JEE Maths - Basic mathematics

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Question - 1

Suppose, the seed of any positive integer n is defined as follows: Seed(n) = n, if n < l0 = seed(s(n)), otherwise, where s(n) indicates the sum of digits of n. For exarnple, seed(7) =7,seed(248) = seed(2 +4+ 8)= seed(14)= seed(l + 4)= seed(5)=5 etc. How many positive integers n, such that n < 50, will have seed(n) = 9?

  • A 39
  • B 72
  • C 81
  • D 55

Question - 2

Let x, y, z be three positive real numbers such that
x + [y]+ {z) = 13.2
[x] + {y} + z = 14.3
{x}+y+[z] = 15.1
where [a] denotes the greatest integer < a and {b} denotes the fractional part of b, then

  • A xyz = 349.3
  • B x+y+z = 21.3
  • C x+y -z = 4.9
  • D x-y+z = 27.6

Question - 3

Determine the sum of all the real solutions of \(\left[\frac{x}{2}\right]+\left[\frac{2 x}{3}\right]=x\)

  • A More than 5
  • B Less than 6
  • C is an odd number
  • D is less than 3

Question - 4

The units digit of \(\left[\frac{10^{20000}}{10^{100}+3}\right]\) is _______

  • A More than 5
  • B Less than 6
  • C is an odd number 
  • D less than number 3

Question - 5

If 'K' is the total number of integers n between 1 and 10000 (both inclusive) such that n is divisible by[\(\sqrt n\)], then Here [ ] is G.I.F

  • A K is more than 198
  • B K is an even no.
  • C K is less than 300
  • D K is more than 200

Question - 6

If K is the number ordistinct terms in the sequence \(\left[\frac{1^{2}}{1980}\right],\left[\frac{2^{2}}{1980}\right],\left[\frac{3^{2}}{1980}\right], \ldots,\left[\frac{1980^{2}}{1980}\right]\) then K is ________

  • A not divisible by 3
  • B is an even number
  • C is an odd number
  • D is less than 1490

Question - 7

If the number of the first 1000 positive integers can be expressed in the form [2x] +[4x] + [6x] + [8x] where x is a real number is k and [x] denotes the greatest integer less than or equal to x. Then k is _________

  • A divisible by 100
  • B perfect square
  • C an odd number
  • D not divisible by a

Question - 8

For x \(\epsilon\) R, let [x] denote the greatest integer < x. Largest natural number for which \(E=\left[\frac{\pi}{2}\right]+\left[\frac{1}{100}+\frac{\pi}{2}\right]+\left[\frac{2}{100}+\frac{\pi}{2}\right] \ldots+\left[\frac{n}{100}+\frac{\pi}{2}\right]<43 \text { is }\)_____.

  • A 40
  • B 41
  • C 42
  • D None of these

Question - 9

If 'k' is the number of positive integers x which satisfy the condition \(\left[\frac{x}{99}\right]=\left[\frac{x}{101}\right]\), then k is. Here [x] is greatest integer function

  • A a perfect square
  • B 3 an odd number
  • C less than 2500
  • D divisible by 3

Question - 10

Find the minimum natural number n, such that the equation \(\left[\frac{10^{n}}{x}\right]=1989\) 1989 has integer solution x.

  • A 7
  • B 8
  • C 6
  • D None of these
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