Anxiety and eating spicy food together in an otherwise normal human may lead to:

$A$ tower subtends angles $\alpha, 2 \alpha$ and $3 \alpha$ respectively at points $A, B$ and $C$,all lying on a horizontal line through the foot of the tower. Then $\frac{A B}{B C}$ is equal to:

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (2 \hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + 3 \hat{k}) = 1$, then $\overrightarrow{a}$ is equal to :

Two wires of the same material and length but diameters in the ratio $1:2$ are stretched by the same force. The elastic potential energy per unit volume for the wires, when stretched by the same force, will be in the ratio: (in $:1$)

When a current of $2\, A$ flows in a battery from negative to positive terminal, the potential difference across it is $12\, V$. If a current of $3\, A$ flows in the opposite direction, the potential difference across the terminals of the battery is $15\, V$. The $emf$ of the battery is ............... $V$.

The function $f: R \rightarrow R$ is defined by $f(x) = \cos^2 x + \sin^4 x$ for $x \in R$. Then the range $f(R)$ is equal to