NIMCET 2021 — Mathematics PYQ

1. Set up the Multiplication

2. Multiply Rows by Columns

The resulting matrix $C$ will have elements $c_{ij}$ calculated as follows:

• Row 1, Column 1: $(\cos \theta )(\cos \alpha )+(-\sin \theta )(\sin \alpha )+(0)(0)=\cos \theta \cos \alpha -\sin \theta \sin \alpha$

• Row 1, Column 2: $(\cos \theta )(-\sin \alpha )+(-\sin \theta )(\cos \alpha )+(0)(0)=-(\sin \alpha \cos \theta +\cos \alpha \sin \theta )$

• Row 2, Column 1: $(\sin \theta )(\cos \alpha )+(\cos \theta )(\sin \alpha )+(0)(0)=\sin \theta \cos \alpha +\cos \theta \sin \alpha$

• Row 2, Column 2: $(\sin \theta )(-\sin \alpha )+(\cos \theta )(\cos \alpha )+(0)(0)=\cos \theta \cos \alpha -\sin \theta \sin \alpha$

• Row 3, Column 3: $(0)(0)+(0)(0)+(1)(1)=1$

• (All other entries involving the 3rd row/column result in 0)

3. Apply Trigonometric Identities

We use the following addition formulas:

• $\cos (\theta +\alpha )=\cos \theta \cos \alpha -\sin \theta \sin \alpha$

• $\sin (\theta +\alpha )=\sin \theta \cos \alpha +\cos \theta \sin \alpha$

Substituting these into our matrix:

4. Conclusion

The resulting matrix matches the structure of the original function $F$, but with the angle $(\theta +\alpha )$.

Correct Option: (C)