Fichas
WJEC Core 4 Maths - Key Facts
Fichas del conjunto (31)
- 1How would you split the following expression into partial fractions? \[\frac{f(x)}{(ax+b)(cx+d)}\]\[\frac{A}{ax+b}+\frac{B}{cx+d}\]
- 2How would you split the following expression into partial fractions? \[\frac{f(x)}{(ax+b)(cx+d)^2}\]\[\frac{A}{ax+b}+\frac{B}{cx+d}+\frac{C}{(cx+d)^2}\]
- 3\[\sin(A\pm B)=?\] (Given in formulae booklet)\[\sin(A\pm B)=\sin A \cos B \pm \cos A \sin B\]
- 4\[\cos(A\pm B)=?\] (Given in formulae booklet)\[\cos(A\pm B)=\cos A \cos B \mp \sin A \sin B\]
- 5\[\tan(A\pm B)=?\] (Given in formulae booklet)\[\tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
\] - 6\[\sin 2A = ?\]\[\sin 2A = 2\sin A \cos A\]
- 7Give the 3 identities for \(\cos 2A\)\[\begin{align*}
\cos 2A=&\cos^2 A - \sin^2 A \\
\cos 2A=&2\cos^2 A - 1 \\
\cos 2A=&1-2\sin^2 A
\end{align*}
\] - 8\[\tan 2A=?\]\[\tan 2A=\frac{2\tan A}{1-\tan^2 A}\]
- 9How do you find \[\int \sin^2 x \, dx \](1) use the identity \(\cos 2x = 1-2\sin^2 x\)
(2) rearrange to get \(\sin^2 x =\frac{1}{2}(1-\cos 2x)\)
(3) work out \(\int \frac{1}{2}(1-\cos 2x) \, dx
\) - 10How do you find \[\int \cos^2 x \, dx \](1) use the identity \(\cos 2x = 2\cos^2 x-1\)
(2) rearrange to get \(\cos^2 x =\frac{1}{2}(1+\cos 2x)\)
(3) work out \(\int \frac{1}{2}(1+\cos 2x) \, dx
\) - 11How do you find \[\int \tan^2 x \, dx \](1) use the identity \(1+\tan^2 x=\sec^2 x\)
(2) rearrange to get \(\tan^2 x=\sec^2 x-1\)
(3) work out \(\int \sec^2 x-1 \, dx
\) - 12What is meant by a Cartesian equation?An equation where the only variables are \(x\) and/or \(y\)
- 13When applying the \(R, \alpha\) method, if \(R\sin \alpha =a\) and \(R\cos \alpha =b\), how would you find \(R\) and \(\alpha\)?\[R=\sqrt{a^2+b^2}\] \[\alpha = \tan^{-1} \left( \frac{a}{b} \right)\]
- 14What is the greatest value of \(R\sin (x+\alpha)\) and find a value of \(x\) for which this occursGreatest value is \(R\).
Happens when \(x+\alpha=90^{\circ}\),
i.e. when \(x=90^{\circ} -\alpha\) - 15What is the least value of \(R\sin (x+\alpha)\) and find a value of \(x\) for which this occursLeast value is \(-R\).
Happens when \(x+\alpha=270^{\circ}\),
i.e. when \(x=270^{\circ} -\alpha\) - 16If \(x=f(t)\) and \(y=g(t)\), how would you find \(\frac{\text{d}y}{\text{d}x}\)?\[\frac{\text{d}y}{\text{d}x}=\frac{\left(\frac{\text{d}f}{\text{d}t}\right)}{\left(\frac{\text{d}g}{\text{d}t}\right)}\]
- 17If the rate of increase of \(P\) over time is directly proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\).\[\frac{\text{d}P}{\text{d}t}=k\times f(P)\]
- 18If the rate of decrease of \(P\) over time is directly proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\).\[\frac{\text{d}P}{\text{d}t}=-k\times f(P)\]
- 19If the rate of increase of \(P\) over time is inversely proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\).\[\frac{\text{d}P}{\text{d}t}=\frac{k}{f(P)}\]
- 20If the rate of decrease of \(P\) over time is inversely proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\).\[\frac{\text{d}P}{\text{d}t}=-\frac{k}{f(P)}\]
- 21What is the formula for the volume of revolution of a curve about the \(x\)-axis?\[V=\int_a ^b \pi y^2 \, \text{d}x\]
- 22What is the formula for integration by parts? (given in formulae book)\[\int u \frac{\text{d}v}{\text{d}x} \, \text{d}x=uv-\int v \frac{\text{d}u}{\text{d}x} \, \text{d}x\]
- 23\[\int \frac{1}{ax+b} \, \text{d}x=?\]\[\int \frac{1}{ax+b} \, \text{d}x=\frac{1}{a}\ln {|ax+b|}+c\]
- 24When \(n \neq -1\), \[\int (ax+b)^n \, \text{d}x=?\]\[\int (ax+b)^n \, \text{d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+c\]
- 25What is a unit vector?A vector of length \(1\)
- 26What is the magnitude of vector \(a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\)\[|a\mathbf{i}+b\mathbf{j}+c\mathbf{k}|=\sqrt{a^2+b^2+c^2}\]
- 27If vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, then...?\[\mathbf{a}=k\mathbf{b}\] They are multiples of each other
- 28If \(\mathbf{a}\) and \(\mathbf{b}\) are the position vectors of points \(\mathbf{A}\) and \(\mathbf{B}\), how would you find \(\mathbf{AB}\)?\[\mathbf{AB}=\mathbf{b}-\mathbf{a}\]
- 29A line passes through the point with position vector \(\mathbf{a}\) and is in the direction of \(\mathbf{b}\). What is the vector equation of the line?\[\mathbf{r}=a+\lambda \mathbf{b}\] (any parameter will do - here, I used \(\lambda \))
- 30What is the formula for the scalar product of vectors \(\mathbf{a}\) and \(\mathbf{b}\)?\[\mathbf{a} \cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta\] where \(\mathbf{a} \cdot \mathbf{b}=a_1b_1+a_2b_2+a_3b_3\)
- 31If vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular, \(\mathbf{a} \cdot \mathbf{b}=?\)\[\mathbf{a} \cdot \mathbf{b}=0\]
Información
- Descripción
- Key facts and formulae which must be known for the WJEC Core 4 examination.
- Creado por
- Daniel Cox
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