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

01 A student is checking a worked solution. The scalar projection of a onto non-zero b is

A \(\frac{|\vec b|}{|\vec a|}\) This does not satisfy the stated relationship.

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

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

D \(\frac{\vec a\cdot\vec b}{|\vec b|}\) Divide the dot product by the magnitude of b.

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

02 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

03 During a one-minute revision round: 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 \(b\mid a\) This does not satisfy the stated relationship.

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

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

04 A student is checking a worked solution. The number of ways to choose r objects from n without order is

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

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

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

D \(\binom nr\) Combinations ignore order.

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

05 Two non-zero vectors are perpendicular when their dot product is

A 0 The dot product includes cos 90° = 0.

B −1 always This does not satisfy the stated relationship.

C their magnitude product This does not satisfy the stated relationship.

D 1 This does not satisfy the stated relationship.

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

06 A student is checking a worked solution. A connected graph with n vertices and n−1 edges is a

A multigraph necessarily This does not satisfy the stated relationship.

B cycle This does not satisfy the stated relationship.

C tree This is a standard characterisation of a tree.

D complete graph This does not satisfy the stated relationship.

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

07 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

08 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

09 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

10 The modulus of \(3+4i\) is

A 5 Use the Pythagorean theorem.

B 7 This does not satisfy the stated relationship.

C 1 This does not satisfy the stated relationship.

D 25 This does not satisfy the stated relationship.

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

11 During a one-minute revision round: A geometric sequence with ratio r has \(u_n=\)

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

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

C \(u_1+(n-1)r\) 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

12 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

13 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

14 \(i^2=\)

A 0 This does not satisfy the stated relationship.

B 1 This does not satisfy the stated relationship.

C −1 This defines the imaginary unit.

D i This does not satisfy the stated relationship.

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

15 During a one-minute revision round: A square matrix is invertible exactly when its determinant is

A non-zero A zero determinant means singular.

B zero This does not satisfy the stated relationship.

C negative only This does not satisfy the stated relationship.

D one only This does not satisfy the stated relationship.

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

16 Which response best completes this VCE-style item? A square matrix is invertible exactly when its determinant is

A negative only This does not satisfy the stated relationship.

B zero This does not satisfy the stated relationship.

C non-zero A zero determinant means singular.

D one only This does not satisfy the stated relationship.

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

17 During a one-minute revision round: Two non-zero vectors are perpendicular when their dot product is

A 0 The dot product includes cos 90° = 0.

B their magnitude product This does not satisfy the stated relationship.

C −1 always This does not satisfy the stated relationship.

D 1 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 determinant of \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is

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

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

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

D \(ad-bc\) Use ad−bc.

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

19 A student is checking a worked solution. A geometric sequence with ratio r has \(u_n=\)

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

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

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

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

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

20 During a one-minute revision round: 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