That y8 kid 🤣 - bot · guest · Specialist Mathematics · Units 1 & 2

01 Which response best completes this VCE-style item? A geometric sequence with ratio r has \(u_n=\)

A \(u_1r^{n-1}\) There are n−1 ratio multiplications from the first term.

B \(u_1+(n-1)r\) This does not satisfy the stated relationship.

C \(u_1r^n\) This does not satisfy the stated relationship.

D \(nr\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Sequences · answered at 100 ms

02 During a one-minute revision round: The sum of degrees in an undirected graph equals

A twice the number of edges Each edge contributes two degree incidences.

B twice the number of vertices This does not satisfy the stated relationship.

C the number of vertices This does not satisfy the stated relationship.

D the number of edges This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Graph theory · answered at 100 ms

03 The negation of “all integers are positive” is

A at least one integer is positive This does not satisfy the stated relationship.

B no integers exist This does not satisfy the stated relationship.

C all integers are not positive This does not satisfy the stated relationship.

D at least one integer is not positive Negating a universal statement produces an existential counterexample.

Written for icaijy.com · U1–2 · Logic · answered at 100 ms

04 \(\cos(A-B)=\)

A \(\cos A-\cos B\) This does not satisfy the stated relationship.

B \(\sin A\cos B\) This does not satisfy the stated relationship.

C \(\cos A\cos B-\sin A\sin B\) This does not satisfy the stated relationship.

D \(\cos A\cos B+\sin A\sin B\) Use the cosine difference identity.

Written for icaijy.com · U1–2 · Trigonometry · answered at 100 ms

05 A student is checking a worked solution. The negation of “all integers are positive” is

A all integers are not positive This does not satisfy the stated relationship.

B no integers exist This does not satisfy the stated relationship.

C at least one integer is not positive Negating a universal statement produces an existential counterexample.

D at least one integer is positive This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Logic · answered at 100 ms

06 A geometric sequence with ratio r has \(u_n=\)

A \(u_1r^n\) This does not satisfy the stated relationship.

B \(nr\) This does not satisfy the stated relationship.

C \(u_1r^{n-1}\) There are n−1 ratio multiplications from the first term.

D \(u_1+(n-1)r\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Sequences · answered at 100 ms

07 During a one-minute revision round: The conjugate of \(3-2i\) is

A \(3-2i\) This does not satisfy the stated relationship.

B \(2+3i\) This does not satisfy the stated relationship.

C \(3+2i\) Change the sign of the imaginary part.

D \(-3+2i\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Complex numbers · answered at 100 ms

08 The determinant of \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is

A \(ac-bd\) This does not satisfy the stated relationship.

B \(a+d\) This does not satisfy the stated relationship.

C \(ab-cd\) This does not satisfy the stated relationship.

D \(ad-bc\) Use ad−bc.

Written for icaijy.com · U1–2 · Matrices · answered at 100 ms

09 A student is checking a worked solution. If \(a\mid b\) and \(a\mid c\), then

A \(bc\mid a\) This does not satisfy the stated relationship.

B \(a\mid(b+c)\) Divisibility is preserved under integer linear combinations.

C \(a=b+c\) This does not satisfy the stated relationship.

D \(b\mid a\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Number · answered at 100 ms

10 The vector from A to B is

A \(\vec b-\vec a\) Endpoint position minus start position.

B \(\vec a+\vec b\) This does not satisfy the stated relationship.

C \(\vec a\cdot\vec b\) This does not satisfy the stated relationship.

D \(\vec a-\vec b\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Vectors · answered at 100 ms

11 Which response best completes this VCE-style item? The determinant of \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is

A \(ab-cd\) This does not satisfy the stated relationship.

B \(a+d\) This does not satisfy the stated relationship.

C \(ac-bd\) This does not satisfy the stated relationship.

D \(ad-bc\) Use ad−bc.

Written for icaijy.com · U1–2 · Matrices · answered at 100 ms

12 A square matrix is invertible exactly when its determinant is

A one only This does not satisfy the stated relationship.

B negative only This does not satisfy the stated relationship.

C zero This does not satisfy the stated relationship.

D non-zero A zero determinant means singular.

Written for icaijy.com · U1–2 · Matrices · answered at 100 ms

13 During a one-minute revision round: A proof by contradiction begins by assuming

A every lemma is false This does not satisfy the stated relationship.

B the negation of the desired conclusion A contradiction disproves the negation.

C a numerical example This does not satisfy the stated relationship.

D the conclusion itself This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Proof · answered at 100 ms

14 If \(a\mid b\) and \(a\mid c\), then

A \(a=b+c\) This does not satisfy the stated relationship.

B \(b\mid a\) This does not satisfy the stated relationship.

C \(a\mid(b+c)\) Divisibility is preserved under integer linear combinations.

D \(bc\mid a\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Number · answered at 100 ms

15 A student is checking a worked solution. The conjugate of \(3-2i\) is

A \(-3+2i\) This does not satisfy the stated relationship.

B \(3-2i\) This does not satisfy the stated relationship.

C \(3+2i\) Change the sign of the imaginary part.

D \(2+3i\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Complex numbers · answered at 100 ms

16 A proof by contradiction begins by assuming

A every lemma is false This does not satisfy the stated relationship.

B a numerical example This does not satisfy the stated relationship.

C the negation of the desired conclusion A contradiction disproves the negation.

D the conclusion itself This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Proof · answered at 100 ms

17 Which response best completes this VCE-style item? The vector from A to B is

A \(\vec b-\vec a\) Endpoint position minus start position.

B \(\vec a\cdot\vec b\) This does not satisfy the stated relationship.

C \(\vec a-\vec b\) This does not satisfy the stated relationship.

D \(\vec a+\vec b\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Vectors · answered at 100 ms

18 A student is checking a worked solution. The modulus of \(3+4i\) is

A 1 This does not satisfy the stated relationship.

B 25 This does not satisfy the stated relationship.

C 7 This does not satisfy the stated relationship.

D 5 Use the Pythagorean theorem.

Written for icaijy.com · U1–2 · Complex plane · answered at 100 ms

19 Which response best completes this VCE-style item? The sum of degrees in an undirected graph equals

A twice the number of edges Each edge contributes two degree incidences.

B twice the number of vertices This does not satisfy the stated relationship.

C the number of vertices This does not satisfy the stated relationship.

D the number of edges This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Graph theory · answered at 100 ms

20 A student is checking a worked solution. \(\cos(A-B)=\)

A \(\cos A-\cos B\) This does not satisfy the stated relationship.

B \(\sin A\cos B\) This does not satisfy the stated relationship.

C \(\cos A\cos B+\sin A\sin B\) Use the cosine difference identity.

D \(\cos A\cos B-\sin A\sin B\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 100 ms