AQA Level 2 Certificate Further Mathematics Paper 1 (Non-Calculator)

💡 Strategy: Use the laws of indices. Multiplication adds powers, division subtracts powers.

💡 Strategy: Use the multiplication law of indices: \( a^x \times a^y = a^{x+y} \).

💡 Strategy: Use the power law of indices: \( (a^x)^y = a^{xy} \). Make the powers equal.

💡 Strategy: To remove a cube root, we must cube both sides of the equation.

💡 Strategy: Isolate \( x \) by adding 13 to both sides and then dividing by 7.

💡 Strategy: To compare coefficients, we first need to expand the left-hand side (LHS) to look like the right-hand side (RHS).

💡 Strategy: Group the \( x \) terms together and the constant terms (those without \( x \)) together.

💡 Strategy: The identity symbol (\(\equiv\)) means the coefficient of \( x \) on the LHS must equal the coefficient of \( x \) on the RHS, and the constants must also match.

LHS: \( (10a)x + (20 – 3a) \)

RHS: \( 60x + b \)

💡 Strategy: Finding the difference in x and y coordinates between A and B tells us the vector for “5 parts” of the journey.

\( A = (3, 7) \)

\( B = (5, 5.5) \)

💡 Strategy: Since the ratio \( AB : BC \) is \( 5 : 2 \), the vector \( \vec{AB} \) represents 5 parts. Divide by 5 to find the vector for 1 part.

💡 Strategy: The distance from B to C is “2 parts”. Multiply the single part vector by 2.

💡 Strategy: Add the change from B to C to the coordinates of B.

💡 Strategy: Before differentiating, we must write all terms in the form \( ax^n \). The term \( \frac{3}{x^2} \) needs to be rewritten with a negative power.

💡 Strategy: Apply the power rule: multiply by the power, then subtract 1 from the power (\( nx^{n-1} \)).

💡 Strategy: Look for the highest common factor in \( 15x^2y – 5xy^2 \). Both terms share \( 5 \), \( x \), and \( y \).

💡 Strategy: Look for the highest common factor in \( 12x – 4y \). Both terms are divisible by 4.

💡 Strategy: Notice that \( (3x – y) \) appears in both the numerator and denominator. We can cancel it out.

💡 Strategy: A rhombus can be split into two identical triangles (ABC and ADC). The area of a triangle is \( \frac{1}{2}ab\sin C \).

💡 Strategy: We know sides \( AB = BC = 8 \) cm and angle \( B = 60^\circ \).

💡 Strategy: Multiply by 2 for the full rhombus.

💡 Strategy: A cubic graph with a positive \( x^3 \) coefficient starts from bottom-left and goes to top-right. We need to plot the specific points given.

• Maximum \( L(-1, 13) \): In the second quadrant (top-left).
• Minimum \( M(2, -14) \): In the fourth quadrant (bottom-right).
• Y-intercept \( N \): When \( x=0, y=6 \). Point is \( (0, 6) \).

💡 Strategy: Draw a smooth ‘S’ shape curve.

1. Start low on the left.
2. Go up to cross the x-axis, reach peak at \( L \).
3. Come down to cross the y-axis at \( N(0,6) \).
4. Continue down to cross x-axis again, reach trough at \( M \).
5. Go up to cross x-axis a third time.

💡 Strategy: Opposite angles in a cyclic quadrilateral sum to \( 180^\circ \). Let’s look at angles at C and A.

💡 Strategy: To prove \( AC \) is a diameter, we need to show that the angle subtended by it at the circumference (Angle \( ADC \) or \( ABC \)) is \( 90^\circ \) (angle in a semicircle).

Let’s use triangle \( ADC \).

💡 Strategy: State the circle theorem.

💡 Strategy: First, rewrite \( f(x) \) in a simpler form, then substitute \( x = 1/3 \).

💡 Strategy: Set \( f(1/3) = g(1/3) \) and solve for \( p \).

💡 Strategy: The circle touches the y-axis at \( (0, 2) \). This means the tangent at that point is the y-axis (vertical). The radius is perpendicular to the tangent, so the radius to this point is horizontal.

Therefore, the centre of the circle must be directly to the right of \( (0, 2) \). This means the y-coordinate of the centre is 2.

💡 Strategy: The horizontal line \( y = k \) intersects the circle at \( x = 1 \) and \( x = 5 \). A perpendicular bisector from the centre splits a chord in half.

💡 Strategy: The distance from the centre \( (3, 2) \) to the touching point on the y-axis \( (0, 2) \) is the radius.

💡 Strategy: We have two sides and the included angle (cosine given). This requires the Cosine Rule.

\( a^2 = b^2 + c^2 – 2bc \cos A \)

💡 Strategy: Multiply both sides by the denominator \( (w^3 – 2) \).

💡 Strategy: Move all terms containing \( w^3 \) to one side and everything else to the other.

💡 Strategy: Factor out \( w^3 \) and then divide.

💡 Strategy: Multiply numerator and denominator by \( \sqrt{3} – 1 \) (the conjugate of the denominator).

💡 Strategy: 1. Find \( y \)-coordinate of A. 2. Differentiate to find gradient of tangent (\( m_t \)). 3. Use \( m_n = -1/m_t \) for normal gradient. 4. Use \( y – y_1 = m(x – x_1) \).

💡 Strategy: Solve the simultaneous equations of the curve and the normal line.

Line: \( y = \frac{x + 16}{3} \)

Curve: \( y = x^2 + 5x + 8 \)

💡 Strategy: In the binomial expansion of \( (A + B)^n \), the general term is \( \binom{n}{r} A^{n-r} B^r \). We want the term with \( x^4 \).

Here, \( A = 2x \), \( B = a \), and \( n = 6 \). To get \( x^4 \), we need \( (2x)^4 \). This means \( n-r = 4 \), so \( r = 2 \).

💡 Strategy: Look at equations (1) and (3). One has \( -c \) and the other has \( +c \). Adding them will eliminate \( c \) immediately.

💡 Strategy: We need to eliminate \( c \) from another pair. Let’s use Eq (2) and Eq (3). Multiply Eq (3) by 2 so we have \( +2c \) to match the \( -2c \) in Eq (2).

💡 Strategy: Solve Eq (4) and Eq (5). From Eq (4), \( b = 2 – 2a \).

💡 Strategy: A negative power reciprocates the fraction. \( x^{-n} = \frac{1}{x^n} \), or if \( x^{-n} = \frac{A}{B} \), then \( x^n = \frac{B}{A} \).

💡 Strategy: To isolate \( x \), raise both sides to the power of \( \frac{3}{2} \) (the reciprocal of the current power).

💡 Strategy: An increasing function has a positive gradient (\( f'(x) > 0 \)). First, find the derivative.

💡 Strategy: Complete the square to show the quadratic expression is always greater than zero.

💡 Strategy: Explain why the result proves the function is increasing.

💡 Strategy: Substitute the identity from part (a) into the equation to create a quadratic in terms of \( \sin\theta \).