Q.Consider optically denser lenses P, Q, R and S drawn below. According to Cartesian sign convention which of these have positive focal length? (A) Only P (B) Only P and Q (C) Only P and R (D) Only Q and S

A lens is built from the intersection of two spherical surfaces (radii $R_1$, $R_2$) or one sphere and a plane. A CONVEX lens is thicker at the centre (the internal cross-section of two intersecting spheres); a CONCAVE lens is thicker at the edges (the external cross-section). Concavo-convex and convexo-concave (meniscus) shapes are used for positive- and negative-number spectacles respectively. For a lens denser than its surroundings, convex means converging (positive $f$) and concave means diverging (negative $f$) by the Cartesian convention -- this REVERSES for a lens in a rarer surrounding medium (e.g. an air bubble in glass). A THIN lens (thickness at least 50 times smaller than the other relevant distances) lets both surfaces share one common pole.

For any thin lens, $\frac1f=\frac1v-\frac1u$. For several thin lenses of the SAME axis held in CONTACT, focal powers simply add: $\frac1f=\frac1{f_1}+\frac1{f_2}+\dots$, i.e. $P=P_1+P_2+\dots$. For two thin lenses SEPARATED by distance $d$ in air, $\frac1f=\frac1{f_1}+\frac1{f_2}-\frac{d}{f_1f_2}$, i.e. $P=P_1+P_2-dP_1P_2$ -- the correction term $dP_1P_2$ is what makes a two-lens combination's overall power depend on their separation, not just their individual powers, a fact exploited deliberately in eyepiece design and in the classic lens-displacement method for measuring an unknown focal length.