In triangle `A B C ,` if `sinAcosB=1/4` and `3t a n A=t a n B ,t h e ncot^2A` is equal to (a)2 (b) 3 (c) 4 (d) 5.

If tanA.tanB=1/2,t h e n(5-3cos2A)(5-3cos2B)= (a)2 (b) 8 (c) 12 (d) 16

In triangle A B C , if cosA+cosB+cosC=7/4, t h e n R/r is equal to 3/4 (b) 4/3 (c) 2/3 (d) 3/2

If y=(1+t a n A)(1-t a n B), where A-B=pi/4, then (y+1)^(y+1) is equal to 9 (b) 4 (c) 27 (d) 81

Let D be the middle point of the side B C of a triangle A B Cdot If the triangle A D C is equilateral, then a^2: b^2: c^2 is equal to 1:4:3 (b) 4:1:3 (c) 4:3:1 (d) 3:4:1

Let C be incircle of A B Cdot If the tangents of lengths t_1,t_2a n dt_3 are drawn inside the given triangle parallel to sidese a , ba n dc , respectively, the (t_1)/a+(t_2)/b+(t_3)/c is equal to 0 (b) 1 (c) 2 (d) 3

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

In triangle A B C , if r_1=2r_2=3r_2, then a : b is equal to 5/4 (b) 4/5 (c) 7/4 (d) 4/7

In triangle A B C , Medians A D and C E are drawn. If A D=5,/_D A C=pi/8,a n d/_A C E=pi/4 , then the area of the triangle A B C is equal to (25)/9 (b) (25)/3 (c) (25)/(18) (d) (10)/3

In triangle A B C ,poin t sD , Ea n dF are taken on the sides B C ,C Aa n dA B , respectigvely, such that (B D)/(D C)=(C E)/(E A)=(A F)/(F B)=ndot Prove that _(D E F)=(n^2-n+1)/((n+1)^2)_(A B C)dot

In triangle A B C ,/_A=30^0,H is the orthocenter and D is the midpoint of B Cdot Segment H D is produced to T such that H D=D Tdot The length A T is equal to a. 2B C b. 3B C c. 4/2B C d. none of these