Maths AA SL practice questions

Let $f(x) = 3x$, $g(x) = 2x - 5$ and $h(x) = (f \circ g)(x)$.

(a) Find $h(x)$.[2]

(b) Find $h^{-1}(x)$.[3]

An arithmetic sequence is given by $5, 8, 11, \ldots$

(a) Write down the value of $d$.[1]

(b) Find

(i) $u_{100}$;[3]

(ii) $S_{100}$.

(c) Given that $u_n = 1502$, find the value of $n$.[2]

(a) Find $\int (6x + 7)dx$[3]

(b) Given $f'(x) = 6x + 7$ and $f(1.2) = 7.32$, find $f(x)$[3]

The $n^{\text{th}}$ term of an arithmetic sequence is given by $u_n = 15 - 3n$.

(a) State the value of the first term, $u_1$.[1]

(b) Given that the $n^{\text{th}}$ term of this sequence is $-33$, find the value of $n$.[2]

(c) Find the common difference, $d$.[2]

Two functions, $f$ and $g$, are defined in the following table.

(a) Write down the value of $f(1)$.[1]

(b) Find the value of $(g \circ f)(1)$.[2]

(c) Find the value of $g^{-1}(-2)$.[2]

Consider a function $f(x)$ such that $\int_1^6 f(x)\mathrm{d}x = 8$.

(a) Find $\int_1^6 2f(x)\mathrm{d}x$.[2]

(b) Find $\int_1^6 (f(x)+2)\mathrm{d}x$.[4]

Let $f(x) = 8x + 3$ and $g(x) = 4x$, for $x \in \mathbb{R}$.

(a) Write down $g(2)$.[1]

(b) Find $(f \circ g)(x)$.[2]

(c) Find $f^{-1}(x)$.[2]

The second term of an arithmetic sequence is 10 and the fourth term is 22.

(a) Find the value of the common difference.[2]

(b) Find an expression for $u_n$, the $n$th term.[2]

Write each of the following expressions in the form $\ln k$, where $k \in \mathbb{Z}^+$.

(a) $\ln 3 + \ln 4$[1]

(b) $3 \ln 2$[2]

(c) $-\ln \frac{1}{2}$[2]

Consider the following set of ordered data.

(a) Write down[3]

(i) the mode;

(ii) the range;

(iii) the median.

(b) Find the interquartile range.[3]

In an arithmetic sequence, $u_1 = 2$ and $u_3 = 8$.

(a) Find $d$.[2]

(b) Find $u_{20}$.[2]

(c) Find $S_{20}$.[2]

Let $f(x) = 3x^2$. The graph of $f$ is translated 1 unit to the right and 2 units down. The graph of $g$ is the image of the graph of $f$ after this translation.

(a) Write down the coordinates of the vertex of the graph of $g$. [2 marks]

(b) Express $g$ in the form $g(x) = 3(x - p)^2 + q$. [2 marks]

The graph of $h$ is the reflection of the graph of $g$ in the $x$-axis.

(c) Write down the coordinates of the vertex of the graph of $h$. [2 marks]

In an arithmetic sequence, $u_2 = 5$ and $u_3 = 11$.

(a) Find the common difference.[2]

(b) Find the first term.[2]

(c) Find the sum of the first 20 terms.[2]

The first three terms of an arithmetic sequence are $5, 6.7, 8.4$.

(a) Find the common difference.[2]

(b) Find the $28^{\text{th}}$ term of the sequence.[2]

(c) Find the sum of the first 28 terms.[2]

In an arithmetic sequence, $u_1 = -5$ and $d = 3$.

(a) Find $u_8$.[2]

(b) Find the value of $n$ for which $u_n = 67$.[4]