Year 11 Mathematics: Algebra, Linear Equations, Sequences, and Geometry

Comprehensive review flashcards covering algebraic factorisation, expanding brackets, solving linear and quadratic equations, linear graphs, sequence difference tables, rounding, and geometric applications.

What are the two key steps to remember when factorising an algebraic expression?

First take out any common variables (unknowns), then divide out any common numerical factors.

What is the completely factorised form of $30x^2 - 9x$?

$$3x(10x - 3)$$

How do you factorise the quadratic expression $2x^2 + 7x + 3$?

$$(2x + 1)(x + 3)$$

How do you factorise the quadratic expression $4x^2 + 4x - 3$?

$$(2x - 1)(2x + 3)$$

What formula is used to find the equation of a line given a point $(x_1, y_1)$ and gradient $m$?

$$y - y_1 = m(x - x_1)$$

How is the gradient $m$ of a line calculated from rise and run?

$$m = \frac{\text{rise}}{\text{run}}$$

What is the general gradient-intercept form of a linear equation?

$y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.

In a quadratic sequence $y = ax^2 + bx + c$, how is the coefficient $a$ determined from the second order differences?

$$a = \frac{\text{second order difference}}{2}$$

If a sequence has a constant second order difference of $+4$, what is the value of $a$ in the quadratic rule?

$$a = \frac{4}{2} = 2$$

What indicates that a numerical sequence is quadratic when examining its difference table?

The first order differences change at a steady rate, producing a constant second order difference.

What is the rule when multiplying or dividing both sides of an inequality by a negative number?

The inequality sign flips (reverses direction).

What is the solution to the linear equation $5x + 7 = 2x + 13$?

Subtract $2x$ to get $3x + 7 = 13$, subtract $7$ to get $3x = 6$, and divide by $3$ to find $x = 2$.

What is the standard formula for the area of a triangle given base $b$ and perpendicular height $h$?

$$\text{Area} = \frac{1}{2} \times b \times h$$

In right-angled triangle trigonometry, how is the sine of an angle $\theta$ defined?

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

When solving quadratic equations for geometric dimensions such as $x^2 + 4x - 45 = 0$, why is the negative root rejected?

A physical measurement of length or dimension cannot be negative; therefore, only the positive solution ($x = 5$) is kept.

How do you solve the quadratic equation $x^2 + x - 12 = 0$ by factorising?

Factorise to $(x - 3)(x + 4) = 0$, setting each factor to zero to obtain $x = 3$ or $x = -4$.

When factorising linear terms that are all negative, what can be done with the negative sign?

A negative sign can be factored out as part of the common factor outside the bracket.

What rule relates the sides and non-right angles (such as $125^\circ$) of a triangle?

The Sine Rule ($\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$) or the Cosine Rule ($c^2 = a^2 + b^2 - 2ab\cos(C)$).

How do you expand the expression $3(2g + 5)$?

$$6g + 15$$

What two conditions must two numbers satisfy to factorise a monic quadratic $x^2 + bx + c$?

They must add to give the coefficient $b$ and multiply to give the constant term $c$.

Which two numbers add to $8$ and multiply to $15$?

$3$ and $5$

How do you factorise $2x^2 + 7x + 5$ using grouping?

Multiply $2 \times 5 = 10$. Find factors of $10$ that add to $7$ ($2$ and $5$). Split the middle term: $2x^2 + 2x + 5x + 5 = 2x(x + 1) + 5(x + 1) = (2x + 5)(x + 1)$.

What is the solution to the linear equation $3x - 4 = x + 5$?

Subtract $x$ to obtain $2x - 4 = 5$, add $4$ to obtain $2x = 9$, and divide by $2$ to get $x = 4.5$.

What is the recommended standard order of terms when preparing to factorise a quadratic expression?

Write the terms in descending powers: $x^2$ first, followed by the $x$ term, and then the constant numerical term.

What is the factorised form of $x^2 + 13x + 36$?

$$(x + 9)(x + 4)$$

What is the area of the region inside a circle of radius $r$ but outside an inscribed square whose corners touch the circle?

$r^2(\pi - 2)$ (calculated by subtracting the square's area, $2r^2$, from the circle's area, $\pi r^2$).

A cuboid has length $3x - 2\,\text{cm}$, width $x + 4\,\text{cm}$, and height $5\,\text{cm}$. What is its total surface area when $x = 6$?

The length is $16\,\text{cm}$, width is $10\,\text{cm}$, and height is $5\,\text{cm}$. Total surface area is $2(16 \times 10 + 16 \times 5 + 10 \times 5) = 2(160 + 80 + 50) = 580\,\text{cm}^2$.

What are the coordinates of point $P$, the intersection of the lines $2x + 3y = 12$ and $y - 2x = 4$?

$$(0, 4)$$

Solve the linear equation: $3(x - 4) = 5(x - 2)$.

Expand to get $3x - 12 = 5x - 10$. Rearrange to get $-2 = 2x$, giving $x = -1$.

What is the factorised form of the difference of squares $t^2 - 1$?

$$(t - 1)(t + 1)$$

Simplify the algebraic fraction: $\frac{x^2 - 16}{2x - 8}$.

$$\frac{(x - 4)(x + 4)}{2(x - 4)} = \frac{x + 4}{2}$$

Solve the linear inequality: $2 - 5x < 12$.

Subtract $2$ to get $-5x < 10$. Divide by $-5$ and reverse the sign to get $x > -2$.

Find $n$ if $s^4 \times s^n = s^6$.

Using exponent rules, $4 + n = 6$, so $n = 2$.

Simplify the algebraic fraction: $\frac{5x^3 + 25x^2}{x^2 + 5x}$.

$$\frac{5x^2(x + 5)}{x(x + 5)} = 5x$$

Solve the quadratic equation: $x^2 = x + 20$.

Rearrange to $x^2 - x - 20 = 0$, factorise to $(x - 5)(x + 4) = 0$, giving solutions $x = 5$ and $x = -4$.

What is $3.523$ rounded to $2$ decimal places?

$$3.52$$

What is $0.0493$ rounded to $2$ significant figures?

$$0.049$$

What is the fundamental rule regarding terms when adding or subtracting algebraic expressions?

You can only add or subtract like terms.

Simplify the term: $4y^3 \div 4y^2$.

$$y$$

What are the first order differences and the constant second order difference for the sequence $1, 7, 17, 31, 49, 71, 97$?

The first order differences are $+6, +10, +14, +18, +22, +26$, and the constant second order difference is $+4$.

What is the linear rule generating the sequence $6, 11, 16, 21, 26, 31, 36$ in the form $t = mx + c$?

$t = 5x + 1$ (since the common difference is $m = 5$ and $6 = 5(1) + c \implies c = 1$).

What is the linear rule generating the sequence $27, 24, 21, 18, 15, 12, 9$ in the form $t = mx + c$?

$t = -3x + 30$ (since the common difference is $m = -3$ and $27 = -3(1) + c \implies c = 30$).