Ishu (My Wife) CBSE Class 12 Math has 2 Case Studies = 8 Marks. These are compulsory! Practice all types here and those 8 marks are yours!
Section D in Board Paper
8 Marks Compulsory
2 Case Studies 4 MCQs each
4 choices per MCQ
Chapter 6 Application of Derivatives
The Garden Fence Problem
Case Study 14 Marks
Read the Scenario
A farmer Ramu wants to fence a rectangular garden of area 800 sq. m using a wall on one side and wire on the other three sides. The cost of wire is 5 per metre. He wants to minimize the cost of fencing. Let the length of the side parallel to the wall be x metres and the other side be y metres.
Answer the following questions:
Q1
The total length of wire needed (three sides) is: [1M]
(a) x + 2y
(b) 2x + y
(c) x + y
(d) 2x + 2y
Answer
(b) 2x + y The wall covers one length (x), so wire covers: x (opposite) + y + y (two widths) = x + 2y. Wait Let side to wall = x, other side = y. Wire = x + 2y. Correct: (a) x + 2y
Q2
If area = 800 m, express y in terms of x: [1M]
(a) y = 800x
(b) y = 400/x
(c) y = 800/x
(d) y = 800/(2x)
Answer
(c) y = 800/x Area = xy = 800 y = 800/x
Q3
The cost function C(x) in terms of x is: [1M]
(a) 5(x + 800/x)
(b) 5(x + 1600/x)
(c) 5(2x + 800/x)
(d) 5x + 800
Answer
(b) 5(x + 1600/x) Wire length = x + 2y = x + 2(800/x) = x + 1600/x. Cost C = 5 (x + 1600/x)
Q4
The value of x that minimizes cost is: [1M]
(a) 20 m
(b) 30 m
(c) 40 m
(d) 45 m
Answer + Solution
(c) 40 m C'(x) = 5(1 1600/x) = 0 x = 1600 x = 40. C''(40) = 5 3200/x > 0 (Minimum)
Board Strategy: In Case Studies, read the scenario TWICE. Every question uses either y=800/x or C(x) from previous parts. Build on each answer they are connected!
Chapter 13 Probability
The Insurance Problem
Case Study 24 Marks
Read the Scenario
An insurance company found that 20% of people in a city are smokers. The probability that a smoker will die within 10 years is 0.5 and for a non-smoker it is 0.2. A person is chosen at random from the city and dies within 10 years. Based on this, answer the following questions.
Answer the following questions:
Q1
P(person is a smoker) = ? [1M]
(a) 0.2
(b) 0.5
(c) 0.8
(d) 0.3
Answer
(a) 0.2 Given directly: 20% are smokers P(S) = 0.20
Q2
P(dies within 10 years) using Total Probability Theorem = ? [1M]
Board Strategy: Probability case studies ALWAYS use Bayes' Theorem. Find P(A), then P(E|A). Q4 is often 1 - Q3 answer. Easy 4 marks if you know Bayes'!
Chapter 8 Application of Integrals
The Arch Bridge Problem
Case Study 34 Marks
Read the Scenario
An arch of a bridge is in the form of a semi-ellipse. The span (width) of the bridge is 10 m and the maximum height (at the centre) is 4 m. Vehicles can pass through the arch if they are at most 3 m high. The arch is placed with its centre at the origin, so the equation is $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ (upper half only, y 0).
Answer the following questions:
Q1
The semi-major axis (a) and semi-minor axis (b) of the ellipse are: [1M]
(a) 20π sq m Area of full ellipse = πab = π 5 4 = 20π sq. m
Q3
The area under the arch (semi-ellipse) is: [1M]
(a) 10π sq m
(b) 20π sq m
(c) 40π sq m
(d) 5π sq m
Answer
(a) 10π sq m Semi-ellipse area = πab = 20π = 10π sq. m
Q4
How wide is the road at height y=3 m? (i.e., what is 2x when y=3?) [1M]
(a) 4 m
(b) 5 m
(c) 7 m
(d) 10 m
Answer + Working
(c) 7 m At y=3: x/25 + 9/16 = 1 x/25 = 7/16 x = 175/16 x = 57/4. Width = 2x = 57/2 6.61 7 m
Board Strategy: Ellipse case studies: remember Area = πab. Semi-ellipse = πab/2. For "width at height h" substitute y=h in equation, solve for x, then width = 2x.
Chapter 12 Linear Programming
The Factory Production Problem
Case Study 44 Marks
Read the Scenario
A factory produces two products A and B. Each unit of A requires 2 hours of machine time and 1 hour of labour. Each unit of B requires 1 hour of machine time and 2 hours of labour. Total machine time available = 10 hours/day. Total labour hours = 8 hours/day. Profit: 40 per unit of A and 30 per unit of B.
Answer the following questions:
Q1
If x = units of A, y = units of B, the machine constraint is: [1M]
(a) x + 2y 10
(b) 2x + y 10
(c) x + y 10
(d) 2x + 2y 10
Answer
(b) 2x + y 10 A needs 2 hrs machine, B needs 1 hr machine. Total 10 2x + y 10.
Q2
The objective function to maximize profit Z is: [1M]
(a) Z = 30x + 40y
(b) Z = 40x + 30y
(c) Z = x + y
(d) Z = 2x + y
Answer
(b) Z = 40x + 30y Profit = 40 per unit of A + 30 per unit of B Z = 40x + 30y
Q3
The corner points of the feasible region are: [1M]
(a) (0,0),(5,0),(4,2),(0,4)
(b) (0,0),(10,0),(0,8)
(c) (0,0),(5,0),(0,8)
(d) (4,2) only
Answer
(a) (0,0),(5,0),(4,2),(0,4) Solve 2x+y=10 and x+2y=8 simultaneously: x=4, y=2. Other corners: (0,0), (5,0), (0,4).
Q4
Maximum profit per day is: [1M]
(a) 160
(b) 200
(c) 220
(d) 240
Answer + Table
(c) 220 Z at (0,0)=0, Z at (5,0)=200, Z at (4,2)=404+302=160+60=220 , Z at (0,4)=120. Maximum = 220 at (4,2).
Board Strategy: LPP Case Study: (1) Define variables (2) Write all constraints (3) Find corner points (4) Evaluate Z at each corner. SHOW the Z-table it's mandatory!
Chapter 3 & 4 Matrices & Determinants
The Shopping Problem
Case Study 54 Marks
Read the Scenario
Three friends A, B, C go shopping. A buys 2 pens, 1 notebook, 1 eraser. B buys 1 pen, 2 notebooks, no eraser. C buys 1 pen, 1 notebook, 2 erasers. Total money spent: A=50, B=40, C=45. Let price of pen=x, notebook=y, eraser=z.
Answer the following questions:
Q1
The system of equations in matrix form AX=B, matrix A is: [1M]
For the system to have a unique solution, |A| must be: [1M]
(a) = 0
(b) 0
(c) > 0
(d) < 0
Answer
(b) 0 A exists |A| 0 unique solution to AX=B. This is a fundamental theorem!
Q4
If x=10 is found, the price of a notebook (y) given x+2y=40 is: [1M]
(a) 12
(b) 15
(c) 18
(d) 20
Answer
(b) 15 x+2y=40 10+2y=40 2y=30 y=15
Board Strategy: Matrix case studies: write system as AX=B. Check |A|0. Always show cofactor working. x,y,z values come from X=AB.
Chapter 11 3D Geometry
The Flight Path Problem
Case Study 64 Marks
Read the Scenario
An aeroplane is flying along the line $\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(2\hat{i}-\hat{j}+\hat{k})$. A monitoring station is at point P(3, -4, 5). Answer the following based on this scenario.
Answer the following questions:
Q1
The direction ratios of the line of flight are: [1M]
(a) (1,-1,1)
(b) (2,-1,1)
(c) (1,2,-1)
(d) (-1,2,1)
Answer
(b) (2,-1,1) From $\vec{r}=\vec{a}+\lambda\vec{b}$, direction vector $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$. DRs = (2,-1,1)
Q2
A point on the line of flight is: [1M]
(a) (2,-1,1)
(b) (0,0,0)
(c) (1,-1,1)
(d) (-1,1,-1)
Answer
(c) (1,-1,1) At λ=0, position vector = $\hat{i}-\hat{j}+\hat{k}$ = point (1,-1,1)
Q3
The Cartesian equation of the flight path is: [1M]
(a) (x-1)/1=(y+1)/(-1)=(z-1)/1
(b) (x-1)/2=(y+1)/(-1)=(z-1)/1
(c) x/2=y/(-1)=z/1
(d) (x+1)/2=(y-1)/(-1)=(z+1)/1
Answer
(b) (x-1)/2 = (y+1)/(-1) = (z-1)/1 Point (1,-1,1), DRs (2,-1,1): Cartesian form (x-x)/a=(y-y)/b=(z-z)/c
Q4
The direction cosines of the flight path are (l,m,n) where denominator = $\sqrt{4+1+1}$: [1M]