Consider the function $ f(x)= \begin{cases} \dfrac{a\,(7x-12-x^{2})}{\,b\,\lfloor x^{2}-7x+12\rfloor\,}, & x<3,\\[6pt] \dfrac{\sin(x-3)}{2^{\,x-1}}, & x>3,\\[6pt] b, & x=3, \end{cases} $ where $\lfloor x\rfloor$ denotes the greatest integer $\le x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is:
Previous 10 Questions — JEE Main 2024 (27 January Morning Shift)
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The length of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$,
whose midpoint is $\left(1,\dfrac{9}{5}\rig…
Topic: JEE Main 2024 (27 January Morning Shift)
The distance of the point $(7,-2,11)$ from the line
$\dfrac{x-6}{1}=\dfrac{y-4}{0}=\dfrac{z-8}{3}$
along the line
$\dfr…
Topic: JEE Main 2024 (27 January Morning Shift)
Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations
$\dfrac{dx}{dt}+ax=0$ and $\dfrac{dy}{dt}+by=0$ re…
Topic: JEE Main 2024 (27 January Morning Shift)
If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and
$I_1=\displaystyle\i…
Topic: JEE Main 2024 (27 January Morning Shift)
If the shortest distance of the parabola $y^2=4x$ from the centre of the circle
$x^2+y^2-4x-16y+64=0$ is $d$, then $d^2…
Topic: JEE Main 2024 (27 January Morning Shift)
Let $S=\{1,2,3,\ldots,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation
$R=\{(A,B): A\cap B\ne…
Topic: JEE Main 2024 (27 January Morning Shift)
If $S = \{ z \in \mathbb{C} : |z - i| = |z + i| = |z - 1| \}$, then $n(S)$ is:
Topic: JEE Main 2024 (27 January Morning Shift)
If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}}\,dx = a + b\sqrt{2} + c\sqrt{3}$, where $a, b, c$ are rational numbers, th…
Topic: JEE Main 2024 (27 January Morning Shift)
Next 10 Questions — JEE Main 2024 (27 January Morning Shift)
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If $a=\displaystyle\lim_{x\to 0}\dfrac{\sqrt{\,1+\sqrt{\,1+x^{2}\,}\,}-\sqrt{2}}{x^{2}}$
and $b=\displaystyle\lim_{x\to…
Topic: JEE Main 2024 (27 January Morning Shift)
Consider the matrix
$f(x)=\begin{bmatrix}
\cos x & -\sin x & 0\\
\sin x & \cos x & 0\\
0 & 0 & 1
\end{bmatrix}$.
Give…
Topic: JEE Main 2024 (27 January Morning Shift)
Four distinct points $(2k,3k)$, $(1,0)$, $(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to:
Topic: JEE Main 2024 (27 January Morning Shift)
If the shortest distance between the lines
$\dfrac{x-4}{2}=\dfrac{y+1}{3}=\dfrac{z-\lambda}{2}$
and
$\dfrac{x-2}{…
Topic: JEE Main 2024 (27 January Morning Shift)
The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the …
Topic: JEE Main 2024 (27 January Morning Shift)
${}^{n-1}C_r \;=\; (k^2-8)\, {}^{n}C_{r+1}$ if and only if:
Topic: JEE Main 2024 (27 January Morning Shift)
The number of common terms in the progressions
$4,\,9,\,14,\,19,\ldots,$ up to $25^{\text{th}}$ term and
$3,\,6,\,9,\,1…
Topic: JEE Main 2024 (27 January Morning Shift)
Let $a_1,a_2,\ldots,a_{10}$ be $10$ observations such that
$\displaystyle \sum_{k=1}^{10} a_k = 50$
and
$\displaystyle …
Topic: JEE Main 2024 (27 January Morning Shift)
Let $\vec a=\hat i+2\hat j+\hat k$, $\quad \vec b=3(\hat i-\hat j+\hat k)$.
Let $\vec c$ be the vector such that $\vec …
Topic: JEE Main 2024 (27 January Morning Shift)
If $A$ denotes the sum of all the coefficients in the expansion of $(1-3x+10x^2)^n$
and $B$ denotes the sum of all the …
Topic: JEE Main 2024 (27 January Morning Shift)
