SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]
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4 Marks
Q1. Case Study: Sports Day Running Track Layout
During an annual school sports day, students are arranged in a rectangular sports ground mapped out on a coordinate grid. Four students are positioned at points $A(1, 2)$, $B(4, 2)$, $C(4, 6)$, and $D(1, 6)$ to form a formation.
a) State the distance formula used to calculate the distance between any two coordinate points (1 Marks)
b) Calculate the distance between students positioned at $A(1, 2)$ and $B(4, 2)$. (1 Marks)
c) Determine the type of quadrilateral formed by joining the points $A, B, C,$ and $D$ in order by calculating all four side lengths and diagonals (2 Marks)
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4 Marks
Q2. Case Study: Archaeological Excavation Site Grid
Archaeologists use a coordinate grid to map out historical artifacts. An artifact is found at position $P(3, 4)$, and another is found at $Q(8, 9)$. A trench line is being dug along the straight line joining $P$ and $Q$.
a) Find the distance between artifacts $P$ and $Q$Find the coordinates of the mid-point of the trench line $PQ$ (1 Marks)
b) Find the coordinates of the mid-point of the trench line $PQ$ (1 Marks)
c) If a monitoring sensor is placed at point $R$ dividing the trench line $PQ$ in the ratio $2:3$ internally, find the coordinates of $R$ (2 Marks)
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4 Marks
Q3. Case Study: City Park Fountain and Bench Placement
A city municipal corporation plans to install a circular fountain in a park. On the park's coordinate map, the fountain's center is located at $C(2, -3)$, and a bench is placed at $Q(5, 7)$. Another bench is located at $P(-1, y)$ such that it is equidistant from the fountain center
a) Using the distance formula, write an equation equating the distance from center $C$ to $P$ and $Q$ (1 Marks)
b) Solve for the possible value(s) of $y$. (2 Marks)
c) Calculate the radius of the circular path around the fountain. (1 Marks)
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4 Marks
Q4. Case Study: Drone Delivery Flight Path
An e-commerce company tests a delivery drone. The drone flies in a straight line from a warehouse located at $A(-3, 10)$ to a delivery destination at $B(6, -8)$. An emergency landing checkpoint is located at $C(-1, 6)$.
a) State the section formula used to find the coordinates of a point dividing a line segment in a given ratio (1 Marks)
b) Find the ratio in which the checkpoint $C(-1, 6)$ divides the drone flight path $AB$. (2 Marks)
c) If the drone travels at a constant speed, what fraction of the total journey from $A$ to $B$ has been completed upon reaching $C$? (1 Marks)
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4 Marks
Q5. Case Study: Triangular Land Surveying
A land surveyor measures a triangular plot of land whose vertices are marked on a GPS coordinate map as $A(1, -1)$, $B(-4, 6)$, and $C(-3, -5)$.
a) State the formula for finding the area of a triangle given its three vertices (1 Marks)
b) Substitute the given coordinates into the area formula and compute the step-by-step intermediate expression (2 Marks)
c) Find the total area of the triangular land plot. (1 Marks)
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4 Marks
Q6. Case Study: Highway Planning and Collinear Toll Booths
Three towns are planned along a straight national highway. On the state coordinate map, their locations are represented by points $A(2, 3)$, $B(4, k)$, and $C(6, -3)$.
a) What mathematical condition must be satisfied by three points for them to lie on a straight highway (collinear)? (1 Marks)
b) Set up the area of the triangle formed by points $A, B,$ and $C$ equal to zero to form an equation in terms of $k$ (2 Marks)
c) Solve for the value of $k$ (1 Marks)
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4 Marks
Q7. Case Study:Triangular Garden Boundary Fencing
A school science club creates a triangular flower bed. The vertices of the flower bed on a school grid are $P(0, -1)$, $Q(2, 1)$, and $R(0, 3)$.
a) Find the coordinates of the mid-points of sides $PQ$, $QR$, and $RP$ (1 Marks)
b) Calculate the area of the inner triangle formed by joining these mid-points (2 Marks)
c) Find the ratio of the area of the mid-point triangle to the area of the original triangle $\triangle PQR$. (1 Marks)
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4 Marks
Q8. Case Study: Telecommunication Tower Coverage
A telecom tower is installed at the origin $(0, 0)$ of a city grid. Three residential houses are located at $A(3, 4)$, $B(5, -12)$, and $C(-6, 8)$.
a) Calculate the distance of house $A(3, 4)$ from the tower (1 Marks)
b) Calculate the distance of house $B(5, -12)$ from the tower (1 Marks)
c) Which house is closest to the telecom tower, and what is its distance? (2 Marks)
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4 Marks
Q9. Case Study: Bridge Pillar Alignment Check
An engineering team checks the alignment of four vertical pillars of a bridge built at coordinates $A(-4, -1)$, $B(-2, -4)$, $C(4, 0)$, and $D(2, 3)$.
a) Calculate the lengths of diagonals $AC$ and $BD$(2 Marks)
b) Calculate the lengths of adjacent sides $AB$ and $BC$ (1 Marks)
c) Deduce whether the pillar layout forms a rectangle or a parallelogram (1 Marks)
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4 Marks
Q10. Case Study: Treasure Hunt Grid Challenge
Students participate in a school treasure hunt organized on a large courtyard coordinate grid. Clues are hidden at $A(-2, 4)$, $B(0, 0)$, and $C(4, 2)$. A median path is drawn from vertex $B$ to meet side $AC$ at $D$.
a) Find the coordinates of point $D$, the mid-point of $AC$ (2 Marks)
b) Calculate the length of the median $BD$. (1 Marks)
c) Verify if median $BD$ divides $\triangle ABC$ into two triangles of equal areas (1 Marks)
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4 Marks
Q11. Case Study: District Boundary Division
A government redraws district boundaries. A boundary line passes through the points $P(1, -5)$ and $Q(-4, 5)$.
a) Write the general coordinate form of a point that lies on the $x$-axis (1 Marks)
b) Find the ratio in which the $x$-axis divides the boundary line segment $PQ$ (2 Marks)
c) Find the coordinates of the point of division on the $x$-axis. (1 Marks)
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4 Marks
Q12. Case Study: Triangular Roof Design Layout
An architect designs a triangular roof truss with vertices $A(7, -3)$, $B(5, 3)$, and $C(3, -1)$.
a) Find the coordinates of the centroid of the triangular roof truss. (2 Marks)
b) State the formula used to compute the centroid coordinates. (1 Marks)
c) Find the length of the median drawn from vertex $A$ to the opposite side (1 Marks)
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4 Marks
Q13. Case Study: River Crossing Coordinate Mapping
Two observation posts are set up across a river at $A(2, -2)$ and $B(3, 7)$. A surveillance boat path intersects this line segment at a point whose coordinates satisfy the equation $2x + y - 4 = 0$.
a) State the section formula in terms of an unknown ratio $k:1$ (1 Marks)
b) Find the ratio in which the line equation divides the segment $AB$ (2 Marks)
c) Find the exact coordinates of the intersection point. (1 Marks)
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4 Marks
Q14. Case Study: Quad-copter Flight Area Calculation
A quad-copter aerial photography survey covers a region shaped like a quadrilateral with vertices $A(-4, -2)$, $B(-3, -5)$, $C(3, -2)$, and $D(2, 3)$.
a) Draw or visualize the diagonal $AC$ dividing the quadrilateral into two triangles (1 Marks)
b) Calculate the area of $\triangle ABC$ (1 Marks)
c) Calculate the total area of quadrilateral $ABCD$ (2 Marks)
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4 Marks
Q15. Case Study: River Bridge Span Proportions
Engineers planning a bridge divide a support beam into four equal segments. The endpoints of the beam are given as $A(-2, 2)$ and $B(2, 8)$.
a) Find the mid-point of the entire beam segment $AB$ (1 Marks)
b) Find the coordinates of the points that divide the segment $AB$ into four equal parts (2 Marks)
c) What is the distance between consecutive division points? (1 Marks)
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4 Marks
Q16. Case Study: Equilateral Land Plot Verification
A real estate developer claims a triangular plot with vertices $A(a, a)$, $B(-a, -a)$, and $C(-\sqrt{3}a, \sqrt{3}a)$ is equilateral.
a) Write the distance formula expression between points $A$ and $B$ (1 Marks)
b) Calculate the squared lengths of sides $AB$, $BC$, and $CA$. (2 Marks)
c) Conclude whether the developer's claim is valid with a brief justification. (1 Marks)
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4 Marks
Q17. Case Study: Water Pipeline Connection Point
A main water pipeline is laid along a straight line connecting houses at $(3, 5)$ and $(7, 9)$. A secondary connection valve is installed to divide this segment internally in the ratio $3:2$.
a) State the internal section formula. (1 Marks)
b) Calculate the $x$-coordinate of the connection valve. (1 Marks)
c) Calculate the $y$-coordinate of the connection valve and state the final coordinate point. (2 Marks)
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4 Marks
Q18. Case Study: Security Camera Range Intersection
A security camera coverage zone forms a triangle with vertices $(0, 0)$, $(3, 0)$, and $(0, 4)$ on a factory floor grid.
a) Calculate the lengths of all three sides of the triangular coverage zone using the distance formula (2 Marks)
b) Name the specific type of triangle formed. (1 Marks)
c) Find the perimeter of the security camera coverage zone (1 Marks)
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4 Marks
Q19. Case Study: Sailboat Navigation Triangle
A sailboat navigates according to GPS coordinates forming $\triangle PQR$ with vertices $P(1, k)$, $Q(4, -3)$, and $R(-9, 7)$. Due to weather conditions, the effective navigation area is recorded as $15$ square units.
a) Write the area expression for $\triangle PQR$ in terms of $k$. (1 Marks)
b) Formulate the absolute value equation equating the area to $15$ (2 Marks)
c) Solve for the possible values of $k$. (1 Marks)
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4 Marks
Q20. Case Study: Triangular Plaza Centroid Fencing
A city plans a triangular public plaza with vertices at $(-1, 4)$, $(5, 2)$, and a third vertex to be determined such that the monument placed at the centroid is at $(0, -3)$.
a) State the coordinate relations connecting the three vertices to the centroid. (1 Marks)
b) Set up equations to solve for the coordinates of the third vertex $(x_3, y_3)$ (2 Marks)
c) Find the coordinates of the third vertex (1 Marks)
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