Functions

Step 1: Check if the function is one-one
A function is one-one if distinct inputs give distinct outputs, i.e., . For : Thus, is one-one.
Step 2: Check if the function is onto
A function is onto if every element in the codomain has a preimage in the domain . The function outputs values of the form .
However, not all integers can be expressed in this form.
For example, there is no such that , as implies , which does not hold for any integer .
Thus, is not onto.
Step 3: Conclusion
The function is one-one but not onto.

Step 1: Understand the greatest integer function
The greatest integer function, denoted by [x], returns the greatest integer less than or equal to x. It is also called the floor function.

Step 2: Behavior of the greatest integer function
The function is constant within intervals between integers but jumps abruptly at integer points. For example, [1.5] = 1, [2.3] = 2, but at x = 2, the function jumps from 1 to 2.

Step 3: Differentiability and continuity
Differentiability requires the function to be continuous and smooth at the point. However, the greatest integer function has jump discontinuities at all integer points, making it non-differentiable there.

Step 4: Analyze the interval 1 < x < 3
Within the interval, the function jumps at integers 2 and 3. Hence, the points where the function is not differentiable are x = 2 and x = 3.

Step 5: Conclusion
Since the question asks where the function is not differentiable within 1 < x < 3, the answer is x = 2 (as 3 is excluded if we consider open interval).

Final Answer: (C) 2

Step 1: Understand the function
The given function is f(x) = x³. This is a cubic function that is continuous and differentiable everywhere.

Step 2: Find the first derivative
Calculate the first derivative to check for stationary points:
f'(x) = 3x²
At x = 0, f'(0) = 0, so x = 0 is a critical point.

Step 3: Find the second derivative
Calculate the second derivative to determine the nature of the critical point:
f''(x) = 6x
At x = 0, f''(0) = 0, so the second derivative test is inconclusive.

Step 4: Analyze the change of concavity
Check the sign of f''(x) around 0:
- For x < 0, f''(x) < 0 (concave down)
- For x > 0, f''(x) > 0 (concave up)
The concavity changes from concave down to concave up as x passes through zero.

Step 5: Conclusion
Since the concavity changes at x = 0, the point is an inflection point, where the curve changes curvature but is neither a local maxima nor minima.

Final Answer: (D) inflexion

Step 1: Represent the system in matrix form.

The given system of equations can be written as:

where

Step 2: Find . The inverse of a matrix is given by:

where is the determinant of and is the adjugate of .

(a) Compute :

Expanding along the first row:

Compute the minors: Substitute back:

(b) Compute : The adjugate of is the transpose of the cofactor matrix.

Compute the cofactors for each element of :
Thus:

(c) Compute : as .

Thus: Step 3: Solve for .

Using the formula: compute the product:

Perform the multiplication:

Simplify each term:

Final Answer:

To evaluate the transformation, let .

Then: When , ; and when , .

The integral becomes:

Reversing the limits of integration, the negative sign is removed:

The given function is:

Step 1: Check for continuity at
For to be continuous at , the following condition must hold: Here, Since , the function is discontinuous at .

Step 2: Check for differentiability at
For , , which is a polynomial function. Polynomial functions are differentiable everywhere on , so is differentiable for all .

Step 3: Verify other points of continuity and differentiability
Since for , it is both continuous and differentiable for . Hence, the function is continuous and differentiable .

For the function to be continuous at , we need to check whether the left-hand limit ( ), the right-hand limit ( ), and the function value at are all equal.

Step-by-Step Evaluation:

1. Left-hand limit:
   
2. Right-hand limit:
   
3. Function value at :
   

Condition for Continuity:

For to be continuous at , the following must hold:

So we equate:

Conclusion:

The given assertion claims continuity at for , which is incorrect. Hence:

• Assertion (A) is false
• Reason (R) is true (it correctly states the definition of continuity)

Final Answer: Assertion is false, but Reason is true.

To analyze whether the function is increasing or decreasing, we calculate the derivative of . Given: Differentiate with respect to : Now observe the sign of : - For all , ⟹ ⟹ So, for all . This implies the function is decreasing for all .

A function is bijective if it is both injective and surjective.
1. Injective: A function is injective (one-to-one) if different elements in the domain map to different elements in the codomain. In this case, since each student has a unique roll number, no two students will have the same roll number.
Hence, is injective.
2. Surjective: A function is surjective (onto) if every element in the codomain has a preimage in the domain.
Here, since the set has 30 students, and the natural numbers are infinite, is not surjective because not every natural number corresponds to a roll number of a student.
Therefore, is not surjective. Thus, is not bijective.