In a frequency table, the upper boundary of each class-interval has a constant ratio to the lower boundary. Show that the geometric mean (G) may be expressed as `log G = A + k/n sum_(i=1)^r f_i (i-1)`.

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (a) ( b ) (c) y=( d ) e^(( e ) (f) K(( g ) (h) x-1( i ))( j ))( k ) (l) (m) (b) ( n ) (o) y=( p ) e^(( q ) (r) K e (s))( t ) (u) (v) (c) ( d ) (e) y=( f ) e^(( g ) (h) K(( i ) (j) x-2( k ))( l ))( m ) (n) (o) (d) None of these

According to Newton's law of gravitation , everybody in this universe attracts every other body with a force, which is directly proportional to the product of their masses and the inversely proportional to the square of the distance between their centres, i.e., F prop (m_1m_2)/(r^2) or F=G(m_1m_2)/(r^2) where G is universal gravitational constant =6.67xx10^(-11)N*m^2*kg^(-2) . Read the above passage and answer the following questions : (i) What is the value of G on the surface of moon ?

According to Newton's law of gravitation , everybody in this universe attracts every other body with a force, which is directly proportional to the product of their masses and the inversely proportional to the square of the distance between their centres, i.e., F prop (m_1m_2)/(r^2) or F=G(m_1m_2)/(r^2) where G is universal gravitational constant =6.67xx10^(-11)N*m^2*kg^(-2) . Read the above passage and answer the following questions : (ii) How is the gravitational force between two bodies affected when distance between them is halved ?

In a frequency distribution, mean is 8.1, sum f_ix_i = 132+5k and sumf_i = 20 . Find the value of k.

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

(i) The pressure and temperature of 0.8 g of He gas are 1 atm and 298 K respectively. If 400 J of heat is applied to the gas at constant volume, what will be change in internal energy and the final temperature of the gas ? (ii) If the same amount of heat is applied to the gas at constant pressure, then what will be the change in internal energy and the final temperature of the gas ? For both the cases, assume He gas behaves ideally and its C_(V)=(3)/(2)R .

If d_i is the deviation of a class mark y_i from 'a' the ' assumed mean ' and F _1 is the frequency, then m_g= x+(1)/(sumf_1)(sumf_i d_i) then x is

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)^("th") " to " (2n)^("th") term is 1/r^n .