Question 1 |
Which one of the following is the closed form for the generating function of the sequence \{a_n \}_{n \geq 0} defined below?
a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.
a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.
\frac{x(1+x^2)}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{x(3-x^2)}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{2x}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{x}{(1-x^2)^2}+\frac{1}{1-x} |
Question 1 Explanation:
Question 2 |
Consider the following sets, where n\geq 2:
S1: Set of all nxn matrices with entries from the set \{a, b, c\}
S2: Set of all functions from the set\{0,1,2,...,n^2-1\} to the set \{0, 1, 2 \}
Which of the following choice(s) is/are correct?
[MSQ]
S1: Set of all nxn matrices with entries from the set \{a, b, c\}
S2: Set of all functions from the set\{0,1,2,...,n^2-1\} to the set \{0, 1, 2 \}
Which of the following choice(s) is/are correct?
[MSQ]
There does not exist a bijection from S1 to S2 | |
There exists a surjection from S1 to S2 | |
There exists a bijection from S1 to S2 | |
There does not exist an injection from S1 to S2 |
Question 2 Explanation:
Question 3 |
If the ordinary generating function of a sequence \{a_{n}\}_{n=0}^{\infty} \; is \; \frac{1+z}{(1-z)^{3}} then a_{3}-a_{0} is equal to ______.
16 | |
17 | |
15 | |
14 |
Question 3 Explanation:
Question 4 |
If g(x)=1-x and h(x)=\frac{x}{x-1}, then \frac{g(h(x))}{h(g(x))} is:
\frac{h(x)}{g(x)} | |
\frac{-1}{x} | |
\frac{g(x)}{h(x)} | |
\frac{x}{(1-x)^{2}} |
Question 4 Explanation:
Question 5 |
Let A be a finite set having x elements and let B be a finite set having y elements. What is the number of distinct functions mapping B into A.
x^{y} | |
2^{(x+y)} | |
y^{x} | |
y ! /(y-x) ! |
Question 5 Explanation:
Question 6 |
Consider the set of all functions f:{0,1,...,2014}\rightarrow{0,1...,2014} such that f(f(i))=i, for 0\leqi\leq2014 . Consider the following statements.
P. For each such function it must be the case that for every i, f(i) = i,
Q. For each such function it must be the case that for some i,f(i) = i,
R. Each such function must be onto.
Which one of the following is CORRECT?
P. For each such function it must be the case that for every i, f(i) = i,
Q. For each such function it must be the case that for some i,f(i) = i,
R. Each such function must be onto.
Which one of the following is CORRECT?
P, Q and R are true | |
Only Q and R are true | |
Only P and Q are true | |
Only R is true |
Question 6 Explanation:
Question 7 |
Let S denote the set of all functions f:{\{0,1\}}^{4} \rightarrow \{0,1\}. Denote by N the number of functions from S to the set {0,1}. The value of log_{2} log_{2}N is______.
4 | |
8 | |
16 | |
32 |
Question 7 Explanation:
Question 8 |
How many onto (or surjective) functions are there from an n-element ( n \geq 2 ) set to a 2-element set?
2^{n} | |
2^{n}-1 | |
2^{n}-2 | |
2(2^{n}-2) |
Question 8 Explanation:
Question 9 |
Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a \in A. Which of the following statements is always true for all such functions f and g?
g is onto \implies h is onto | |
h is onto \implies f is onto | |
h is onto \implies g is onto | |
h is onto \implies f and g are onto |
Question 9 Explanation:
Question 10 |
Let f: B \rightarrow C and g: A \rightarrow B be two functions and let h = f\cdotg. Given that h is an onto function which one of the following is TRUE?
f and g should both be onto functions | |
f should be onto but g need to be onto | |
g should be onto but f need not be onto | |
both f and g need to be onto |
Question 10 Explanation:
There are 10 questions to complete.
Q1 is from calculus, kindly move it from here to the calculus section
We have moved Q1 to calculus.