Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\dfrac{1+ai}{\,b+i\,}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\dfrac{1+[a]}{4b}$, where $[\,\cdot\,]$ is the greatest integer function, is:

1

The number of integral values of $k$ for which one root of the equation $2x^{2}-8x+k=0$ lies in the interval $(1,2)$ an…

Topic: JEE Main 2023 (1 February Evening Shift)

2

The area of the region given by $\{(x,y):\, xy\le 8,\ 1\le y\le x^{2}\}$ is:

Topic: JEE Main 2023 (1 February Evening Shift)

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If $y(x)=x^{x},\ x>0$, then $y''(2)-2y'(2)$ is equal to:

Topic: JEE Main 2023 (1 February Evening Shift)

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The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:

Topic: JEE Main 2023 (1 February Evening Shift)

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For the system of linear equations $\alpha x+y+z=1,\quad x+\alpha y+z=1,\quad x+y+\alpha z=\beta$, which one of the fol…

Topic: JEE Main 2023 (1 February Evening Shift)

6

Let $\vec a=5\hat{\imath}-\hat{\jmath}-3\hat{k}$ and $\vec b=\hat{\imath}+3\hat{\jmath}+5\hat{k}$ be two vectors. Then …

Topic: JEE Main 2023 (1 February Evening Shift)

7

The sum of the absolute maximum and minimum values of the function $f(x)=\lvert x^{2}-5x+6\rvert-3x+2$ in the interval …

Topic: JEE Main 2023 (1 February Evening Shift)

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Let $P(S)$ denote the power set of $S=\{1,2,3,\ldots,10\}$. Define the relations $R_{1}$ and $R_{2}$ on $P(S)$ as $A\…

Topic: JEE Main 2023 (1 February Evening Shift)

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Let $9=x_{1} < x_{2} < \ldots < x_{7}$ be in an A.P. with common difference d. If the standard deviation of $x_{1}, x_{…

Topic: JEE Main 2023 (1 February Evening Shift)

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If $A=\dfrac12\begin{bmatrix}1 & \sqrt{3}\\ -\sqrt{3} & 1\end{bmatrix}$, then:

Topic: JEE Main 2023 (1 February Evening Shift)

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Two dice are thrown independently. Let \(A\) be the event that the number on the \(1^{\text{st}}\) die is less than t…

Topic: JEE Main 2023 (1 February Evening Shift)

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Let \(\vec a = 2\hat i - 7\hat j + 5\hat k\), \(\vec b = \hat i + \hat k\) and \(\vec c = \hat i + 2\hat j - 3\hat k\) …

Topic: JEE Main 2023 (1 February Evening Shift)

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Let $\alpha x = \exp(x^{\beta} y^{\gamma})$ be the solution of the differential equation $2x^{2}y\,dy - (1 - xy^{2})\…

Topic: JEE Main 2023 (1 February Evening Shift)

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Let $f:\mathbb{R}-\{0,1\}\to\mathbb{R}$ be a function such that $f(x)+f\!\left(\frac{1}{1-x}\right)=1+x.$ Then $f(2)$…

Topic: JEE Main 2023 (1 February Evening Shift)

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Let $S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }…

Topic: JEE Main 2023 (1 February Evening Shift)