cameroon gce advanced level 2025 mathematics 3

Based on the image you provided, here is a breakdown of the questions from Section A.

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Question 1

 

Given two functions, g(x)=x3+x2+2x+2 and f(x)=g(x)x+2​.

• (a) Show that $x+1$ is a factor of $g(x)$.
• (b) Factorize $g(x)$ completely.
• (c) Solve in $\mathbb{R}$, the equation $g(x)=0$.
• (d) Deduce in $\mathbb{R}$, the solution of the equation e3x+e2x+2ex+2=0.
• (e) Determine the function $f(x)$.
• (f) Express $f(x)$ into partial fractions.

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Question 2

 

The study of a function $f$ gives the following table of variations.

• (a) Copy and complete the table of variations.
• (i) Give the set of definition of $f$.
• (ii) Determine the limits of $f$ at the boundaries of its set of definition.
• (iii) Verify that the curve (Cf​) of $f$ has a vertical asymptote (D) and give its equation.
• (iv) Give the number of solutions of the equation $f(x)=0$. Justify your answer.

Given that f(x)=x−1x2+x+2​

• (i) Determine three real numbers $a$, $b$ and $c$ for which $f(x)$ can be expressed in the form f(x)=ax+b+x−1c​.
• (ii) Show that the curve (Cf​) of $f$ has an oblique asymptote (D) and give an equation of the latter.
• (iii) Calculate f′(x).
• (iv) Draw (Cf​) and its asymptotes in the same orthonormal reference system (O,i” />,j” />​) in which a unit length on each axis is 1cm.
• (v) $\alpha$ is a real parameter. Discuss in terms of $\alpha$ the existence and sign of the roots of the equation x2−(1−α)x+2+α=0.

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Question 3

 

Given the equation (E):16(x2−18x2+81)−81y4.

• (a) Show that the equation (E) represents the union of two conics and determine their nature.
• (b) Determine the characteristics elements of each conic (vertices, eccentricity, foci, directrices and asymptotes).
• (c) Sketch the curves of both conics on the same orthonormal reference system (O,i” />,j” />​). Unit 1cm.