List of top Mathematics Questions on Area of a Triangle - by Heron’s Formula asked in JEE Main

Let $f: [0, 3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g: [0, \infty) \to B$ be defined by $g(x) = \frac{x}{x^{2025} + 1}$. If both functions are onto and $S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$, then $n(S)$ is equal to:

Let $ A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta > 0. $ If $ B = P A P^T $, $ C = P^T B P $, and the sum of the diagonal elements of $ C $ is $ \frac{m}{n} $, where gcd(m, n) = 1, then $ m + n $ is:

Consider the region \[ R = \left\{ (x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}. \] The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $ R $ is:

Let A, B, C be three points in the xy-plane, whose position vectors are given by \[ \sqrt{3} \hat{i} + \hat{j}, \quad \hat{i} + \sqrt{3} \hat{j}, \quad \text{and} \quad a\hat{i} + (1-a) \hat{j} \] respectively with respect to the origin $ O $. If the distance of the point $ C $ from the line bisecting the angle between the vectors $ \overrightarrow{OA} $ and $ \overrightarrow{OB} $ is $ \frac{9}{\sqrt{2}} $, then the sum of all possible values of $ a $ is:

Let $ f $ be a real-valued continuous function defined on the positive real axis such that $ g(x) = \int_0^x t f(t) \, dt $. If $ g(x^3) = x^6 + x^7 $, then the value of $ \sum_{r=1}^{15} f(r^3) $ is:

The area of the region bounded by the curves $ x(1 + y^2) = 1 $ and $ y^2 = 2x $ is:

The maximum area of a triangle whose one vertex is at $ (0, 0) $ and the other two vertices lie on the curve $ y = -2x^2 + 54 $ at points $ (x, y) $ and $ (-x, y) $ where $ y > 0 $ is:

Let $\Delta$ be the area of the region $\{(x,y)∈R^2:x^2+y^2≤21,y^2≤4x,x≥1\}$. Then $\frac{1}{2}(\Delta-21\text{ sin}^{-1} (\frac{2}{\sqrt7}))$ is equal to

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0, 2, \alpha)$ and the other two vertices lie on the line $x+\frac{\alpha}{5}=y-\frac{1}{2}=z+\frac{4}{3}$ For $\alpha∈Z$, if the area of $\Delta ABC$ is 21 sq. units and the line segment BC has length $2\sqrt{21}$ units, then α² is equal to ___________.

A rectangle ABCD with ABCD with AB = 2 and BC = 4 is inscribed in rectangle PQRS such that vertices of ABCD lie on sides of PQRS then maximum possible area(in sq. unit) of rectangle PQRS is :