The value of $\dfrac{{}^{100}C_{50}}{51} + \dfrac{{}^{100}C_{51}}{52} + \cdots + \dfrac{{}^{100}C_{100}}{101}$ is:
Previous 10 Questions — JEE Main 2026 (23 January Morning Shift)
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Among the statements:
I :
\(
\begin{vmatrix}
1 & \cos\alpha & \cos\beta \\
\cos\alpha & 1 & \cos\gamma \\
\cos\beta & …
Topic: JEE Main 2026 (23 January Morning Shift)
The value of the integral
$\displaystyle \int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt{3}\tan 2x}$
is:
Topic: JEE Main 2026 (23 January Morning Shift)
Let \( f(x) =
\begin{cases}
\dfrac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \ne -\frac{3}{2}, \frac{1}{2} \\
b, & x = -\frac…
Topic: JEE Main 2026 (23 January Morning Shift)
A building construction work can be completed by two masons $A$ and $B$ together in $22.5$ days. Mason $A$ alone can co…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function$f(\theta) = 4\left(\sin^4\l…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $S = {z : 3 \le |2z - 3(1 + i)| \le 7}$ be a set of complex numbers. Then
$\min_{z \in S} \left|z + \frac{1}{2}(5 +…
Topic: JEE Main 2026 (23 January Morning Shift)
Number of solutions of
$\sqrt{3}\cos2\theta + 8\cos\theta + 3\sqrt{3} = 0,; \theta \in [-3\pi, 2\pi]$ is:
Topic: JEE Main 2026 (23 January Morning Shift)
The vertices $B$ and $C$ of a triangle $ABC$ lie on the line
$\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$.
The …
Topic: JEE Main 2026 (23 January Morning Shift)
The sum of all possible values of $n \in \mathbb{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion…
Topic: JEE Main 2026 (23 January Morning Shift)
If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation$(-2+\sqrt{3})\left(|\sqrt{x}-3|\right) + (x…
Topic: JEE Main 2026 (23 January Morning Shift)
Next 10 Questions — JEE Main 2026 (23 January Morning Shift)
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Let the mean and variance of $8$ numbers
$-10,; -7,; -1,; x,; y,; 9,; 2,; 16$
be $\dfrac{7}{2}$ and $\dfrac{293}{4}$,…
Topic: JEE Main 2026 (23 January Morning Shift)
Let the direction cosines of two lines satisfy the equations:
$4\ell + m - n = 0$ and $2mn + 10n\ell + 3\ell m = 0$.
…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $|A| = 6$, where $A$ is a $3 \times 3$ matrix. If
$|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot…
Topic: JEE Main 2026 (23 January Morning Shift)
Let the area of the region bounded by the curve
$y = \max{\sin x, \cos x}$, lines $x = 0,; x = \frac{3\pi}{2}$, and th…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $f$ be a twice differentiable non-negative function such that$(f(x))^2 = 25 + \int_0^x \big((f(t))^2 + (f'(t))^2\bi…
Topic: JEE Main 2026 (23 January Morning Shift)
From the first $100$ natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If …
Topic: JEE Main 2026 (23 January Morning Shift)
The number of $4$-letter words, with or without meaning, which can be formed using the letters $PQRPRSTUVP$, is ______.
Topic: JEE Main 2026 (23 January Morning Shift)
If $f(x) = \begin{cases} \dfrac{a|x| + x^2 - 2(\sin x)(\cos x)}{x}, & x \ne 0 \ b, & x = 0 \end{cases}$
is cont…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $\frac{\pi}{2} < \theta < \pi$ and $\cot\theta = -\frac{1}{2\sqrt{2}}$. Then the value of$\sin\left(\frac{150…
Topic: JEE Main 2026 (23 January Morning Shift)
Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k},; \vec{b} = 2\hat{i} + \hat{j} - \hat{k},; \vec{c} = \lambda\hat{i} + \hat…
Topic: JEE Main 2026 (23 January Morning Shift)
