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Maths Question 1 – AP-EAMCET 2026

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[x] represents the greatest integer less than or equal to x. If a function f:R→N is defined by f(x)=[x]2+[x]+1, then {x∈R/f−1(7)=x}=

The greatest integer function, denoted by [x] or ⌊x⌋, gives the greatest integer less than or equal to x. For example, [3.7]=3, [−2.1]=−3. If [x]=k for some integer k, then k≤x<k+1.

Step 1: Set up the equation for f(x)=7✦ Active

The given function is f(x)=[x]2+[x]+1, where [x] represents the greatest integer less than or equal to x. We need to find the set of x values for which f(x)=7. We set the expression for f(x) equal to 7:

[x]2+[x]+1=7
Step 2: Solve the quadratic equation for [x]○ Expand

Let y=[x]. Substituting y into the equation from Step 1, we get a quadratic equation in terms of y:

y2+y+1=7

Rearrange the equation into the standard quadratic form ay2+by+c=0:

y2+y−6=0

Factor the quadratic equation:

(y+3)(y−2)=0

This gives two possible integer values for y:

y=−3ory=2

Therefore, we have two possibilities for [x]:

[x]=−3or[x]=2
💡 Teacher's Secret Hint

Remember that [x] must always be an integer. If you get non-integer solutions for y, they would be disregarded.

Step 3: Determine the intervals for x based on the values of [x]○ Expand

Recall the definition of the greatest integer function: if [x]=k, then k≤x<k+1 for any integer k.

For the first case, [x]=−3:

−3≤x<−3+1⟹−3≤x<−2

This interval is [−3,−2).

For the second case, [x]=2:

2≤x<2+1⟹2≤x<3

This interval is [2,3).

The set of all x values for which f(x)=7 is the union of these two intervals:

x∈[−3,−2)∪[2,3)
💡 Teacher's Secret Hint

Pay close attention to the inequality signs for the greatest integer function: k≤x<k+1. The right endpoint is always strictly less than k+1.

Step 4: Compare the result with the given options○ Expand

We need to find the option that represents the set [−3,−2)∪[2,3). Let's analyze option 3:

[−3,3)−[−2,2)

The interval [−3,3) represents all x such that −3≤x<3. The interval [−2,2) represents all x such that −2≤x<2. Subtracting [−2,2) from [−3,3) means taking all elements in [−3,3) that are not in [−2,2). This effectively removes the middle part of the interval.

So, we are looking for x such that x∈[−3,3) AND xotin[−2,2). This means:

(−3≤x<−2)or(2≤x<3)

This precisely corresponds to the union of intervals we found: [−3,−2)∪[2,3). Thus, option 3 matches our derived solution.

💡 Teacher's Secret Hint

Understand set difference notation. A−B means the elements in A that are not in B. Visualizing intervals on a number line can be very helpful for set operations.

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