CGT SYCS

What does the principle of mathematical induction state

• if the base case (for n = 1) is true and inductive step is true, then the theorem holds for all positive integers

• if the base case (for n = 1) is true and inductive step is false, then the theorem holds for all positive integers

• if the base case (for n = 1) is false and inductive step is true, then the theorem holds for all positive integers

• if the base case (for n = 1) is false and inductive step is false, then the theorem holds for all positive integers

What must be proven in the inductive step?

• Prove that the base case is true

• Prove that k is in the same domain as n

• Prove that, assuming k is true, k+1 is true

• Prove that the process of induction is beyond our understanding and that only the math gods can help us now

According to principle of mathematical induction, if P(k+1) = m^(k+1) + 5 is true then _____ must be true.

• P(k) = 3m^(k)

• P(k) = m^(k) + 5

• P(k) = m^(k+2) + 5

• P(k) = m^(k)

Answer explanation

By the principle of mathematical induction, if a statement is true for any number m = k, then for its successor m = k + 1, the statement also satisfies, provided the statement is true for m = 1. So, the required answer is p(k) = m^k + 5.

For any positive integer m ______ is divisible by 4.

• 5m² + 2

• 3m + 1

• m² + 3

• m³ + 3m

Answer explanation

The required answer is, m³ + 3m. Now, by induction hypothesis, we have to prove for m=k, k³+3k is divisible by 4. So, (k + 1)³ + 3 (k + 1) = k³ + 3 k² + 6 k + 4
= [k³ + 3 k] + [3 k² + 3 k + 4] = 4M + (12k² + 12k) – (8k² + 8k – 4), both the terms are divisible by 4. Hence (k + 1)³ + 3 (k + 1) is also divisible by 4 and hence it is proved for any integer m.

The ordinary enumerator of the selection of r objects out of n objects with unlimited repetition is

• C( n + r - 1, r)

• C(n, r)

• P(n, r)

• S (n, r)

C(n, 0) + C(n, 1) + C(n, 2) + . . . + C(n, n) =

• 2 ^ n

• n ^ 2

• n !

• n^n

Combination concerns itself with the number of _______.

• counting techniques

• selections

• arrangements

• events

The number of ways of selecting six coins from a penny, a nickel, a dime and a quarter with unlimited repetitions is -----------

• 80

• 83

• 84

• None of the above

How many different ways can the letters in the word "trial" be arranged such that the word will begin with a vowel?

• 24

• 48

• 72

• 120

In how many ways can the letters of the word BEAUTY be arranged in which the vowels are always together?

• 144

• 720

• 72

• 24