The value of the sum C1n2 + C2n2 + C3n2&

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51.

The value of the sum C1n2 + C2n2 + C3n2 + ... + Cnn2 is

  • Cn2n2

  • Cn2n

  • Cn2n + 1

  • Cn2n - 1


D.

Cn2n - 1

We know that

1 + xn = C0n + C1nx + C2nx2 + ... + Cnn      ...(i)and x + 1n = C0nxn + C1nxn - 1 + C2nxn - 2 + ... + Cnn On multiplying Eqs. (i) and (ii), we get 1 + x2n = C0n + C1nx + C2nx2 + ... + Cnn × C0n + C1nx + C2nx2 + ... + CnnCoefficient of xn  in RHS = C0n2 + C1n2 + ... + Cnn2and coefficient of xn  in LHS =  Cn2n C0n2 + C1n2 + ... + Cnn2 = 2n!n! n! C1n2 + ... + Cnn2 = 2n!n! n! - 1                                       = Cn2n   - 1   


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52.

The remainder obtained when 1! + 2! + 3! + ... + 11! is divided by 12 is

  • 9

  • 8

  • 7

  • 6


53.

Let S = 21C0n + 222C1n + 233C2n + ... + 2n + 1n + 1Cnn. Then, S equals

  • 2n + 1 - 1n + 1

  • 3n + 1 - 1n + 1

  • 3n - 1n

  • 2n - 1n


54.

If a, b and c are positive numbers in a GP, then the roots of the quadratic equation

logeax2 - 2logebx + logec = 0 are

  • - 1 and logeclogea

  • 1 and - logeclogea

  • 1 and logac

  • - 1 and logca


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55.

The sum of the series n = 1sinn! π720 is

  • sinπ180 + sinπ360 + sinπ540

  • sinπ6 + sinπ30 + sinπ120 + sinπ360

  • sinπ6 + sinπ30 + sinπ120 + sinπ360 + sinπ720

  • sinπ180 + sinπ360


56.

The coefficient of x3 in the infinite series  expansion of 

21 - x2 - x, for x < 1, is

  • - 116

  • 158

  • - 18

  • 1516


57.

For every real number x, 

let f(x) = x1! + 32!x2 + 73!x3 + 154!x4 + ...  Then, the equation f(x) = 0 has

  • no real solution

  • exactly one real solution

  • exactly two real solutions

  • infinite number of real solutions


58.

Let S denote the sum of the infinite series 1 + 82! + 213! + 404! + 655! + ...

  • S < 8

  • S > 12

  • 8 < S < 12

  • S = 8


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59.

Let tn denotes the nth term of the infinite series 11! + 102! + 213! + 344! + 495! +... . Then, limntn is

  • e

  • 0

  • e2

  • 1


60.

Five numbers are in HP. The middle term is 1 and the ratio of the second and the fourth terms is 2 : 1. Then, the sum of the first three terms is

  • 112

  • 5

  • 2

  • 143


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