The 2019 paper, half of it
35 of the 70 maths questions, re-typed from a solution video. Use them to see what the examiners like to ask, not as a mock test.
The answers are the ones marked in the video, worked through again here. The video changes the option letters and hides most question numbers, so they can’t be matched to the commission’s 2019 key. Two answers are left open because the video and the maths disagree.
Q31Principal value of log(-1 + i*sqrt3)ComplexC
Answer: log2 + i*2pi/3
Q32Integral of x dz over ellipse x^2/a^2+y^2/b^2=1Complex?
Presenter mark falls on (b) pi*ab; math gives (a) i*pi*ab - CONFLICT, treat unreliable
Q33Name of "bounded entire function is constant" resultComplexB
Answer: Liouville
Implied: presenter braces the other three.
Q34Radius of convergence of series with coefficient (3+4i)^n (variable symbol garbled as 2^n on screen)ComplexC
Answer: 1/5
Q?For f=e^z/(e^z-1), which statement is NOT true (entire; only poles; infinitely many poles on imaginary axis; each pole simple)ComplexA
Answer: f entirely analytic
Crossed out.
Q?When is cross-ratio of four points of C realComplexC
Answer: all four lie on a circle
Q?Where do all zeros of z^4+6z+3 lie (|z|<2, <1, 1<|z|<2, >2)ComplexC
Answer: 1<|z|<2
Q?Which statement NOT true: zeros isolated, identity theorem, poles isolated, zeros of infinite orderComplexD
Answer: nonzero analytic fn can have zeros of infinite order
Q?Two norms equivalent iff exist alpha,beta with ordering of norms (a: alpha||x||1<=||x||2<=beta||x||1)AnalysisA
Q?Which is true: IPS is Banach/Hilbert, Hilbert = complete IPS, Banach is HilbertAnalysisC
Answer: Hilbert space is a complete inner product space
Q?Which is true: l2 not topological; l_p (p!=2) Hilbert; not Banach; l2 HilbertAnalysisD
Answer: l2 is Hilbert
Math; presenter brackets (b),(c) as wrong.
Q?Norm of identity operator (options labelled e-h on screen: 1,0,10,2)AnalysisA
Answer: = 1
Q?Null space N(T) of bounded linear T between normed spaces: {0}, open, X, closedAnalysisD
Answer: N(T) is closed
Q?Which of compactness, connectedness, openness are NOT preserved by continuous fTopology and DGD
Answer: (iii) openness
Q?Infinite set with discrete topology: connected, compact, every map continuous, not T1Topology and DGC
Answer: every function to another space is continuous
Q?What does Tychonoff assert (product of compact is compact etc.)Topology and DGA
Answer: product of compact spaces is compact
Q?Minimum distance between circle x^2+y^2=1 and line x+y=4D
Answer: 2*sqrt2 - 1
Math; presenter sketch shows distance from centre 2sqrt2 minus radius 1.
Q?Extremal of integral_0^1 [(y')^2+12xy]dx, y(0)=0,y(1)=1 (y=x^4, x^3, x^2, x)MethodsB
Answer: y=x^3
Presenter reaches y''=6x; final tick not seen.
Q?Reflection of (1,2,3) in the xz-planeC
Answer: (1,-2,3)
Q?Centre size of non-abelian group of order p^3AlgebraB
Answer: p
Q?Degree of Q(sqrt2, sqrt3) over QAlgebraD
Answer: 4
Presenter chain Q<Q(sqrt2)<..., 2 x 2; option mark not seen.
Q?V of dim n: which statement NOT true (any n vectors a basis; n+1 dependent; n spanning; n independent)Linear algebraA
Q?A^2=A, A!=0, A!=I: Ker A!={0}; V=Ker A (+) Range A; I+A invertibleLinear algebraD
Answer: all three
Math; presenter circles eigenvalues 0,1.
Q?4x4 invertible: which NOT necessarily true (rows basis; null space zero; 4 distinct eigenvalues)Linear algebraC
Answer: (iii)
Math; presenter crosses the (iii) line.
Q?Det of A_n with entries 2d_ij - d_{i+1,j} - d_{i,j+1}Linear algebraB
Answer: n+1
Math; no presenter answer visible.
Q?Finite-dim inner product space, S nonempty, 0 not in S: which is true about S perpLinear algebraB
Answer: S subset S^perp perp
Math; no presenter answer visible.
Q?Matrix of T(x,y)=(x-2y,y) in standard basisLinear algebraD
Answer: [[1,-2],[0,1]]
Q?Nullity of [[A_n,C],[0,B_m]] with A,B invertibleLinear algebraA
Answer: 0
Math; presenter answer not seen.
Q?Bounded alpha on [a,b]: BV; continuous; constant; sup|alpha| existsAnalysisD
Answer: sup exists
Q?Lebesgue measure of Q: infinity,1,0,2AnalysisC
Answer: 0
Q?Measure of irrationals in [0,1]AnalysisA
Answer: 1
Q?Indicator of rationals on [0,1]: continuous; Riemann integrable; Lebesgue integrable; non-measurableAnalysisC
Answer: Lebesgue integrable
Q?Dual of max Z=c^T X, AX=b, X>=0 (four dual forms; Y free vs Y>=0)OR and numerical?
Presenter mark at (d) (min b^T Y, A^T X>=C, Y>=0); standard theory gives Y free i.e. (c) - CONFLICT, treat unreliable
Q99W a non-1 cube root of unity: sum 1+w+...+w^(3n-1)AlgebraA
Answer: 0
Q100Which of conj z, polynomial, tan z, cosh z are analytic on all of CComplexD
Answer: (ii) and (iv)