Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 2

Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 2

Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 2

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1. (a) Find the value of the constant k for which the two equations f(x)=kx2+2x+1=0 and g(x)=x2+2x+k=0 have a common root. (6marks) (b) With k < 0, find the factors of f(x) and hence solve the inequality $f(x)\ge 0$. (4marks)

2. A polynomial V is defined on R the set of real numbers by V(x)=2x3+kx2−20x−75. a. Given that $(x+5)$ is a factor of V(x), show that k=9 (1mark) b. Factorize V(x) completely and hence solve the equation V(x)=0 (4marks)

The diagram above shows a cuboid of length <span class="math-inline">\(2x \+ 1\)cm</span>, width <span class="math-inline">\(x \+ 6\)cm</span> and height <span class="math-inline">\(x \- 2\)cm</span>.
i. Find the surface area of the box in descending powers of x. (1mks)
ii. Given that the volume of the cuboid is <span class="math-inline">63cm^3</span>, show that x satisfies the equation <span class="math-inline">V\(x\) \= 0</span> (3marks)
iii. Hence find the total surface area of the box. (2mks)

3. a) Find the value of x if $\ln (4x+2)-\ln (x-1)=\ln 10$ (2mks) b) Express $f(\theta )=\cos \theta +$

3​sinθ in the form $R\cos (\theta -\alpha )$ where $R>0$ and $0<\alpha <90°$. Hence state the general solution for which f(θ)=3​ (6mks)

4. The equation of two lines are given by: l1​:r=i+aj−k+λ(i+2j+2k) l2​:r=ai+2k+μ(−j+3k) Where a and λ are scalar and that l1​ and l2​ intersect, find: a. The value of the constant a. (4marks) b. The position vector of the point of intersection. (1marks) c. The equation of the plane containing the two lines. (4marks)