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3 maths questions
Q3
Taylor series
Construct series for a composite or related function
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Find the Taylor series expansion of $f ( x ) = e ^ { x } \sin ( x )$ centered at $x = \frac { \pi } { 4 }$.
Q6
Number Theory
Prime Number Properties and Identification
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Find all prime numbers $p$ such that $p ^ { 2 } + 2$ is also a prime number.
Q9
Simultaneous equations
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Solve the system of equations:
$$\begin{aligned} & x _ { 1 } + 2 x _ { 2 } - 3 x _ { 3 } = 5 \\ & 2 x _ { 1 } - x _ { 2 } + 4 x _ { 3 } = 7 \\ & 3 x _ { 1 } + 4 x _ { 2 } + x _ { 3 } = - 2 \end{aligned}$$