Pure Maths
Area of a sector $= \dfrac{1}{2}r^2\theta$ Arc length $= r\theta$ $\theta$ must be in radians Degrees to radians: multiply by $\pi$, divide by $180$ Radians to degrees: multiply by $180$, divide by $\pi$ Question 1 Show Answers a) $3\pi$ cm b) $5\pi$ cm c)
Pure Maths
To integrate: raise the power by one, then divide the coefficient by the new power. $\displaystyle\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C$ Using limits gives the area between the curve and the $x$-axis. Question 1 Integrate the following expressions with respect to $x$
Pure Maths
Sequences $n$: the position of a term · $a$: first term · $d$: common difference · $r$: common ratio Question 1 Write down the first four terms of each sequence. a) $u_n = u_{n-1} + 4, \quad n > 1, \quad u_1 = 3$ b) $u_n = 3u_{n-1} + 1, \quad
Pure Maths
To differentiate $y = ax^n$: multiply the coefficient by the power and subtract one from the power, giving $\displaystyle\frac{dy}{dx} = nax^{n-1}$. Question 1 Differentiate the following expressions with respect to $x$. a) $y = x^6 - 7x^2$ b) $y = 1 - 6x^{\frac{5}{2}}$ c) $y
Pure Maths
Question 1 $(a + b)^n = a^n + \dbinom{n}{1}a^{n-1}b + \cdots + \dbinom{n}{r}a^{n-r}b^r + \cdots + b^n$ Find the first four terms in the expansion in ascending powers of $x$ of (a) $(1 + x)^{10}$ (b) $(1 + 2x)^8$ (c) $\left(
Pure Maths
Question 1 Divide the cubic by the linear factor, and fully factorise the cubic if possible. (a) $x^3+2x^2-x-2$ by $x+1$ (b) $x^3+2x^2-9x+2$ by $x-2$ (c) $20+x+3x^2+x^3$ by $x+4$ (d) $2x^3-x^
Pure Maths
Worked Example $\sin x° = 0.4$, $0 \leq x \leq 360°$ $x = \sin^{-1}(0.4)$ $x = 23.6°, 156.4°$ Question 1 Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate. a) $\sin x° = 0.27$
Pure Maths
Key Formulae $\displaystyle\int e^{kx}\,dx = \frac{1}{k}e^{kx} + c$ $\displaystyle\int \sin(kx)\,dx = -\frac{1}{k}\cos(kx) + c$ $\displaystyle\int \cos(kx)\,dx = \frac{1}{k}\sin(kx) + c$ $\displaystyle\int \frac{1}{kx+c}\,dx = \frac{1}{k}\ln|kx+c| + C$ Question
Pure Maths
Sketching Quadratics Question 1 Sketch each curve showing the coordinates of any points of intersection with the coordinate axes. (a) $y=x^2-3x+2$ (b) $y=x^2+5x+6$ (c) $y=x^2-9$ (d) $y=x^2-2x$ (e) $y=x^2-10x+25$ (f) $y=2x^
Pure Maths
Co-ordinates Gradient $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$ Worked Example Find the gradient of the line segment joining $(2,3)$ and $(6,11)$. $m = \dfrac{11-3}{6-2} = \dfrac{8}{4} = 2$ Question 1 Find the gradient of the line segment joining each
Pure Maths
Straight-line graphs The equation of a straight line is always in the form y=mx+c m: the gradient c: the y-intercept To sketch a straight-line graph, draw a line through the y-intercept and the x-intercept. Circles The equation of a circle is always in
Pure Maths
Question 1 Question 2 Question 3 Question 4 Question 5