Cambridge International A & AS Level (Pure Mathematics 1)

Learn sign rules for integers: two same signs yield a positive result; two different signs yield a negative result, and apply these to add, subtract, multiply, and divide integers.

Solve past paper problems on indices, converting decimals to powers of ten, applying exponent rules for multiplication, division, zero and negative powers, and solving for unknowns by equating exponents.

Learn to convert statements into algebraic expressions by assigning a number x and applying operations like nine times x, subtracting, adding, multiplying, dividing, and using brackets.

Multiply algebraic expressions by applying distributive multiplication of a monomial with a binomial, multiplying coefficients and variables separately. Manage signs and negatives carefully as you practice more problems.

Group terms to factor by regrouping, then extract a common factor from each group and apply the distributive property to form factors such as x+1 or x-2.

Apply the middle term split method to factor quadratics, using ac and b to pick two numbers, split the middle term, and factor by grouping into linear factors.

Derive the quadratic formula from ax^2+bx+c=0, learn the discriminant b^2-4ac, and identify conditions for two real roots, one repeated root, or no real roots.

Sketch quadratic graphs by completing the square: convert to completed square form, identify the vertex (h, k), axis of symmetry x = h, and intercepts.

Graph quadratic inequalities by sketching the curve, identifying the vertex and intercepts; determine the solution sets for y>0 and y≤0, using three examples.

Apply the wavy curve method to solve quadratic inequalities by factoring x^2 - a^2 into (x - a)(x + a), and analyzing the intervals divided by -a and a.

Explore completing the square and discriminant methods for quadratics. Determine k for a single root and find x-values where f(x) is less than nine.

Explore vertical and horizontal lines, axis-parallel forms like x=a and y=b, and lines such as y=x and y=-x. Apply algebraic intercept methods using x-intercepts and y-intercepts.

Solve problems on the equation of the straight line by using slope-intercept and point-slope forms to find lines through given points, and determine parallel or perpendicular lines via gradient relationships.

Tackle past paper style coordinate geometry problems for Cambridge International A & AS Level Pure Mathematics 1, covering gradients, distance formula, coordinates, perpendicular bisectors, and line equations.

Compute the reflection of a point across a line by using the perpendicular, intersection, and midpoint, then find k and contact point for the tangent to y = x^2/4.

Learn to find the number of terms in an arithmetic progression using the formula when the last term is known, illustrated with examples and given a and d values.

Apply S_n = n/2 (2a + (n-1)d) or S_n = n/2 (a + l) to sum an arithmetic progression; a is first term, d is difference, l is last term.

Solve practice questions on the sum of n terms in an arithmetic progression using the standard formula, then check your solutions with the video lectures.

Master the sum to n terms of an AP using the standard formula, including that the sum of the first n odd numbers is n^2.

Prove that if the sums of m and n terms of an AP are equal, the sum of m plus n terms is zero, and analyze the reciprocal-term case.

Find nth term from end in an arithmetic progression using l minus (n minus 1) d, with l as last term and d as the common difference; reverse the sequence.

Learn to find the first negative term in a decreasing arithmetic progression using the general term a+(k-1)d, with d negative, by solving a+(k-1)d<0 and checking the previous term.

Explore geometric progressions by finding common ratios and first terms, counting terms, and computing finite sums and sum to infinity. Use examples like 8, 16, 2048 and 48, 768.

Convert recurring decimals to fractions using geometric progression (GP) and the sum-to-infinity formula, illustrated with examples like 0.55, 0.3565656..., and 0.231231231.

Explore practice problems on geometric progressions, determining the common ratio from given terms and sums. Compute sum to infinity with and without the first term, and find the first term.

derive gp concepts: identify first term and ratio, derive nth term, calculate sums to n and to infinity, and apply to a 54 and 1/3 sequence to find n.

Explore problems on arithmetic and geometric progressions, derive that equal initial terms force r=1 and d=0, and compute common ratio, sums to infinity, and first ten terms.

Solve past papers problems on arithmetic and geometric progressions, deriving d in terms of a, and evaluating sums to 100 and 200 as well as sum to infinity.

Solve past paper problems on arithmetic and geometric progressions, finding r and d, and calculating nth and fourth terms, plus sums to infinity and sums of first n terms.

Solve past paper problems on arithmetic and geometric progressions, including sums to infinity, finding gp unknowns, and proving r^n exceeds 0.1 for 90% of the sum to infinity.

Find the 10th term in the binomial expansion of (2x^2 + 1/x)^12 using a summation method, and derive the 6th and 7th terms by combining exponents of x.

Demonstrate binomial expansion in ascending and decreasing powers of x, obtain the first three terms of (1+2x)^8 with x=0.01, and approximate 1.02^8; then expand (2+5x)^12 in ascending powers.

Work through past paper problems on binomial expansions in pure mathematics, focusing on coefficients and ascending powers of x. Determine p from the zero coefficient of x^2 in (1 - x)(1 - p x)^6.

Apply the vertical line test to determine if a curve represents a function. Ensure every x has a unique y; multiple intersections mean not a function.

Learn to draw basic rational functions by identifying vertical and horizontal asymptotes, using denominator zeros and degree comparisons, with examples.

Find inverse functions graphically and test existence with the horizontal line test, then view inverses as reflections across y = x, valid only for one-to-one functions.

Solve practice problems on inverse functions, domains and ranges, and function composition, including f inverse x for f(x)=3x and g(x)=9x^2+3, and find the range of g(f inverse x).

Practice problems on functions f and g: determine f’s range on [0,4], find k (k=5) from g(f(2))=16, and assess f’s inverse possibilities.

Learn how to find the slope of lines and curves, using secant and tangent lines, and define the derivative as the limit of a secant slope.

Learn the power rule for derivatives, derived from first principles, and apply it to constants, k x^n forms, and real-number exponents.

Apply power rule to determine a and b in f(x)=ax^3+bx+4 from (2,14) and gradient 21; differentiate to get 12a+b=21, solve a=2 and b=-3, sketch y=1/x and y=1/x+2 with x-intercept -1/2.

Use dy/dx to obtain tangent and normal lines for the cubic at x=2 and x=3, then analyze tangents and normals for a quadratic, including intersection-based square geometry.