NTA JEE Main 26th July 2022 Shift 2

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______.

If $$\displaystyle\sum_{k=1}^{10} \dfrac{k}{k^4 + k^2 + 1} = \dfrac{m}{n}$$, where $$m$$ and $$n$$ are co-prime, then $$m + n$$ is equal to ______.

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.

If the sum of solutions of the system of equations $$2\sin^2\theta - \cos 2\theta = 0$$ and $$2\cos^2\theta + 3\sin\theta = 0$$ in the interval $$[0, 2\pi]$$ is $$k\pi$$, then $$k$$ is equal to ______.

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $$\sigma$$ is the standard deviation of the data after omitting the two wrong observations from the data, then $$38\sigma^2$$ is equal to ______.

Let 𝐴 = {1, 2, 3, 4, 5, 6, 7} and 𝐵 = {3, 6, 7, 9}. Then the number of elements in the set $$C \subseteq A : C \cap B \neq \phi$$ is 

The number of matrices $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$, where $$𝑎, 𝑏, 𝑐, d ∈ -1, 0, 1, 2, 3, … … , 10,$$ such that $$A=A^{T}$$, is______.

Suppose $$𝑦 = 𝑦𝑥$$ be the solution curve to the differential equation $$\frac{dy}{dx}-y=2-e^{-x}$$ such that $$\lim_{x \rightarrow \infty} yx$$ If $$𝑎$$ and $$𝑏$$ are respectively the $$𝑥 -$$ and $$𝑦 -$$ intercept of the tangent to the curve at $$𝑥 = 0$$, then the value of $$𝑎 - 4𝑏$$ is equal to _______.

The largest value of $$𝑎$$, for which the perpendicular distance of the plane containing the lines $$\vec{r} = \hat{i} + \hat{j} + \lambda \hat{i} + a \hat{j} - \hat{k} \quad \text{and} \quad \vec{r} = \hat{i} + \hat{j} + \mu \hat{i} + \hat{j} - a \hat{k}$$ from the point 2, 1, 4 is $$\sqrt{3}$$, is 

The plane passing through the line $$L:l 𝑥 - 𝑦 + 31 - 𝑙 𝑧 = 1, 𝑥 + 2𝑦 - 𝑧 = 2$$ and perpendicular to the plane $$3𝑥 + 2𝑦 + 𝑧 = 6$$ is $$3𝑥 - 8𝑦 + 7𝑧 = 4$$. If $$\theta$$ is the acute angle between the line $$𝐿$$ and the 𝑦-axis, then $$415 \cos^{2}\theta $$ is equal to_______.