Cambridge International A & AS Level (Pure Mathematics 1)

Explore the concept of integers, including positive and negative numbers on the number line, and master addition, subtraction, multiplication, and division rules for signs.

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.

Explore powers and roots through examples of squares, cubes, and higher roots, learn to use the radical sign, and apply prime factorization and the zero exponent rule.

Practice problems on powers and roots cover squaring numbers, cubing values, and handling negative bases; apply even/odd exponent rules and zero exponents for clear, quick calculations.

learn to find square roots, cube roots, and higher nth roots using prime factorization, with examples like 49, 121, 256, 196, 729, 625, 64, 2187, and 243.

Learn to simplify powers and roots by multiplying and dividing under a single radical, using prime factorization to extract factors. Addition and subtraction under a square root do not distribute.

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.

Work through past paper problems part 2 in pure mathematics 1, applying exponent rules to simplify expressions and solve for x, p, and m.

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.

Master the order of operations with Bodmas, Bidmas, and Pemdas, including brackets and powers. See how division, multiplication, addition, and subtraction follow a left-to-right rule across these variants.

Learn to add algebraic expressions using horizontal and column methods, grouping like terms, and combining coefficients, with practice on variables x, y, and z for clear simplification.

Explore subtraction of algebraic expressions using horizontal and column methods, open brackets, change signs, and collect like terms across x, x^2, and constants.

Learn to multiply algebraic expressions with one term by multiplying numerical coefficients and variables separately, then combine like bases by adding exponents and handle negative coefficients.

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.

Practice multiplying algebraic expressions, including binomial by binomial and binomial by polynomial, using step-by-step multiplication and combining like terms to obtain simplified results.

Divide any algebraic expression by a monomial using laws of indices, applying term-by-term division and sign handling. Practice with examples of dividends, divisors, and simplified results.

Learn to divide algebraic expressions by a binomial divisor using long division, arranging terms in descending powers, computing quotients, and handling remainders step by step.

Covers division of algebraic expressions by polynomials, including one- and two-variable cases, with divisor and dividend ordered by decreasing powers, sign changes, remainders, and worked examples.

Apply the distributive property to remove brackets by multiplying a factor across terms inside parentheses, keeping the internal signs, with examples showing distribution over sums and handling negative signs.

Take out the greatest common factor to factor expressions by forming a product of factors. Use examples with two terms and exponents to illustrate the inside remainder.

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.

Master factoring by difference of two squares with the identity a^2 − b^2 = (a−b)(a+b). Apply this to examples such as 49x^2 − 36 to obtain 7x−6 and 7x+6.

Master factorization with the identities (a+b)^2 = a^2+2ab+b^2 and (a-b)^2 = a^2-2ab+b^2, recognizing perfect squares and choosing the correct form to factor complex expressions.

Master factorization using the identity (a + b + c)^2 by expanding a^2 + b^2 + c^2 + 2ab + 2bc + 2ac and recognizing perfect squares. Apply sign rules and product terms through worked examples.

Learn factorization by the middle-term split for quadratics, guiding you to rewrite in standard form, compute ac, split the middle term, and factor by grouping.

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.

Explore quadratic equations in the standard form x^2 + bx + c = 0 with a not equal to zero, and learn to determine which given equations are quadratic.

Learn to solve quadratic equations by factorization, turning them into zero products of linear factors and extracting roots from each factor, with examples including fractional coefficients.

Learn to rewrite ax^2 + bx + c in completing the square form and convert to vertex form y = a(x - h)^2 + k, with worked examples and practice.

Learn to solve quadratics by completing the square, rewriting 5x^2+4x-2 as 5(x-h)^2+k, and finding roots -0.4±√0.56, with shortcut methods illustrated.

Master the direct completing the square method: factor the x^2 term, halve the x-term, add and subtract its square, and write expressions as a square plus constants.

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.

Learn to solve quadratic equations using the quadratic formula. The discriminant determines whether you obtain a single repeated root, two real roots, or no real roots, with examples.

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

Explore open, closed, and half-open intervals in interval notation, including endpoints, infinity, and the three subintervals: minus infinity to a, a to b, and b to infinity.

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.

Factor quadratic expressions and solve inequalities algebraically using a number line, open endpoints for strict cases, and identify intervals with positive and negative values.

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 how the discriminant b^2 minus 4ac decides the nature of quadratic roots. See examples where D>0 yields two real roots, D=0 a repeated root, and D<0 nonreal roots.

Explore the nature of roots by applying the discriminant to quadratic equations. Identify k values that yield a repeated root, two real roots, or no real roots.

Reduce non-quadratic equations to quadratic forms through substitutions such as y = x^3 or y = x^2, then solve by factorization or the quadratic formula to find x.

Learn how to transform equations into quadratics and solve them with factorization or the quadratic formula, using substitutions such as y = (x-1)/(2x+3)^2 or y = 3^x.

solve nonlinear simultaneous equations by substituting a linear expression for x into the quadratic equation and applying the quadratic formula.

Learn to complete the square and direct method to rewrite 4x^2 -24x + p as 4(x-3)^2 + p-36, then apply discriminant analysis to determine p for no real roots.

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.

Derive the distance formula from coordinates by using a rectangle with points (x1,y1) and (x2,y2) and the pythagorean theorem. Define origin, abscissa, and ordinate, and show r = sqrt((x2−x1)^2+(y2−y1)^2).

Apply the distance formula to find distances between points and from origin. Explore equidistant points on the x- and y-axes and calculate the area of an isosceles right triangle.

Learn the midpoint formula to find the midpoint coordinates as (x1+x2)/2 and (y1+y2)/2, with worked examples. Then apply the distance formula to the triangle median, yielding 5 units.

Learn how to find the gradient of a line via rise over run and two points, using (y2 - y1)/(x2 - x1), with zero, positive, negative, and infinite gradients.

Determine when lines are parallel or perpendicular using gradients: parallel when gradients are equal, and perpendicular when their product equals minus one; derive perpendicular gradients as the negative reciprocal.

Derive gradients from two points, use perpendicular slope relationships, and apply collinearity to verify parallelogram structures and solve a right triangle's sides and area.

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.

Derive a straight line's equation in standard form y = mx + c from gradient and y-intercept, and apply the one-point form or the two-point form using given points.

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.

Explore graphing lines by using gradient m and y intercept, convert equations to y = mx + c, and trace rise and run from the y intercept.

Learn how to find intersection points of a line and a parabola using substitution and the discriminant, and classify outcomes as two points, a tangent, or no intersection, with examples.

Solve past paper questions on the intersection of a line and a parabola, find midpoints of intersection points, and determine tangency using discriminant analysis to locate touchpoints.

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.

Compute A and C coordinates, find midpoint m and perpendicular bisector through B, then prove AB is perpendicular to BC. Determine D for square abcd and calculate AD.

Solve past paper coordinate geometry problems: find x as intersection of cx and ab with cx perpendicular to ab, then compute ax:xp and determine c via ac and bc intersections.

Define a sequence as numbers in order following a rule, with the kth term uk as u1, u2, u3. Use sigma notation to show how these terms form a series.

Learn how arithmetic progression works, with fixed common difference d, identify arithmetic sequences, and use first term, last term, and general term u k in examples.

Examine arithmetic sequences by testing a kth term for linearity and finding the first term. Then find the common difference; for linear terms, it equals the coefficient of k.

Derive the general term of an arithmetic progression as a plus k minus one times d, and use L equals a plus n minus one times d to find terms.

Determine whether a given term belongs to an arithmetic progression and, if so, which term it is, using the first term and common difference in a + (k-1)d.

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.

Determine the first term and common difference of an arithmetic progression from two known terms by solving simultaneous equations. Write the resulting sequence using a and d.

Use the condition for three terms in arithmetic progression: twice the middle equals the sum of the outer terms; apply to solve for x and determine the common difference.

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.

Derive the sum to n terms for an arithmetic progression with t_n = 2n+1, yielding a_1=3 and S_n = n^2+2n. Explain zero and double-answer for a_1=18, d=-2 and a_1=20, d=-2/3.

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 arithmetic progression through practice problems by counting odd numbers between 2 and 100 divisible by three and summing sequences like multiples of eight and even numbers.

Solve arithmetic progression problems with first term 3000 and tenth term 1200; determine the common difference, the sum of 20 terms, and when the sum becomes negative.

Solve arithmetic progression problems using first term and common difference to find a term and the sum to n terms, illustrated by salary growth and jogging distance.

Solve arithmetic progression problems, including cutting a string into arithmetic progression pieces with given end terms and total length, and find a, d, and n using sum and nth-term formulas.

Explore geometric progressions, where each term multiplies by a constant common ratio r; derive the general term u_k = a r^{k-1} and last term l = a r^{n-1}, with examples.

Explore sums to n terms of a GP, using the formulas for |r|<1 and |r|>1 with examples from 3,6,12 and 0.15,0.015,0.0015.

Learn to evaluate sums in a gp using sigma notation, separate constants from the geometric part, and apply the sum-to-n terms formula with 2+3^k and 4^k examples.

Explore how to compute the sum to infinity of a geometric progression using the formula a/(1−r) for |r|<1, and distinguish convergence from divergence with practical examples.

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.

Define factorial as the product of numbers from 1 to n, with 0! = 1 and factorials defined only for whole numbers; show how n! relates to (n-1)! by expansion.

Explore n choose r, the number of ways to select r objects from n when n and r are non-negative integers, using the binomial coefficient notation and the factorial formula.

Explore properties of n choose r, including symmetry nCr = nC(n-r) and the sum identity nCr + nCr+1 = (n+1)Cr+1, with examples like 10C4 and 11C5.

Summarize the binomial theorem for positive index, using binomial coefficients and Pascal's triangle to expand expressions. Apply the general term n choose r, x^{n−r} a^{r}, with (x+2)^5 and (1+x)^n, (1−x)^n.

Apply the binomial theorem with Pascal's triangle to expand (2x+3y)^3 and extract coefficients. Extend the method to fast expansions of (2x-3)^4 and (3x^2-2/x)^5, yielding full term expressions.

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.

Find the fourth term from the end in the binomial expansion of (3/x^2 - x^3/6)^7 using the end-term formula, yielding 35/48 x^6.

Learn to find coefficients for given exponents in binomial expansions using the binomial theorem and summation form. Apply to x^5 in (1+x)^8 and x^7 in (1-2x)^15.

Use the binomial theorem in summation form to find the term independent of x by setting the x-exponent to zero, solving for r, and computing the (r+1)th term.

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.

Develop skills in binomial expansions and coefficient extraction using Pascal's triangle. Practice computing coefficients in the products and truncated expansions of (1+2x)^6(1−x)^7, (2+x)^6, (1−2x)^6, and (4−x^2−4x)^6.

Explore binomial expansions: derive the first three terms of (4x - k/x^2)^6, use the independent term to find k and x, and extract the x^3 coefficient in (x^2 - 1/x)^6.

Explore using binomial expansion to approximate powers and compute terms up to x^2, including estimating 1.0304^4, expanding (3x+5)^3 − (3x−5)^3, and deriving t^2 coefficients in (1+αt)^5(1−βt)^8.

Identify the term independent of x in binomial expansions by using summation form and binomial coefficients, and determine coefficients such as x^2 and x^8.

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.

Explore binomial expansions by expanding (2−y)^5, substituting y=2x−x^2 to extract the coefficient of x^2, then determine k in (k+x)^8 when x^2 and x^3 coefficients match.

This lecture explains that a function assigns every input x a unique output f(x) from a codomain, defining domain and range with arrow diagrams and examples.

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 find the value of a function at a given point both graphically and algebraically, using a piecewise f(x)=3x-2 for x<0, f(0)=1, and f(x)=4x+1 for x>0.

Derive the domain and range of functions algebraically by excluding undefined input values, handling denominators, square roots, and logarithms, and illustrate with rational and square-root examples.

Explore how to determine domain and range from graphs, including open and closed intervals, infinite extensions, and points not included, using x-values left to right and y-values bottom to top.

Discover how even functions are symmetric about the y-axis and satisfy f(-x)=f(x), while odd functions are symmetric about the origin with f(-x) = -f(x).

Determine when a function is one-to-one. Use the horizontal line test to decide if the graph is 1 to 1 and whether the inverse exists.

Understand composite functions by combining two functions under domain–range conditions. See how f∘g exists when g’s range lies in f’s domain and compute examples such as f(x)=x^2+1 and g(x)=x−2.

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

Draw basic rational functions using vertical and horizontal asymptotes. Learn to plot f(x)=1/x and variants like x+2, 1/x^2, and -1/x by identifying asymptotes and sample points.

Learn to find inverses algebraically by switching x and y and making y subject, then replace y with f inverse x, with examples of linear, rational, and cubic functions.

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.

Demonstrates derivative is the slope of a curve at a point. Derives f'(x) for y = x^2, showing f'(x) = 2x and slopes 6 at x=3 and 10 at x=5.

Compute derivatives from the first principle using the limit of the difference quotient, applying to f(x) = -6x^2 + 7x - 9 and f(x) = 3x^2 + 6x + 6.

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

Apply the power rule to differentiate expressions such as x^2 and x^-2, simplify derivatives of root x and 1/x, and find tangent slopes.

Practice problems on power rule 2 guide computing dy/dx and gradients for polynomials, sketching parabolas, locating x-intercepts, and identifying turning points and stationary points.

sketch parabola y = 4 - x^2 and line y = 2x + 5 on [-3, 3], verify (-1, 3) lies on both, and differentiate to show tangent slope 2.

Solve power-rule practice problems on cubic and quadratic curves by sketching intercepts, differentiating, and identifying parallel tangents; include vertex calculations and verifying tangency to a given line.

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.

This lecture shows how to find tangent and normal lines to a curve by differentiating for the tangent slope and using the negative reciprocal for the normal, with point-slope equations.

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.

Find the intersection points of the parabola y=2x^2-9x and the line y=x-8, derive the tangents at those points, and compute their intersection; also analyze tangent and normal to y=√x.

Find the tangent and normal to y = 1/x at (2, 1/2) and the y-axis area, then the tangent to y = 9/√x at (9, 3) and AB length.

Practice tangents and normals to a quadratic curve. Find intersection points, tangent slope, acute angle with a line, and non-intersecting line conditions.