Advanced level 2025 South West Regional Mock TVE mathematics 3

Advanced level 2025 South West Regional Mock TVE mathematics 3

Advanced level 2025 South West Regional Mock TVE mathematics 3

 

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SECTION A

FOR ALL INDUSTRIAL CANDIDATES, EXCEPT CLOTHING INDUSTRY (CLIN)

ANSWER ALL FOUR QUESTIONS

1. a) Consider the functions $f$ and $g$ in the interval $[0,+\infty [$ defined by f(x)=x+22​−lnx and g(x)=−x+22​−lnx. i) Calculate the derivative of $g$ and deduce that it is the primitive of $f(x)$ in the stated interval. ii) Hence, calculate the exact integral ∫13​f(x)dx.

b) Express A(x)=x2+x−22x2+4x−12​ in the form A(x)=a+x+2b​+x−1c​, stating the values of $a,b$ and $c$.

c) Hence, calculate the integral ∫01​A(x)dx.

2. a) Consider two sequences (un​) and (vn​) defined as follows: u0​=e2 un+1​=21​un​ and vn​=lnun​, $\forall n\in \mathbb{N}$. i) Show that (vn​) is a geometric sequence and state its common ratio and first term. ii) Express vn​ and un​ as functions of $n$. iii) Determine limn→+∞​vn​. iv) Deduce that un​ converges to e0=1.

b) The first three terms of an arithmetic series are $p$, $5p-8$, and $3p+8$ respectively. i) Show that $p=4$. ii) Find the value of the 40th term of this series. iii) Prove that the sum of the first $n$ terms of the series is a perfect square.

3. Given the differential equation (E):y′′+81y=0. a) Determine the general solution of $(E)$. b) Determine the solution of $(E)$ given that y(2π​)=−1 and y′(2π​)=−9. c) Express the solution in (b) into the form $r\cos (9x+\alpha )$ where $r\in \mathbb{R}$ and $\alpha$ is an acute angle. d) Solve in $[0,\pi ]$ the equation $y(x)=1$.

4. a) According to Kirchhoff’s law, the currents I1​,I2​,I3​ is obtained from each node yields the following equations: I1​+I2​+I3​=9 I1​−I2​+2I3​=11 2I1​+3I2​+I3​=20 i) Solve for I1​,I2​ and I3​ in $\mathbb{R}$. ii) Find the product of I1​,I2​ and I3​.

b) i) Express z=1+2​i−3+2​i​ in complex trigonometric form. ii) Find another complex number $Z$, such that Z3=1+2​i−3+2​i​=11​(cos125π​+isin125π​). Hence find $Z$.

c) A certain technical college employs 3 men and 4 women, 3 of the employees are to be selected at random to meet with the principal of the college for a talk. i) In how many ways can the three employees be selected? ¹ Find the probability that: ii) All those selected are men iii) All those selected are women