Differential Calculus NAT Level - 2


10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Differential Calculus NAT Level - 2


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This mock test of Differential Calculus NAT Level - 2 for Physics helps you for every Physics entrance exam. This contains 10 Multiple Choice Questions for Physics Differential Calculus NAT Level - 2 (mcq) to study with solutions a complete question bank. The solved questions answers in this Differential Calculus NAT Level - 2 quiz give you a good mix of easy questions and tough questions. Physics students definitely take this Differential Calculus NAT Level - 2 exercise for a better result in the exam. You can find other Differential Calculus NAT Level - 2 extra questions, long questions & short questions for Physics on EduRev as well by searching above.
*Answer can only contain numeric values
QUESTION: 1

Maximum area of a rectangle which can be inscribed in a circle of given radius R is given by αR2. Find the value of α.


Solution:

Let rectangle has width b and height h.
Area = h·b
Also,  b2 + h2 = (2R)2 = 4R2

Area is maximum when A2 is maxima
A2 = h2(4R2 – h2)]
f(h) = h2(4R2 – h2)
For maxima,  
⇒   = h2(–2h) + (4R2 – h2)2h = 0

h2 + 4R2 – h2 = 0
h2 = 2R2
h = √2

From physical nature of problem, it is clear that this should be maximum area since minimum area will tend towards zero.

Hence, value of α = 2
The correct answer is: 2

*Answer can only contain numeric values
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 cm3/s. The value of 'n' is equal to :


Solution:


h = αr + c
α = 3
h = 3r + c
h = 6, r = 1
c = 3


= 33 cm3/second
The correct answer is: 33

*Answer can only contain numeric values
QUESTION: 3

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


Solution:


⇒ 
For maxima or minima of y.

⇒ 
and 

Hence, y attains maximum value at 
The value of λ = 1.

The correct answer is: 1

*Answer can only contain numeric values
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


Solution:

f(x) is decreasing in  and increasing in 

The correct answer is: 7

*Answer can only contain numeric values
QUESTION: 5

The ratio of absolute maxima and minima of  is


Solution:

Absolute Maxima = 3
and The Absolute Minima = 1/3

The correct answer is: 9

*Answer can only contain numeric values
QUESTION: 6

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


Solution:


⇒ 
⇒ 
⇒ 
Hence,    value of α = 1
The correct answer is: 1

*Answer can only contain numeric values
QUESTION: 7

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


Solution:


Area of 
S = r2sin θ
⇒ 
Area of circle = 
Least area = πS

So,               Value of α = 1

The correct answer is: 1

*Answer can only contain numeric values
QUESTION: 8

Let  f(x) = 2x3 + ax2 + bx - 3cos2 x is an increasing function for all  x∈R  such that  ma2 + nb + 18 < 0 then the value of m + n + 7 is


Solution:

Given   


D < 0

m = 1, n = – 6
m + n + 7 = 1 – 6 + 7
= 8 – 6 = 2

The correct answer is: 2

*Answer can only contain numeric values
QUESTION: 9

If the maximum value of the function f(x) = (sin-1 x)3 + (cos-1 x)3, -1 << 1 is α and minimum value is β and α - β is of the form n · π3. Find the value of n.


Solution:

Let  

π/4  is point of minima

The correct answer is: 0.844

*Answer can only contain numeric values
QUESTION: 10

If a, b, c, d are real numbers such that then the equation ax3 + bx2 + cx + d = 0  has at least one root in (0, α). Find the value of α.


Solution:

⇒  6a + 4b + 3c + 3d = 0
Let  

f(0) = e

Since f(x) is continuous and differentiable in (0,2) and f(0) = f(2) = e

Hence, according to Rolle's Theorem, the equation

 has at least one root in (0,2). Thus, value of α = 2.

The correct answer is: 2

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