Further Maths Quiz (Entire course revision)

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Questions and Answers

What is the first step when dividing mixed numbers?

Write as improper fractions (correct)

Multiply by the original fraction

Convert both to decimals

Subtract the numerators

To find 15% of £200, what operation should you perform?

Divide £200 by 15

Multiply £200 by 0.15 (correct)

Multiply £200 by 15

Multiply £200 by 1.15

If a coat is marked down by 20% and the sale price is £50, what was the original price?

£70

£55

£62.50 (correct)

£40

How is compound interest calculated after 5 years with a 10% annual interest rate on £1000?

£1000 × 1.1^5 Signup and view all the answers

To share £60 in the ratio 3:2, how much does the first person receive?

£36 Signup and view all the answers

What is the ratio of sugar to flour if there are twice as much sugar as flour?

2:1 Signup and view all the answers

Which of the following is a surd?

√5 Signup and view all the answers

What is the correct simplification for √36?

6 Signup and view all the answers

What is the result of expanding the expression $3x^4y(2xy - 5x^3)$?

$6x^5y^2 - 15x^7y$ Signup and view all the answers

What is the simplified form of $3x(2 - 5x) - 7(x - 5)$?

$-15x^2 - x + 35$ Signup and view all the answers

Which of the following correctly describes the process of multiplying two binomials $(a + b)(c + d)$?

Multiply each term in the first bracket by each term in the second bracket Signup and view all the answers

When fully factoring the expression $28x^3y^2 - 12x^5y^7$, what is the correct factorization?

$4x^3y^2(7y^5 - 3x^2)$ Signup and view all the answers

What does $y = ax^2 + bx + c$ represent in graphing terms when $a > 0$?

An upward-opening parabola Signup and view all the answers

How many distinct real solutions does the equation $ax^2 + bx + c = 0$ have if it crosses the x-axis at two points?

Two distinct solutions Signup and view all the answers

To factor a quadratic with the form $x^2 + bx + c$, what must be true about the numbers that you find to go alongside x in the factors?

They must multiply to $c$ and add to $-b$. Signup and view all the answers

What is true about a quadratic that does not touch the x-axis?

It has no real solutions. Signup and view all the answers

What does the factor theorem state about a polynomial P(x) and a value a?

(x - a) is a factor of P(x) if P(a) = 0 Signup and view all the answers

Which step is NOT part of the process to factor a cubic polynomial?

Use synthetic division to find all roots Signup and view all the answers

In the example provided, what was the first root found for the polynomial P(x) = 2x³ - x² - 16x + 15?

1 Signup and view all the answers

When sketching a positive quadratic function, what characteristic is expected?

The graph will have a minimum turning point Signup and view all the answers

What is the y-intercept of a quadratic function y = ax² + bx + c?

(0, c) Signup and view all the answers

What term describes the values of x where the graph of a function crosses the y-axis?

Y-intercepts Signup and view all the answers

From the example provided, what form did the polynomial take after applying the factor theorem?

(x - 1)(2x² + x - 15) Signup and view all the answers

How does the factor theorem relate roots and factors of a polynomial?

A root indicates the polynomial can be split into linear components Signup and view all the answers

What is the center of the circle defined by the equation 𝑥² + 2𝑥 + 𝑦² − 6𝑦 = 25?

(-1, 3) Signup and view all the answers

What is the gradient of the tangent line at the point (2, 7) according to the provided information?

-1 Signup and view all the answers

How is the gradient of a curve defined at a specific point?

As the gradient of the tangent to the curve at that point. Signup and view all the answers

What notation is used to represent the differential of y with respect to x?

dy/dx Signup and view all the answers

In the differentiation process, what happens to the exponent when differentiating the term 𝑦 = 𝑥ⁿ?

It is multiplied into the coefficient and decreases by one. Signup and view all the answers

What is the relationship between the gradient of chords and the gradient of the tangent as the points approach each other?

The gradient of the tangent converges to the gradient of the chord. Signup and view all the answers

What does the notation 𝛿𝑦/𝛿𝑥 represent in calculus?

The average rate of change of y with respect to x. Signup and view all the answers

Which of the following statements about differentiation is correct?

It provides the rate of change of y with respect to x. Signup and view all the answers

What is the limit of the sequence defined by $T(n) = \frac{4n + 20}{8n - 4}$ as $n$ approaches infinity?

2 Signup and view all the answers

Which formula correctly represents the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?

$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Signup and view all the answers

What is the midpoint of the line segment connecting the points $(2, 4)$ and $(6, 8)$?

(4, 6) Signup and view all the answers

If a point is $rac{1}{3}$ of the way between the points $(1, 2)$ and $(4, 6)$, what is the coordinate of that point?

$(1 + \frac{1}{3}(4 - 1), 2 + \frac{1}{3}(6 - 2))$ Signup and view all the answers

What does the 'm' represent in the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$?

The gradient of the line Signup and view all the answers

Which of the following statements is true about finding a point that is a specific proportion between two points?

You add a fraction of the difference to the starting point. Signup and view all the answers

What is the average of the x-coordinates $(x_1, x_2)$ used to find the midpoint between two points?

$\frac{x_1 + x_2}{2}$ Signup and view all the answers

When finding the gradient 'm' of a line, which calculation is performed?

$\frac{y_2 - y_1}{x_2 - x_1}$ Signup and view all the answers

What is the result of simplifying the expression $\sqrt{12} - \sqrt{18} + \sqrt{8}$?

$-3\sqrt{2}$ Signup and view all the answers

Which of the following is the correct expression for rationalizing the denominator of $\frac{5}{\sqrt{3} + \sqrt{2}}$?

$\frac{5(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}$ Signup and view all the answers

What is the length of the missing side of a right triangle where the other two sides are $12 cm$ and $8 cm$, using the Pythagorean theorem?

$\sqrt{208}$ Signup and view all the answers

How does the expression $\sqrt{9} + \sqrt{16}$ differ from $\sqrt{9 + 16}$?

The second expression equals $5$. Signup and view all the answers

Which of the following is a property of multiplying surds?

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a} \times \sqrt{b}$ Signup and view all the answers

What simplification process is used to transform $\frac{6}{3 - 2\sqrt{5}}$ into a rational denominator?

Multiply by $3 + 2\sqrt{5}$ Signup and view all the answers

What is the final result when simplifying $\sqrt{80}$?

$4\sqrt{5}$ Signup and view all the answers

When simplifying the fraction $\frac{a \sqrt{b}}{\sqrt{b}}$, what is the correct outcome?

$a$ Signup and view all the answers

To perform the operation $a(b + c)$, what is the resulting expression?

$ab + ac$ Signup and view all the answers

Which of the following steps is incorrect when using the difference of squares to rationalize $\frac{a}{b + \sqrt{c}}$?

Multiply only the numerator by $a$ Signup and view all the answers

Study Notes

AQA Level 2 Certificate in Further Mathematics - Need-To-Know Booklet

This booklet is a revision aid for Further Maths Level 2, AQA.
It is not a replacement for textbooks or classroom teaching, but rather a summary of key formulae, results, and methods.
Candidates are expected to have achieved, or be expecting to achieve, an A or A* GCSE Maths grade.

Section 1 - Number

Fractions: Add or subtract fractions by finding a common denominator and then performing the operation on the numerators. Simplify the answer if possible. If mixed numbers, add whole number and fraction parts separately.
Multiplying fractions: Multiply numerators and denominators directly. For mixed numbers, convert to improper fractions first. Pre-cancel for large numbers.
Dividing fractions: Multiply by the reciprocal of the fraction you want to divide by. Convert mixed numbers to improper fractions first.
Percentage calculations: Convert to decimal form and multiply to find percentages of amounts. To increase or decrease by a percentage, multiply by the appropriate decimal.

Section 2 - Algebra

Multiplying out brackets: Multiply every term inside the bracket by the multiplier outside.
Simplifying algebraic expressions: Collect like terms after multiplying out brackets.
Multiplying brackets together: Multiply each term in one bracket by each term in the other bracket. For more than two brackets, multiply pairs at a time.
Factorising: Identify common factors (numbers, letters, or combinations) of all terms in the expression and factor them out.

Section 3 - Coordinate Geometry

Distance between two points: √((x₂ - x₁)² + (y₂ - y₁)²).
Midpoint: ((x₁ + x₂)/2 , (y₁ + y₂)/2).
Gradient: (y₂ - y₁)/(x₂ - x₁).
Parallel lines: Same gradient (m₁ = m₂).
Perpendicular lines: Gradients multiply to -1 (m₁m₂ = -1).
Equation of a line: y - y₁ = m(x - x₁).
Equation of a circle, centre (0, 0): x² + y² = r².
Equation of a circle, centre (a, b): (x - a)² + (y - b)² = r².

Section 4 - Calculus

Gradient of a curve: The limit of the gradient of a chord as the points on the curve get closer together.
Differentiation: Finding the gradient of a curve at any given point.
Differentiation rules:
- The power rule: y=xn -> dy/dx=nx^(n-1)
- The sum rule: y = f(x) + g(x) -> dy/dx= f’(x) + g’(x) -Constants differentiate to zero
Gradient at a particular point: Substitute the x-coordinate of the point into the derivative.
Finding points on a curve with a given gradient: Set the derivative equal to the given gradient and solve for x. Substitute the x-values into the original equation to find the corresponding y-values.
Tangents to a curve: Find the gradient at a given point, then use the gradient and the coordinates of the point to calculate the equation of the tangent using y - y₁ = m(x - x₁).

Section 5 - Matrix Transformations

Matrix: A rectangular array of numbers.
Order of a matrix: m × n (rows × columns).
Adding and subtracting matrices: Correspond elements are added or subtracted.
Multiplying a matrix by a constant: Multiply every element by the constant.
Matrix multiplication: Calculate the product of two matrices according to the rules (number of columns of the first matrix must equal the number of rows of the second).
Identity matrix: A matrix that, when multiplied by other compatible matrices, does not change them.
The matrix transformations (eg. reflections in x, y axis) can be defined by particular matrices

Section 6 - Geometry

Area of a triangle: (1/2) * base * height, or (1/2) * a * b * sin(C) (where a and b are sides and C is the included angle).
Pythagorean theorem: a² + b² = c² (for right-angled triangles).
Sine rule: a/sin A = b/sin B = c/sin C.
Cosine rule: a² = b² + c² – 2bc cos A.
Angles in a triangle: Add up to 180°
Circle theorems.

Appendix – Formula Sheet

Provides a list of key formulae for geometry, trigonometry, and calculus. These are to be given in the exam.

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