JEE Main: Vectors, Integration, Functions & More

Vectors $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ are coplanar with vector $\vec{c}$. Given that $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, determine $|\vec{c}|$.

Evaluate $I(9, 14) + I(10, 13)$, given $I(m, n) = \int_{0}^{1} x^{m-1}(1-x)^{n-1} dx$, with $m, n > 0$.

A function $f: \mathbb{R} \setminus {0} \rightarrow \mathbb{R}$ satisfies $f(x) - 6f(\frac{1}{x}) = \frac{35}{3x} - \frac{5}{2}$. Given that $\lim_{x \to \infty} [x + f(x)] = \beta$, what is the value of $\alpha + 2\beta$ given $\lim_{x \to 0} f(x) = \alpha$?

Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an arithmetic progression (AP) with a first term of $-p$ and common difference $p$ is $\sqrt{2026}S_{2025}$, what is the absolute difference between the $20^{th}$ and $15^{th}$ terms of the AP?

For $f(x) = \frac{2^{x+2} + 16}{2^{2x} + 2^{x+4} + 32}$, what is the value of $f(\frac{1}{15}) + f(\frac{2}{15}) + \ldots + f(\frac{59}{15})$?

If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z + 2i = 0$ where $i = \sqrt{-1}$, find the imaginary part of $\frac{\alpha^{19} + \beta^{19}}{\alpha^{15} + \beta^{15}} + \frac{\alpha^{11} + \beta^{11}}{\alpha^{7} + \beta^{7}}$

Evaluate $\lim_{x \to 0} \frac{\operatorname{cosec} x \left(\sqrt{2 \cos ^{2} x} + 3 \cos x - \sqrt{\cos ^{2} x + \sin x + 4}\right)}{\sqrt{2 \cos ^{2} x} + \sqrt{\cos ^{2} x + \sin x + 4}}$

In triangle $\triangle ABC$, the length of side $AC$ is 6, vertex $B$ is $(1, 2, 3)$, where $A$ and $C$ lie on line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Find the area of $\triangle ABC$.

Solve the differential equation $(1 + x^2)dy + (xy - 5x^2\sqrt{1+x^2})dx = 0$, with initial condition $y(0) = 0$, and then determine the value of $y(\sqrt{3})$.

Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$, the product of the focal distances of the point $(\frac{\sqrt{3}}{2}, \frac{1}{2})$on the ellipse is $\frac{7}{4}$. Find the absolute difference of the eccentricities of the two such ellipses .

A and B alternatively throw a traditional six-sided die. A wins if he throws a sum of 5 before B throws a sum of 8. B wins if he throws a sum of 8 before A throws a sum of 5. If A throws first, what is the probability A will win?

Determine the area of the largest rectangle with sides parallel to the coordinate axes inscribed in the region $R = {(x,y) : x \le y \le 9 - \frac{11}{3}x, x \ge 0}$.

Given a statistical data set $x_1, x_2, \dots, x_{10}$ with 10 values, a student finds the mean to be $5.5$ and $\sum_{i=1}^{10} x_i^2 = 371$. Later, it is discovered that two values were incorrectly recorded as 4 and 5 instead of the correct values 6 and 8. What is the variance of the corrected data?

Let circle C be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ about the line $2x - 3y + 5 = 0$, where the point $A$ on $C$ is such that $OA$ is parallel to the x-axis, with $A$ to the right of the center $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, is on $C$ such that the arc length $AB$ is $\frac{1}{6}$th of the perimeter of $C$, what is the value of $\beta - \sqrt{3}\alpha$?

For $n \neq 10$, given that the coefficients of the $5^{th}$, $6^{th}$, and $7^{th}$ terms in the binomial expansion of $(1 + x)^{n+4}$ form an arithmetic progression, find the largest coefficient in the expansion of $(1 + x)^{n+4}$.