MHTCET 5 May 2025 Shift 2 Paper

Concept:If one vector is a positive scalar multiple of another, they are in the same direction; if the scalar is negative, they are in opposite directions.Explanation:Given: $\overline{c}=5\overline{a}+6\overline{b}$ and $3\overline{c}=\overline{a}-4\overline{b}$.Substitute the expression for $\overline{c}$ into the second equation:$$3(5\overline{a}+6\overline{b})=\overline{a}-4\overline{b}$$$$15\overline{a}+18\overline{b}=\overline{a}-4\overline{b}$$$$14\overline{a}=-22\overline{b}$$$$\overline{a}=-\frac{11}{7}\overline{b}$$Thus, $\overline{a}$ and $\overline{b}$ are in opposite directions.Now write $\overline{b}$ in terms of $\overline{a}$:$$\overline{b}=-\frac{7}{11}\overline{a}$$Substitute this into $\overline{c}=5\overline{a}+6\overline{b}$:$$\overline{c}=5\overline{a}+6\left(-\frac{7}{11}\overline{a}\right)$$$$\overline{c}=5\overline{a}-\frac{42}{11}\overline{a}=\frac{13}{11}\overline{a}$$Since $\frac{13}{11}$ is positive, $\overline{a}$ and $\overline{c}$ are in the same direction.Therefore, $\overline{a}, \overline{c}$ are in the same direction, but $\overline{a}, \overline{b}$ are in the opposite direction.Answer:Option C is correct.