Maths4U Original · Short practice

Let \(f(x)=e^{x} \cos{\left(x \right)}\). Write \(f(x)=1+Ax+Bx^2+Cx^3+O(x^4)\).

Find A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

Let \(f(x)=e^{2 x} \cos{\left(2 x \right)}\). Write \(f(x)=1+Ax+Bx^2+Cx^3+O(x^4)\).

Find A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(\displaystyle A=\left[\begin{matrix}15 & 1 & -1\\- \frac{3}{2} & \frac{35}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{33}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.

Find det A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(\displaystyle A=\left[\begin{matrix}16 & 1 & -1\\- \frac{3}{2} & \frac{37}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{35}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.

Find det A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(y''+8y'+15y=17e^{-3x},\ y(0)=15,\ y'(0)=16\). Write \(y=(A+Px)e^{-3x}+Be^{-5x}\).

Find B.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(y''+4y'+3y=18e^{-1x},\ y(0)=16,\ y'(0)=17\). Write \(y=(A+Px)e^{-1x}+Be^{-3x}\).

Find B.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(\displaystyle z= 2e^{\frac{i\pi }{3}}\) and \(S=\sum_{k=0}^{5}z^k\).

Find the real part of the sum from \(\displaystyle k=1\) to \(\displaystyle k=6\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Maths4U Original · Short practice

\(\displaystyle z= 3e^{\frac{i\pi }{4}}\) and \(S=\sum_{k=0}^{2}z^k\).

Find the real part of the sum from \(\displaystyle k=1\) to \(\displaystyle k=3\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.