Q1: For x ∈ R, let g(x) be a solution of the differential equation such that y(2) = 7. Then the maximum value of the function y(x) is : [JEE Advanced 2023 Paper 2]
Ans: 16
Q2: Let be a differentiable function such that f(1) = 1/3 and . Let e denote the base of the natural logarithm. Then the value of f(e) is :
(a)
(b)
(c)
(d) [JEE Advanced 2023 Paper 2]
Ans: (c)
1. We are given a relationship between the integral of the function f(t) and the function f(x) :
2. Differentiate both sides of the above equation with respect to x :
3. We can rewrite this as a first order linear differential equation:
4. The general form of a first order linear differential equation is:
5. The integrating factor is obtained by exponentiating the integral of p(x) with respect to x:
6. ∴ Solution of D.E :
∴ The general solution is :
Here, y(x) is the same as f(x), i.e. the function we're trying to find.
7. Given that f(1) = 1 / 3, let's use this condition to find the constant c :
Therefore, the function f(x) we are looking for is:
We want to find f(e), so substituting x = e into the function, we get :
So, the correct answer is Option C.
Q1: If y(x) is the solution of the differential equation and the slope of the curve y = y(x) is never zero, then the value of 10y(√2) is: [JEE Advanced 2022 Paper 2]
Ans: 8
Integrating both side, we get
Given, y (1) = 2
Now,
Q2: For x ∈ R, let the function y(x) be the solution of the differential equation
Then, which of the following statements is/are TRUE ?
(a) y(x) is an increasing function
(b) y(x) is a decreasing function
(c) There exists a real number β such that the line y = β intersects the curve y = y(x) at infinitely many points
(d) y(x) is a periodic function [JEE Advanced 2022 Paper 2]
Ans: (c)
Given,
This is a linear differential equation.
Where P = 12 and
Solution of differencial equation is
Given, y(0) = 0
y(x) is not a periodic function as e^{−12x} is a nonperiodic function.
∴ Option (D) is a wrong statement.
Now,
Values of varies between
So, f(x) is neither increasing nor decreasing.
∴ Option (A) and (B) are wrong statements.
For some β ∈ R, y = β intersects y = f(x) at infinitely many points.
So option C is correct.
Q1: Let Γ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the yaxis at Y_{P}. If PY_{P} has length 1 for each point P on I`, then which of the following options is/are correct?
(a)
(b)
(c)
(d)
Ans: (a) & (c)
Let a point P(h, k) on the curve y = y(x), so equation of tangent to the curve at point P is
Now, the tangent (i) intersect the Yaxis at Y_{p}, so coordinates Y_{p} is
Where,
So, PYp = 1 (given)
[on replacing h by x]
On putting x = sinθ, dx = cosθdθ, we get
[on rationalization]
∵ The curve is in the first quadrant so y must be positive, so
As curve passes through (1, 0), so
so required curve is
and required differential equation is
Hence, options (a) and (c) are correct.
Q1: Let f : R → R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation , then the value of is [JEE Advanced 2018 Paper 2]
Ans: 0.4
We have,
On integrating both sides, we get
when x = 0 ⇒ y = 0, then A = 1
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