Let H be orthocenter of ΔABC and O
its circumcenter. Prove that the vectors
OA^(→) , OB^(→) , OC^(→) and OH^(→) satisfy the
following equality:
OA^(→) + OB^(→) + OC^(→) = OH^(→)
Find distinct natural numbers from 1
to 9 such that these six equations are
satisfied simultaneously:
(1) a + bc = 20
(2) d + e + f = 20
(3) g − hi = −20
(4) adg = 20
(5) b + eh = 20
(6) c + f − i = 10
The horizontal range of a projectile
is R and the maximum height attained
by it is H. A strong wind now begins to
blow in the direction of horizontal
motion of projectile, giving it a constant
horizontal acceleration equal to g.
Under the same conditions of projection,
the new range will be
(g = acceleration due to gravity)
[Answer: R + 4H]
Three consecutive terms of an A.P form the three consecutive terms of a G.P,
If the common ratio of the G.P forms the common difference of the A.P by
adding the first term of the G.P to itself. Find the sum of the fifth term of the G.P.
Related to Q16675:
Find the number of intersection points
of graph sin x=(x/(10)).
Let′s see sin x = (x/n) with n>1.
For n≤1 there is one intersection point.
Let x=2kπ+t with k∈N ∧ t∈[0,2π]
sin x=sin t
cos x=cos t
we find the point on f(x)=sin x where its
tangent is g(x)=(x/n).
f′(x)=cos x=cos t
g′(x)=(1/n)
cos t=(1/n)
t=cos^(−1) (1/n)
sin t=(n/(√(n^2 +1)))
so that f(x) intersects with g(x),
((sin x)/x)≥(1/n)
⇒n sin x≥x
⇒n sin t≥2kπ+t
⇒k≤((n sin t −t)/(2π))=(((n^2 /(√(n^2 +1)))−cos^(−1) (1/n))/(2π))
k_(max) =⌊(((n^2 /(√(n^2 +1)))−cos^(−1) (1/n))/(2π))⌋
number of intersecting points is
m=2×2(k_(max) +1)−1=4k_(max) +3
for n=10
k_(max) =⌊(((n^2 /(√(n^2 +1)))−cos^(−1) (1/n))/(2π))⌋
=⌊((((10^2 )/(√(10^2 +1)))−cos^(−1) (1/(10)))/(2π))⌋=⌊1.35⌋=1
⇒m=4×1+3=7
for n=20
k_(max) =⌊((((20^2 )/(√(20^2 +1)))−cos^(−1) (1/(20)))/(2π))⌋=⌊2.94⌋=2
⇒m=4×2+3=11
A ball is dropped vertically from a
height d above the ground. It hits the
ground and bounces up vertically to a
height (d/2). Neglecting subsequent
motion and air resistance its velocity
V varies with height h above the
ground is
A body is at rest at x = 0. At t = 0, it
starts moving in the positive x-direction
with a constant acceleration. At the
same instant another body passes
through x = 0 moving in the positive
x-direction with a constant speed. The
position of the first body is given by
x_1 (t) after time ′t′ and that of the
second body by x_2 (t) after the same
time interval. Which of the following
graphs correctly describes (x_1 − x_2 ) as
a function of time ′t′?
Consider a rubber ball freely falling
from a height h = 4.9 m onto a
horizontal elastic plate. Assume that
the duration of collision is negligible
and the collision with the plate is
totally elastic. Then the velocity as a
function of time and the height as
function of time will be