If $$f\left(1\right)=1,f^{\prime }\left(1\right)=2,$$ then \mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}} is

$$f$$ is defined in $$\left[-5,5\right]$$ as $$f\left(x\right)=x$$ if $$x$$ is rational $$=-x$$ if $$x$$ is irrational. Then

f(x) and g(x) are two differentiable functions on [0, 2] such that f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9 then f(x) - g(x) at x = {3 \over 2} is

If f(x + y) = f(x).f(y) $$\forall$$ x, y and f(5) = 2, f'(0) = 3, then f'(5) is

$${\cot }^{-1}\left(\sqrt{\cos \alpha }\right)-{\tan }^{-1}\left(\sqrt{\cos \alpha }\right)=x,$$ then sin x is equal to :

The number of solution of $$\tan x+\sec x=2\cos x$$ in $$\left[0,2\pi \right]$$ is

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is :

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

The sum of integers from 1 to 100 that are divisible by 2 or 5 is :

Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are :

Product of real roots of equation $$t^2{x}^2+\left|x\right|+9=0$$

If $$\alpha \ne \beta$$ but $${\alpha }^2=5\alpha -3$$ and $${\beta }^2=5\beta -3$$ then the equation having $$\alpha /\beta$$ and $$\beta /\alpha$$ as its roots is

If $$a,b,c$$ are distinct $$+ve$$ real numbers and $$a^2+b^2+c^2=1$$ then $$ab+bc+ca$$ is

Difference between the corresponding roots of $$x^2+ax+b=0$$ and $$x^2+bx+a=0$$ is same and $$a\ne b,$$ then

If $$p$$ and $$q$$ are the roots of the equation $$x^2+px+q=0,$$ then

The sides of a triangle are $$3x+4y,$$ $$4x+3y$$ and $$5x+5y$$ where $$x$$, $$y>0$$ then the triangle is :

In a triangle with sides $$a,b,c,$$ $$r_1>r_2>r_3$$ (which are the ex-radii) then :

A plane which passes through the point $$(3,2,0)$$ and the line {{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \over 4} is :

The $$d.r.$$ of normal to the plane through $$(1,0,0),(0,1,0)$$ which makes an angle $$\pi /4$$ with plane $$x+y=3$$ are :

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?