JEE Mains Previous Year Questions (PYQ)

JEE Main 2024 29th Jan Evening Shift

If $ \sin\!\left( \dfrac{y}{x} \right) = \log_{e}|x| + \dfrac{\alpha}{2} $ is the solution of the differential equation $ x \cos\!\left( \dfrac{y}{x} \right) \dfrac{dy}{dx} = y \cos\!\left( \dfrac{y}{x} \right) + x $ and $ y(1) = \dfrac{\pi}{3} $, then $ \alpha^{2} $ is equal to:

The solution of the differential equation $ (x^{2}+y^{2})dx-5xy\,dy=0, \; y(1)=0 $, is:

7.

If the solution of the differential equation $ (2x + 3y - 2)\,dx + (4x + 6y - 7)\,dy = 0 $, $ y(0) = 3 $, is $ \alpha x + \beta y + 3 \log_{e} |2x + 3y - \gamma| = 6 $, then $ \alpha + 2\beta + 3\gamma $ is equal to ____________.

8.

Let $ Y=Y(x) $ be a curve lying in the first quadrant such that the area enclosed by the line $ Y-y=Y'(x)(X-x) $ and the co-ordinate axes, where $ (x,y) $ is any point on the curve, is always $ \dfrac{-y^2}{2Y'(x)}+1 ,$$ Y'(x)\neq 0) $. If $ Y(1)=1 $, then $ 12Y(2) $ equals _______.

9.

Let $ y = y(x) $ be the solution of the differential equation $ \sec x \, dy + \{2(1 - x)\tan x + x(2 - x)\}dx = 0 $ such that $ y(0) = 2 $. Then $ y(2) $ is equal to:

10.

The solution curve of the differential equation $ y \frac{dx}{dy} = x(\log_{e} x - \log_{e} y + 1), \, x>0, y>0 $ passing through the point $ (e, 1) $ is:

11.

If $ y = y(x) $ is the solution curve of the differential equation $ (x^{2}-4)\,dy-(y^{2}-3y)\,dx=0, \, x>2, \, y(4)=\frac{3}{2} $ and the slope of the curve is never zero, then the value of $ y(10) $ equals:

12.

A function $ y = f(x) $ satisfies $ f(x)\sin 2x + \sin x - (1+\cos^2 x)f'(x) = 0 $ with condition $ f(0) = 0 $. Then, $ f\!\left(\frac{\pi}{2}\right) $ is equal to:

13.

If the solution $ y(x) $ of the given differential equation $ (e^{y}+1)\cos x\,dx+e^{y}\sin x\,dy=0 $ passes through the point $ \left(\frac{\pi}{2},0\right) $, then the value of $ e^{\,y(\frac{\pi}{6})} $ is equal to ______.

14.

The differential equation of the family of circles passing through the origin and having centre at the line $ y = x $ is:

15.

If the solution curve, of the differential equation $\displaystyle \frac{dy}{dx}=\frac{x+y-2}{x-y} $ passing through the point $ (2,1) $ is
$\displaystyle \tan^{-1}\!\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta}\log_{e}\!\left(a+\left(\frac{y-1}{x-1}\right)^{2}\right)=\log_{e}|x-1|, $ then $ 5\beta+\alpha $ is equal to:

16.

Let $ x = x(t) $ and $ y = y(t) $ be solutions of the differential equations $ \frac{dx}{dt} + a x = 0 $ and $ \frac{dy}{dt} + b y = 0 $ respectively, $ a, b \in \mathbb{R} $. Given that $ x(0) = 2 $; $ y(0) = 1 $ and $ 3y(1) = 2x(1) $. The value of $ t $, for which $ x(t) = y(t) $, is:

17.

Let $y = y(x)$ be the solution of the differential equation $\left(1 - x^2\right)dy = \left[ xy + \left(x^3 + 2\right)\sqrt{3\left(1 - x^2\right)} \right] dx$, $-1 < x < 1, y(0) = 0$. If $y\left(\frac{1}{2}\right) = \frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m + n$ is equal to _____.

18.

Let $ y=y(x) $ be the solution of the differential equation $ \left(x+y+2\right)^{2} \, dx = dy, \; y(0)=-2 $. Let the maximum and minimum values of the function $ y=y(x) $ in $ \left[0,\tfrac{\pi}{3}\right] $ be $ \alpha $ and $ \beta $, respectively.
If $ (3\alpha+\pi)^{2}+\beta^{2}=\gamma+\delta\sqrt{3},\;\gamma,\delta\in\mathbb{Z} $, then $ \gamma+\delta $ equals ____________.

19.

Let $ f $ be a differentiable function in the interval $ (0, \infty) $ such that $ f(1) = 1 $ and $ \lim_{t \to x} \frac{t^{2} f(x) - x^{2} f(t)}{t - x} = 1 $ for each $ x > 0 $. Then $ 2 f(2) + 3 f(3) $ is equal to ____________.

20.

Let a conic $ C $ pass through the point $ (4,-2) $ and $ P(x,y) $, $ x \ge 3 $, by any point on $ C $. Let the slope of the line touching the conic $ C $ only at a single point $ P $ be half the slope of the line joining the points $ P $ and $ (3,-5) $. If the focal distance of the point $ (7,1) $ on $ C $ is $ d $, then $ 12d $ equals:

21.

If $ \sin\!\left( \dfrac{y}{x} \right) = \log_{e}|x| + \dfrac{\alpha}{2} $ is the solution of the differential equation $ x \cos\!\left( \dfrac{y}{x} \right) \dfrac{dy}{dx} = y \cos\!\left( \dfrac{y}{x} \right) + x $ and $ y(1) = \dfrac{\pi}{3} $, then $ \alpha^{2} $ is equal to:

22.

Let $ y=y(x) $ be the solution of the differential equation $ \dfrac{dy}{dx}=2x(x+y)^{3}-x(x+y)-1 $, $ y(0)=1 $. Then, $ \left(\dfrac{1}{\sqrt{2}}+y\!\left(\dfrac{1}{\sqrt{2}}\right)\right)^{2} $ equal:

23.

Let $\displaystyle \int_{0}^{x}\sqrt{1-\left(y'(t)\right)^{2}}\,dt=\int_{0}^{x}y(t)\,dt,\ 0\le x\le 3,\ y\ge 0,\ y(0)=0 $. Then at $ x=2 $, $ y''+y+1 $ is equal to

24.

For a differentiable function $ f:\mathbb{R} \to \mathbb{R} $, suppose $ f'(x) = 3f(x) + \alpha $, where $ \alpha \in \mathbb{R} $, $ f(0) = 1 $ and $\displaystyle \lim_{x \to -\infty} f(x) = 7 $. Then $ 9 f(\ln 3) $ is equal to ________.

25.

If the solution curve $ y=y(x) $ of the differential equation $ (1+y^{2})(1+\log_{e} x)\,dx + x\,dy = 0,\ x>0 $ passes through the point $ (1,1) $ and $ y(e)=\frac{\alpha-\tan\!\left(\frac{3}{2}\right)}{\beta+\tan\!\left(\frac{3}{2}\right)} $, then $ \alpha+2\beta $ is ____________.

26.

If $ y = y(x) $ is the solution of the differential equation $ \dfrac{dy}{dx} + 2y = \sin(2x), \ y(0) = \dfrac{3}{4} $, then $ y\!\left(\dfrac{\pi}{8}\right) $ is equal to:

27.

Let $ y = y(x) $ be the solution of the differential equation
$ \frac{dy}{dx} + \frac{2x}{(1 + x^2)^2} y = x e^{\frac{1}{(1 + x^2)}}; \quad y(0) = 0 $, $ y(0)=0 $.
Then the area enclosed by the curve $ f(x) = y(x) e^{-\frac{1}{(1 + x^2)}} $ and the line $ y-x=4 $ is $ \_\_\_\_\_\_\_ $.

28.

Let $ y = y(x) $ be the solution of the differential equation, $ (1 + y^{2}) e^{\tan x} \, dx + \cos^{2} x \, (1 + e^{2 \tan x}) \, dy = 0 $, $ y(0) = 1 $. Then $ y\!\left(\frac{\pi}{4}\right) $ is equal to:

29.

Let $ y = y(x) $ be the solution of the differential equation $ (1 + x^2)\frac{dy}{dx} + y = e^{\tan^{-1} x}, \; y(1) = 0 $. Then $ y(0) $ is:

30.

Let $ f(x) $ be a positive function such that the area bounded by $ y = f(x) $, $ y = 0 $ from $ x = 0 $ to $ x = a > 0 $ is $ e^{-a} + 4a^{2} + a - 1 $. Then the differential equation, whose general solution is $ y = c_{1} f(x) + c_{2} $, where $ c_{1} $ and $ c_{2} $ are arbitrary constants, is:

31.

Let $ y = y(x) $ be the solution of the differential equation $ (x^{2} + 4)^{2} \, dy + (2x^{3}y + 8xy - 2) \, dx = 0 $.
If $ y(0) = 0 $, then $ y(2) $ is equal to:

32.

Let the solution $ y = y(x) $ of the differential equation $ \frac{dy}{dx} - y = 1 + 4\sin x $ satisfy $ y(\pi) = 1 $. Then $ y\!\left(\frac{\pi}{2}\right) + 10 $ is equal to __________.

33.

The solution curve, of the differential equation $ 2y\frac{dy}{dx}+3=5\frac{dy}{dx} $, passing through the point $ (0,1) $, is a conic, whose vertex lies on the line:

34.

Suppose the solution of the differential equation $ \frac{dy}{dx}=\frac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta y-4\alpha)} $ represents a circle passing through origin. Then the radius of this circle is:

35.

If $\displaystyle \frac{dx}{dy} = \frac{1 + x - y^2}{y} , \; x(1) = 1 $, then $ 5x(2) $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

36.

If $ x=x(t) $ is the solution of the differential equation $ (t+1)\,dx=(2x+(t+1)^{4})\,dt $, $ x(0)=2 $, then, $ x(1) $ equals ______.

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