IGDTUW 2026 — Mathematics PYQ

Step 1: Find the derivative of the function

The given function is:

$$f(x)=2\sin x+\sin 2x$$

To find the critical points, we differentiate $f(x)$ with respect to $x$:

$$f^{\prime }(x)=\frac{d}{dx}(2\sin x+\sin 2x)$$

$$f^{\prime }(x)=2\cos x+2\cos 2x$$

Step 2: Find the critical points

Set $f^{\prime }(x)=0$ to look for critical values in the given interval $\left[0,\frac{3\pi }{2}\right]$:

$$2\cos x+2\cos 2x=0$$

$$\cos x+\cos 2x=0$$

Using the trigonometric identity $\cos 2x=2{\cos }^{2}x-1$:

$$2{\cos }^{2}x+\cos x-1=0$$

This is a quadratic equation in terms of $\cos x$. Factoring it gives:

$$2{\cos }^{2}x+2\cos x-\cos x-1=0$$

$$2\cos x(\cos x+1)-1(\cos x+1)=0$$

$$(2\cos x-1)(\cos x+1)=0$$

This yields two cases:

1. $2\cos x-1=0⟹\cos x=\frac{1}{2}$

2. $\cos x+1=0⟹\cos x=-1$

Now, find the values of $x$ that lie within the interval $\left[0,\frac{3\pi }{2}\right]$:

• For $\cos x=\frac{1}{2}⟹x=\frac{\pi }{3}$

• For $\cos x=-1⟹x=\pi$

Step 3: Evaluate the function at critical points and boundary points

To find the absolute maximum (greatest value), we evaluate $f(x)$ at the critical points ($x=\frac{\pi }{3},\pi$) and the endpoints ($x=0,\frac{3\pi }{2}$).

1. At $x=0$:

   $$f(0)=2\sin (0)+\sin (0)=0$$

2. At $x=\frac{\pi }{3}$:

   $$f\left(\frac{\pi }{3}\right)=2\sin \left(\frac{\pi }{3}\right)+\sin \left(\frac{2\pi }{3}\right)$$

   $$f\left(\frac{\pi }{3}\right)=2\left(\frac{\sqrt{3}}{2}\right)+\frac{\sqrt{3}}{2}=\sqrt{3}+\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}$$

3. At $x=\pi$:

   $$f(\pi )=2\sin (\pi )+\sin (2\pi )=2(0)+0=0$$

4. At $x=\frac{\pi \cdot 3}{2}=\frac{3\pi }{2}$:

   $$f\left(\frac{3\pi }{2}\right)=2\sin \left(\frac{3\pi }{2}\right)+\sin (3\pi )$$

   $$f\left(\frac{3\pi }{2}\right)=2(-1)+0=-2$$

Step 4: Compare the values

Comparing the computed values:

• $f(0)=0$

• $f\left(\frac{\pi }{3}\right)=\frac{3\sqrt{3}}{2}\approx 2.598$

• $f(\pi )=0$

• $f\left(\frac{3\pi }{2}\right)=-2$

The greatest value among these is $\frac{3\sqrt{3}}{2}$.

Correct Answer

The correct option is (d).