Start learning →

AP Intermediate 2nd Year Mathematics — practice by topic

Browse 1,105 questions across 17 Mathematics topics. Choose what you want to revise, then open a question to check its answer and worked solution.

Start learning AP Intermediate 2nd Year with Quill →

Mathematics topics

Vector Algebra and 3D Geometry (218)Continuity, Differentiability & Derivatives (165)Integrals and Applications (121)Linear Programming (97)Relations, Functions & Inverse Trigonometry (92)Differential Equations (88)Determinants (87)Matrices and Determinants (87)Probability (51)Application of Derivatives (17)Inverse Trigonometric Functions (16)Integrals (15)Application of Integrals (14)Matrices (13)Relations and Functions (12)Probability & Statistics (8)Statistics and Probability (4)

Try a Mathematics question

Find $\displaystyle\int x^4\,dx$.Read the answer and worked solution →Find $\displaystyle\int \sin x\,dx$.Read the answer and worked solution →Evaluate $\displaystyle\int_{0}^{\pi/2} \sin x\,dx$.Read the answer and worked solution →Find $\displaystyle\int 2x \cos(x^2)\,dx$.Read the answer and worked solution →

Worked practice questions

Find $\displaystyle\int x^4\,dx$.

  1. $5x^5 + C$
  2. $\dfrac{x^5}{5} + C$
  3. $x^5 + C$
  4. $4x^3 + C$
Show answer and solution

Correct answer: B. $\dfrac{x^5}{5} + C$

Power rule for integration: $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Plug $n = 4$: $\dfrac{x^5}{5} + C$. Option A is the *derivative* of $x^4$, not the integral. Always check by differentiating your answer.

Open this question →

Find $\displaystyle\int \sin x\,dx$.

  1. $\cos x + C$
  2. $-\cos x + C$
  3. $-\sin x + C$
  4. $\sin x + C$
Show answer and solution

Correct answer: B. $-\cos x + C$

$\dfrac{d}{dx}(-\cos x) = \sin x$, so $\displaystyle\int \sin x\,dx = -\cos x + C$. Mnemonic: differentiating gives a negative on the sine term ($\dfrac{d}{dx} \cos x = -\sin x$). To reverse, you keep the sign that makes the derivative *positive* $\sin x$, which is the $-\cos x$ choice.

Open this question →

Evaluate $\displaystyle\int_{0}^{\pi/2} \sin x\,dx$.

  1. $-1$
  2. $1$
  3. $\pi/2$
  4. $0$
Show answer and solution

Correct answer: B. $1$

$\displaystyle\int_{0}^{\pi/2} \sin x\,dx = [-\cos x]_{0}^{\pi/2} = -\cos(\pi/2) + \cos(0) = -0 + 1 = 1$. Geometric interpretation: this is the area under one quarter of the sine curve from $0$ to $\pi/2$, which equals exactly $1$ in standard units.

Open this question →

How to practise

  1. Choose a Mathematics question and try to work out your answer.
  2. Open the question to check the correct answer and read its worked solution.
  3. Continue in the app for bite-sized lessons, explanations and review with Quill. You can start without creating an account.