Advanced level 2026 Littoral regional mock replaced further mathematics 3

Given the vectors $p = 3i + 4j - k$ , $q = -i + 2j + 2k$ and $r = i + 4j + \lambda k$ , where $\lambda$ is a scalar parameter.

Determine:

(a) $p \times q$ (2 marks)

(b) $p \times q \cdot r$ (2 marks)

Hence,

(c) determine the value of $\lambda$ if $p, q$ and $r$ are linearly dependent. (2 marks)

(d) equation of the plane which contains $p, q$ (2 marks)

10. The curve C has equation $f(x) = \frac{px^2 + 4x + 1}{x + 1}$ where p is a positive constant and $p \ne 3$ .

(a) Express $y$ in the form $f(x) = ax + b + \frac{c}{x+1}$ (2 marks)

(b) Hence, state the equations of the asymptotes of C (2 marks)

(c) Find the value of p for which the x-axis is a tangent to C, sketch C in this case (4 marks)

(d) For the case $p = 1$ :

(i) State the domain of definition of C (1 mark)

(ii) State the equations of the asymptotes to C (1 mark)

(iii) Show that C has no turning points (3 marks)

(iv) State the coordinates of points of intersection of C with the coordinate axes (2 marks)

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