Differential Calculus MCQ Level - 1

# Differential Calculus MCQ Level - 1

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Differential Calculus MCQ Level - 1 for IIT JAM 2023 is part of Topic wise Tests for IIT JAM Physics preparation. The Differential Calculus MCQ Level - 1 questions and answers have been prepared according to the IIT JAM exam syllabus.The Differential Calculus MCQ Level - 1 MCQs are made for IIT JAM 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Differential Calculus MCQ Level - 1 below.
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Differential Calculus MCQ Level - 1 - Question 1

### The functIon u(x, y) = x2 + xy + 3x + 2y + 5 has a point of :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 1

Let z = x2 + xy + 3x + 2y + 5 For max. or min of z, we must have at (–2, 1)
rt – s2 = –1 < 0
∴  neither max. nor min.
The correct answer is: neither max nor min

Differential Calculus MCQ Level - 1 - Question 2

### is maximum at x= :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 2  We get, x = ±cos x  ∴ f(x) has maximum where x = cos x.

The correct answer is: cos x

Differential Calculus MCQ Level - 1 - Question 3

### The points on  z2 = xy + 1  nearest to the origin are :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 3

Let l be the distance from (0, 0, 0) to (x, y, z) on the surface
z2 = xy + 1 Solving, we get
x = 0, y = 0
r = 2, s = 1, t = 2
rt – s2 = 3 > 0 and r > 0
Thus, minimum at (0, 0)
When, x = 0, y = 0, z = ±1
So, the points are (0, 0, 1) and (0, 0, –1)

The correct answer is: (0, 0, 1) and (0, 0, –1)

Differential Calculus MCQ Level - 1 - Question 4

The values of x1 and x2 with x1 < x2  such that has the largest value are :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 4 Since, x1 < x2
So, x1 = –2  &  x2 = 3

The correct answer is: –2, 3

Differential Calculus MCQ Level - 1 - Question 5

The sum of two number is k. The minima value of the sum of their squares is :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 5

Let the two numbers be x and y and let z be the sum of squares. x2 + (k – x)2   Hence, z is minimum at x = k/2 and its minimum value is k2/2.

Differential Calculus MCQ Level - 1 - Question 6

The maximum value of f(x) = (1/x)x is

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 6  = –x log x i.e., 1 + log x = 0
i.e., log x = –1  Hence, y is maximum where x = 1/e and maximum value is e1/e.

Differential Calculus MCQ Level - 1 - Question 7

For the function x2 + y2,  (0, 0) is a point of :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 7

Let f(x, y) = x2 + y2 For stationary points, ⇒ (0, 0) is the only stationary point
r = fxx = 2, t = fyy = 2, s = fxy = 0
rt – s2 = 4 – 0 = 4 > 0 and r > 0
Thus, (0, 0) is a point of minima.

Differential Calculus MCQ Level - 1 - Question 8

If  dy/dx = (x - a)2n (x - b)2p + 1,  where n and p are positive integers, then at x = a, x = b, f(x) has respectively :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 8

For stationary points, put dy/dx = 0
i.e. (x – a)2n(x – b)2p+1 = 0
i.e. x = ab
Nature at x = a : Since 2n is even, dy/dx does not change its sign while x passes through a.

∴  Neither maximum nor minima at x = a.

Nature at x = b : when x is slightly less than is negative as 2p + 1 is odd.

Also, when x is slightly more than is positive. Thus, y is minimum at x = b.

The correct answer is: neither maxima nor minima, minima

Differential Calculus MCQ Level - 1 - Question 9

The maximum value of (x – 1)(x – 2)(x – 3) is :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 9

f(y) = (x – 1)(x – 2)(x – 3)
= x3 – 6x2 + 11x – 6 Hence, f(x) has minima at and f(x) has maxima at ∴  the maximum value of f(x) is The correct answer is: Differential Calculus MCQ Level - 1 - Question 10

The saddle point of the function z = x2y – y2x – x + y is :

Detailed Solution for Differential Calculus MCQ Level - 1 - Question 10

For max. or min., Solving, we get
x = 1, y = 1
x = –1, y = –1 at (1, 1)     r = 2, s = 0, t = –2
rt – s2 = –4 < 0

∴ neither max nor min at (1, 1)
at (–1, –1)  r = –2, s = 0, t = 2
rt – s2 = –4 < 0
Thus, (–1, –1) is also a saddle point.

The correct answer is: both (1, 1) and (–1, –1)

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