Advanced level 2026 North west regional mock mathematics with mechanics 2

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Question 1

• (i) Given that the roots of the equation $3x^2 – 4x + 2 = 0$ are $\alpha$ and $\beta$.

  • a) Show that $\alpha^3 + \beta^3 = -\frac{8}{27}$.

  • b) Hence, find the equation with integral coefficients whose roots are $\frac{1}{\alpha^3}$ and $\frac{1}{\beta^3}$.

• (ii) Find the set of real values of $x$ for which $|\frac{x+3}{x-5}| = \frac{x+3}{x-5}$. (3, 2, 5 Marks)

Question 2

• (i) A curve is defined by the equation $y = x^2e^{2x}$.

  • a) Find the coordinates of the stationary points of the curve and determine their nature.

  • b) Show that the curve has points of inflection where $x^2 – 4x + 2 = 0$.

• (ii) Find the numerical value of the term independent of $x$ in the expression $(2x – \frac{1}{x^2})^9$. (3, 3, 4 Marks)

Question 3

• a) Solve the equation $\sin 2\theta + \cos 2\theta + 1 = 0$, for $0^\circ \leq \theta \leq 360^\circ$.

• b) Prove the identity $\frac{\cos A}{\sec A – \tan A} = 1 + \sin A$.

• c) Express $5\cos x – 12\sin x$ in the form $R\cos(x+\alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Hence, find the maximum value of $\frac{1}{5\cos x – 12\sin x + 15}$ and the smallest positive value of $x$ for which this maximum occurs. (3, 3, 4 Marks)

Question 4

The functions $f$ and $g$ are defined by:

• a) Find $f^{-1}(x)$ and state its domain.

• b) Determine the composite function $fg(x)$ and find its range.

• c) Show that $f$ is injective. (3, 4, 3 Marks)

Question 5

The expression $y = ax^2 + bx$ is an approximation to a relation connecting two variables $x$ and $y$, where $a$ and $b$ are constants. By using the values in the following table, draw a suitable straight-line graph and use it to estimate the values of $a$ and $b$.

Question 6

• (i) a) Evaluate $\sum_{r=0}^{50} (4r + 2)$.

• (ii) A school committee of 5 is to be formed from 7 male teachers, 6 female teachers, and the principal.

  • a) In how many ways can the committee be formed if it must include exactly 3 male teachers and 2 female teachers?

  • b) In how many ways can the committee be formed if it must include the principal and at least 2 female teachers? (4, 2, 3 Marks)

Question 7

The points A, B and C have position vectors $\mathbf{a} = 2i + j – 3k$, $\mathbf{b} = 4i – j + 2k$, and $\mathbf{c} = i + 3j – k$ respectively.

• a) Find the vector $\vec{AB} \times \vec{AC}$.

• b) Hence, find the area of the triangle ABC.

• c) Find the Cartesian equation of the plane ABC.

• d) Find the acute angle between the plane ABC and the plane with the equation $x – 2y + z = 5$. (3, 3, 3, 4 Marks)

Question 8

Given $Z_1 = 2 + 3i$ and $Z_2 = 2 – i$.

• a) Express $\frac{Z_1}{Z_2}$ in the form $a + bi$ where $a, b \in \mathbb{R}$.

• b) Find the modulus and argument of $Z_1 \times Z_2$.

• c) Given also that $Z_3$ is a complex number such that $|Z_3| = 5$ and the argument of $Z$ is $\frac{\pi}{3}$, express $\frac{Z_1 \times Z_2}{Z_3}$ in polar form. (3, 5, 2 Marks)

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