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Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced

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9 Questions MCQ Test Mathematics For JEE | Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced

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Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 1

The order of the differential equation whose general solution is given by y = (C1 + C2) cos (x+C3) – C4ex+C5,  where C1, C2, C3, C4, C5, are arbitrary constants, is

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 1

The given solution of D.E. is

[Taking c1 + c2 = A,c4 ec5 = B] Thus, there are actually three arbitrary constants and hence this differential equation should be of order 3.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 2

The differential equation representing the family of curves  where c is a positive parameter, is of

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 2

2 yy1 =2c ⇒ c = yy1
Eliminating, c, we get,

It involves only Ist order derivative, its order is 1 but its degree is 3 as y13 is there.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 3

A curve y = f (x) passes through (1, 1) and at P(x, y), tangent cuts the x–axis and y–axis at A and B respectively such that BP : AP = 3 : 1, then

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 3

Tangent to the curve y = f (x) at (x, y) is

∴ cur ve is x3y = 1 , which also passes through

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 4

If y (x) satisfies the differential equation y ' – y tan x = 2x secx and y(0) = 0, then

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 4

The given differential equation is

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 5

A curve passes through the point  Let the slope of the curve at each point (x, y) be

Then the equation of the curve is

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 5

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 6

Let y (x) be a solution of the differential equation (1 + ex) y ' + yex = 1 . If y(0) = 2, then which of the following statement is (are) true?

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 6

Its solution will be intersection point of y = x + 3 and y = e–x​

Clearly there is a critical point in (–1, 0).

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 7

Consider the family of all circles whose centers lie on the straight line y = x. If this family of circle is represented by the differential equation Py '' + Qy' + 1=0 , where P, Q are functions of x, y and y'    then which of the following statements is (are) true?

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 7

Let the equation of circle be x2 + y2 + 2gx + 2gy + c = 0
⇒ 2x + 2yy' + 2g + 2gy' = 0
⇒ x + yy' + g + gy' = 0          ...(i)

Differentiating again,

Substituting value of g in eqn. (i)

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 8

be a differentiable function such that   for all x ∈ (0, ∞) and f(1) ≠ 1. Then

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 8

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 9

A solution curve of the differen tial equation  passes through thepoint (1, 3). Then the solution curve

Detailed Solution for Test: MCQs (One or More Correct Option): Differential Equations | JEE Advanced - Question 9

As it passes through (1, 3) ⇒ C  = –1 – log 3

..(1)

Intersection of (1) and y = x + 2

∴ (1, 3) is the only intersection point.
Intersection of (1) and y = (x + 2)2

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