Questions on vector arithmetic and vector operations, including dot and cross product, as well as the vector equations of planes and lines.

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.

Express various integers and rationals mod $\mathbb{Z}_3, \;\mathbb{Z}_5,\;\mathbb{Z}_7,\;\mathbb{Z}_{11}$

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Calculate definite integrals: $\int_0^\infty\;e^{-ax}\,dx$, $\int_1^2\;\frac{1}{x^{b}}\,dx$, $\; \int_0^{\pi}\;\cos\left(\frac{x}{2n}\right)\,dx$

Evaluate $\int_1^{\,m}(ax ^ 2 + b x + c)^2\;dx$, $\int_0^{p}\frac{1}{x+d}\;dx,\;\int_0^\pi x \sin(qx) \;dx$, $\int_0^{r}x ^ {2}e^{t x}\;dx$

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$. Input constant of integration as $C$.

Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$. 

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx $ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx $ 

$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$  for a polynomial $g(x)$. Find $g(x)$.

Evaluate $\int_0^{\,m}e^{ax}\;dx$, $\int_0^{p}\frac{1}{bx+d}\;dx,\;\int_0^{\pi/2} \sin(qx) \;dx$. 

No solutions given in Advice to parts a and c.

Tolerance of 0.001 in answers to parts a and b. Perhaps should indicate to the student that a tolerance is set. The feedback on submitting an incorrect answer within the tolerance says that the answer is correct - perhaps there should be a different feedback in this case if possible for all such questions with tolerances.

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$.

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Find an integral by choosing a suitable substitution.

Find the integral of an improper fraction.

Questions which rely on knowledge of standard integrals.

Inputting ratios of algebraic expressions.

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx $ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

Integrating by parts.

Find $ \int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$