A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :

$$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$$ is :

If a₁, a₂, a₃, ..... are in A.P. such that a₁ + a₇ + a₁₆ = 40, then the sum of the first 15 terms of this A.P. is :

A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
$$\left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr } } \right| = 0$$, is :