Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\dfrac{\pi}{6}$ and $\dfrac{\pi}{4}$ respectively with the positive $x$-axis. If $27\displaystyle\int_{1}^{3}\big((f'(t))^{2}+1\big)f'''(t),dt=\alpha+\beta\sqrt{3}$, where $\alpha,\beta$ are integers, then the value of $\alpha+\beta$ equals:
Next 10 Questions — JEE Main 2024 (30 January Evening Shift)
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For $\alpha,\beta\in(0,\dfrac{\pi}{2})$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such th…
Topic: JEE Main 2024 (30 January Evening Shift)
If $z$ is a complex number, then the number of common roots of
$z^{1985}+z^{100}+1=0$ and $z^{3}+2z^{2}+2z+1=0$ is equa…
Topic: JEE Main 2024 (30 January Evening Shift)
Let $f:\mathbb{R}\setminus{0}\to\mathbb{R}$ satisfy $f!\left(\dfrac{x}{y}\right)=\dfrac{f(x)}{f(y)}$ for all $x,y$ with…
Topic: JEE Main 2024 (30 January Evening Shift)
Let $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix…
Topic: JEE Main 2024 (30 January Evening Shift)
If $x^{2}-y^{2}+2hxy+2gx+2fy+c=0$ is the locus of a point which is always equidistant from the lines $x+2y+7=0$ and $2x…
Topic: JEE Main 2024 (30 January Evening Shift)
$a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq …
Topic: JEE Main 2024 (30 January Evening Shift)
Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\l…
Topic: JEE Main 2024 (30 January Evening Shift)
Let $L_1:\ \vec r=(\hat i-\hat j+2\hat k)+\lambda(\hat i-\hat j+2\hat k),\ \lambda\in\mathbb R,$
$L_2:\ \vec r=(\hat …
Topic: JEE Main 2024 (30 January Evening Shift)
Let $\vec a=\hat i+\alpha\hat j+\beta\hat k,\ \alpha,\beta\in\mathbb R$.
Let $\vec b$ be such that the angle between …
Topic: JEE Main 2024 (30 January Evening Shift)
Let $P$ be a point on the hyperbola $H:\ \dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, in the first quadrant, such that the area of…
Topic: JEE Main 2024 (30 January Evening Shift)
