Let $C:\ x^{2}+y^{2}=4$ and $C':\ x^{2}+y^{2}-4\lambda x+9=0$ be two circles. If the set of all values of $\lambda$ for which the circles $C$ and $C'$ intersect at two distinct points is $\mathbb{R}\setminus [a,b]$, then the point $(\,8a+12,\ 16b-20\,)$ lies on the curve:

1

Let $3,a,b,c$ be in A.P. and $3,\,a-1,\,b+1,\,c+9$ be in G.P. Then, the arithmetic mean of $a,b,c$ is:

Topic: JEE Main 2024 (1 February Morning Shift)

2

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1, \quad y(0) = 1.…

Topic: JEE Main 2024 (1 February Morning Shift)

3

For $0 < \theta < \dfrac{\pi}{2}$, if the eccentricity of the hyperbola $x^2 - y^2 \csc^2\theta = 5$ is $\sqrt{7}…

Topic: JEE Main 2024 (1 February Morning Shift)

4

Let $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\ a > b$ be an ellipse, whose eccentricity is $\dfrac{1}{\sqrt{2}}$ and …

Topic: JEE Main 2024 (1 February Morning Shift)

5

Let $f:\mathbf{R}\rightarrow\mathbf{R}$ be defined as $ f(x)= \begin{cases} \dfrac{a - b\cos 2x}{x^2}, & x < 0, \\[6…

Topic: JEE Main 2024 (1 February Morning Shift)

6

If the system of equations $ \begin{aligned} 2x + 3y - z &= 5, \\ x + \alpha y + 3z &= -4, \\ 3x - y + \beta z &= …

Topic: JEE Main 2024 (1 February Morning Shift)

7

Let $f:\mathbf{R}\rightarrow\mathbf{R}$ and $g:\mathbf{R}\rightarrow\mathbf{R}$ be defined as $ f(x)= \begin{cases} …

Topic: JEE Main 2024 (1 February Morning Shift)

8

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to:

Topic: JEE Main 2024 (1 February Morning Shift)

9

Let $S=\Big\{x\in\mathbb{R}:(\sqrt3+\sqrt2)^{x}+(\sqrt3-\sqrt2)^{x}=10\Big\}$. Then the number of elements in $S$ is:

Topic: JEE Main 2024 (1 February Morning Shift)

10

Let $\vec a=-5\hat i+\hat j-3\hat k$, $\vec b=\hat i+2\hat j-4\hat k$ and $\vec c=\big(((\vec a\times\vec b)\times\hat …

Topic: JEE Main 2024 (1 February Morning Shift)

1

If $5f(x)+4f\!\left(\frac{1}{x}\right)=x^{2}-2,\ \forall x\ne 0$ and $y=9x^{2}f(x)$, then $y$ is strictly increasing …

Topic: JEE Main 2024 (1 February Morning Shift)

2

If the shortest distance between the lines \[ \frac{x-\lambda}{2}=\frac{y-2}{1}=\frac{z-1}{1} \quad\text{and}\quad \…

Topic: JEE Main 2024 (1 February Morning Shift)