Formulae and changing the subject

(a) 2 hours

1. $T = 40 \times 2.5 + 20 = 100 + 20 = 120$ minutes.
2. 120 minutes is 2 hours.

• M1 Working out $40 \times 2.5 + 20 = 120$.
• A1 2 hours (120 minutes).

(a) $x = (y + 8)/5$

1. $$y+8=5x$$
2. Add 8 to both sides: y + 8 = 5x.
3. Divide both sides by 5: x = (y + 8)/5.

• P1 Establishing $y+8=5x$ or an equivalent valid method.
• A1 Correct answer: $x = (y + 8)/5$

(a) $a=P/(2+b)$

1. Factor a from both terms.
   $$P=a(2+b)$$
2. Therefore $a=P/(2+b)$.

• M1 Factor a from both terms.
• A1 Correct answer: $a=P/(2+b)$

(b) $7$

1. Substitute into the rearranged formula.
   $$49/(2+5)$$
2. Therefore $7$.

• M1 Substitute into the rearranged formula.
• A1 Correct answer: $7$

(a) $r=\sqrt{\frac{V}{\pi \times h}}$

1. Divide by the complete coefficient of r squared.
   $$r^{2}=V/(\pi \times h)$$
2. Therefore $r=\sqrt{\frac{V}{\pi \times h}}$.

• M1 Divide by the complete coefficient of r squared.
• A1 Correct answer: $r=\sqrt{\frac{V}{\pi \times h}}$

(b) $2$

1. Cancel pi and the height before taking the positive square root.
   $$\sqrt{20/5}$$
2. Therefore $2$.

• M1 Cancel pi and the height before taking the positive square root.
• A1 Correct answer: $2$

(a) $s = \dfrac{v^2 - u^2}{2a}$

1. Subtract $u^2$, then divide by $2a$.
   $$v^2 - u^2 = 2as \Rightarrow s = \frac{v^2 - u^2}{2a}$$

• M1 Subtracting $u^2$: $v^2 - u^2 = 2as$.
• A1 $s = \frac{v^2 - u^2}{2a}$.

(b) $r = \sqrt{\dfrac{A}{\pi}}$

1. Divide by $\pi$, then take the square root.
   $$r^2 = \frac{A}{\pi} \Rightarrow r = \sqrt{\frac{A}{\pi}}$$

• M1 Writing $r^2 = \frac{A}{\pi}$.
• A1 $r = \sqrt{\frac{A}{\pi}}$.

(a) $t = p/(3 + r)$

1. $$p=t(3+r)$$
2. Factorise the right side: p = t(3 + r).
3. Divide by 3 + r, which is non-zero: t = p/(3 + r).

• P1 Establishing $p=t(3+r)$ or an equivalent valid method.
• A1 Correct answer: $t = p/(3 + r)$

(a) $x=\frac{4+2y}{1-y}$

1. Multiply by the denominator and collect the x terms.
   $$yx+2y=x-4$$
2. Factor x and divide by its coefficient.
   $$x(1-y)=4+2y$$
3. Therefore $x=\frac{4+2y}{1-y}$.

• M1 Multiply by the denominator and collect the x terms.
• M1 Factor x and divide by its coefficient.
• A1 Correct answer: $x=\frac{4+2y}{1-y}$

(b) If y were 1, the original equation would require $x-4=x+2$, which is impossible.

1. If y were 1, the original equation would require $x-4=x+2$, which is impossible.

• C1 Correct conclusion with supporting reasoning: If y were 1, the original equation would require $x-4=x+2$, which is impossible.

(a) x = $\sqrt{}$((4y + 2)/(3 - y)), with −1/2 < y < 3.

1. $$(3-y)x^{2}=4y+2$$
2. $$3-14/(x^{2}+4)$$
3. Multiply by $x^{2}$ + 4: $yx^{2}$ + 4y = $3x^{2}$ $-$ 2.
4. Collect the $x^{2}$ terms: (3 $-$ y)$x^{2}$ = 4y + 2. Since x > 0, x = $\sqrt{}$[(4y + 2)/(3 $-$ y)].
5. Rewrite the original formula as y = 3 $-$ 14/($x^{2}$ + 4). For x > 0, the denominator is greater than 4, so 0 < 14/($x^{2}$ + 4) < 7/2.
6. Therefore $-1$/2 < y < 3. Conversely every y in this interval gives a positive x through the rearranged formula.

• P1 Establishing $(3-y)x^{2}=4y+2$ or an equivalent valid method.
• P1 Establishing $3-14/(x^{2}+4)$ or an equivalent valid method.
• A1 Correct answer: x = $\sqrt{}$((4y + 2)/(3 - y)), with −1/2 < y < 3.