Scottish National 5 (Nat 5) Mathematics Flashcards

24 sample cards from the Mathematics deck

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

1. How do you use the converse of Pythagoras to check if a triangle is right-angled?

   If $a^{2} + b^{2} = c^{2}$ (longest side $c$), the triangle is right-angled; if $a^{2}+b^{2} \neq c^{2}$ it is not

2. State the formula for the length of an arc subtending angle $\theta$ in a circle of radius $r$.

   Arc length $= \frac{\theta}{360} \times 2\pi r$ (i.e. $\pi d$)

3. State the formula for the area of a sector with angle $\theta$ and radius $r$.

   Sector area $= \frac{\theta}{360} \times \pi r^{2}$

4. State the formula for the volume of a sphere of radius $r$.

   $V = \frac{4}{3}\pi r^{3}$

5. State the formula for the volume of a cone with base radius $r$ and height $h$.

   $V = \frac{1}{3}\pi r^{2} h$

6. State the formula for the volume of a cylinder with radius $r$ and height $h$.

   $V = \pi r^{2} h$

7. State the formula for the volume of a pyramid with base area $A$ and height $h$.

   $V = \frac{1}{3} A h$

8. What is the relationship between a tangent to a circle and the radius at the point of contact?

   The tangent is perpendicular ($90^{\circ}$) to the radius drawn to the point of contact

9. What angle is in a semicircle (angle subtended by a diameter)?

   An angle in a semicircle is a right angle, $90^{\circ}$

10. In a circle, how does a perpendicular from the centre relate to a chord?

    The perpendicular from the centre to a chord bisects the chord (cuts it exactly in half)

11. For linear scale factor $k$, what are the area and volume scale factors of similar shapes?

    Area scale factor $= k^{2}$; volume scale factor $= k^{3}$

12. What condition makes two triangles similar?

    Their corresponding angles are equal (equiangular), so corresponding sides are in the same ratio

13. In a right-angled triangle, define $\sin\theta$, $\cos\theta$ and $\tan\theta$ (SOH CAH TOA).

    $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$

14. How do you find an unknown angle when you know two sides, e.g. opposite and hypotenuse?

    Use the inverse: $\theta = \sin^{-1}\!\left(\frac{\text{opp}}{\text{hyp}}\right)$ (similarly $\cos^{-1}$ or $\tan^{-1}$)

15. State the Sine Rule.

    $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

16. State the Cosine Rule for finding a side.

    $a^{2} = b^{2} + c^{2} - 2bc\cos A$

17. State the Cosine Rule rearranged to find an angle.

    $\cos A = \frac{b^{2} + c^{2} - a^{2}}{2bc}$

18. State the formula for the area of a triangle using two sides and the included angle.

    Area $= \frac{1}{2}ab\sin C$

19. For $y = a\sin(bx)$, what do $a$ and $b$ represent?

    $a$ is the amplitude (max height); $b$ is the number of complete cycles (periods) in $360^{\circ}$

20. State the two key trigonometric identities for National 5.

    $\sin^{2}x + \cos^{2}x = 1$ and $\tan x = \frac{\sin x}{\cos x}$

21. Which two averages and measures of spread are used to compare data sets at Nat 5?

    Compare a measure of centre (mean or median) and a measure of spread (range, interquartile range, or standard deviation)

22. State a formula for the standard deviation of a sample of $n$ values.

    $s = \sqrt{\dfrac{\sum (x - \bar{x})^{2}}{n - 1}}$ (equivalently $s = \sqrt{\dfrac{\sum x^{2} - \frac{(\sum x)^{2}}{n}}{n-1}}$)

23. On a scattergraph, what do positive, negative and no correlation look like?

    Positive: points rise left to right; negative: points fall left to right; no correlation: no clear pattern

24. How do you add the vectors $\vec{u} = \begin{pmatrix} a \\ b \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} c \\ d \end{pmatrix}$, and find the magnitude of $\vec{u}$?

    $\vec{u} + \vec{v} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$; magnitude $|\vec{u}| = \sqrt{a^{2} + b^{2}}$

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