JEE Main 2025 April 4 Maths Question Paper is available for download. NTA conducted JEE Main 2025 Shift 1 B.Tech Exam on 4 April 2025 from 9:00 AM to 12:00 PM and for JEE Main 2025 B.Tech Shift 2 appearing candidates from 3:00 PM to 6:00 PM. The JEE Main 2025 3rd April B.Tech Question Paper was Moderate to Tough.
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JEE Main 2025 April 4 Shift 1 Maths Question Paper with Solutions
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JEE Main 2025 Mathematics Questions with Solutions
Question 1:
Let \(f, g: (1, \infty) \rightarrow \mathbb{R}\) be defined as \(f(x) = \frac{2x + 3}{5x + 2}\) and \(g(x) = \frac{2 - 3x}{1 - x}\). If the range of the function \(fog: [2, 4] \rightarrow \mathbb{R}\) is \([\alpha, \beta]\), then \(\frac{1}{\beta - \alpha}\) is equal to
Consider the sets \(A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}\), \(B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}\), \(C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}\), and \(D = A \cap B\). The total number of one-one functions from the set \(D\) to the set \(C\) is:
Let \(A = \{1, 6, 11, 16, \ldots\}\) and \(B = \{9, 16, 23, 30, \ldots\}\) be the sets consisting of the first 2025 terms of two arithmetic progressions. Then \(n(A \cup B)\) is
For an integer \(n \geq 2\), if the arithmetic mean of all coefficients in the binomial expansion of \((x + y)^{2n-3}\) is 16, then the distance of the point \(P(2n-1, n^2-4n)\) from the line \(x + y = 8\) is:
The probability of forming a 12 persons committee from 4 engineers, 2 doctors, and 10 professors containing at least 3 engineers and at least 1 doctor is:
Let the shortest distance between the lines \(\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}\) and \(\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}\) be \(3\sqrt{30}\). Then the positive value of \(5\alpha + \beta\) is
If \(\lim_{x \to 1} \frac{(x-1)(6+\lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1\), where \(\lambda, \mu \in \mathbb{R}\), then \(\lambda + \mu\) is equal to
Let \(f: [0, \infty) \to \mathbb{R}\) be a differentiable function such that \(f(x) = 1 - 2x + \int_0^x e^{x-t} f(t) \, dt\) for all \(x \in [0, \infty)\). Then the area of the region bounded by \(y = f(x)\) and the coordinate axes is
Let \(A\) and \(B\) be two distinct points on the line \(L: \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}\). Both \(A\) and \(B\) are at a distance \(2\sqrt{17}\) from the foot of perpendicular drawn from the point \((1, 2, 3)\) on the line \(L\). If \(O\) is the origin, then \(\overrightarrow{OA} \cdot \overrightarrow{OB}\) is equal to:
Let \(f: \mathbb{R} \to \mathbb{R}\) be a continuous function satisfying \(f(0) = 1\) and \(f(2x) - f(x) = x\) for all \(x \in \mathbb{R}\). If \(\lim_{n \to \infty} \left\{ f(x) - f\left( \frac{x}{2^n} \right) \right\} = G(x)\), then \(\sum_{r=1}^{10} G(r^2)\) is equal to
1 + 3 + \(5^2\) + 7 + \(9^2\) + \(\ldots\) upto 40 terms is equal to
In the expansion of \(\left( \sqrt{5} + \frac{1}{\sqrt{5}} \right)^n\), \(n \in \mathbb{N}\), if the ratio of \(15^{th}\) term from the beginning to the \(15^{th}\) term from the end is \(\frac{1}{6}\), then the value of \(^nC_3\) is:
Considering the principal values of the inverse trigonometric functions, \(\sin^{-1} \left( \frac{\sqrt{3}}{2} x + \frac{1}{2} \sqrt{1-x^2} \right)\), \(-\frac{1}{2} < x < \frac{1}{\sqrt{2}}\), is equal to
Consider two vectors \(\vec{u} = 3\hat{i} - \hat{j}\) and \(\vec{v} = 2\hat{i} + \hat{j} - \lambda \hat{k}\), \(\lambda > 0\). The angle between them is given by \(\cos^{-1} \left( \frac{\sqrt{5}}{2\sqrt{7}} \right)\). Let \(\vec{v} = \vec{v}_1 + \vec{v}_2\), where \(\vec{v}_1\) is parallel to \(\vec{u}\) and \(\vec{v}_2\) is perpendicular to \(\vec{u}\). Then the value \(|\vec{v}_1|^2 + |\vec{v}_2|^2\) is equal to
Let the three sides of a triangle are on the lines \(4x - 7y + 10 = 0\), \(x + y = 5\), and \(7x + 4y = 15\). Then the distance of its orthocenter from the orthocenter of the triangle formed by the lines \(x = 0\), \(y = 0\), and \(x + y = 1\) is
The value of \(\int_{-1}^{1} \frac{(1 + \sqrt{|x| - x})e^x + (\sqrt{|x| - x})e^{-x}}{e^x + e^{-x}} \, dx\) is equal to
The length of the latus-rectum of the ellipse, whose foci are \((2, 5)\) and \((2, -3)\) and eccentricity is \(\frac{4}{5}\), is
Consider the equation \(x^2 + 4x - n = 0\), where \(n \in [20, 100]\) is a natural number. Then the number of all distinct values of \(n\), for which the given equation has integral roots, is equal to
A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let \(X\) denote the number of defective pens. Then the variance of \(X\) is
If \(10 \sin^4 \theta + 15 \cos^4 \theta = 6\), then the value of \(\frac{27 \csc^6 \theta + 8 \sec^6 \theta}{16 \sec^8 \theta}\) is:
If the area of the region \(\{ (x, y) : |x - 5| \leq y \leq 4\sqrt{x} \}\) is \(A\), then \(3A\) is equal to
Let \(A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta
0 & 1 & 0
\sin \theta & 0 & \cos \theta \end{bmatrix}\). If for some \(\theta \in (0, \pi)\), \(A^2 = A^T\), then the sum of the diagonal elements of the matrix \((A + I)^3 + (A - I)^3 - 6A\) is equal to
Let \(A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \}\), \(B = \{ z \in \mathbb{C} : Re(z - iz) = 2 \}\), and \(S = A \cap B\). Then \(\sum_{z \in S} |z|^2\) is equal to
Let \(C\) be the circle \(x^2 + (y - 1)^2 = 2\), \(E_1\) and \(E_2\) be two ellipses whose centres lie at the origin and major axes lie on the \(x\)-axis and \(y\)-axis respectively. Let the straight line \(x + y = 3\) touch the curves \(C\), \(E_1\), and \(E_2\) at \(P(x_1, y_1)\), \(Q(x_2, y_2)\), and \(R(x_3, y_3)\) respectively. Given that \(P\) is the mid-point of the line segment \(QR\) and \(PQ = \frac{2\sqrt{2}}{3}\), the value of \(9(x_1 y_1 + x_2 y_2 + x_3 y_3)\) is equal to
Let \(m\) and \(n\) be the number of points at which the function \(f(x) = \max \{ x, x^3, x^5, \ldots, x^{21} \}\) is not differentiable and not continuous, respectively. Then \(m + n\) is equal to
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