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?

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.