NTA JEE Mains 04th April 2024 Shift 1

Let $$a = 1 + \frac{^2C_2}{3!} + \frac{^3C_2}{4!} + \frac{^4C_2}{5!} + \ldots$$, $$b = 1 + \frac{^1C_0 + ^1C_1}{1!} + \frac{^2C_0+^2C_1+^2C_2}{2!} + \frac{^3C_0+^3C_1+^3C_2+^3C_3}{3!} + \ldots$$. Then $$\frac{2b}{a^2}$$ is equal to ______.

Let the length of the focal chord PQ of the parabola $$y^2 = 12x$$ be 15 units. If the distance of PQ from the origin is p, then $$10p^2$$ is equal to ______.

Let A be a square matrix of order 2 such that |A| = 2 and the sum of its diagonal elements is −3. If the points (x, y) satisfying $$A^2 + xA + yI = O$$ lie on a hyperbola whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then $$81(e^4 + l^2)$$ is equal to ______.

If $$\lim_{x\to 1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$$, where gcd(m, n) = 1, then $$8m + 12n$$ is equal to ______.

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $$m + n$$ is equal to ______.

Let A be a 3×3 matrix of non-negative real elements such that $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = 3\begin{bmatrix}1\\1\\1\end{bmatrix}$$. Then the maximum value of det(A) is ______.

If $$\int_0^{\pi/4}\frac{\sin^2 x}{1+\sin x\cos x}dx = \frac{1}{a}\log_e\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}}$$, where $$a, b \in N$$, then $$a + b$$ is equal to ______.

Let the solution $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ satisfy $$y(\pi) = 1$$. Then $$y\left(\frac{\pi}{2}\right) + 10$$ is equal to ______.

Let ABC be a triangle of area $$15\sqrt{2}$$ and the vectors $$\overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}$$, $$\overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}$$ and $$\overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}$$, d > 0. Then the square of the length of the largest side of the triangle ABC is ______.

If the shortest distance between the lines $$\frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4}$$ and $$\frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}$$ is $$\frac{38}{3\sqrt{5}}k$$, and $$\int_0^k [x^2]dx = \alpha - \sqrt{\alpha}$$, where [x] denotes the greatest integer function, then $$6\alpha^3$$ is equal to ______.