ZIMSEC A Level Additional Mathematics Paper 1 Past Papers

Topics in this paper

Series and Proof by Induction Integration and Reduction Formulae Vectors Groupsdifferential equationsmatricesmatrices and transformationspartial fractionshyperbolic functions

Explained questions

The position vectors of the points $A$ , $B$ and $C$ relative to the origin $O$ are $a = 3i - j$ , $b = 2i + j + 3k$ and $c = i - j + 6k$ . Find the value of $a \cdot (b \times c)$ .
For three non-zero vectors $a$ , $b$ and $c$ , the scalar triple product $a \cdot (b \times c)$ measures which quantity?
Let $y = \sinh^{-1} 2x$ . Writing $\sinh y$ in exponential form and clearing the fraction gives which equation in $e^y$ ?
Given that $y = \sinh^{-1} x$ , express $y$ in logarithmic form.
The standard result is $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$ . Which expression gives $\sum_{r=k+1}^{3k} r^3$ ?
Using $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$ , write $\sum_{r=1}^{3k} r^3$ in terms of $k$ in its simplest form.
Given that $\sum_{r=k+1}^{3k} r^3 = k^2(4k+1)(5k+2)$ for $k \geq 1$ , evaluate $21^3 + 22^3 + 23^3 + \ldots + 60^3$ .
The set of positive rational numbers forms a group under the operation $a * b = \frac{ab}{10}$ . Find the identity element.
Under the operation $a * b = \frac{ab}{10}$ on the positive rational numbers, write the inverse of an element $a$ in terms of $a$ .
The positive rational numbers form a group under $a * b = \frac{ab}{10}$ . Which group axiom fails if the set is narrowed to the positive integers?
The set $G = \{0, 1, 2\}$ forms a group under addition modulo 3. State the identity element.
In the group $G = \{0, 1, 2\}$ under addition modulo 3, find the inverse of the element 2.
The set $G = \{0, 1, 2\}$ under addition modulo 3 is shown to be abelian. Which observation establishes that?
Find the value of $10^n + 12\left(4^{n+1}\right) + 5$ when $n = 1$ .
For $f(n) = 10^n + 12\left(4^{n+1}\right) + 5$ , the difference $f(n+1) - f(n)$ simplifies to which expression?
An induction proof shows that $10^n + 12\left(4^{n+1}\right) + 5$ is divisible by 9 for every positive integer $n$ . What does the inductive step assume?
For $I_n = \int \sin^n ax\, dx$ , where $n$ is a positive integer and $a$ is a constant, which reduction formula is correct?
Write $\sin^2 \theta$ in terms of $\cos 2\theta$ .
Find $\int \sin^2 3x\, dx$ .
Find $\int \sin^4 3x\, dx$ .
Given that $y = \sinh^{-1} x$ , find $\frac{dy}{dx}$ .
Express $4x^2 - 4x + 5$ in the form $(ax + b)^2 + c$ , where $a$ , $b$ and $c$ are integers.
A line has direction $d = 3i - k$ and passes through the point $(8, 1, -1)$ . A plane has equation $r \cdot n = 5$ , where $n = -i + 10j - 3k$ . Which pair of checks shows that the line lies in the plane?
Find the angle between the planes $r \cdot (-i + 10j - 3k) = 5$ and $r \cdot (i + j + 3k) = 6$ .

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A Level Additional Mathematics Paper 1 Past Papers, 2010 to 2010

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