Let $\alpha\in(0,1)$ and $\beta=\log_{e}(1-\alpha)$. Let $P_{n}(x)=x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}+\cdots+\dfrac{x^{n}}{n},\ x\in(0,1)$. Then the integral $\displaystyle \int_{0}^{\alpha}\frac{t^{50}}{1-t}\,dt$ is equal to

1

A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of a…

Topic: JEE Main 2023 (31 January Morning Shift)

2

$ \text{If the domain of } f(x)=\dfrac{\lfloor x\rfloor}{1+x^{2}},\ \text{where } \lfloor x\rfloor \text{ is greatest i…

Topic: JEE Main 2023 (31 January Morning Shift)

3

$ \textbf{Q:}$ For the system of linear equations $x + y + z = 6,\ \alpha x + \beta y + 7z = 3,\ x + 2y + 3z = 14$, whi…

Topic: JEE Main 2023 (31 January Morning Shift)

4

Let $\mathrm{R}$ be a relation on $\mathbb{N}\times\mathbb{N}$ defined by $(a,b)\,\mathrm{R}\,(c,d)$ if and only if $ad…

Topic: JEE Main 2023 (31 January Morning Shift)

5

If $\sin^{-1}\!\left(\dfrac{\alpha}{17}\right)+\cos^{-1}\!\left(\dfrac{4}{5}\right)-\tan^{-1}\!\left(\dfrac{77}{36}\rig…

Topic: JEE Main 2023 (31 January Morning Shift)

6

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of c…

Topic: JEE Main 2023 (31 January Morning Shift)

7

A bag contains $6$ balls. Two balls are drawn from it at random and both are found to be black. The probability that th…

Topic: JEE Main 2023 (31 January Morning Shift)

8

The number of real roots of the equation $\sqrt{x^{2}-4x+3}+\sqrt{x^{2}-9}=\sqrt{4x^{2}-14x+6}$ is:

Topic: JEE Main 2023 (31 January Morning Shift)

1

Let the shortest distance between the lines $L:\ \dfrac{x-5}{-2}=\dfrac{y-\lambda}{0}=\dfrac{z+\lambda}{1},\ \lambda\ge…

Topic: JEE Main 2023 (31 January Morning Shift)

2

Let $\vec a=2\hat{\imath}+\hat{\jmath}+\hat{k}$, and $\vec b,\vec c$ be two nonzero vectors such that $\left\lvert \vec…

Topic: JEE Main 2023 (31 January Morning Shift)

3

Let $A=\begin{pmatrix}1&0&0\\[2pt]0&4&-1\\[2pt]0&12&-3\end{pmatrix}$. Then the sum of the diagonal elements of the matr…

Topic: JEE Main 2023 (31 January Morning Shift)

4

For all $z\in\mathbb{C}$ on the curve $\mathcal{C}_1:\ |z|=4$, let the locus of the point $z+\dfrac{1}{z}$ be the curve…

Topic: JEE Main 2023 (31 January Morning Shift)

5

$ \text{The value of } \displaystyle \int_{\pi/3}^{\pi/2} \frac{2+3\sin x}{\sin x\,(1+\cos x)}\,dx \text{ is equal to:}…

Topic: JEE Main 2023 (31 January Morning Shift)