Advanced level 2025 South West Regional Mock TVE mathematics 2

Advanced level 2025 South West Regional Mock TVE mathematics 2

Advanced level 2025 South West Regional Mock TVE mathematics 2

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1. a) P is a polynomial function of real variables $x$ defined by P(x)=3x3+2x2−19x+6. i) Show that $(3x-1)$ is a factor of P. ii) Find three real numbers $a,b,$ and $c$ such that P(x)=(3x−1)(ax2+bx+c). iii) Hence, or otherwise solve in $\mathbb{R}$ the equation $P(x)=0$, i.e., 3x3+2x2−19x+6=0.

b) i) Solve the equation 3e3x+2e2x−19ex+6=0. ii) Solve the equation 3(lnx)3+2(lnx)2−19(lnx)+6=0.

c) Consider the quadratic function: T(x)=4x2−6x+10. i) Show that $T(x)>0,\forall x\in \mathbb{R}$. ii) Solve the equation $T(x)=2$. iii) Solve the equation $T(x)=-3$.

2. A capital U0​=3,500,000 Frs is placed in a bank at 3% per year of compound interests. Un​ is the capital obtained after $n$ years. a) i) Determine the nature of the sequence (Un​) and express it in terms of $n$. ii) Calculate the capital obtained after 10 years. iii) After how many years will the capital be doubled? b) Suppose that at the beginning of each year, an additional amount of 50,000 Frs is deposited in the bank. Let Vn​ be the capital obtained after $n$ years. i) Establish a relation between Vn+1​ and Vn​. ii) Calculate the capital obtained after 5 years.