TIFR PHD CS & SS 2012
Question 1 
Amar and Akbar both tell the truth with probability 3/4 and lie with probability 1/4. Amar watches a test match and talks to Akbar about the outcome. Akbar, in turn, tells Anthony, “Amar told me that India won”.
What probability should Anthony assign to India’s win?
What probability should Anthony assign to India’s win?
9/16  
6/16  
7/16  
10/16  
None of the above 
Question 2 
If Mr.M is guilty, then no witness is lying unless he is afraid. There is a witness who is afraid. Which of the following statements is true?
(Hint: Formulate the problem using the following predicates
G – Mr. M is guilty
W(x) – x is a witness
L(x) – x is lying
A(x) – x is afraid )
(Hint: Formulate the problem using the following predicates
G – Mr. M is guilty
W(x) – x is a witness
L(x) – x is lying
A(x) – x is afraid )
Mr.M is guilty  
Mr.M is not guilty  
From these facts one cannot conclude that Mr.M is guilty  
There is a witness who is lying  
No witness is lying. 
Question 3 
Long ago, in a planet far far away, there lived three races of intelligent inhabitants: the Blues (who always tell the truth), the Whites (who always lie), and the Pinks (who, when asked a series of questions, start with a lie and then tell the truth and lie alternately). To three creatures, chosen from the planet and seated facing each other at A, B and C (see figure), the following three questions are put:
(i) What race is your lefthand neighbour?
(ii) What race is your righthand neighbour?
(iii) What race are you?
Here are their answers:
A: (i) White (ii) Pink (iii) Blue
B: (i) Pink (ii) Pink (iii) Blue
C: (i) White (ii) Blue (iii) Blue
What is the actual race of each of the three creatures?
(i) What race is your lefthand neighbour?
(ii) What race is your righthand neighbour?
(iii) What race are you?
Here are their answers:
A: (i) White (ii) Pink (iii) Blue
B: (i) Pink (ii) Pink (iii) Blue
C: (i) White (ii) Blue (iii) Blue
What is the actual race of each of the three creatures?
A is Pink, B is White, C is Blue  
A is Blue, B is Pink, C is White  
A is Pink, B is Blue, C is Pink  
A is White, B is pink, C is blue  
Cannot be determined from the above data 
Question 4 
Let ABC be a triangle with n distinct points inside. A triangulation of ABC with respect to the n points is obtained by connecting as many points as possible such that no more line segment can be added without intersecting other line segments. In other words, ABC has been partitioned into triangles with end points at the n points or at the vertices A, B, C. For example, the following figure gives one possible triangulation of ABC with two points inside it.
Although there are many different ways to triangulate ABC with the n points inside, the number of triangles depends only on n. In the above figure it is five. How many triangles are there in a triangulation of ABC with n points inside it?
Although there are many different ways to triangulate ABC with the n points inside, the number of triangles depends only on n. In the above figure it is five. How many triangles are there in a triangulation of ABC with n points inside it?
3n − 1  
n^{2} + 1  
n + 3
 
2n + 1  
4n − 3 
Question 5 
What is the maximum number of points of intersection between the diagonals of a convex octagon (8vertex planar polygon)? Note that a polygon is said to be convex if the line segment joining any two points in its interior lies wholly in the interior of the polygon. Only points of intersection between diagonals that lie in the interior of the octagon are to be considered for this problem.
55  
60  
65  
70  
75 
Question 6 
A certain pair of used shoes can be repaired for Rs. 1250 and will last for 1 year. A pair of the same kind of shoes can be purchased new for Rs. 2800 and will last for 2 years. The average cost per year of the new shoes is what percent greater than the cost of repairing the used shoes?
5%  
12%  
15%  
3%  
24% 
Question 7 
It is required to divide the 2n members of a club into n disjoint teams of 2 members each. The teams are not labelled. The number of ways in which this can be done is:
a  
b  
c  
d  
e 
Question 8 
How many pairs of sets (A, B) are there that satisfy the condition A, B ⊆ {1, 2, . . . , 5}, A ∩ B = {}?
125  
127  
130  
243  
257 
Question 9 
The probability of throwing six perfect dices and getting six different faces is
1 − 6!/6^{6}  
6!/6^{6}  
6^{6}  
16^{6}  
None of the above 
Question 10 
In how many different ways can r elements be picked from a set of n elements if
(i) Repetition is not allowed and the order of picking matters?
(ii) Repetition is allowed and the order of picking does not matter?
(i) Repetition is not allowed and the order of picking matters?
(ii) Repetition is allowed and the order of picking does not matter?
a  
b  
c  
d  
e 
Question 11 
Let N be the sum of all numbers from 1 to 1023 except the five primes numbers: 2, 3, 11, 17, 31. Suppose all numbers are represented using two bytes (sixteen bits). What is the value of the least significant byte (the least significant eight bits) of N?
00000000  
10101110  
01000000  
10000000  
11000000 
Question 12 
For the polynomial p(x) = 8x^{10} − 7x^{3} + x − 1 consider the following statements (which may be true or false)
(i) It has a root between [0, 1].
(ii) It has a root between [0, −1].
(iii) It has no roots outside (−1, 1).
Which of the above statements are true?
(i) It has a root between [0, 1].
(ii) It has a root between [0, −1].
(iii) It has no roots outside (−1, 1).
Which of the above statements are true?
Only (i)  
Only (i) and (ii)  
Only (i) and (iii)  
Only (ii) and (iii)  
All of (i), (ii) and (iii) 
Question 13 
The maximum value of the function
f(x, y, z) = (x − 1/3)^{2} + (y − 1/3)^{2} + (z − 1/3)^{2}
subject to the constraints
x + y + z = 1, x ≥ 0, y ≥ 0, z ≥ 0
f(x, y, z) = (x − 1/3)^{2} + (y − 1/3)^{2} + (z − 1/3)^{2}
subject to the constraints
x + y + z = 1, x ≥ 0, y ≥ 0, z ≥ 0
1/3  
2/3  
1  
4/3  
4/9 
Question 15 
Consider the differential equation dx/dt = (1−x)(2−x)(3−x). Which of its equilibria is unstable?
x = 0  
x = 1  
x = 2  
x = 3  
None of the above 
Question 16 
Walking at 4/5 is normal speed a man is 10 minute too late. Find his usual time in minutes
81  
64  
52  
40  
It is not possible to determine the usual time from given data 
Question 17 
A spider is at the bottom of a cliff, and is n inches from the top. Every step it takes brings it one inch closer to the top with probability 1/3, and one inch away from the top with probability 2/3, unless it is at the bottom in which case, it always gets one inch closer. What is the expected number of steps for the spider to reach the top as a function of n?
It will never reach the top  
Linear in n  
Polynomial in n  
Exponential in n  
Double exponential in n 
Question 18 
A large community practices birth control in the following peculiar fashion. Each set of parents continues having children until a son is born; then they stop. What is the ratio of boys to girls in the community if, in the absence of birth control, 51% of the babies are born male?
51:49  
1:1  
49:51  
51:98  
98:51

Question 19 
An electric circuit between two terminals A and B is shown in the figure below, where the numbers indicate the probabilities of failure for the various links, which are all independent.
What is the probability that A and B are connected?
What is the probability that A and B are connected?
6/25  
379/400  
1/1200  
1199/1200  
59/60 
Question 20 
There are 1000 balls in a bag, of which 900 are black and 100 are white. I randomly draw 100 balls from the bag. What is the probability that the 101st ball will be black?
9/10  
More than 9/10 but less than 1  
Less than 9/10 but more than 0  
0  
1 
Question 21 
For x, y ∈ {0, 1}^{n}, let x ⊕ y be the element of {0, 1}^{n} obtained by the componentwise exclusiveor of x and y. A Boolean function F : {0, 1}
n → {0, 1} is said to be linear if F(x ⊕ y) = F(x) ⊕ F(y), for all x and y. The number of linear functions from {0, 1}^{n} to {0, 1} is
2^{2n}  
2^{n+1}  
2^{n1}+1  
n!  
2^{n} 
Question 22 
In a graph, the degree of a vertex is the number of edges incident (connected) on it. Which of the following is true for every graph G?
There are even number of vertices of even degree  
There are odd number of vertices of even degree  
There are even number of vertices of odd degree  
There are odd number of vertices of odd degree  
All the vertices are of even degree 
Question 23 
For a person p, let w(p), A(p, y), L(p) and J(p) denote that p is a woman, p admires y, p is a lawyer and p is a judge respectively. Which of the following is the correct translation in first order logic of the sentence: “All women who are lawyers admire some judge”?
∀x : [(w(x) ∧ L(x)) ⇒ (∃y : (J(y) ∧ w(y) ∧ A(x, y)))]  
∀x : [(w(x) ⇒ L(x)) ⇒ (∃y : (J(y) ∧ A(x, y)))]  
∀x∀y : [(w(x) ∧ L(x)) ⇒ (J(y) ∧ A(x, y))]  
∃y∀x : [(w(x) ∧ L(x)) ⇒ (J(y) ∧ A(x, y))]  
∀x : [(w(x) ∧ L(x)) ⇒ (∃y : (J(y) ∧ A(x, y)))] 
Question 24 
Let ∧, ∨ denote the meet and join operations of a lattice. A lattice is called distributive if for all x, y, z,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
It is called complete if meet and join exist for every subset. It is called modular if for all x, y, z,
z ≤ x ⇒ x ∧ (y ∨ z) = (x ∧ y) ∨ z
The positive integers under divisibility ordering i.e. p ≤ q if p divides q forms a
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
It is called complete if meet and join exist for every subset. It is called modular if for all x, y, z,
z ≤ x ⇒ x ∧ (y ∨ z) = (x ∧ y) ∨ z
The positive integers under divisibility ordering i.e. p ≤ q if p divides q forms a
complete lattice  
modular, but not distributive lattice  
distributive lattice  
lattice but not a complete lattice  
under the give ordering positive integers do not form a lattice 
Question 27 
A bag contains 16 balls of the following colors: 8 red, 4 blue, 2 green, 1 black, and 1 white. Anisha picks a ball randomly from the bag, and messages Babu its color using a string of zeros and ones. She replaces the ball in the bag, and repeats this experiment, many times. What is the minimum expected length of the message she has to convey to Babu per experiment?
3/2  
log 5  
15/8  
31/16  
2 
Question 28 
Consider the parse tree
Assume that * has higher precedence than +, − and operators associate right to left (i.e. a + b + c =(a + (b + c))). Consider
(i) 2 + a − b
(ii) 2 + a − b ∗ a + b
(iii) (2 + ((a − b) ∗ (a + b)))
(iv) 2 + (a − b) ∗ (a + b)
The parse tree corresponds to
Assume that * has higher precedence than +, − and operators associate right to left (i.e. a + b + c =(a + (b + c))). Consider
(i) 2 + a − b
(ii) 2 + a − b ∗ a + b
(iii) (2 + ((a − b) ∗ (a + b)))
(iv) 2 + (a − b) ∗ (a + b)
The parse tree corresponds to
Expression (i)  
Expression (ii)  
Expression (iv) only  
Expression (ii), (iii) and (iv)  
Expressions (iii) and (iv) only 
Question 29 
Consider the concurrent program
x:=1;
cobegin
x := x + x + 1  x := x + 2
coend;
Reading and writing of a variable is atomic, but evaluation of an expression is not atomic. The set of possible values of variable x at the end of the execution of the program is
x:=1;
cobegin
x := x + x + 1  x := x + 2
coend;
Reading and writing of a variable is atomic, but evaluation of an expression is not atomic. The set of possible values of variable x at the end of the execution of the program is
{3}  
{7}  
{3,5,7}  
{3,7}  
{3,5} 
Question 30 
Consider the blockedset semaphore where the signaling process awakens any one of the suspended proceses; i.e.,
Wait(S): If S > 0 then S ← S − 1, else suspend the execution of this process.
Signal(S): If there are processes that have been suspended on semaphore S, then wake any one of them,
else S ← S + 1.
Consider the following solution of mutual exclusion problem using blockedset semaphores.
S :=1;
cobegin
P(1)  P(2)  ...  P(N)
coend
where the task body P(i) is
begin
while true do
begin
Wait(S)
Signal(S)
end
end
Here N is the number of concurrent processors. Which of the following is true?
Wait(S): If S > 0 then S ← S − 1, else suspend the execution of this process.
Signal(S): If there are processes that have been suspended on semaphore S, then wake any one of them,
else S ← S + 1.
Consider the following solution of mutual exclusion problem using blockedset semaphores.
S :=1;
cobegin
P(1)  P(2)  ...  P(N)
coend
where the task body P(i) is
begin
while true do
begin
Wait(S)
Signal(S)
end
end
Here N is the number of concurrent processors. Which of the following is true?
The program fails to achieve mutual exclusion of critical regions  
The program achieves mutual exclusion; but starvation freedom is ensured only for N ≤ 2  
The program does not ensure mutual exclusion if N ≥ 3  
The program achieves mutual exclusion, but allows starvation for any N ≥ 2  
The program achieves mutual exclusion and starvation freedom for any N ≥ 1 
Question 31 
Consider the following three versions of the binary search program. Assume that the elements of type T can be compared with each other; also assume that the array a is sorted.
i,j,k : integer;
a : array [1..N] of T;
x :T;
Program 1 : i := 1; j := N;
repeat
k := (i+j)div 2;
if a[k] < x then i := k else j := k
until (a[k] = x) or (i > j)
Program 2 : i := 1; j := N;
repeat
k := (i+j) div 2;
if x < a [k] then j := k1;
if a[k] < x then i := k+1;
until i > j
Program 3 : i := 1; j := N;
repeat
k := (i+j) div 2;
if x < a[k] then j := k else i := k+1
until i > j
A binary search program is called correct provided it terminates with a[k] = x whenever such an element exists, or it terminates with a[k] 6= x if there exists no array element with value x.
Which of the following statements is correct?
i,j,k : integer;
a : array [1..N] of T;
x :T;
Program 1 : i := 1; j := N;
repeat
k := (i+j)div 2;
if a[k] < x then i := k else j := k
until (a[k] = x) or (i > j)
Program 2 : i := 1; j := N;
repeat
k := (i+j) div 2;
if x < a [k] then j := k1;
if a[k] < x then i := k+1;
until i > j
Program 3 : i := 1; j := N;
repeat
k := (i+j) div 2;
if x < a[k] then j := k else i := k+1
until i > j
A binary search program is called correct provided it terminates with a[k] = x whenever such an element exists, or it terminates with a[k] 6= x if there exists no array element with value x.
Which of the following statements is correct?
Only Program 1 is correct  
Only Program 2 is correct  
Only Programs 1 and 2 are correct  
Both Programs 2 and 3 are correct  
All the three programs are wrong 
Question 32 
Let A be a matrix such that A^{k} = 0. What is the inverse of I − A?
0  
I  
A  
1 + A + A^{2} + · · · + Asup>k−1  
Inverse is not guaranteed to exist 
Question 33 
An array A contains n integers. We wish to sort A in ascending order. We are told that initially no element of A is more than a distance k away from its final position in the sorted list. Assume that n and k are large and k is much smaller than n. Which of the following is true for the worst case complexity of sorting A?
A can be sorted with constant · kn comparisons but not with fewer comparisons  
A cannot be sorted with less than constant · n log n comparisons  
A can be sorted with constant · n comparisons  
A can be sorted with constant · n log k comparisons but not with fewer comparisons  
A can be sorted with constant · k^{2}n comparisons but not fewer 
Question 34 
Consider the quicksort algorithm on a set of n numbers, where in every recursive subroutine of the algorithm, the algorithm chooses the median of that set as the pivot. Then which of the following statements is TRUE?
The running time of the algorithm is Θ(n).  
The running time of the algorithm is Θ(n log n)  
The running time of the algorithm is Θ(n^{1.5}).  
The running time of the algorithm is Θ(n^{2})  
None of the above. 
Question 35 
Let T be a tree on n nodes. Consider the following algorithm, that constructs a sequence of leaves u1, u2, . . .. Let u1 be some leaf of tree. Let u2 be a leaf that is farthest from u1. Let u3 be the leaf that is farthest from u2, and, in general, let u_{i+1} be a leaf of T that is farthest from ui (if there are many choices for u_{i+1}, pick one arbitrarily). The algorithm stops when some u_{i} is visited again. What can you say about the distance between u_{i} and u_{i+1}, as i = 1, 2, ...?
For some trees, the distance strictly reduces in each step  
For some trees, the distance increases initially and then decreases  
For all trees, the path connecting u2 and u3 is a longest path in the tree.  
For some trees, the distance reduces initially, but then stays constant.
 
For the same tree, the distance between the last two vertices visited can be different, based on the choice of the first leaf u1. 
Question 36 
Consider a complete binary tree of height n, where each edge is a one Ohm resistor. Suppose all the leaves of the tree are tied together. Approximately how much is the effective resistance from the root to this bunch of leaves for very large n?
Exponential in n  
Cubic in n  
Linear in n  
Logarithmic in n  
Of the order square root of n 
Question 36 Explanation:
Question and answer is wrong.
Question 37 
Which of the following correctly describes LR(k) parsing?
The input string is alternately scanned left to right and right to left with k reversals.  
Input string is scanned once Left to Right with rightmost derivation and k symbol lookahead  
LR(k) grammers are expressively as powerful as contextfree grammers.  
Parser makes k lefttoright passes over input string  
Input string is scanned from left to right once with k symbol to the right as lookahead to give leftmost
derivation. 
Question 38 
Let a^{i} denote a sequence a · a · · · a with i letters and let ℵ be the set of natural numbers {1, 2, · · · }
Let L1 = {a^{i}b^{2i}  i ∈ ℵ} and L2 = {a^{i}b^{i2}  i ∈ ℵ} be two langauges.
Which of the following is correct?
Let L1 = {a^{i}b^{2i}  i ∈ ℵ} and L2 = {a^{i}b^{i2}  i ∈ ℵ} be two langauges.
Which of the following is correct?
Both L1 and L2 are context free languages.  
L1 is contextfree and L2 is recursive but not contextfree  
Both L1 and L2 are recursive but not contextfree  
L1 is regular and L2 is contextfree.  
Complement of L2 is contextfree 
Question 39 
Which of the following statements is TRUE?
Every Turing machine recognizable language is recursive  
The complement of every recursively enumerable language is recursively enumerable  
The complement of a recursive language is recursively enumerable.  
The complement of a context free language is context free  
The set of turing machines which do not halt on empty input forms a recursively enumerable set 
Question 40 
This question concerns the classes P and NP. If you are familiar with them, you may skip the definitions and go directly to the question.
Let L be a set. We say that L is in P if there is some algorithm which given input x decides if x is in L or not in time bounded by a polynomial in the length of x. For example, the set of all connected graphs is in P, because there is an algorithm which, given a graph graph, can decide if it is connected or not in time roughly proportional to the number of edges of the graph.
The class NP is a superset of class P. It contains those sets that have membership witnesses that can be verified in polynomial time. For example, the set of composite numbers is in NP. To see this take the witness for a composite number to be one of its divisors. Then the verification process consists of performing just one division using two reasonable size numbers. Similarly, the set of those graphs that have a Hamilton cycle, i.e. a cycle containing all the vertices of the graph, is in in NP. To verify that the graph has a Hamilton cycle we just check if the witnessing sequence of vertices indeed a cycle of the graph that passes through all the vertices of the graph. This can be done in time that is polynomial in the size of the graph.
More precisely, if L is a set in P consisting of elements of the form (x, w), then the set
Let L be a set. We say that L is in P if there is some algorithm which given input x decides if x is in L or not in time bounded by a polynomial in the length of x. For example, the set of all connected graphs is in P, because there is an algorithm which, given a graph graph, can decide if it is connected or not in time roughly proportional to the number of edges of the graph.
The class NP is a superset of class P. It contains those sets that have membership witnesses that can be verified in polynomial time. For example, the set of composite numbers is in NP. To see this take the witness for a composite number to be one of its divisors. Then the verification process consists of performing just one division using two reasonable size numbers. Similarly, the set of those graphs that have a Hamilton cycle, i.e. a cycle containing all the vertices of the graph, is in in NP. To verify that the graph has a Hamilton cycle we just check if the witnessing sequence of vertices indeed a cycle of the graph that passes through all the vertices of the graph. This can be done in time that is polynomial in the size of the graph.
More precisely, if L is a set in P consisting of elements of the form (x, w), then the set
a  
b  
c  
d  
e 
Question 41 
a  
b  
c  
d  
e 
Question 41 Explanation:
Given question and answer is wrong.
Question 42 
Let α1, α2, · · · , αk be complex numbers. Then
is
is
0  
∞  
α_{k}
 
α1  
max_{j} α_{j}  
Question 43 
a  
b  
c  
d  
None of the above statements are true 
Question 44 
a  
b  
c  
d  
e 
Question 45 
Consider a periodic square wave f(t) with a period of 1 second such that f(t) = 1 for t ∈ [0, 1/2) and f(t) = −1 for t ∈ [1/2, 1). It is passed through an ideal lowpass filter with cutoff at 2 Hz. Then the output is
sin(2πt)  
cos(2πt)  
sin(2πt) − sin(6πt)/3 + sin(10πt)/5 − ...  
sin(2πt) − cos(2πt)  
None of the above 
Question 47 
A linear timeinvariant system has a transfer function H(s) = 1/(1 + s). If the input to the system is cos(t), the output is
(e^{jt} + e^{jt})/2 where j =√−1  
cos(t)/2  
(cos(t) + sin(t))/2  
sin(t)/2.  
The system is unstable and the output is not welldefined 
Question 48 
The input to a series RLC circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude frequency response of this system?
Dependending on the values of R, L and C, a steady state may not exist, and the magnitude frequency
response is not welldefined.  
It is lowpass with 3 dB bandwidth 1/(2π√(LC)).  
It is highpass with 3 dB bandwidth 1/(2π√(LC)).  
It is lowpass and the 3dB bandwidth depends on all: R, L, C  
It is 0 at DC, decays to zero as frequency increases to infinity, and has a maximum at 1/(2π√LC) 
Question 49 
resample y(t) at 16 kHz and sinc interpolate using T =1/8 ms  
resample y(t) at 8 kHz and sinc interpolate using T =1/16 m  
send y(t) over a low pass filter of bandwidth 4 kHz  
any of the above  
none of the above 
Question 50 
Suppose three dice are rolled independently. Each dice can take values 1 to 6 with equal probability. Find the probability that the second highest outcome equals the average of the other two outcomes. Here, the ties may be resolved arbitrarily.
1/6  
1/9  
39/216  
7/36  
43/216 
Question 51 
A Poisson random variable X is given by Pr{X = k} = e^{−λ}λ^{k}/k!, k = 0, 1, 2, ... for λ > 0. The variance of X scales as
λ  
λ^{2}
 
λ^{3}
 
√λ  
None of the above 
Question 52 
In modelling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all nonnegative integer up to 5,
where pn denotes the probability that a health insurance policy holder files n claims during this period. The analyst assumes that no individual files more than 5 claims in this period. Under these assumptions, what is the probability that a policy holder files more than two claims in this period?
where pn denotes the probability that a health insurance policy holder files n claims during this period. The analyst assumes that no individual files more than 5 claims in this period. Under these assumptions, what is the probability that a policy holder files more than two claims in this period?
7/31  
29/125  
1/3  
13/125  
None of the above 
Question 53 
Consider a single amoeba that at each time slot splits into two with probability p or dies otherwise with probability 1 − p. This process is repeated independently infinitely at each time slot, i.e. if there are any amoebas left at time slot t, then they all split independently into two amoebas with probability p or die with probability 1 − p. Which of the following is the expression for the probability that the race of amoeba becomes extinct
a  
b  
c  
d  
None of the above 
Question 55 
Consider a string of length 1 m. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the probability that the three parts can form the sides of a triangle?
1/4  
1/3  
1/2  
2/3  
3/4 
Question 56 
Let P be a n × n matrix such that P^{k} = 0, for some k ∈ N and where 0 is an all zeros matrix. Then at least how many eigenvalues of P are zero
1  
n1  
n  
0  
None of the above 
Question 57 
Let A = UΛU† be a n × n matrix, where UU† = I. Which of the following statements is TRUE
The matrix I + A has nonnegative eigen values  
The matrix I + A is symmetic  
det(I + A) = det(I + Λ)  
(a) and (c)  
(a), (b) and (c) 
Question 58 
Under a certain coordinate transformation from (x, y) to (u, v) the circle x^{2} + y^{2} = 1 shown below on the left side was transformed into the ellipse shown on the right side.
A1 only
A2 only
A1 or A2
A1 or A3
A2 or A3
Question 59 
X and Y are two 3 by 3 matrices. If
then
then
X has rank 2  
at least one of X, Y is not invertible  
X can’t be an invertible matrix  
X and Y could both be invertible  
None of the above 
There are 60 questions to complete.