IB Maths AA HL Topic 2: Functions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 1, no calculatorMedium7 marks

The graph in Figure 1 shows the straight line $\ell$

(a)

Write down the $y$ -intercept of $\ell$ and determine its gradient.

[2]

(b)

The line $m$ is parallel to $\ell$ and passes through $P(2,1)$ Determine the equation of $m$ in the form $y=mx+c$

[2]

(c)

The line $n$ is perpendicular to $\ell$ and passes through the point where $m$ intersects the $y$ -axis. Determine the equation of $n$ in the form $Ax+By+C=0$ where $A,B,C$ are integers with no common factor and $A>0$

[3]

Question 2

Paper 1, no calculatorMedium6 marks

Two analysts model the annual demand for a service in a region. The models predict demand, in thousands of requests per year, $t$ years from now:

A(t)=8+e^{t/5},\qquad B(t)=6e^{t/10},\qquad t\geq 0.

The following values are given, rounded to two decimal places.

$t$ (years)	0	5	7	8	10	13	14	15
$A(t)$	9.00	10.72	12.06	12.95	15.39	21.46	24.44	28.09
$B(t)$	6.00	9.89	12.08	13.35	16.31	22.02	24.33	26.89

(a)

At $t=0$ state which analyst predicts the greater annual demand.

[1]

(b)

Sketch the two models on the same axes for $0\leq t\leq15$ Use your sketch to estimate, to the nearest year, when the forecasts agree.

[2]

(c)

Determine analytically the exact times at which the two forecasts agree. Give your answers in years.

[3]

Question 3

Paper 1, no calculatorMedium6 marks

For real $p$ consider the equation

x^2-2px+p+2=0,\qquad x\in\mathbb R.

(a)

Determine the values of $p$ for which the equation has (i) two distinct real roots, (ii) two equal real roots, and (iii) no real roots.

[4]

(b)

Hence, when $p=3$ solve

x^2-2px+p+2\leq 0.

[2]

Question 4

Paper 1, no calculatorMedium7 marks

Let $f(x)=x^2-4x+3$ The graph $y=f(x)$ is stretched horizontally by scale factor $2$ Let the resulting graph be $y=F(x)$

(a)

Write down $F(x)$

[2]

(b)

The graph $y=F(x)$ is reflected in the $x$ -axis and then translated upwards by $3$ units, giving $y=g(x)$ Determine $g(x)$ in completed-square form and state the coordinates of its turning point.

[3]

(c)

Instead, starting from $y=F(x)$ translate it upwards by $3$ units and then reflect it in the $x$ -axis, giving $y=h(x)$ Hence, or otherwise, determine whether $h$ has any $x$ -intercepts. Give a reason.

[2]

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