JAMIA 2026 — Mathematics PYQ

Step 1: Perform the Substitution

Let us substitute the denominator:

$$t=3+4\sin x$$

Differentiate both sides with respect to $x$:

$$dt=4\cos xdx$$

$$\cos xdx=\frac{dt}{4}$$

Step 2: Change the Limits of Integration

Since we changed the variable from $x$ to $t$, we must find the corresponding limits for $t$:

• Lower limit: When $x=0$

  $$t=3+4\sin (0)=3+4(0)=3$$

• Upper limit: When $x=\frac{\pi }{3}$

  $$t=3+4\sin \left(\frac{\pi }{3}\right)=3+4\left(\frac{\sqrt{3}}{2}\right)=3+2\sqrt{3}$$

Step 3: Evaluate the Integral

Now substitute the values into the integral:

$$I={\int }_3^{3+2\sqrt{3}}\frac{1}{t}\left(\frac{dt}{4}\right)$$

$$I=\frac{1}{4}{\int }_3^{3+2\sqrt{3}}\frac{1}{t}dt$$

The integral of $\frac{1}{t}$ is $\log t$:

$$I=\frac{1}{4}[\log t{]}_3^{3+2\sqrt{3}}$$

Apply the upper and lower limits:

$$I=\frac{1}{4}\left(\log (3+2\sqrt{3})-\log 3\right)$$

Using the logarithmic property $\log a-\log b=\log \left(\frac{a}{b}\right)$:

$$I=\frac{1}{4}\log \left(\frac{3+2\sqrt{3}}{3}\right)$$

$$I=\frac{1}{4}\log \left(\frac{3}{3}+\frac{2\sqrt{3}}{3}\right)$$

$$I=\frac{1}{4}\log \left(1+\frac{2}{\sqrt{3}}\right)$$

Step 4: Compare with the Given Expression

We are given that:

$$I=\frac{1}{4}\log k$$

Comparing this with our calculated value:

$$\frac{1}{4}\log \left(1+\frac{2}{\sqrt{3}}\right)=\frac{1}{4}\log k$$

By direct comparison:

$$k=1+\frac{2}{\sqrt{3}}$$

Correct Answer:

(a) $\left(1+\frac{2}{\sqrt{3}}\right)$