Let p, q, r be three mutually perpendicular
vectors of the same magnitude. If a vector
x satisfies the equation
p ×{x−q)×p}+q×{x−r)×q}
+ r×{x−p)×r}=0, then x is
given by
A semicircle is tangent to both legs of a
right triangle and has its centre on the
hypotenuse. The hypotenuse is
partitioned into 4 segments, with lengths
3, 12, 12, and x, as shown in the figure.
Determine the value of ′x′.
If
f(x)= determinant (((sin x+sin2x+sin 3x),(sin 2x),(sin 3x)),(( 3+4 sin x),( 3),(4 sin x)),(( 1+sin x),( sin x),( 1)))
then the value of ∫_( 0) ^(π/2) f(x) dx is
A racing car travels on a track (without
banking) ABCDEFA. ABC is a circular
arc of radius 2R. CD and FA are
straight paths of length R and DEF is
a circular arc of radius R = 100 m. The
co-efficient of friction on the road is μ =
0.1. The maximum speed of the car is
50 ms^(−1) . Find the minimum time for
completing one round.
Figure shows (x, t), (y, t) diagram of a
particle moving in 2-dimensions. If the
particle has a mass of 500 g, find the
force (direction and magnitude) acting
on the particle.
Let ABC be an acute-angled triangle
with AC ≠ BC and let O be the
circumcenter and F be the foot of
altitude through C. Further, let X and Y
be the feet of perpendiculars dropped
from A and B respectively to (the
extension of) CO. The line FO intersects
the circumcircle of ΔFXY, second time
at P. Prove that OP < OF.
A polynomial f(x) with rational
coefficients leaves remainder 15, when
divided by x − 3 and remainder 2x + 1,
when divided by (x − 1)^2 . Find the
remainder when f(x) is divided by
(x − 3)(x − 1)^2 .