Prove by induction that for n ∈ N, n2 + n is an even in

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 Multiple Choice QuestionsMultiple Choice Questions

1.

If the number of terms in the expansion of open parentheses 1 minus 2 over straight x space plus 4 over straight x squared close parentheses to the power of straight n comma straight x not equal to 0 comma is 28, then the sum of the coefficients of all the terms in this expansion is

  • 64

  • 2187

  • 243

  • 243

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2.

Statement − 1: For every natural number n ≥ 2 fraction numerator 1 over denominator square root of 1 end fraction space plus space fraction numerator 1 over denominator square root of 2 end fraction space plus space..... space plus space fraction numerator 1 over denominator square root of straight n end fraction space greater than space square root of straight n

Statement −2: For every natural number n ≥ 2,straight n greater or equal than 2 comma space square root of straight n left parenthesis straight n plus 1 right parenthesis space end root space less than space straight n plus 1

  • Statement −1 is false, Statement −2 is true

  • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

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3.

If A = open square brackets table row 1 0 row 1 1 end table close square brackets and I = open square brackets table row 1 0 row 0 1 end table close square brackets , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

  • An = nA – (n – 1)I

  • An = 2n-1A – (n – 1)I

  • An = nA + (n – 1)I

  • An = nA + (n – 1)I

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4.

Let S(K) = 1 +3+5+..... (2K-1) = 3+K2. Then which of the following is true?

  • S(1) is correct

  • Principle of mathematical induction can be used to prove the formula

  • S(K) ≠S(K+1)

  • S(K) ≠S(K+1)

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5.

Maximum sum of coefficient in the expansion of (1 – x sinθ + x2 )n is

  • 1

  • 2n

  • 3n

  • 3n

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6.

For positive integer n, n3 + 2n is always divisible by

  • 3

  • 7

  • 5

  • 6


7.

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8 m/s2. The time taken by the particle to move the second metre is

  • (√2-1)/2 S

  • (√2+1)/2 S

  • (1 + √2)S

  • (√2-1)S


 Multiple Choice QuestionsShort Answer Type

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8.

Prove by induction that for n  N, n2 + n is an even integer (n  1)


n = 1, n2 + n = 2 is an even integer

Let for n = k, k2 + k is even

Now for n = k + 1.

(k + 1)2 + (k + 1) - (k2 + k)

= k2 + 2k + 1 + k + 1 - k2 - k = 2k + 2

which is even integer, also k2 + k is integer

Hence (k + 1)2 + (k + 1) is also an even integer.

Hence n2 + n is even integer for all n  N.


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 Multiple Choice QuestionsMultiple Choice Questions

9.

A particle is moving in a straight line. At time t, the distance between the particle from its starting point is given by x = t - 6t2 + t3. Its acceleration will be zero at

  • t = 1 unit time

  • t = 2 unit time

  • t = 3 unit time

  • t = 4 unit time


10.

For each n N, 23n - 1is divisible by

  • 7

  • 8

  • 6

  • 16


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