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JEE Advanced · Mathematics · Differential Equations

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1
Let y:(−∞,∞)→(0,∞)y:(-\infty,\infty)\to(0,\infty) be the solution of dydx=e5xy3+y3ex+exy4\frac{dy}{dx}=\frac{e^{5x}y^3+y^3}{e^x+e^xy^4} satisfying y(0)=1/2y(0)=1/\sqrt2. Find y(log⁡e2)y(\log_e2).
Not attempted
A
5+352\sqrt{\dfrac{5+\sqrt{35}}{2}}
B
7+532\sqrt{\dfrac{7+\sqrt{53}}{2}}
C
7+532\dfrac{7+\sqrt{53}}{2}
D
5+352\dfrac{5+\sqrt{35}}{2}
2
Let y=f(x)y=f(x) be the real-valued function on (0,∞)(0,\infty) satisfying y(1)=0y(1)=0 and xdydx=y−x3x\frac{dy}{dx}=y-x^3. Then which of the following statements is (are) TRUE?
Not attempted
A
The function ff has a local minimum at x=1/3x=1/\sqrt3
B
The function ff has a local maximum at x=1/3x=1/\sqrt3
C
The function ff is increasing in the interval (1,2)(1,2)
D
If g(x)=4x3−5x2+32xg(x)=4x^3-5x^2+\frac32x for x>0x>0, the number of elements in {x∈(0,∞):f(x)=g(x)}\{x\in(0,\infty): f(x)=g(x)\} is 2
3
Let y(x)y(x) be the solution of the differential equation x2dydx+xy=x2+y2,x>1e,x^2\dfrac{dy}{dx}+xy = x^2+y^2,\qquad x>\dfrac1e, satisfying y(1)=0y(1)=0. Then the value of 2(y(e))2y(e2)\dfrac{2(y(e))^2}{y(e^2)} is ___________.
Not attempted
Your answer: —
Correct answer: 0.75
4
For all x>0x>0, let y1(x),y2(x)y_1(x), y_2(x), and y3(x)y_3(x) be the functions satisfying dy1dx−(sin⁡x)2 y1=0,y1(1)=5,\dfrac{dy_1}{dx} - (\sin x)^2\,y_1 = 0, \quad y_1(1)=5, dy2dx−(cos⁡x)2 y2=0,y2(1)=13,\dfrac{dy_2}{dx} - (\cos x)^2\,y_2 = 0, \quad y_2(1)=\dfrac13, dy3dx−(2−x3x3)y3=0,y3(1)=35e,\dfrac{dy_3}{dx} - \left(\dfrac{2-x^3}{x^3}\right)y_3 = 0, \quad y_3(1)=\dfrac{3}{5e}, respectively. Then lim⁡x→0+y1(x)y2(x)y3(x)+2xe3xsin⁡x\displaystyle\lim_{x\to0^+}\frac{y_1(x)y_2(x)y_3(x)+2x}{e^{3x}\sin x} is equal to ______________.
Not attempted
Your answer: —
Correct answer: 2
5
Let f(x)f(x) be a continuously differentiable function on the interval (0,∞)(0,\infty) such that f(1)=2f(1)=2 and lim⁡t→xt10f(x)−x10f(t)t9−x9=1\displaystyle\lim_{t\to x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1 for each x>0x>0. Then, for all x>0x>0, f(x)f(x) is equal to
Not attempted
A
3111x−911x10\dfrac{31}{11x}-\dfrac{9}{11}x^{10}
B
911x+1311x10\dfrac{9}{11x}+\dfrac{13}{11}x^{10}
C
−911x+3111x10\dfrac{-9}{11x}+\dfrac{31}{11}x^{10}
D
1311x+911x10\dfrac{13}{11x}+\dfrac{9}{11}x^{10}
6
For x∈Rx\in\mathbb{R}, let y(x)y(x) be a solution of the differential equation (x2−5)dydx−2xy=−2x(x2−5)2(x^2-5)\dfrac{dy}{dx}-2xy=-2x(x^2-5)^2 such that y(2)=7y(2)=7. Then the maximum value of the function y(x)y(x) is
Not attempted
Your answer: —
Correct answer: 16
7
Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be a differentiable function such that f(1)=13f(1)=\dfrac{1}{3} and 3∫1xf(t) dt=xf(x)−x333\int_{1}^{x} f(t)\,dt = xf(x) - \dfrac{x^3}{3} for x∈[1,∞)x\in[1,\infty). Let ee denote the base of the natural logarithm. Then the value of f(e)f(e) is
Not attempted
A
e2+43\dfrac{e^2+4}{3}
B
log⁡e4+e3\dfrac{\log_e 4 + e}{3}
C
4e23\dfrac{4e^2}{3}
D
e2−43\dfrac{e^2-4}{3}
8
For x∈Rx\in\mathbb{R}, let the function y(x)y(x) be the solution of the differential equation dydx+12y=cos⁡(π12x),y(0)=0\dfrac{dy}{dx}+12y=\cos\left(\dfrac{\pi}{12}x\right),\qquad y(0)=0. Then, which of the following statements is/are TRUE?
Not attempted
A
y(x)y(x) is an increasing function
B
y(x)y(x) is a decreasing function
C
There exists a real number β\beta such that the line y=βy=\beta intersects the curve y=y(x)y=y(x) at infinitely many points
D
y(x)y(x) is a periodic function
9
If y(x)y(x) is the solution of the differential equation x dy−(y2−4y) dx=0x\,dy - (y^2-4y)\,dx = 0 for x>0x>0, y(1)=2\quad y(1)=2, and the slope of the curve y=y(x)y=y(x) is never zero, then the value of 10 y(2)10\,y(\sqrt2) is _______.
Not attempted
Your answer: —
Correct answer: 8
10
For any real numbers α\alpha and β\beta, let yα,β(x)y_{\alpha,\beta}(x), x∈Rx\in\mathbb{R}, be the solution of the differential equation dydx+αy=xeβx\frac{dy}{dx}+\alpha y = xe^{\beta x}, y(1)=1y(1)=1. Let S={yα,β(x):α,β∈R}S=\{y_{\alpha,\beta}(x) : \alpha,\beta\in\mathbb{R}\}. Then which of the following functions belong(s) to the set SS?
Not attempted
A
f(x)=x22e−x+(e−12)e−xf(x) = \frac{x^2}{2}e^{-x} + \left(e-\frac{1}{2}\right)e^{-x}
B
f(x)=−x22e−x+(e+12)e−xf(x) = -\frac{x^2}{2}e^{-x} + \left(e+\frac{1}{2}\right)e^{-x}
C
f(x)=ex2(x−12)+(e−e24)e−xf(x) = \frac{e^x}{2}\left(x-\frac{1}{2}\right) + \left(e-\frac{e^2}{4}\right)e^{-x}
D
f(x)=ex2(12−x)+(e+e24)e−xf(x) = \frac{e^x}{2}\left(\frac{1}{2}-x\right) + \left(e+\frac{e^2}{4}\right)e^{-x}
11
Let bb be a nonzero real number. Suppose f:R→Rf:\mathbb{R}\to\mathbb{R} is a differentiable function such that f(0)=1f(0)=1. If the derivative f′f' of ff satisfies the equation f′(x)=f(x)b2+x2f'(x) = \dfrac{f(x)}{b^2+x^2} for all x∈Rx \in \mathbb{R}, then which of the following statements is/are TRUE?
Not attempted
A
If b>0b>0, then ff is an increasing function
B
If b<0b<0, then ff is a decreasing function
C
f(x)f(−x)=1f(x)f(-x)=1 for all x∈Rx \in \mathbb{R}
D
f(x)−f(−x)=0f(x)-f(-x)=0 for all x∈Rx \in \mathbb{R}
12
Let Γ\Gamma denote a curve y=y(x)y=y(x) which is in the first quadrant and let the point (1,0)(1,0) lie on it. Let the tangent to Γ\Gamma at a point PP intersect the yy-axis at YPY_P. If PYPPY_P has length 11 for each point PP on Γ\Gamma, then which of the following options is/are correct?
Not attempted
A
y=log⁡e ⁣(1+1−x2x)−1−x2y=\log_e\!\left(\dfrac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}
B
xy′−1−x2=0xy'-\sqrt{1-x^2}=0
C
y=−log⁡e ⁣(1+1−x2x)+1−x2y=-\log_e\!\left(\dfrac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}
D
xy′+1−x2=0xy'+\sqrt{1-x^2}=0
13
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function with f(0)=0f(0)=0. If y=f(x)y=f(x) satisfies the differential equation dydx=(2+5y)(5y−2)\frac{dy}{dx}=(2+5y)(5y-2), then the value of lim⁡x→−∞f(x)\lim_{x\to-\infty}f(x) is _____.
Not attempted
Your answer: —
Correct answer: 0.4
14
Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be two non-constant differentiable functions. If f′(x)=(e(f(x)−g(x)))g′(x)f'(x) = \left(e^{(f(x)-g(x))}\right) g'(x) for all x∈Rx\in\mathbb{R}, and f(1)=g(2)=1f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE?
Not attempted
A
f(2)<1−log⁡e2f(2) < 1-\log_e 2
B
f(2)>1−log⁡e2f(2) > 1-\log_e 2
C
g(1)>1−log⁡e2g(1) > 1-\log_e 2
D
g(1)<1−log⁡e2g(1) < 1-\log_e 2
15
Let f:[0,∞)→Rf:[0,\infty)\to\mathbb{R} be a continuous function such that f(x)=1−2x+∫0xex−tf(t) dtf(x) = 1-2x+\displaystyle\int_0^x e^{x-t}f(t)\,dt for all x∈[0,∞)x\in[0,\infty). Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
The curve y=f(x)y=f(x) passes through the point (1,2)(1,2)
B
The curve y=f(x)y=f(x) passes through the point (2,−1)(2,-1)
C
The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\{(x,y)\in[0,1]\times\mathbb{R} : f(x)\le y\le\sqrt{1-x^2}\} is π−24\dfrac{\pi-2}{4}
D
The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\{(x,y)\in[0,1]\times\mathbb{R} : f(x)\le y\le\sqrt{1-x^2}\} is π−14\dfrac{\pi-1}{4}
16
If y=y(x)y = y(x) satisfies the differential equation 8x(9+x)dy=(4+9+x)−1dx8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right) dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1} dx, x>0x > 0, and y(0)=7y(0) = \sqrt{7}, then y(256)=y(256) =
Not attempted
A
33
B
99
C
1616
D
8080
17
If f:R→Rf: \mathbb{R} \to \mathbb{R} is a differentiable function such that f′(x)>2f(x)f'(x) > 2f(x) for all x∈Rx \in \mathbb{R}, and f(0)=1f(0) = 1, then
Not attempted
A
f(x)f(x) is increasing in (0,∞)(0, \infty)
B
f(x)f(x) is decreasing in (0,∞)(0, \infty)
C
f(x)>e2xf(x) > e^{2x} in (0,∞)(0, \infty)
D
f′(x)<e2xf'(x) < e^{2x} in (0,∞)(0, \infty)
18
A solution curve of the differential equation (x2+xy+4x+2y+4)dydx−y2=0\left(x^2+xy+4x+2y+4\right)\dfrac{dy}{dx}-y^2=0, x>0x>0, passes through the point (1,3)(1,3). Then the solution curve
Not attempted
A
intersects y=x+2y=x+2 exactly at one point
B
intersects y=x+2y=x+2 exactly at two points
C
intersects y=(x+2)2y=(x+2)^2
D
does NOT intersect y=(x+3)2y=(x+3)^2
19
Let f:(0,∞)→Rf:(0,\infty)\to\mathbb{R} be a differentiable function such that f′(x)=2−f(x)xf'(x)=2-\dfrac{f(x)}{x} for all x∈(0,∞)x\in(0,\infty) and f(1)≠1f(1)\ne1. Then
Not attempted
A
lim⁡x→0+f′ ⁣(1x)=1\displaystyle\lim_{x\to0^+}f'\!\left(\dfrac1x\right)=1
B
lim⁡x→0+xf ⁣(1x)=2\displaystyle\lim_{x\to0^+}xf\!\left(\dfrac1x\right)=2
C
lim⁡x→0+x2f′(x)=0\displaystyle\lim_{x\to0^+}x^2f'(x)=0
D
∣f(x)∣≤2|f(x)|\le2 for all x∈(0,2)x\in(0,2)
20
Let y(x)y(x) be a solution of the differential equation (1+ex)y′+yex=1(1+e^x)y'+ye^x=1. If y(0)=2y(0)=2, then which of the following statements is (are) true?
Not attempted
A
y(−4)=0y(-4)=0
B
y(−2)=0y(-2)=0
C
y(x)y(x) has a critical point in the interval (−1,0)(-1,0)
D
y(x)y(x) has no critical point in the interval (−1,0)(-1,0)
21
Consider the family of all circles whose centers lie on the straight line y=xy=x. If this family of circles is represented by the differential equation Py′′+Qy′+1=0Py''+Qy'+1=0, where P,QP,Q are functions of x,yx,y and y′y' (here y′=dydx, y′′=d2ydx2y'=\dfrac{dy}{dx},\ y''=\dfrac{d^2y}{dx^2}), then which of the following statements is (are) true?
Not attempted
A
P=y+xP=y+x
B
P=y−xP=y-x
C
P+Q=1−x+y+y′+(y′)2P+Q=1-x+y+y'+(y')^2
D
P−Q=x+y−y′−(y′)2P-Q=x+y-y'-(y')^2
22
The function y=f(x)y=f(x) is the solution of the differential equation dydx+xyx2−1=x4+2x1−x2\dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}} in (−1,1)(-1,1) satisfying f(0)=0f(0)=0. Then ∫−3232f(x) dx\displaystyle\int_{-\frac{\sqrt3}{2}}^{\frac{\sqrt3}{2}} f(x)\,dx is
Not attempted
A
π3−32\dfrac{\pi}{3}-\dfrac{\sqrt3}{2}
B
π3−34\dfrac{\pi}{3}-\dfrac{\sqrt3}{4}
C
π6−34\dfrac{\pi}{6}-\dfrac{\sqrt3}{4}
D
π6−32\dfrac{\pi}{6}-\dfrac{\sqrt3}{2}
23
A curve passes through the point (1,π6)\left(1,\dfrac{\pi}{6}\right). Let the slope of the curve at each point (x,y)(x,y) be yx+sec⁡(yx)\dfrac{y}{x}+\sec\left(\dfrac{y}{x}\right), x>0x>0. Then the equation of the curve is
Not attempted
A
sin⁡(yx)=log⁡x+12\sin\left(\dfrac{y}{x}\right)=\log x+\dfrac12
B
cosec(yx)=log⁡x+2\text{cosec}\left(\dfrac{y}{x}\right)=\log x+2
C
sec⁡(2yx)=log⁡x+2\sec\left(\dfrac{2y}{x}\right)=\log x+2
D
cos⁡(2yx)=log⁡x+12\cos\left(\dfrac{2y}{x}\right)=\log x+\dfrac12
This is homogeneous in x,yx,y; substitute v=y/xv=y/x so dydx=v+xdvdx\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}, which collapses to a separable equation in vv and xx.
Deconstruction\textbf{Deconstruction}: Solve a first-order ODE with a homogeneous right-hand side and a given point on the curve. Strategy\textbf{Strategy}: Substitute v=y/xv=y/x to reduce the homogeneous ODE to a separable one, integrate, then use the initial point to fix the constant. Execution\textbf{Execution}: dydx=yx+sec⁡(yx)\dfrac{dy}{dx}=\dfrac{y}{x}+\sec\left(\dfrac{y}{x}\right). Let y=vxy=vx, so dydx=v+xdvdx\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}. - v+xdvdx=v+sec⁡v⇒xdvdx=sec⁡v⇒cos⁡v dv=dxxv+x\dfrac{dv}{dx}=v+\sec v \Rightarrow x\dfrac{dv}{dx}=\sec v \Rightarrow \cos v\,dv=\dfrac{dx}{x}. - Integrating: sin⁡v=ln⁡x+C\sin v=\ln x+C, i.e. sin⁡(yx)=ln⁡x+C\sin\left(\dfrac{y}{x}\right)=\ln x+C. - At (1,π/6)(1,\pi/6): v=π/6v=\pi/6, so sin⁡(π/6)=ln⁡1+C⇒12=0+C⇒C=12\sin(\pi/6)=\ln 1+C \Rightarrow \dfrac12=0+C \Rightarrow C=\dfrac12. - Curve: sin⁡(yx)=ln⁡x+12\sin\left(\dfrac{y}{x}\right)=\ln x+\dfrac12. Verification\textbf{Verification}: Substituting x=1,y=π/6x=1,y=\pi/6 back into sin⁡(y/x)=log⁡x+12\sin(y/x)=\log x+\frac12 gives sin⁡(π/6)=12=0+12\sin(\pi/6)=\frac12=0+\frac12, which checks out exactly, confirming the constant of integration. Final Answer\textbf{Final Answer}: (A) sin⁡(yx)=log⁡x+12\sin\left(\dfrac{y}{x}\right)=\log x+\dfrac12