0:00
A Level Pure 1
Select the skills to practice, and then click Go!
Surds and indices
Simplify a surd
\[ \sqrt{72} = 6\sqrt{2} \]
Simplify a surd by extracting the largest square factor.
Add or subtract surds
\[ 3\sqrt{5} + 2\sqrt{5} – \sqrt{5} \]
Add or subtract surds with the same root.
Rationalise single-term denominator
\[ \frac{6}{\sqrt{3}} = 2\sqrt{3} \]
Remove a single surd from a denominator.
Rationalise two-term denominator
\[ \frac{1}{3 + \sqrt{2}} \]
Use the conjugate to rationalise.
Evaluate a fractional index
\[ 27^{2/3} = 9 \]
Calculate the value of a fractional power.
Apply index laws to simplify
\[ \frac{x^4 \times x^3}{x^2} \]
Combine terms using laws of indices.
Write expressions using indices
\[ \frac{1}{x^3} = x^{-3} \]
Convert roots and reciprocals to index form.
Quadratics
Factorise a quadratic (a=1)
\[ x^2 – 5x + 6 \]
Factorise when the x² coefficient is 1.
Factorise a quadratic (a≠1)
\[ 2x^2 + 5x – 3 \]
Factorise when the x² coefficient is not 1.
Complete the square
\[ x^2 + 6x + 5 = (x+3)^2 – 4 \]
Rewrite a quadratic in completed square form.
Use the discriminant
\[ b^2 – 4ac \]
Determine the number of roots using b² – 4ac.
Solve using quadratic formula
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Apply the quadratic formula to find solutions.
Coordinate geometry
Find gradient between two points
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Calculate the gradient using the formula.
Find the equation of a line
\[ y – y_1 = m(x – x_1) \]
Write the equation given gradient and a point.
Find perpendicular gradient
\[ m_1 \times m_2 = -1 \]
Find the gradient of a perpendicular line.
Find midpoint of two points
\[ \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]
Calculate the midpoint using the formula.
Find distance between two points
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
Calculate distance using Pythagoras.
Circles
Write circle equation
\[ (x-a)^2 + (y-b)^2 = r^2 \]
Write the equation from centre and radius.
Find centre and radius
\[ (x-3)^2 + (y+4)^2 = 25 \]
Identify centre and radius from equation.
Complete square for circles
\[ x^2 + y^2 – 6x + 4y – 12 = 0 \]
Convert expanded form to standard form.
Verify point lies on circle
\[ \text{Show } (5,0) \text{ lies on…} \]
Substitute coordinates to verify.
Differentiation
Differentiate x^n
\[ \frac{d}{dx}(3x^5) = 15x^4 \]
Apply the power rule to a single term.
Differentiate a polynomial
\[ \frac{d}{dx}(2x^3 – 5x^2 + 3x – 7) \]
Differentiate each term of a polynomial.
Differentiate negative/fractional powers
\[ \frac{d}{dx}(4x^{-2}) = -8x^{-3} \]
Apply the power rule to any index.
Find gradient at a point
\[ \text{Find } \frac{dy}{dx} \text{ when } x = 2 \]
Evaluate the derivative at a given x-value.
Find stationary points
\[ \frac{dy}{dx} = 0 \]
Set derivative to zero and solve.
Find equation of tangent
\[ y = mx + c \]
Use differentiation to find the tangent line.
Integration
Integrate x^n
\[ \int 4x^3 \, dx = x^4 + C \]
Apply the power rule for integration.
Integrate a polynomial
\[ \int (3x^2 – 2x + 5) \, dx \]
Integrate each term of a polynomial.
Integrate negative/fractional powers
\[ \int x^{-2} \, dx = -x^{-1} + C \]
Apply integration to any index.
Evaluate a definite integral
\[ \int_1^3 2x \, dx \]
Calculate F(b) – F(a).
Find area under a curve
\[ \text{Area} = \int_a^b y \, dx \]
Use definite integration for area.
Logarithms
Convert index and log form
\[ 2^3 = 8 \Leftrightarrow \log_2(8) = 3 \]
Rewrite between index and logarithmic form.
Evaluate a logarithm
\[ \log_2(32) = 5 \]
Find the value of a logarithm.
Use laws of logarithms
\[ \log(3) + \log(4) = \log(12) \]
Apply log laws to simplify.
Solve exponential equations
\[ 3^x = 20 \]
Take logarithms of both sides.
Solve equations with logarithms
\[ \log_2(x) + \log_2(x-2) = 3 \]
Use log laws then convert to index form.
Binomial expansion
Expand (1 + x)^n
\[ (1 + x)^4 \]
Expand using Pascal’s triangle or nCr.
Find a specific coefficient
\[ \text{Coef of } x^3 \text{ in } (1+2x)^5 \]
Use the binomial formula for one term.
Expand (a + bx)^n
\[ (2 + 3x)^5 \]
Expand binomials where neither term is 1.
Timer (Optional)
0:00
Question