Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Maths 35 Years JEE Mains & Advance Past year Papers Class 12

JEE : Previous year Questions (2016-20) - Differential Equations Notes | EduRev

The document Previous year Questions (2016-20) - Differential Equations Notes | EduRev is a part of the JEE Course Maths 35 Years JEE Mains & Advance Past year Papers Class 12.
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Q.1. If y = y (x) is the solution of the differential equationPrevious year Questions (2016-20) - Differential Equations Notes | EduRevsuch that y (0) = 0, then y (1) is equal to    (2020)
(1) 1 + loge2
(2) 2 + loge2
(3) 2e
(4) loge2
Ans. 
(1)
Solution.
We have,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Let ey = t ⇒ ex
Now, the differential equation becomesPrevious year Questions (2016-20) - Differential Equations Notes | EduRev
Integrating factor of equation isPrevious year Questions (2016-20) - Differential Equations Notes | EduRev
So, the solution of differential equation isPrevious year Questions (2016-20) - Differential Equations Notes | EduRev
Now, at x = 0, y = 0, then c =1. Therefore,Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Substituting x =1, we get y(1) = 1 + loge 2

Q.2. Let y = y (x) be the solution curve of the differential equationPrevious year Questions (2016-20) - Differential Equations Notes | EduRevsatisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is    (2020)
(1) 2 – e
(2) −e
(3) 2
(4) 2 + e
Ans. 
(1)
Solution.

The given differential equation isPrevious year Questions (2016-20) - Differential Equations Notes | EduRev
This is a linear differential equation.
Therefore,Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, the solution of differential equation is given by
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Given: y (0) =1
So, 0 = e + c ⇒ c = -e.
Therefore,Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Hence, the abscissa of point of intersection with the x-axis is
x = (0 − 0 + 2) - e = 2 - e

Q.3. Let y = y (x) be a solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRev thenPrevious year Questions (2016-20) - Differential Equations Notes | EduRevis equal to    (2020)
(1)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2)Previous year Questions (2016-20) - Differential Equations Notes | EduRev 
(3)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. 
(3)
Solution.

We havePrevious year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Integrating both sides of equation, we get sin-1 x + sin-1 y = c
Now,Previous year Questions (2016-20) - Differential Equations Notes | EduRev
sin-1 x = cos-1y
At x = -1/√2 , we get y = sinPrevious year Questions (2016-20) - Differential Equations Notes | EduRev

Q.4. The differential equation of the family of curvesPrevious year Questions (2016-20) - Differential Equations Notes | EduRev, is    (2020)
(1)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3) xy'' = y'
(4)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. 
(1)
Solution.

The equation of the family of curve is x= 4b (y+b) .... (i)
Differentiating Eq. (1) w.r.t. x, we get
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Substituting the value of b in Eq. (1), we get
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.5. If forPrevious year Questions (2016-20) - Differential Equations Notes | EduRevis the solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRevthen y(3) is equal to ________.    (2020)
Ans. 
(4.00)
Solution.

We have
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev.....(1)
This is a linear differential equation of first order. So,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Therefore, the solution of differential equation is
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Given, y (2) = 0 ⇒ c + 2 + 1 = 0 ⇒ c = -3.
Therefore,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Substituting x = 3 in above equation, we get
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.6. IfPrevious year Questions (2016-20) - Differential Equations Notes | EduRevy(1) = 1; then a value of x satisfying y(x) = e is    (2020)
(1)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. 
(4)
Solution.

We havePrevious year Questions (2016-20) - Differential Equations Notes | EduRev..... (1)

SubstitutingPrevious year Questions (2016-20) - Differential Equations Notes | EduRevin Eq. (1), we get
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Integrating both sides of the equation, we get
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, y(1) = 1 ⇒Previous year Questions (2016-20) - Differential Equations Notes | EduRev
So,Previous year Questions (2016-20) - Differential Equations Notes | EduRev
For y(x) = e, we havePrevious year Questions (2016-20) - Differential Equations Notes | EduRev

Q.7. If y = y(x) is the solution of the differential equation, Previous year Questions (2016-20) - Differential Equations Notes | EduRev satisfying y(1) = 1, thenPrevious year Questions (2016-20) - Differential Equations Notes | EduRevis equal to:    (2019)
(1) 7/64
(2) 1/2
(3) 49/16
(4) 13/16

Ans. (3)
Solution.
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Solution of differential equation is:
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev     ...(1)
∵ y(1) = 1
∴ C = 3/4
Then, from equation (1)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.8. Let f : [0, 1] → R be such that f(xy) = f(x).f(y), for all x, y ∈ [0, 1], and f(0) ≠ 0. If y = y(x) satisfies the differential equation, dy/dx = f(x) with y(0) = 1, then Previous year Questions (2016-20) - Differential Equations Notes | EduRev     (2019)
(1) 3
(2) 4
(3) 2
(4) 5
Ans. (1)
Solution.
f(xy) = f(x).f(y)   ...(1)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
By first principle derivative formula,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.9. Previous year Questions (2016-20) - Differential Equations Notes | EduRevand Previous year Questions (2016-20) - Differential Equations Notes | EduRev     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (1)
Solution.

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev   ...(i)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.10. The curve amongst the family of curves represented by the differential equation, (x2 - y2) dx + 2xydy = 0 which passes through (1, 1), is:     (2019)
(1) a circle with centre on the x-axis.
(2) an ellipse with major axis along they-axis.
(3) a circle with centre on the y-axis.
(4) a hyperbola with transverse axis along the x-axis.
Ans. (1)
Solution.
y2dx - 2xydy = x2dx
2xydy - y2dx = -x2dx
d(xy2) = -x2dx
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev    ...(1)
Since, the above curve passes through the point (1,1)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, the curve (1) becomes
y2 = -x2 + 2x
⇒ y2 = -(x-1)2+1
(x - 1)2 + y2 = 1
The above equation represents a circle with centre (1, 0) and centre lies on x-axis.

Q.11. If y(x) is the solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRev 

Previous year Questions (2016-20) - Differential Equations Notes | EduRev     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (3)
Solution. 
Given differential equation is,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Complete solution is given by
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Differentiate both sides with respect to x,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.12. The solution of the differential equation, dy/dx = (x - y)2, when y(1) = 1, is:     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (2)
Solution. The given differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRev      ...(1)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, from equation (1)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.13. Let y = y(x) be the solution of the differential equation, Previous year Questions (2016-20) - Differential Equations Notes | EduRev If 2y(2) = loge 4 - 1, then y(e) is equal to:     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (3)
Solution. Consider the differential equation,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.14. Let y = y(x) be the solution of the differential equation, Previous year Questions (2016-20) - Differential Equations Notes | EduRev such that y(0) = 0. If Previous year Questions (2016-20) - Differential Equations Notes | EduRev , then the value of 'a' is:     (2019)
(1) 1/4
(2) 1/2
(3) 1
(4) 1/16
Ans. (4)
Solution.
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Since, the above differential equation is a linear differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Then, the solution of the differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
If x = 0 then y = 0 (given)
⇒ 0 = 0 + c
⇒ c = 0
Then, equation (1) becomes,
⇒ y (1 + x2) = tan-1 x
Now put x = 1 in above equation, then
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.15. The solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRev (x ≠ 0) with y(1) = 1, is:     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (3)
Solution.
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Since, the above differential equation is the linear differential equation, then Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, the solution of the linear differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.16.Previous year Questions (2016-20) - Differential Equations Notes | EduRev and Previous year Questions (2016-20) - Differential Equations Notes | EduRev then Previous year Questions (2016-20) - Differential Equations Notes | EduRev is equal to:     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (3)
Solution.

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Now, put y = π/6 in the above equation,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.17. If y = y(x) is the solution of the differential equation dy/dx = (tan x - y)sec2 x, Previous year Questions (2016-20) - Differential Equations Notes | EduRev such that y (0) = 0, then Previous year Questions (2016-20) - Differential Equations Notes | EduRev     (2019)

(1) e - 2
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (1)
Solution.

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Given equation is linear differential equation.
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Hence, solution of differential equation,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.18. Let y = y(x) be the solution of the differential equation, Previous year Questions (2016-20) - Differential Equations Notes | EduRev such that y (0) = 1.       (2019)
Then:
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (4)
Solution.
Given differential equation is.
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.19. Consider the differential equation,Previous year Questions (2016-20) - Differential Equations Notes | EduRev value of y is 1 when x = 1, then the value of x for which y = 2, is:     (2019)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. (2)
Solution.
Consider the differential equation,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.20. The general solution of the differential equation (y2 - x3) dx - xydy = 0 (x ≠ 0) is :     (2019)
(1) y2 - 2x2 + cx3 = 0
(2) y2 + 2x3 + cx2 = 0
(3) y2 + 2x2 + cx3 = 0
(4) y2 - 2x3 + cx2 = 0

(where c is a constant of integration)
Ans. (2)
Solution. 
Given differential equation can be written as,
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.21. Let y = y(x) be the solution of the differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRevPrevious year Questions (2016-20) - Differential Equations Notes | EduRev     (2018)
(1)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2)Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans. 
(3)
Solution:

Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Integrating both sides we get y sin x = 2x2+ C
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.22. Let y = y(x) be the solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRev where f(x) Previous year Questions (2016-20) - Differential Equations Notes | EduRev. If y(0) = 0, then y(3/2) is:    (2018)
(1) 1/2e
(2) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans.
Solution. 
dy/dx + 2y = f(x)  is a linear differential equation
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
solution of the above equation is
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.23. The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is:    (2018)
(1) xyy' - y2 + 9 = 0  
(2) xyy" + x (y')2 - yy' = 0
(3) xyy' + y2 - 9 = 0
(4) x + yy" = 0
Ans.
(2)
Solution.

Equation of ellipse
Previous year Questions (2016-20) - Differential Equations Notes | EduRev since it passes through (0, 3)⇒b2=9
∴ Equation of ellipse becomes
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev......(2)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.24. If ( 2 + sinx) dy/dx + (y + 1)cos x = 0 and y(0) = 1, then y(π/2) is equal to    (2017)
(1) 4/3
(2) 1/3
(3) -2/3
(4) -1/3
Ans.
(2)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
ln| y + 1 |+ ln (2 + sinx ) = lnC
(y + 1) (2+ sinx ) = C
Put x = 0, y = 1
(1 + 1)× 2 = C ⇒ C = 4
Now, ( y +1) (2+ sinx ) = 4 
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
( y +1) (2+ 1) = 4 
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.25. The curve satisfying the differential equation, ydx – (x + 3y2)dy = 9 and passing through the point (1, 1), also passes through the point:    (2017)
(1) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans.
(2)
ydx - xdy - 3y2dy = 0
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Passes through (1, 1) ∴ 1 = 3 + c ; c = -2
x = 3y2 - 2y
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.26. If Previous year Questions (2016-20) - Differential Equations Notes | EduRev then λ+k is equal to:    (2017)
(1) 26
(2) -24
(3) -23
(4) -26
Ans. 
(2)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.27. If a curve y = f(x) passes through the point (1, -1) and satisfies the differential equation, y(1 + xy) dx = x dy, then Previous year Questions (2016-20) - Differential Equations Notes | EduRevis equal to:    (2016)
(1) -2/5
(2) -4/5
(3) 2/5
(4) 4/5
Ans. 
(4)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
We have to find Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Put x = -1/2
 Previous year Questions (2016-20) - Differential Equations Notes | EduRev

Q.28. The solution of the differential equation Previous year Questions (2016-20) - Differential Equations Notes | EduRev  is given by    (2016)
(1) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(2) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(3) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
(4) Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Ans.
(4)
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev
Previous year Questions (2016-20) - Differential Equations Notes | EduRev 

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