Cameroon GCE advanced level June 2025 mathematics with mechanics 2

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1. (i) When the polynomial P(x)=2×3+ax2+bx−12 is divided by (x+3) the remainder is 18.

   Given that (x+1) is a factor of P(x),

   (a) find the values of the constants a and b. (9 marks)

   (b) factorise P(x) completely.

   (ii) Find the set of real values of k, for which the roots of the quadratic equation x2+kx+3+k=0 are real and distinct. (3 marks)

2. The table below shows some particular values of two variables $x$ and $y$.

   | x | 2 | 13 | 24 | 37 | 52 |

   |—|—|—-|—-|—-|—-|

   | y | 3.1 | 5.4 | 6.5 | 7.6 | 8.2 |

   It is known that $x$ and $y$ satisfy a relation of the form y=axn. By drawing a suitable linear graph relating $\log y$ and $\log x$, find the values of the constants $a$ and $n$, giving the answer to one decimal place. (8 marks)

3. (i) Show that,

   cos3θ+cosθsin3θ+sinθ​=tan2θ. (3 marks)

   (ii) Given that f(x)=cosx+3​sinx,

   (a) Express f(x) in the form Rcos(x−λ), where R>0 and 0<λ<2π​.

   Hence, find

   (b) the minimum value of 1+∣f(x)∣1​

   (c) the general solution of f(x)=3​. (8 marks)

4. (i) The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms of the progression is 57.

   Find the number of terms in the progression. (4 marks)

   (ii) Find the numerical value of the term independent of x in the expansion of (3×2−x31​)9. (4 marks)

5. (i) Express the complex number z=1+2i−5+10i​ in the form a+bi, where a,b∈R.

   Hence or otherwise, find the modulus and argument the of z3. (6 marks)

   (ii) Given that f(x)=x3−2x−11, show that the equation f(x)=0 has a root between 2 and 3.

   Taking 2 as a first approximation to the root of the equation f(x)=0, use one iteration of the Newton-Raphson procedure to obtain a second approximate root, giving the answer to one decimal place. (6 marks)

6. (i) The function f is periodic with period 4. Given that f(x)={x2−2 for 0≤x<24−x for 2≤x≤4​,

   (a) evaluate f(19) and f(−43).

   (b) sketch the graph of f(x) in the range −4≤x≤8. (6 marks)

   (ii) A relation R is defined on Z, the set of integers, by aRb if and only if (a+b) is even.

   Show that R is an equivalence relation. (6 marks)

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7. (i) Two statements p and q are defined as:

   p: Kelvin will go to Kribi

   q: Kelvin will visit the seaport

   Translate into ordinary English, the statements:

   (a) p⇒q

   (b) p∧q

   (c) ∼p∨∼q (3 marks)

   (ii) The position vectors of the points A, B and C are 2i−3j−3k,i+j−4k and 7i+2j+k respectively.

   Find

   (a) the vector equation of the line AB.

   (b) the cartesian equation of the plane ABC.

   (c) the sine of the angle between line AB and plane ABC. (8 marks)

8. (i) Given that 3×2−4xy+2y2=0, show that dxdy​=2x−2y3x−2y​. (3 marks)

   (ii) Given that f(x)=x2−x−12x+10​, express f(x) in partial fractions.

   Hence, show that ∫02​f(x)dx=ln(536​). (7 marks)

9. (i) A matrix M is given by M=​121​−111​1−31​​

   Find

   (a) ∣M∣, the determinant of matrix M.

   (b) M−1, the inverse of matrix M. (5 marks)

   (ii) Prove by mathematical induction that for all n∈N,

   ∑r=1n​r(3r−1)=n2(n+1) (5 marks)

10. (a) Solve the differential equation (x−1)dxdy​=1−y, given that y=−2 when x=2, expressing your result in the form y=f(x). (5 marks)

    (b) Sketch the curve y=x−1x−4​, showing clearly the points where the curve crosses the coordinate axes and the behaviour of the curve near its asymptotes. (4 marks)