Scholarship Mathematics

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Question 1

What is the value of this limit?

cal1
.



  • 0

  • 1

  • ∞

  • 1/6

Question 2

Let A be a 3×3 matrix with real entries such that: A3=A and A is neither the zero matrix nor the identity matrix.

What are ALL possible values of det⁡(A))?


  • only 0

  • only 1

  • only −1

  • 0, 1, or −1

Question 3

 Let A,B∈Rn×n such that: AB=I

Then:

  • BA=I always

  • BA≠I always

  • Depends on A

  • Cannot be determined

Question 4

If eigenvalues of a matrix are 2,3,5, then determinant is:

  • 10

  • 30

  • 5

  • 0

Question 5

Two events  A and B are independent and mutually exclusive with P(A)>0 and  P(B)>0 This is:


  • Possible

  • Impossible

  • Only if P(A)=P(B)

  • Only if P(A)+P(B)=1

Question 6

A test for a disease is 99% accurate. The disease affects 1 in 10,000 people. You test positive. What is the probability you actually have the disease?


  • 99%

  • 50%

  •  ~1%

  •  ~0.1%  

Question 7

 Let X be a random variable. Which of the following are always true?

  • E[X2]≥(E[X])2

  •  E[∣X∣]≥∣E[X]∣


  • Var(2X)=2Var(X)


  • Var(X+c)=Var(X) for constant  c


Question 8

 Let X1,X2,…,Xn ∼Bernoulli(p). Define:

cal7
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As n→∞n, what is the distribution of Tn​?

  •  χ2(1)


  • N(0,1)


  •  t(n−1)


  • Bernoulli(p)  


Question 9

Let A and B be n×n matrices. Which is always true?


  • rank(A+B)=rank(A)+rank(B)

  • rank(AB)≥rank(A)+rank(B)−n

  • rank(AB)=rank(A)⋅rank(B)

  • rank(AB)≤rank(A)−rank(B)

Question 10

 Let A be m×n matrix with singular value decomposition:

A=UΣVT

Define the pseudoinverse:

A+=VΣ+UT

If b∉Col(A) what does x∗=A+b represent

  • Exact solution to Ax=b


  • Solution with minimum ∥Ax−b∥ and minimum ∥x∥


  • Solution with maximum ∥x∥


  • Solution to ATAx=0


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