Number System

Step 1: Removing radicals and fractions.
Square both sides:
Step 2: Identifying order.
The highest order derivative present is . Hence, the order is .
Step 3: Identifying degree.
The highest power of the highest order derivative is . Hence, the degree is .
Step 4: Conclusion.
The order and degree are and respectively.

Step 1: Using formulas for binomial distribution.
For ,
Step 2: Substituting given values.

Step 3: Solving for .

Step 4: Conclusion.
The value of is .

Step 1: Writing the given vectors.

Step 2: Using coplanarity condition.
Three vectors are coplanar if
Step 3: Evaluating the determinant.

Step 4: Conclusion.
Since , Thus, is the geometric mean of and .

Step 1: Understanding the greatest integer function.
The function takes constant integer values in each interval .
Step 2: Breaking the interval.

Step 3: Evaluating each integral.

Step 4: Conclusion.

Step 1: Performing algebraic division.

Step 2: Using partial fractions.

Step 3: Finding constants.
Putting : Putting :
Step 4: Integrating termwise.

Step 5: Conclusion.

Step 1: Applying Lagrange’s Mean Value Theorem.
According to L.M.V.T., there exists a point such that
Step 2: Computing .

Step 3: Evaluating the right-hand side.

Step 4: Solving for .
Since ,
Step 5: Conclusion.
The value of is .

Step 1: Writing the given equation in standard form.
This represents a pair of straight lines through the origin.
Step 2: Using the formula for angle bisectors.
For , the joint equation of bisectors is
Step 3: Substituting values.
Here , , . Taking square root and simplifying:
Step 4: Conclusion.
The joint equation of the angle bisectors is .

Step 1: Using the given condition .

Step 2: Writing .

Step 3: Integrating to find .

Step 4: Using the condition .

Step 5: Conclusion.