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Differential Calculus NAT Level - 2 - Question 1

Maximum area of a rectangle which can be inscribed in a circle of given radius R is given by αR^{2}. Find the value of α.

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Differential Calculus NAT Level - 2 - Question 2

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is 1 *cm* the altitude is 6 *cm*. When the radius is 6 *cm*, the volume is increasing at the rate of 1 *cm*/*s*. When the radius is 36 *cm*, the volume is increasing at a rate of *n* cm^{3}/s. The value of '*n*' is equal to :

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Differential Calculus NAT Level - 2 - Question 3

The maximum value of is given as (λ/e). The value of λ is

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Differential Calculus NAT Level - 2 - Question 4

Consider the function If α is the length of interval of decrease and β be the length of interval of increase, then β/α is

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Differential Calculus NAT Level - 2 - Question 6

If the interval of monotonicity of the function Find the value of α?

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Differential Calculus NAT Level - 2 - Question 7

The least area of a circle circumscribing any right triangle of area *S* is given as απS. Find the value of α.

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Differential Calculus NAT Level - 2 - Question 8

Let f(x) = 2x^{3} + ax^{2} + bx - 3cos^{2} x is an increasing function for all x∈R such that ma^{2} + nb + 18 < 0 then the value of m + n + 7 is

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Differential Calculus NAT Level - 2 - Question 9

If the maximum value of the function f(x) = (sin^{-1} x)^{3} + (cos^{-1} x)^{3}, -1 __<__ x __<__ 1 is α and minimum value is β and α - β is of the form n · π^{3}. Find the value of n**.**

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Differential Calculus NAT Level - 2 - Question 10

If a, b, c, d are real numbers such that then the equation ax^{3} + bx^{2} + cx + d = 0 has at least one root in (0, α). Find the value of α.

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