NTA JEE Main 8th April 2023 Shift 2

The negation of $$(p \wedge (-q)) \vee (-p)$$ is equivalent to

Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m + n$$ is equal to

Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the relation $$R = \{(x, y) \in A \times A : x + y = 7\}$$ is

If $$A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$$, $$A^{-1} = \alpha A + \beta I$$ and $$\alpha + \beta = -2$$, then $$4\alpha^2 + \beta^2 + \lambda^2$$ is equal to:

Let $$S$$ be the set of all values of $$\theta \in [-\pi, \pi]$$ for which the system of linear equations
$$x + y + \sqrt{3}z = 0$$
$$-x + (\tan\theta)y + \sqrt{7}z = 0$$
$$x + y + (\tan\theta)z = 0$$
has non-trivial solution. Then $$\frac{120}{\pi}\sum_{\theta \in S} \theta$$ is equal to

If domain of the function $$\log_e\left(\frac{6x^2 + 5x + 1}{2x - 1}\right) + \cos^{-1}\left(\frac{2x^2 - 3x + 4}{3x - 5}\right)$$ is $$(\alpha, \beta) \cup (\gamma, \delta)$$, then $$18(\alpha^2 + \beta^2 + \gamma^2 + \delta^2)$$ is equal to

The integral $$\int\left(\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x\right) \log_2 x \, dx$$ is equal to

Let the vectors $$\vec{u}_1 = \hat{i} + \hat{j} + a\hat{k}$$, $$\vec{u}_2 = \hat{i} + b\hat{j} + \hat{k}$$, and $$\vec{u}_3 = c\hat{i} + \hat{j} + \hat{k}$$ be coplanar. If the vectors $$\vec{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$$, $$\vec{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$$ and $$\vec{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$$ are also coplanar, then $$6(a + b + c)$$ is equal to

Let $$P$$ be the plane passing through the line $$\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+5}{7}$$ and the point $$(2, 4, -3)$$. If the image of the point $$(-1, 3, 4)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to

If the probability that the random variable $$X$$ takes values $$x$$ is given by $$P(X = x) = k(x + 1)3^{-x}$$, $$x = 0, 1, 2, 3, \ldots$$, where $$k$$ is a constant, then $$P(X \geq 2)$$ is equal to