The line $x=\frac{\pi}{4}$ divides the area of the region bounded by $y=\sin x$,$y=\cos x$ and the $x$-axis $(0 \leq x \leq \frac{\pi}{2})$ into two regions of areas $A_1$ and $A_2$. Then $A_1 : A_2$ equals (in $: 1$)

Identify the plant shown in the given figure.

If $A(2,-1)$ and $B(6,5)$ are two points,the ratio in which the foot of the perpendicular from $P(4,1)$ to $AB$ divides $AB$ is:

The expression $[x + (x^3 - 1)^{1/2}]^5 + [x - (x^3 - 1)^{1/2}]^5$ is a polynomial of degree :

If the slope of one line is twice the slope of the other in the pair of straight lines $ax^2 + 2hxy + by^2 = 0$,then $8h^2$ is equal to (in $ab$)

An object $2.4 \, m$ in front of a lens forms a sharp image on a film $12 \, cm$ behind the lens. $A$ glass plate $1 \, cm$ thick,of refractive index $1.50$,is interposed between the lens and the film with its plane faces parallel to the film. At what distance (from the lens) should the object be shifted to be in sharp focus on the film? (in $m$)