If the number of terms in the expansion of is 28, then the sum of the coefficients of all the terms in this expansion is
64
2187
243
243
Statement − 1: For every natural number n ≥ 2
Statement −2: For every natural number n ≥ 2,
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.
If A = and I = , 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
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)
D.
S(K) ≠S(K+1)
S(K) = 1 + 3 + 5 + ...... + (2K - 1) = 3 + K2
Put K = 1 in both sides
∴ L.H.S = 1 and R.H.S. = 3 + 1 = 4 ⇒ L.H.S. ≠ R.H.S.
Put (K + 1) on both sides in the place of K L.H.S. = 1 + 3 + 5 + .... + (2K - 1) + (2K + 1)
R.H.S. = 3 + (K + 1)2 = 3 + K2 + 2K + 1
Let L.H.S. = R.H.S.
1 + 3 + 5 + ....... + (2K - 1) + (2K + 1) = 3 + K2 + 2K + 1
⇒ 1 + 3 + 5 + ........ + (2K - 1) = 3 + K2 If S(K) is true, then S(K + 1) is also true. Hence, S(K) ⇒ S(K + 1)
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
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