Number and algebra: 247 IB Maths AA HL practice questions

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]

Consider two consecutive positive integers, $n$ and $n+1$.

Show that the difference of their squares is equal to the sum of the two integers.[4]

Consider the complex numbers $z_1 = \cos \frac{11\pi}{12} + i\sin \frac{11\pi}{12}$ and $z_2 = \cos \frac{\pi}{6} + i\sin \frac{\pi}{6}$.

(a)(i) Find $\frac{z_1}{z_2}$[1]

(ii) Find $\frac{z_2}{z_1}$[3]

(b) $0, \frac{z_1}{z_2}$ and $\frac{z_2}{z_1}$ are represented by three points O, A and B respectively on an Argand diagram. Determine the area of the triangle OAB.[2]

Consider the arithmetic sequence $8, 26, 44, \ldots$

(a) Find an expression for the $n^{\text{th}}$ term.[1]

(b) Write down the sum of the first $n$ terms using sigma notation.[1]

(c) Calculate the sum of the first 15 terms.[2]

A set of positive integers $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is used to form a pack of nine cards. Each card displays one positive integer without repetition from this set. Grace wishes to select four cards at random from this pack of nine cards.

(a) Find the number of selections Grace could make if the largest integer drawn among the four cards is either a 5, a 6 or a 7.[3]

(b) Find the number of selections Grace could make if at least two of the four integers drawn are even.[4]

When the polynomial $3x^3 + ax + b$ is divided by $(x - 2)$, the remainder is 2, and when divided by $(x + 1)$, it is 5. Find the value of $a$ and the value of $b$.[5]

Consider the infinite series $$\sum_{k=0}^{\infty}\left(\frac{53}{1000}\right)\left(\frac{1}{100}\right)^k$$

(a) Find the exact value of $S_{\infty}$.[3]

Let $0.\overline{253}$ represent the repeating decimal $0.253253253\ldots$

(b) Use your answer from part (a) to express $0.\overline{253}$ as a fraction such that the numerator and denominator have no common factors.[2]

Prove that $(3n + 2)^2 - (3n - 2)^2$ is a multiple of 12 for all $n \in \mathbb{Z}^+$.[4]

Bob invests 1000 dinar in an account which pays a nominal annual interest rate of 4% compounded quarterly.

The amount of money in the account after one complete year can be written as $1000(1 + k)^4$ where $k \in \mathbb{Q}$.

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

(b) Expand and simplify $(1 + x)^4$.[2]

(c) Hence or otherwise, find the amount of money in the account after one complete year, giving your answer correct to the nearest dinar.[4]

(a) Write down and simplify the expansion of $(2 + x)^4$ in ascending powers of $x$.[3]

(b) Hence find the exact value of $(2.1)^4$.[3]

The following system of equations represents three planes in space.

$$x + 3y + z = -1$$

$$x + 2y - 2z = 15$$

$$2x + y - z = 6$$

Find the coordinates of the point of intersection of the three planes.[6]

Solve the equation $\log_2(x + 3) + \log_2(x - 3) = 4$.[5]

It is given that $z = 5 + qi$ satisfies the equation $z^2 + iz = -p + 25i$, where $p, q \in \mathbb{R}$.

Find the value of $p$ and the value of $q$.[5]

Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3$, $x - y + z = 5$ and $x + y + 2z = 6$.[5]

The sum of the first $n$ terms of an arithmetic sequence is given by $S_n = pn^2 - qn$, where $p$ and $q$ are positive constants.

It is given that $S_4 = 40$ and $S_5 = 65$.

(a) Find the value of $p$ and the value of $q$.[5]

(b) Find the value of $u_5$.[2]