section B — Case Study Based Questions
(2 marks each)
Q11. Business Optimization
A manufacturer models revenue by:
$R(x)= -3x^2 + 72x$
where (x) is number of products sold in hundreds.
Answer:
Why can revenue not increase forever?
Determine number of products for maximum revenue.
Q12. Sports Analytics
A football is kicked such that its height is given by:
$h(t)= -5t^2 + 20t + 1$
Answer:
Find maximum height attained.
Explain practical significance of vertex.
Q13. Agriculture Model
A farmer uses fencing to create a rectangular field of area 300 m².
Length
is 5 m more than breadth.
Answer:
Form the quadratic equation.
Determine dimensions of the field.
Q14. Environmental Study
Pollution level in a lake over time is modeled as:
$P(t)= t^2 - 10t + 21$
Answer:
Find the times when pollution level becomes zero.
Interpret these values in real-life context.
Section C — Competency & HOTS Questions
(4 marks each)
Q15.
A company launches an online campaign. Number of daily users visiting the app is modeled by:
$$U(x)= -2x^2 + 28x + 64$$
where (x) is number of days after launch.
Answer:
Determine day on which users are maximum.
Find maximum users.
Explain why user growth decreases after a certain stage.
Q16.
The sum of two numbers is 20 and their product is maximum.
Answer:
Form a quadratic model.
Determine the numbers.
Explain mathematically why the product becomes maximum.
Q17.
A student claims:
“Every quadratic equation with positive discriminant has integer roots.”
Is the statement always true? Justify using suitable examples.
Q18.
A toy rocket follows trajectory:
$h(x)= -x^2 + 12x + 20$
Answer:
Find maximum height.
Determine horizontal distance when rocket hits ground.
Suggest one real-life factor ignored in this model.
Q19.
A school auditorium plans seating arrangement. Number of seats per row exceeds number of rows by 8. Total seats = 240.
Answer:
Form quadratic equation.
Find possible arrangement.
-
Discuss why quadratic modelling is useful in planning problems.
Section D — Long Analytical Questions
(6 marks each)
Q20.
A company’s monthly profit function is:
$P(x)= -x^2 + 40x - 300$
where (x) is number of units sold.
Answer:
Find break-even points.
Determine maximum profit.
Find number of units needed for maximum profit.
Interpret graph economically.
Q21.
A bridge arch is modeled by:
$y = -x^2 + 14x - 24$
where (y) represents height above road level.
Answer:
Find maximum height of arch.
Determine points where arch meets road.
Sketch graph.
Explain importance of quadratic functions in architecture.
Q22.
A scientist models spread of a chemical reaction using:
$R(t)= -4t^2 + 32t + 12$
Answer:
Find time for maximum reaction intensity.
Determine maximum intensity.
Find when reaction stops.
Explain limitations of mathematical modelling.
Q23.
A rectangular garden is designed such that diagonal is 13 m and length exceeds breadth by 7 m.
Answer:
Form quadratic equation using Pythagoras theorem.
Find dimensions of garden.
Verify solution geometrically.
-
Explain how algebra simplifies practical engineering calculations.
Internal Choice
Attempt any ONE:
Q24(A).
A drone’s height is modeled by:
$h(t)= -16t^2 + 64t + 80$
Find:
Maximum height
Time to reach maximum height
Time when drone touches ground
Q24(B).
The roots of equation:
$x^2 - (k+3)x + 2k = 0$
are consecutive integers.
Find:
Value of (k)
The roots
Verify discriminant condition
End of Question Paper
ISC Class XI — Applied Mathematics (885)
Chapter: Trigonometry & Applications
NEP-Based HOTS Question Paper
Time: 3 Hours
Maximum Marks: 80
General Instructions
-
This question paper contains 4 Sections — A, B, C and D.
All questions are compulsory unless stated otherwise.
Use of calculator is permitted where necessary.
-
The paper is competency-based and aligned with NEP assessment pattern.
-
Questions assess:
Conceptual understanding
Real-life application
Mathematical reasoning
Analytical and critical thinking
Interpretation and modelling skills
Draw neat diagrams wherever necessary.
Section A — Conceptual & Analytical MCQs
(1 mark each)
Q1.
A surveyor measures the angle of elevation of a tower from a point on the
ground as $45^\circ$.
If the distance from the tower is doubled, the
angle of elevation becomes:
A. Greater than $45^\circ$
B. Equal to $45^\circ$
C. Less than
$45^\circ$
D. Cannot be determined
Q2.
A drone moves upward maintaining constant horizontal distance from an observer. Which trigonometric ratio best represents the changing situation?
A. $\sin \theta$
B. $\cos \theta$
C. $\tan \theta$
D. $\sec
\theta$
Q3.
If:
$\sin \theta = \frac{3}{5}$
and $\theta$ lies in first quadrant, then:
A. $\cos \theta = \frac{3}{5}$
B. $\tan \theta = \frac{4}{3}$
C.
$\tan \theta = \frac{3}{4}$
D. $\sec \theta = \frac{3}{4}$
Q4.
A student claims:
“$\sin \theta + \cos \theta = 1$ for all angles.”
The statement is:
A. Always true
B. True only at $45^\circ$
C. True for some
specific angles only
D. Impossible for any angle
Q5.
A ladder leaning against a wall forms a right triangle. If the ladder slips downward, the angle with ground:
A. Increases
B. Decreases
C. Remains same
D. Becomes
$90^\circ$
Q6.
If:
$\tan \theta = 1$
then the angle could be:
A. $30^\circ$
B. $45^\circ$
C. $60^\circ$
D. $90^\circ$
Q7.
Which trigonometric function is undefined at (90^\circ)?
A. $\sin \theta$
B. $\cos \theta$
C. $\tan \theta$
D. $\csc
\theta$
Q8.
A ship observes a lighthouse at angle of elevation $30^\circ$. As the ship moves closer, the angle becomes $60^\circ$. This implies:
A. Distance from lighthouse increased
B. Height of lighthouse
decreased
C. Distance from lighthouse decreased
D. Lighthouse
tilted
Q9.
The identity:
$1+\tan^2\theta = \sec^2\theta$
is derived using:
A. Pythagoras theorem
B. Distance formula
C. Quadratic formula
D.
Midpoint theorem
Q10.
If:
$\sin \theta = \cos \theta$
then:
A. $\theta = 30^\circ$
B. $\theta = 45^\circ$
C. $\theta =
60^\circ$
D. $\theta = 90^\circ$
Section B — Case Study Based Questions
(2 marks each)
Q11. Architecture Application
An architect designs a ramp making an angle of $12^\circ$ with the ground for wheelchair access.
Answer:
Why are small angles preferred in ramp design?
If ramp length is 15 m, estimate vertical rise.
Q12. Aviation Model
A plane takes off making angle of elevation (25^\circ). After travelling 800 m horizontally, determine approximate altitude.
Answer:
Calculate altitude.
Explain importance of trigonometry in aviation safety.
Q13. Solar Energy Study
The angle of elevation of the Sun changes from $30^\circ$ to $60^\circ$.
Answer:
How does shadow length change?
Explain relevance in solar panel positioning.
Q14. Disaster Management
A rescue helicopter spots a stranded person at angle of depression (40^\circ). Helicopter is flying at height 500 m.
Answer:
Find horizontal distance from victim.
Explain practical importance of such calculations.
Section C — Competency & HOTS Questions
(4 marks each)
Q15.
A lighthouse 80 m high is observed from a ship. Angle of elevation changes from (30^\circ) to (60^\circ) after the ship moves toward lighthouse.
Answer:
Find initial distance of ship.
Find distance travelled by ship.
-
Explain why angular measurement is more practical than direct distance measurement over water.
Q16.
A student says:
“If $\sin \theta$ increases, then $\cos \theta$ must also increase.”
Answer:
Is the statement always true?
Justify using unit circle or trigonometric reasoning.
Provide numerical examples.
Q17.
A tower casts shadow of 20 m when angle of elevation of Sun is (45^\circ). Later shadow becomes 10 m.
Answer:
Determine new angle of elevation.
Explain relationship between shadow length and angle.
Discuss real-life applications in time estimation.
Q18.
A drone camera records a stadium from height 150 m. Angle of depression to a car entering stadium is $35^\circ$.
Answer:
Find distance of car from base of drone.
Suggest assumptions made in trigonometric modelling.
Explain limitations of such models.
Q19.
Prove:
$\frac{1-\cos^2\theta}{\sin\theta} = \sin\theta$
Then explain how identities help simplify engineering calculations.
Section D — Long Analytical Questions
(6 marks each)
Q20.
Two buildings stand on opposite sides of a road. From top of first building, angle of depression to base of second building is $35^\circ$. From top of second building, angle of elevation to top of first building is $50^\circ$. Distance between buildings is 40 m.
Answer:
Find heights of both buildings.
Compare the heights.
Draw labelled diagram.
-
Explain importance of angular measurements in urban planning.
Q21.
A surveillance drone flies horizontally at height 120 m. It observes two vehicles moving toward it at angles of depression $30^\circ$ and $60^\circ$.
Answer:
Determine distances of both vehicles from drone’s base.
Find distance between vehicles.
Explain how trigonometry supports surveillance systems.
Q22.
A mountain climber observes the top of a peak at angle $28^\circ$. After climbing 500 m toward the mountain, angle becomes $46^\circ$.
Answer:
Find height of mountain.
Determine initial distance from mountain.
-
Discuss how surveyors use similar methods in inaccessible regions.
Q23.
A Ferris wheel has radius 20 m. A rider’s height above ground after rotation is modelled using trigonometric concepts.
Answer:
Explain how sine and cosine functions model circular motion.
-
Determine maximum and minimum heights if wheel’s center is 25 m above ground.
-
Explain practical applications of trigonometric modelling in engineering and animation.
Internal Choice
Attempt any ONE:
Q24(A).
A person standing on top of a 60 m building observes two objects on ground on opposite sides at angles of depression $30^\circ$ and $45^\circ$.
Find:
Distances of objects from building.
Distance between objects.
Which object is nearer and why.
Q24(B).
Prove:
$\frac{\sec\theta - \cos\theta}{\tan\theta} = \sin\theta$
Then answer:
Why are identities important in higher mathematics?
-
Mention one real-life field where trigonometric identities are extensively used.