$\Rightarrow \vec{v}=(\lambda+\mu) \hat{i}+(\lambda-\mu) \hat{j}+(\lambda+\mu) \hat{k}$
$\text { Projection of } \vec{v} \text { on } \vec{c}=\frac{\vec{v} \cdot \vec{c}}{|\vec{c}|}=\frac{1}{\sqrt{3}}$
$\Rightarrow \frac{(\lambda+\mu)-(\lambda-\mu)-(\lambda+\mu)}{\sqrt{3}}=\frac{1}{\sqrt{3}}$
$\Rightarrow \mu-\lambda=1$
$\text { or } \mu=\lambda+1$
$\Rightarrow \vec{v}=(2 \lambda+1) \hat{i}-\hat{j}+(2 \lambda+1) \hat{k}$
$\text { For } \lambda=1, \vec{v}=3 \hat{i}-\hat{j}+3 \hat{k}$
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$(A)$ $\hat{j}-\hat{k}$ $(B)$ $-\hat{i}+\hat{j}$ $(C)$ $\hat{i}-\hat{j}$ $(D)$ $-\hat{j}+\hat{k}$
($A$) There are infinitely many functions from $S$ to $T$
($B$) There are infinitely many strictly increasing functions from $\mathrm{S}$ to $\mathrm{T}$
($C$) The number of continuous functions from $\mathrm{S}$ to $\mathrm{T}$ is at most $120$
($D$) Every continuous function from $\mathrm{S}$ to $\mathrm{T}$ is differentiable