Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Would you have fallen for these?

Each is a real option from a previous-year paper. Read it cold and decide before opening it.

• Option C

  There is a bijective function from X to Y.

  Why this is wrong, and why it looks right

  False. A bijection would need |Y| = |X|, but |Y| = 2^|X| and $2^n>n$ for every n ≥ 1 — even the smallest case fails, since |X| = 1 gives |Y| = 2 (SymPy: $2^1-1=1$, so the gap is already positive). For |X| = 3 the sets have 3 and 8 elements.

  From this question

  Let X be a non-empty finite set and Y = {$f^{-1}(0)$ : f is a real-valued function on X}. Which one of the following statements is true?

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  Part B · december 2023 · free

• Option D

  There is a surjective function from X to Y.

  Why this is wrong, and why it looks right

  False. A surjection X → Y needs |X| ≥ |Y|, that is $n\ge 2^n$, which holds for no n ≥ 1. The inequality runs the other way: the power set is always strictly larger.

  From this question

  Let X be a non-empty finite set and Y = {$f^{-1}(0)$ : f is a real-valued function on X}. Which one of the following statements is true?

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  Part B · december 2023

• Option A

  A_a is positive definite for all a < 3.

  Why this is wrong, and why it looks right

  False. Sylvester's criterion needs all leading principal minors positive: they are 2, 3, and det A_a, which SymPy evaluates as 3a − 2. So A_a is positive definite exactly when a > 2/3. The range a < 3 includes a = 0, where the determinant is −2 (verified) — not positive definite.

  From this question

  For $a\in \mathbb{R}$, let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?

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  Part B · december 2023

• Option C

  A_a is positive definite for all a ≥ −2.

  Why this is wrong, and why it looks right

  False. a ≥ −2 includes a = −2, where the determinant is 3(−2) − 2 = −8 (verified), so the matrix is indefinite there. The threshold is 2/3, well above −2.

  From this question

  For $a\in \mathbb{R}$, let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?

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  Part B · december 2023

• Option C

  45

  Why this is wrong, and why it looks right

  False, and this is the engineered near-miss: 45 is exactly the answer if the first digit is forbidden from being 0, giving 9 + 8 + ⋯ + 1 = 45 (verified). The 10 extra PINs are those starting 0, from 000 up to 099. Everything about the calculation is right except the reading of 'PIN'.

  From this question

  The number of three digit PINs, in which the third digit is the sum of the first two digits, is

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  Part A · september 2022

• Option A

  88

  Why this is wrong, and why it looks right

  False here, but it is the right answer to a neighbouring question — which is exactly why it is offered. The power sum $\sum k^{m}modp$ equals −1 ≡ p − 1 precisely when (p − 1) | m, and 0 otherwise. SymPy computes both cases directly: $\sum k^{88}mod89=88$, while $\sum k^{37}mod89=0$. Since 88 ∤ 37 (verified: 37 mod 88 = 37), we are in the second case.

  From this question

  Which one of the following is equal to $1^{37}+2^{37}+3^{37}+\cdots +88^{37}$ in $\mathbb{Z}/89\mathbb{Z}$?

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  Part B · december 2023

• Option C

  8

  Why this is wrong, and why it looks right

  False. 8 is p itself, not p − 1. Every source of variation in a Latin square — rows, columns, treatments — carries p − 1 degrees of freedom, one fewer than the number of levels.

  From this question

  In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is

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  Part B · december 2023

• Option B

  inf U = 4.

  Why this is wrong, and why it looks right

  False — 4 is the supremum of U, not the infimum. SymPy gives the roots as {3, 6} and {3, 4}, so U = [3, 4]: 3 is the left endpoint and 4 the right. This option is what correct algebra plus a misread of which endpoint was asked for produces, which is why it is the most-picked wrong answer.

  From this question

  Consider the following subset of $\mathbb{R}:U=$ {$x\in \mathbb{R}:x^2-9x+18\le 0,x^2-7x+12\le 0$}. Which one of the following statements is true?

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  Part B · december 2023

• Option A

  F(x) has exactly one real root.

  Why this is wrong, and why it looks right

  False, because 'all antiderivatives' means the whole family F + C, and the root count changes with C. Take $f(x)=x^3+x$, whose only real root is 0 and is simple (SymPy: roots 0, ±i). Then $F=x^4/4+x^2/2+C$, and SymPy gives two real roots at C = −1, exactly one at C = 0, and none at C = +1. So 'exactly one' holds for a single value of C, not for all.

  From this question

  Let f(x) be a cubic polynomial with real coefficients. Suppose that f(x) has exactly one real root and that this root is simple. Which one of the following statements holds for ALL antiderivatives F(x) of f(x)?

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  Part B · december 2023

• Option C

  |$I_k$| ≤ |$I_k$₊${}_1$| for every integer k.

  Why this is wrong, and why it looks right

  False. The moduli are $2\pi /(k-1)$!, which decrease once the factorial starts biting. SymPy gives |$I_2$| $=2\pi$ and |$I_3$| $=\pi$, so |$I_2$| > |$I_3$| and the claimed inequality fails at k = 2. The first two values being equal (|$I_1$| = |$I_2$| $=2\pi$, since 0! = 1! = 1) is what makes the sequence look non-decreasing at a glance.

  From this question

  For an integer k, consider the contour integral $I_k={\int }_{|z|=1}$ eᶻ/zᵏ dz. Which of the following statements are true?

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  Part C · december 2023

• Option D

  parabolic in {(x, y) : xy ≠ 0}

  Why this is wrong, and why it looks right

  False. The region xy ≠ 0 is exactly where the discriminant is non-zero, so it is precisely where the equation is not parabolic. Parabolic behaviour occurs only on the degenerate set xy = 0, where all three second-order coefficients collapse.

  From this question

  The partial differential equation $x^2{u}_{xx}-2$xy $u_x$ᵧ $-3y^2{u}_{y}y+u_x-u_y=0$ is

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  Part B · december 2023

• Option A

  $\lim P(1/2<Y_n<3/2)=1$

  Why this is wrong, and why it looks right

  False, and it is ruled out before any limit is taken. The distribution is supported on $[5,\infty )$ — SymPy confirms $F(5)=1-e^0=0$ — so every $X_i$, and hence every $Y_n$, is at least 5. The event {$1/2<Y_n<3/2$} is impossible, with probability 0 for every n.

  From this question

  Let $X_1,X_2,\dots$ be i.i.d. with CDF F(x) = 0 for x < 5 and 1 − $e^{5-x}$ for x ≥ 5. Define $Y_n=\min${$X_1,\dots ,X_n$} and $Z_n=\sqrt{n}(Y_n-5)$, and let Z be standard normal. Which one of the following statements is true?

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  Part B · december 2023

• Option B

  The series is convergent.

  Why this is wrong, and why it looks right

  Not the intended answer, and it is worth being clear why: this statement is literally true — the series does converge. But 'conditionally convergent' in (C) is strictly more informative, and the examiner marks only the most precise true statement. When two options are nested like this, always choose the sharper one. If you found (B) attractive, your mathematics was right and your reading of the instruction was not.

  From this question

  Consider the series ${\sum }_{n=1}^{\infty }$ $a_n$, where $a_n=$ $(-1{)}^{n+1}(\sqrt{n+1}-\sqrt{n})$. Which of the following statements is true?

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  Part B · june 2023

• Option B

  $e^{-6t}$|x(t)| → 0 as $t\to \infty$, for all $x_0\ne 0$.

  Why this is wrong, and why it looks right

  False for the initial conditions lying along the dominant eigendirection. If $x_0$ is an eigenvector for $\lambda =6$, then |x(t)| = |$x_0|e^{6t}$ exactly, so $e^{-6t}$|x(t)| = |$x_0$|, a non-zero constant. The claim is 'for all $x_0\ne 0^{\prime }$, and this one starting point defeats it — damping at exactly the top growth rate leaves a constant, not zero.

  From this question

  Suppose x(t) is the solution of the initial value problem in ${\mathbb{R}}^2$: ẋ = Ax, $x(0)=x_0$, where A = [[5, 4], [1, 2]]. Which of the following statements is true?

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  Part B · june 2023

• Option C

  $e^{-t}$|x(t)| $\to \infty$ as $t\to \infty$, for all $x_0\ne 0$.

  Why this is wrong, and why it looks right

  False, defeated by the other eigendirection. Along the $\lambda =1$ eigenvector, |x(t)| = |$x_0|e^t$, so $e^{-t}$|x(t)| = |$x_0$| — constant, not tending to infinity. The slowest mode is what breaks this one, just as the fastest broke (B).

  From this question

  Suppose x(t) is the solution of the initial value problem in ${\mathbb{R}}^2$: ẋ = Ax, $x(0)=x_0$, where A = [[5, 4], [1, 2]]. Which of the following statements is true?

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  Part B · june 2023

• Option C

  There exists an integer n ≥ 1 such that n $e^{-n}$ ≥ x.

  Why this is wrong, and why it looks right

  False — and the whole question is built around this option. n·e^(−n) is bounded: the function t·e^(−t) peaks at t = 1 with value 1/e ≈ 0.3679 and decays thereafter. Sampled across [0.5, 40], the largest value SymPy finds is exactly that peak. So for any x > 1/e there is no such n. The mistake: reading ${}^{\prime }n\to {\infty }^{\prime }$ off the leading factor and assuming the product must grow. e^(−n) decays faster than n grows, and comparing growth rates — not merely spotting an n — is what is being tested.

  From this question

  Let x be a real number. Which of the following statements are true?

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  Part C · december 2023

• Option C

  There exists $\lambda$ such that $I_{\lambda },K_{\lambda }$ both converge.

  Why this is wrong, and why it looks right

  False, and it is the option the question is built to catch. Both converging would need $\lambda <1$ and $\lambda >1$ at once. The two conditions are not merely different, they are complementary — there is no overlap window, and the single point $\lambda =1$ they share is where both diverge.

  From this question

  For a real number $\lambda$, consider the improper integrals $I_{\lambda }={\int }_0^1$ dx$/(1-x{)}^{\lambda }$ and $K_{\lambda }={\int }_1^{\infty }$ dx$/x^{\lambda }$. Which of the following statements are true?

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  Part C · december 2023

• Option A

  0

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  Part B · june 2023

• Option C

  2

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  Part B · june 2023

• Option A

  16th from the top and 26th from the bottom

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  Part A · september 2022

• Option C

  15th from the top and 27th from the bottom

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  Part A · september 2022

• Option D

  16th from the top and 27th from the bottom

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  Part A · september 2022

• Option B

  S ≠ ∅ and S is a proper subset of $(1,\infty )$

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  Part B · december 2023

• Option C

  $(2,\infty )$ is a proper subset of S

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  Part B · december 2023

• Option D

  S ∩ (0, 1) ≠ ∅

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  Part B · december 2023

• Option A

  f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

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  Part C · december 2023

• Option A

  $99C_0$

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  Part A · september 2022

• Option C

  $S_1$ has exactly one element and $S_2$ has exactly two elements.

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  Part C · december 2023

• Option C

  f has a Taylor series expansion $f(z)=\sum a_n{z}^n$, where $a_0=1,a_1=0$ and $a_n$₊${}_2=a_n+a_n$₊${}_1$ for n ≥ 0.

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  Part B · june 2023

• Option D

  K = [−1, 1]

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  Part C · december 2023

• Option B

  For all $n\ge 1,p_n$ has a zero in the set {$z\in \mathbb{C}$ : |z| ≤ 2023}.

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  Part C · december 2023

• Option C

  P(Y < −1) < P(Y > 1)

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  Part C · december 2023

• Option C

  ${\chi }_{np}^2$

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  Part B · december 2023

• Option B

  f need not be one-one.

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  Part C · december 2023

• Option B

  2

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  Part B · december 2023

• Option D

  Var(X) > 2/3

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  Part C · december 2023

• Option C

  $\mu +{\alpha }_1+{\alpha }_2$ is an estimable linear parametric function

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  Part C · december 2023

• Option C

  $[\bar{X}-(S/\sqrt{n})$ $t_{n-1,0.9}$, $\infty )$

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  Part C · december 2023

• Option C

  C > 180/19

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  Part C · december 2023

• Option A

  $limsu{p}_n\exp (\cos ((n\pi +(-1{)}^n\cdot 2e)/(2n)))>1$.

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  Part B · june 2023

• Option D

  $li{m}_n\exp (\tan ((n{\pi }^2+(-1{)}^n{e}^2)/(7n)))$ does not exist.

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  Part B · june 2023

• Option A

  P{$Y_n\in [0,2n]$}

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  Part C · june 2023

• Option C

  P{$Y_n\in [2n,4n]$}

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  Part C · june 2023

• Option D

  $X_{(4)}$ is a degenerate random variable.

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  Part B · june 2023

• Option D

  If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $=I_3$.

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  Part B · december 2023

• Option D

  If a = 0, then the sequence {$x_n$} does not converge.

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  Part C · june 2023

• Option D

  The ideal generated by $\psi (X)$ is a prime ideal in R.

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  Part C · december 2023

• Option D

  Quadratic form (c) is positive definite.

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  Part C · june 2023

• Option A

  u changes sign in the interior of S.

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  Part C · june 2023

• Option D

  conj(f(z̄)) + f(z̄) is an entire function.

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  Part C · june 2023

• Option D

  f(ℍ) is a convex subset of $\mathbb{C}$.

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  Part B · december 2023

• Option B

  k = −2

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  Part C · december 2023

• Option B

  P(X ≤ 4) = P(X ≥ 22)

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  Part C · december 2023

Other mistakes that cost marks