EASY
Earn 100

Define the smallest integer function or ceiling function. Find the range of ceiling function by using graph of ceiling function.

Important Questions on Relations and Functions

MEDIUM
Find the number of positive integers n such that the highest power of 7 dividing n! is 8.
MEDIUM

Let x be the greatest integer less than or equal to x, for a real number x. Then the following sum

22020+122018+1+32020+132018+1+42020+142018+1+52020+152018+1+62020+162018+1

is :-

HARD
For any real number x, let [x] denote the integer part of x; {x} be the fractional part of x. x=x-x. Let A denote the set of all real numbers x satisfying x=x+x+x+1220. If S is the sum of all numbers in A, find S.
MEDIUM
Let A={x:x+3+x+43}, B=x:3xr=1310rx-3<3-3x, where [t] denotes greatest integer function. Then,
HARD
The function f :NI defined by fx=x-5x5 , where N is the set of natural numbers and x denotes the greatest integer less than or equal to x, is:
HARD
The equation x24x+[x]+3=x[x], where [x] denotes the greatest integer function, has:
HARD
Consider the sequence of numbers n+2n+12 for n1, where x denotes the greatest integer not exceeding x. If the missing integers in the sequence are n1<n2<n3<.., then find n12.
HARD
Let fx=ax (a>0) be written as fx=f1x+f2x,  where f1(x) is an even function and f2(x) is an odd function. Then f1x+y+f1(x-y) equals:
MEDIUM
Let t denote the greatest integert. Then the equation in x, x2+2x+2-7=0 has :
MEDIUM
02x2dx is equal to, where · represents greatest integer function
HARD
For a real number r let r denote the largest integer less than or equal to r. Let a>1 a real number which is not an integer and k be the smallest positive integer such that ak>ak Then which of the following statement is always true?
HARD
If [x] denotes the greatest integer x, then the system of linear equations [sinθ]x+[-cosθ]y=0[cotθ]x+y=0
MEDIUM
The function f(x)=seclogx+1+x2 is
HARD
If x,y are positive real numbers such that x·x=36 and y·y=71, then x+y equals
HARD
If P(x) be a polynomial with real coefficients such that P(sin2x)=P(cos2x), for all x0,π2. Consider the following statements:

I. P(x) is an even function.

II. P(x) can be expressed as a polynomial in (2x-1)2

III. P(x) is a polynomial of even degree. Then
EASY
Which of the following functions is neither even nor odd?
EASY
If f:RR, such that fx=ex+e-xex-e-x, then f is
EASY
The value of limx0+xpqx is (where · represents greatest integer function)
HARD
For xR,  Let [x] denotes the greatest integer x, then the sum of the series -13+-13-1100+-13-2100+.....+-13-99100 is
HARD
For a real number r we denote by r the largest integer less than or equal to r. If x, y are real numbers with x, y 1 then which of the following statements is always true?