Cambridge A Level Mathematics

Conducting Body: Cambridge Assessment International Education (CAIE)
For: Pre-university students, typically ages 16-19
Papers: Pure Mathematics, Mechanics, and Probability & Statistics
Question Format: Structured questions requiring detailed working
Paper Duration: Varies by paper, approx. 1.5 to 2 hours each
Exam Sessions: Twice a year (June and November series)
Qualifies For: University admission, especially for STEM and economics

Cambridge A Level Mathematics is a pre-university qualification for students aged 16 to 19. It is taken by those planning to pursue degrees in science, technology, engineering, mathematics (STEM), economics, or computer science, as it is a common prerequisite for these courses at universities worldwide. The qualification demonstrates a high level of mathematical ability and is highly regarded by admissions tutors.

The examination consists of several written papers, typically covering Pure Mathematics, Mechanics, and Probability & Statistics. Students must answer structured questions, showing their working clearly to gain full marks. The challenge of A Level Mathematics comes from the need to apply familiar concepts to new and complex problems. Simple recall of formulas is insufficient. Success depends on a deep conceptual understanding, strong algebraic manipulation skills, and the ability to construct logical arguments under time pressure. The step up in abstract thinking and problem-solving from previous qualifications is significant.

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JEE Advanced 2023

Let $f : [0,1] \to [0,1]$ be the function defined by $f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}$ . Consider the square region $S = [0,1]\times[0,1]$ . Let $G = \{(x,y) \in S : y > f(x)\}$ be called the green region and $R = \{(x,y) \in S : y < f(x)\}$ be called the red region. Let $L_h = \{(x,h) \in S : x \in [0,1]\}$ be the horizontal line drawn at a height $h \in [0,1]$ . Then which of the following statements is(are) true?

Applications of Integrals Let

f : [0,1] \to [0,1]

be the function defined by

f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}

. Consider the square region

S = [0,1]\times[0,1]

. Let

G = \{(x,y) \in S : y > f(x)\}

be called the green region and

R = \{(x,y) \in S : y < f(x)\}

be called the red region. Let

L_h = \{(x,h) \in S : x \in [0,1]\}

be the horizontal line drawn at a height

h \in [0,1]

. Then which of the following statements is(are) true?

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$f(x) = \dfrac{x^3}{3} - x^2 + \dfrac{5}{9}x + \dfrac{17}{36}, \quad x \in [0,1]$

$S = [0,1]^2;\quad G: y > f(x)\ \text{(green)};\quad R: y < f(x)\ \text{(red)};\quad L_h: y = h$

$\text{To find: which statements (1)--(4) are true}$

$f'(x) = x^2 - 2x + \dfrac{5}{9} = (x-\tfrac13)(x-\tfrac53)$

$\int_0^1 f(x)\,dx = \dfrac{1}{12} - \dfrac13 + \dfrac{5}{18} + \dfrac{17}{36} = \dfrac12$

$\Rightarrow \operatorname{area}(R) = \operatorname{area}(G) = \dfrac12$

$A_G^< + A_R^< = h, \qquad A_G^> + A_R^> = 1 - h$

$A_G^< + A_G^> = \dfrac12, \qquad A_R^< + A_R^> = \dfrac12$

$A_G^> - A_R^< = \dfrac12 - h, \qquad A_R^> - A_G^< = \dfrac12 - h$

$h = \dfrac12 \Rightarrow A_G^> = A_R^< \text{ and } A_R^> = A_G^< \Rightarrow (3), (4)\ \text{TRUE}$

$(2):\ A_R^> = A_R^< \iff A_R^< = \dfrac14$

$f(x) \ge \dfrac{13}{36} > \dfrac14 \Rightarrow A_R^<\!\left(\dfrac14\right) = \dfrac14 \Rightarrow (2)\ \text{TRUE}$

$(1):\ A_G^> = A_G^< \iff A_G^< = \dfrac14$

$f(x) < \dfrac23 \Rightarrow A_G^<\!\left(\dfrac23\right) = \dfrac23 - \dfrac12 = \dfrac16 < \dfrac14 \Rightarrow (1)\ \text{FALSE}$

$\text{TRUE: } (2),\ (3),\ (4)$

$\boxed{(2),\ (3),\ (4)}$

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