If `|x-a| lt l_(1)` and `|x-b| lt l_(2)`, show that `|a-b| lt l_(1) + l_(2)`.

If |a| lt 1, |b| lt 1 , then show that a(a+b) + a^(2) (a^(2) + b^(2)) + a^(3)(a^(3) + b^(3)) +… = (a^(2))/(1-a^(2)) + (ab)/(1- ab)

If 0 lt x lt pi/2 , show that, sin x lt x lt tan x

If 0 lt x lt pi/2 , show that, cos x gt 1-x^(2)/(2!)

3x-2 lt 2x+1

Let l_(1) and l_(2) be thw two lines intersecting at P. If A_(1), B_(1), C_(1) are points on l_(1), A_(2), B_(2), C_(2), D_(2), E_(2) are points on l_(2) an dif none of these coincides with P, then the number of triangles formed by joining these eight points is-

An aluminium rod (coefficient of linear expansion alpha_(1) ) of length l_(1) " at " 0^(@)C is welded with a steel rod (coefficient of linear expansion alpha_(2) ) of length l_(2) to form a composite rod of length (l_(1)+l_(2)) . If, when heated through t^(@)C , each rod increases in length by the same amount, then the value of l_(1)/(l_(1)+l_(2)) will be

Using Lagrange's mean value theorem prove that, (b-a)sec^(2)a lt tan b-tan a lt (b-a)sec^(2)b when 0 lt a lt b lt (pi)/(2) .

If x = 1 + a + a^(2) +… oo (-1 lt a lt 1) and y = 1 + b + b^(2)+ … oo (-1 lt b lt 1) , prove that, 1+ab + a^(2)b^(2)+ … oo = (xy)/(x+y - 1)

The frequency of the first overtone of a closed pipe of length l_(1) is equal to that of the first overtone of an open pipe of length l_(2) . The ratio of their lengths (l_(1):l_(2) is

The rate of the reaction , A_(3)Brarr(3)/(2)A_(2) +(1)/(2) B_(2), " is " x "mol.L"^(-1).s^(-1) . If the rates of formation of A_(2) and B_(2) " are " x_(1) and x_(2) "mol.L"^(-1).s^(-1) , respectively , then -