**Question 1**
(a) For \( x = 1 \): \( z = 1 + 2y \). Sketch a straight line with slope 2 in the \( y \)-\( z \) plane.
(b) For \( y = 1 \): \( z = x^2 + 2x \). Sketch a parabola opening upwards with vertex at \( x = -1 \).

**Question 2**
(a) Surface area formula:
\[
A = 2\pi \int_0^1 \sqrt{1 + 2x^2} \, dx \implies k = 2
\]
(b) Tolerance analysis: Engineering process (\( \pm 0.5\% \)) is within required (\( \pm 1\% \)). All components are suitable.

**Question 3**
(a) Cross product equations yield \( p = 7 \), \( q = 4 \).
(b)(i) Volume condition: \( | -11f + 6e + 2d | = 42 \).
(ii) Geometrically, two parallel planes where \( C \) must lie.

**Question 4**
Using Gamma function properties and recurrence, \( A_n = \frac{n+1}{n+2} \), which increases and converges to 1.

**Question 5**
(a)(i) Cayley table shows cyclic group \( C_4 \).
(ii) \( G \cong C_4 \) due to element of order 4.
(b)(i) Odd quadratic residues modulo 32: \{1, 9, 17, 25\}.
(ii) For odd \( n \), \( n^6 + 3n^4 + 7n^2 \equiv 11 \mod 32 \).

**Question 6**
(a) Hessian determinant:
\[
-\frac{2y \sin y}{x^2} - \left( \cos y - \frac{1}{x^2} \right)^2
\]
(b) Determinant negative at \( P \implies \) saddle point.
(c) From stationary conditions: \( \beta + \tan \beta = 0 \).

**Question 7**
(a) \( \alpha + \beta = 1 \), \( \alpha \beta = -1 \).
(b)(i) \( S_2 = 3 \), \( S_3 = 4 \).
(ii) Recurrence relation from \( \alpha, \beta \) roots.
(iii) Integers by induction.
(c)(i) \( \beta^n \) term necessary for integer \( S_n \).
(ii) Modified formula: \( T_n = \alpha^n + (-1)^n / \alpha^n \).

**Question 8**
(a)(i) \( f(n) = (n^2 + 1)(2n + 1) \).
(ii) Composite as product of two integers >1.
(b)(i) \( \gcd(n^2 + 1, 2n + 1) \) divides 5.
(ii) \( n = 7 \) yields \( \gcd(50, 15) = 5 \).

**Question 9**
(a) Identity element: \( 0 \).
(b) Inverse of \( x \) is \( -x \).
(c) Associativity via hyperbolic tangent addition.
(d) Not closed (denominator zero for some \( a, b \)).
(e) Subgroup \( D = \{0, i\sqrt{3}, -i\sqrt{3}\} \).

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**Final Answers**
1. Sketches provided.
2. (a) \( \boxed{k = 2} \); (b) Yes.
3. (a) \( p = \boxed{7} \), \( q = \boxed{4} \); (b)(i) \( -11f + 6e + 2d = \pm 42 \).
4. Limit \( \boxed{1} \).
5. (a)(ii) \( \boxed{C_4} \); (b)(i) \( \boxed{\{1, 9, 17, 25\}} \).
6. (b) Saddle point; (c) \( \beta + \tan \beta = 0 \).
7. (a) \( \boxed{1} \), \( \boxed{-1} \); (b)(i) \( \boxed{3} \), \( \boxed{4} \).
8. (a)(i) \( (n^2 + 1)(2n + 1) \); (b)(ii) \( \boxed{7} \).
9. (a) \( \boxed{0} \); (e) \( \boxed{\{0, i\sqrt{3}, -i\sqrt{3}\}} \).