AOI Case Study & Competency PYQs
Practice Class 12 CBSE Board Previous Year Questions (2008-2026)
Q1
2026 Board (Sample)
00:00
A city planning commission wants to design a park bounded by a parabolic path $y = x^2$ and a linear path $y = 2x + 3$. (i) Find the intersection points. (ii) Calculate the total area of the park. 4 Marks
Part (i): Find Intersection Points
$x^2 = 2x + 3 \Rightarrow x^2 - 2x - 3 = 0$.
Factorizing: $(x-3)(x+1) = 0 \Rightarrow x = -1, 3$.
Corresponding y-values: at $x=-1, y=1$; at $x=3, y=9$.
Points: $(-1, 1)$ and $(3, 9)$.
Part (ii): Calculate Area
Step 1: Set up the Integral
Area $= \int_{-1}^3 (y_{upper} - y_{lower}) \, dx = \int_{-1}^3 (2x + 3 - x^2) \, dx$.
Step 2: Integrate
$= [x^2 + 3x - \frac{x^3}{3}]_{-1}^3$.
Step 3: Evaluate
$= (9 + 9 - 9) - (1 - 3 + 1/3) = 9 - (-2 + 1/3) = 9 + 5/3 = 32/3$ sq. units.
32/3 sq. units
Q2
2024 Board
00:00
An artist is designing a logo using the curves $y = \sin x$ and $y = \cos x$ in the interval $0 \le x \le \pi/2$. Find the area of the region bounded by these curves and the x-axis. 4 Marks
Step 1: Analyze Boundaries
The region bounded by $y = \sin x$, $y = \cos x$ and the x-axis in $[0, \pi/2]$ consists of two parts.
Step 2: Find Intersection Point
$\sin x = \cos x \Rightarrow \tan x = 1 \Rightarrow x = \pi/4$.
Step 3: Area Strategy
Total Area $= \int_0^{\pi/4} \sin x \, dx + \int_{\pi/4}^{\pi/2} \cos x \, dx$.
Step 4: Integrate and Evaluate
$= [-\cos x]_0^{\pi/4} + [\sin x]_{\pi/4}^{\pi/2}$.
$= (-\frac{1}{\sqrt{2}} + 1) + (1 - \frac{1}{\sqrt{2}}) = 2 - \frac{2}{\sqrt{2}} = 2 - \sqrt{2}$ sq. units.
2 - √2 sq. units