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Maths Question 2 – AP-EAMCET 2026

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A and B are respectively the domain and range of a real valued function f(x)=[Sin−1(10x+63)]. then the number of integers in A∪B is ([y] represents the greatest integer less than or equal to y)

Understand that for a function f(x)=g(h(x)), the domain is determined by the conditions for h(x) to be defined, and the range is determined by the possible values of g after h(x) is evaluated.

Step 1: Determine the Domain (A) of the function✦ Active

The function is f(x)=[Sin−1(10x+63)]. For the inverse sine function, Sin−1(u), to be defined, its argument u must be in the interval [−1,1]. Here, u=10x+63.

−1≤10x+63≤1

Multiply all parts by 3:

−3≤10x+6≤3

Subtract 6 from all parts:

−3−6≤10x≤3−6−9≤10x≤−3

Divide all parts by 10:

−910≤x≤−310−0.9≤x≤−0.3

Thus, the domain A=[−0.9,−0.3]. There are no integers in this interval.

Step 2: Determine the Range (B) of the function○ Expand

First, consider the range of Sin−1(10x+63). Since the argument 10x+63 spans the entire domain of Sin−1 (i.e., [−1,1]), the range of Sin−1(10x+63) is its principal value range, which is [−π2,π2]. Numerically, π2≈1.57.

−1.57≤Sin−1(10x+63)≤1.57

Next, apply the greatest integer function [y] (also known as the floor function ⌊y⌋) to this interval. The greatest integer function returns the largest integer less than or equal to y. For y∈[−1.57,1.57], the possible integer values are:

{−2if y∈[−1.57,−1)−1if y∈[−1,0)0if y∈[0,1)1if y∈[1,1.57]

Therefore, the set of all possible integer values for f(x), which is the range B, is {−2,−1,0,1}.

💡 Teacher's Secret Hint

Remember that the greatest integer function always rounds down to the nearest integer. For example, [−1.57]=−2 and [1.57]=1.

Step 3: Find the number of integers in A∪B○ Expand

We have the domain A=[−0.9,−0.3] and the range B={−2,−1,0,1}.

The union of these two sets is A∪B=[−0.9,−0.3]∪{−2,−1,0,1}.

To find the number of integers in A∪B, we list the integers from both sets. Set A contains no integers. Set B contains the integers {−2,−1,0,1}.

The integers in A∪B are thus {−2,−1,0,1}.

There are 4 distinct integers in A∪B.

💡 Teacher's Secret Hint

Carefully identify integers. Integers are whole numbers, positive, negative, or zero. Ensure you don't double-count any elements if there were overlaps.

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