If the system of equations $ \begin{aligned} 2x + 3y - z &= 5, \\ x + \alpha y + 3z &= -4, \\ 3x - y + \beta z &= 7 \end{aligned} $ has infinitely many solutions, then $13\alpha\beta$ is equal to:

1

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)

2

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

Topic: JEE Main 2024 (1 February Morning Shift)

3

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)

4

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)

5

Let the median and the mean deviation about the median of 7 observations $170,125,230,190,210,a,b$ be $170$ and $\dfrac…

Topic: JEE Main 2024 (1 February Morning Shift)

6

Let $S=\left\{\,z\in\mathbb{C}:\ |z-1|=1 \ \text{and}\ \left|(\sqrt2-1)(z+\bar z)-i(z-\bar z)\right|=2\sqrt2\,\right\}$…

Topic: JEE Main 2024 (1 February Morning Shift)

7

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may …

Topic: JEE Main 2024 (1 February Morning Shift)

8

The value of the integral $\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}$

Topic: JEE Main 2024 (1 February Morning Shift)

9

If $\tan A=\dfrac{1}{\sqrt{x^{2}+x+1}},\quad \tan B=\dfrac{\sqrt{x}}{\sqrt{x^{2}+x+1}}$ and $\tan C=\big(x^{-3}+x^{-2}+…

Topic: JEE Main 2024 (1 February Morning Shift)

10

If $A=\begin{bmatrix}\sqrt2&1\\-1&\sqrt2\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^{\mathrm T}$ a…

Topic: JEE Main 2024 (1 February Morning Shift)

1

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)

2

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)

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 $y = y(x)$ 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)

5

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)

6

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…

Topic: JEE Main 2024 (1 February Morning Shift)

7

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)

8

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)