NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

For the vectors $\vec{a} = -4\hat{i} + 2\hat{j}$ , $\vec{b} = 2\hat{i} + \hat{j}$ and $\vec{c} = 2\hat{i} + 3\hat{j}$ , if $\vec{c} = m\vec{a} + n\vec{b}$ , then the value of $m + n$ is:

Choose the correct answer:

Explanation

Solution

Concept:

If two vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are equal, then $a_1 = b_1$ , $a_2 = b_2$ and $c_1 = c_2$ .

Calculation:

We have $\vec{c} = m\vec{a} + n\vec{b}$ .

\Rightarrow 2\hat{i} + 3\hat{j} = m(-4\hat{i} + 2\hat{j}) + n(2\hat{i} + \hat{j})

\Rightarrow 2\hat{i} + 3\hat{j} = (-4m + 2n)\hat{i} + (2m + n)\hat{j}

Equating the scalar coefficients, we get:

-4m + 2n = 2 \quad \dots(1)

2m + n = 3 \quad \dots(2)

Multiplying equation (2) by $2$ and adding to equation (1), we get:

4n = 8

\Rightarrow n = 2

Using either of the equations above, we also get:

m = \frac{1}{2}

Therefore:

\therefore m + n = 2 + \frac{1}{2} = \frac{5}{2}

Correct Option: 3. ( $\frac{5}{2}$ )

Choose the correct answer:

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