CAIE IGCSE Additional Math Calculus Questions

Attempts to find $y=4 x-5 x^{-2}$
at least one term correct

$\frac{\mathrm{d} y}{\mathrm{~d} x}=4+10 x^{-3}$
Correct completion to given answer $\frac{2\left(2 x^{3}+5\right)}{x^{3}}$
Alternative version

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{12 x^{4}-8 x^{4}+10 x}{x^{4}}$ or equivalent
unsimplified form

M1 for an attempt at the quotient rule with correct structure
$$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x^{2}\left(\text { their } 12 x^{2}\right)-\left(4 x^{3}-5\right)(\text { their } 2 x)}{\left(x^{2}\right)^{2}}$$

Correct completion to given answer $\frac{2\left(2 x^{3}+5\right)}{x^{3}}$

[When x=5 ] y=9
$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{3}(5 x+2)^{-\frac{1}{3}} \times 5$ oe
B1 for $\frac{\mathrm{d} y}{\mathrm{~d} x}=k(5 x+2)^{-\frac{1}{3}}, k \neq \frac{10}{3}$$\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}=\frac{10}{9}$
FT their $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}$ providing previous B1
awarded

Equation of normal e.g.
$$y-\text { their } 9=-\frac{1}{\text { their } \frac{10}{9}}(x-5) \text { oe, soi }$$
FT $\frac{-1}{\text { their }\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}}$ and their y-coordinate

Eliminates one variable e.g.
$$11-x-\text { their } 9=-\frac{1}{\text { their } \frac{10}{9}}(x-5)$$
dep on previous M mark

For Q: x=-25, y=36
A1 for each

Coordinates of R:(35,-18)
or ((10-their(-25)), (18-their36))
FT their coordinates of Q

$\frac{dy}{dx}=1-\frac{2x+2}{x^2+2x}$
B1 for $\frac{d(-\ln(x^2+2x))}{dx}=\frac{1}{x^2+2x}\times f(x)$ soi.
Equates their first derivative to 0 and simplifies as far as 2(x+1)=x(x+2) oe
M1 FT their $\frac{dy}{dx}$.
$x^2-2=0$ or $x^2=2$$x=\sqrt{2}$ [$x=-\sqrt{2}$]
Justification that the negative solution should be rejected, e.g. $x=-\sqrt{2}$ gives $y=2-\ln(2-2\sqrt{2})$, which is impossible.

Correct use of $x^{2} h=5$ to find an expression that
can be used to eliminate h
M1 for $x^{2} h=5$ soi

Surface area: $x^{2}+4 x\left(\frac{5}{x^{2}}\right)$ oe
Derivative of the surface area: $2 x-20 x^{-2}$ oe
FT their surface area of form $a x^{2}+\frac{b}{x}$

Equates their $2 x-20 x^{-2}$ to 0 and solves to find
a value of x
FT their derivative of form $a x+\frac{b}{x^{2}}$ oe

x=2.15 or $2.154[\ldots] h=1.08$ or $1.077[\ldots]$ nfww
x=2.15 or $2.154[\ldots]$
h=1.08 or $1.077[\ldots]$ nfww
dep on all previous marks awarded

Alternative

Correct use of $x^{2} h=5$ to find an expression that
can be used to eliminate x
M1 for $x^{2} h=5$ soi

Surface area: $\left(\sqrt{\frac{5}{h}}\right)^{2}+4 h \times \sqrt{\frac{5}{h}}$ oe
Derivative of the surface area: $4 \sqrt{5} \times \frac{1}{2} h^{-\frac{1}{2}}-\frac{5}{h^{2}}$
oe
FT their surface area of form $\frac{a}{h}+b \sqrt{h}$

Equates their $4 \sqrt{5} \times \frac{1}{2} h^{-\frac{1}{2}}-\frac{5}{h^{2}}$ to 0 and solves
to find a value of h
FT their derivative of form $\frac{a}{\sqrt{h}}+\frac{b}{h^{2}}$ oe

h=1.08 or $1.077[\ldots] x=2.15$ or $2.154[\ldots]$ nfww
h=1.08 or $1.077[\ldots]$
x=2.15 or $2.154[\ldots]$ nfww
dep on all previous marks awarded