NIMCET 2013 — Mathematics PYQ

To find the value of $\alpha +\beta$, we can use the trigonometric identity for the tangent of a sum.

Step 1: Use the Tangent Addition Formula

The formula for $\tan (\alpha +\beta )$ is:

Step 2: Substitute the Given Values

Substitute $\tan \alpha =\frac{m}{m+1}$ and $\tan \beta =\frac{1}{2m+1}$ into the formula:

Step 3: Simplify the Numerator and Denominator

• Numerator:

  $$\frac{m(2m+1)+1(m+1)}{(m+1)(2m+1)}=\frac{2m^2+m+m+1}{(m+1)(2m+1)}=\frac{2m^2+2m+1}{(m+1)(2m+1)}$$

• Denominator:

  $$1-\frac{m}{(m+1)(2m+1)}=\frac{(m+1)(2m+1)-m}{(m+1)(2m+1)}$$

  Expanding the denominator terms:

  $$\frac{2m^2+m+2m+1-m}{(m+1)(2m+1)}=\frac{2m^2+2m+1}{(m+1)(2m+1)}$$

Step 4: Final Calculation

Now, putting the simplified numerator and denominator back together:

Step 5: Determine the Angle

Since $\tan (\alpha +\beta )=1$, we know that:

Final Answer: The value of $\alpha +\beta$ is $\frac{\pi }{4}$.