Practice Paper 1 (Cambridge International IGCSE Further Maths)

Write $2x^{2}+3x-4$ in the form $a(x+b)^{2}+c$, where $a,b$ and $c$ are constants.

Extract:

Write 2x² + 3x - 4 in the form a(x + b)² + c, where a, b and c are constants.

Answer: 2(x + 3/4)² - 41/8

Hence write down the coordinates of the stationary point on the curve $y=2x^{2}+3x-4$.

Extract:

2(x + 3/4)² - 41/8

On the axes below, sketch the graph of $y = \left| 2 x^{2} + 3 x - 4 \right|$, showing the exact values of the intercepts of the curve with the coordinate axes.

Extract:

The minimum point on the curve y = 2x² + 3x - 4 is (-3/4, -41/8).

The graph of y = |2x² + 3x - 4| is a parabola where everything below the x-axis is reflected above the x-axis. The x-intercepts are ((-3 + √41)/4, 0) and ((-3 - √41)/4, 0), and the y-intercept is (0, 4).

Find the value of $k$ for which $\left| 2 x^{2} + 3 x - 4 \right| = k$ has exactly 3 values of $x$.

$p(x)=6x^{3}+ax^{2}+12x+b$, where $a$ and $b$ are integers.

$p(x)$ has a remainder of 11 when divided by $x-3$ and a remainder of $-21$ when divided by $x+1$. Given that $p(x)=(x-2)Q(x)$, find $Q(x)$, a quadratic factor with numerical coefficients.

Extract:

p(x) = 6x³ + ax² + 12x + b, where a and b are integers.

p(x) has a remainder of 11 when divided by x - 3 and a remainder of -21 when divided by x + 1.

Given that p(x) = (x - 2)Q(x), find Q(x), a quadratic factor with numerical coefficients.

Using remainder theorem and solving simultaneous equations: a = -23 and b = 20.

Therefore p(x) = 6x³ - 23x² + 12x + 20

Using algebraic long division:

p(x) = (x - 2)(6x² - 11x - 10)

Hence solve $p(x)=0$.

Two functions are given by $\mathrm{f} ( x ) = 1 + 8 x$ where $x\inℝ$ and $\mathrm{g} ( x ) = 2 + \ln x$ where $x>0$.

Find an expression for $\mathrm{f}^{- 1} ( x )$.

State the geometric relationship between the graph of $y = \mathrm{f} ( x )$ and the graph of $y = \mathrm{f}^{- 1} ( x )$.

Find $fg^{2} ( 1 )$.

Explain why the function $gf ( x )$ does not exist.

Write the expression $\mathrm{log}_{a}9+(\mathrm{log}_{a}b)(\mathrm{log}_{\sqrt{b}}9a)$ in the form $c+d\mathrm{log}_{a}9$, where $c$ and $d$ are integers.

Write $9x^{2}-12x+5$ in the form $p(x-q)^{2}+r$, where $p$,$q$ and $r$ are constants.

Extract:

Write 9x² - 12x + 5 in the form p(x - q)² + r, where p, q and r are constants.

Hence write down the coordinates of the minimum point of the curve $y=9x^{2}-12x+5$.

The line $y=5x+6$ meets the curve $xy=8$ at the points $A$ and $B$. Find the coordinates of $A$ and of $B$.

A particle, $P$, moves in a straight line such that its displacement from a fixed point at time $t$ is s.

The acceleration of $P$ is given by $( 2 t + 4 )^{- 1 \frac{1}{2}} , \text{for} t > 0$

(i) Given that $P$ has a velocity of 9 when $t=6$, find the velocity of $P$ at time $t$.

(ii) Given that s = $\frac{1}{3}$ when $t=6$, find the displacement of $P$ at time $t$.

Given that $y=\mathrm{tan}x$, use calculus to find the approximate change in $y$ as $x$ increases from $-\frac{π}{4}$ to $h-\frac{π}{4}$, where $h$ is small.

A curve is such that $\frac{d^{2}y}{dx^{2}}=5\mathrm{cos}2x$. This curve has a gradient of $\frac{3}{4}$ at the point ( - pi / 12 , (5 pi)/(4) ). Find the equation of this curve.

Construct the graph of $x^{2}+y^{2}=9$

Extract:

Construct the graph of space x squared space plus space y squared equals space 9

By drawing the line $x+y=1$ on the grid, solve the equations $x^{2}+y^{2}=9\,x+y=1$

Variables $x$ and $y$ are such that, when fourth root of y is plotted against $\frac{1}{x}$, a straight line graph passing through the points (0.5, 9) and (3, 34) is obtained. Find $y$ as a function of $x$.

The 7th and 10th terms of an arithmetic progression are 158 and 149 respectively.

Find the common difference and the first term of the progression.

Extract:

The 7th and 10th terms of an arithmetic progression are 158 and 149 respectively.

The common difference d = -3 and the first term a = 176.

Find the least number of terms of the progression for their sum to be negative.