Cambridge Pre-U Mathematics (Principal Subject) Flashcards

24 sample cards from the Mathematics (Principal Subject) deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the addition rule of probability for two events $A$ and $B$.

   $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ For mutually exclusive events $P(A\cap B)=0$.

2. Define conditional probability and state the multiplication rule.

   $$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A\mid B)\,P(B)$$ Independent events: $P(A\cap B)=P(A)P(B)$.

3. For a discrete random variable $X$, give the formulas for the expected value $E(X)$ and variance $\operatorname{Var}(X)$.

   $$E(X)=\sum x\,P(X=x),\qquad \operatorname{Var}(X)=E(X^{2})-[E(X)]^{2}=\sum x^{2}P(X=x)-\mu^{2}$$

4. State the mean and variance of a binomial distribution $X\sim B(n,p)$, and its probability function.

   $P(X=r)=\binom{n}{r}p^{r}(1-p)^{n-r}$; mean $E(X)=np$; variance $\operatorname{Var}(X)=np(1-p)$.

5. What are the key properties of the normal distribution $N(\mu,\sigma^{2})$ regarding shape and the 68-95-99.7 rule?

   Bell-shaped, symmetric about $\mu$. Approximately 68% of data lies within $\mu\pm\sigma$, 95% within $\mu\pm 2\sigma$, and 99.7% within $\mu\pm 3\sigma$.

6. How do you standardise a normal variable $X\sim N(\mu,\sigma^{2})$ to the standard normal $Z$?

   $$Z=\frac{X-\mu}{\sigma},\qquad Z\sim N(0,1)$$

7. In hypothesis testing, define the null hypothesis $H_0$, the alternative $H_1$, and the significance level.

   $H_0$: the default/no-effect claim being tested. $H_1$: the alternative claim. The significance level (e.g. 5%) is the probability of rejecting $H_0$ when it is true (Type I error), defining the critical region.

8. Distinguish a Type I error from a Type II error in hypothesis testing.

   Type I error: rejecting $H_0$ when it is actually true (false positive). Type II error: failing to reject $H_0$ when it is actually false (false negative).

9. Define the distribution of the sample mean $\bar{X}$ from a population $N(\mu,\sigma^{2})$ of sample size $n$.

   $$\bar{X}\sim N\!\left(\mu,\frac{\sigma^{2}}{n}\right)$$ so the standard error is $\dfrac{\sigma}{\sqrt{n}}$.

10. Define displacement, velocity and acceleration in terms of derivatives with respect to time.

    Velocity $v=\dfrac{dx}{dt}$ (rate of change of displacement); acceleration $a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{dt^{2}}$.

11. State the four SUVAT equations of motion for constant acceleration.

    $$v=u+at,\quad s=ut+\tfrac{1}{2}at^{2},\quad v^{2}=u^{2}+2as,\quad s=\tfrac{1}{2}(u+v)t$$

12. State Newton's three laws of motion.

    1st: a body remains at rest or moves with constant velocity unless acted on by a resultant force. 2nd: $F=ma$ (resultant force = mass × acceleration). 3rd: every action has an equal and opposite reaction.

13. What is the limiting friction model for a body on the point of moving?

    Friction $F\leq\mu R$, where $\mu$ is the coefficient of friction and $R$ is the normal reaction. At the point of slipping (limiting), $F=\mu R$.

14. For two particles connected by a light inextensible string over a smooth pulley, what is true of their accelerations and the tension?

    Both particles have the same magnitude of acceleration, and the tension is the same throughout the string. Apply $F=ma$ to each particle separately and solve simultaneously.

15. Define linear momentum and state the impulse-momentum relationship.

    Momentum $p=mv$. Impulse $=Ft=$ change in momentum: $$Ft=mv-mu$$

16. State the principle of conservation of momentum for a collision between two bodies.

    Total momentum before = total momentum after (no external forces): $$m_1u_1+m_2u_2=m_1v_1+m_2v_2$$

17. Give the formulas for work done, kinetic energy and gravitational potential energy.

    Work $W=Fs\cos\theta$; kinetic energy $E_k=\tfrac{1}{2}mv^{2}$; gravitational PE $E_p=mgh$.

18. State the work-energy principle and define power.

    Work-energy principle: the work done by the resultant force equals the change in kinetic energy. Power = rate of doing work: $P=\dfrac{W}{t}=Fv$.

19. For a projectile launched with speed $u$ at angle $\theta$, give the horizontal and vertical components of velocity and the equations of motion.

    Horizontal: $u_x=u\cos\theta$ (constant), $x=u\cos\theta\,t$. Vertical: $u_y=u\sin\theta$, $y=u\sin\theta\,t-\tfrac{1}{2}gt^{2}$, with acceleration $-g$.

20. Derive the formulas for the time of flight and range of a projectile launched at angle $\theta$ on level ground with speed $u$.

    Time of flight $T=\dfrac{2u\sin\theta}{g}$; range $R=\dfrac{u^{2}\sin 2\theta}{g}$ (maximum when $\theta=45^{\circ}$).

21. What is the maximum height reached by a projectile launched with speed $u$ at angle $\theta$?

    $$H=\frac{u^{2}\sin^{2}\theta}{2g}$$ reached when the vertical velocity is zero.

22. How can you tell whether a function $f$ has an inverse, and how do their graphs relate?

    $f$ has an inverse if and only if it is one-to-one (injective). The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$, and the domain and range are swapped.

23. Define the modulus function $|x|$ and describe how to solve $|f(x)|=k$.

    $|x|=x$ if $x\geq 0$ and $-x$ if $x<0$. To solve $|f(x)|=k$ ($k\geq0$): solve $f(x)=k$ and $f(x)=-k$, then check solutions.

24. Give the exact derivative results for $\tan x$ and $\sqrt{x}$.

    $\dfrac{d}{dx}\tan x=\sec^{2}x$; $\dfrac{d}{dx}\sqrt{x}=\dfrac{d}{dx}x^{1/2}=\dfrac{1}{2\sqrt{x}}$.

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