NTA JEE Main 29th January 2023 Shift 2

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

Let $$K$$ be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x)^{99}$$. Let $$a$$ be the middle term in the expansion of $$\left(2 + \frac{1}{\sqrt{2}}\right)^{200}$$. If $$\frac{^{200}C_{99}K}{a} = \frac{2^l m}{n}$$, where $$m$$ and $$n$$ are odd numbers, then the ordered pair $$(l, n)$$ is equal to:

The set of all values of $$\lambda$$ for which the equation $$\cos^2 2x - 2\sin^4 x - 2\cos^2 x = \lambda$$

If the tangent at a point P on the parabola $$y^2 = 3x$$ is parallel to the line $$x + 2y = 1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4} + \frac{y^2}{1} = 1$$ are perpendicular to the line $$x - y = 2$$, then the area of the triangle $$PQR$$ is:

The statement $$B \Rightarrow ((\neg A) \vee B)$$ is not equivalent to:

Let $$R$$ be a relation defined on $$\mathbb{N}$$ as $$a R b$$ is $$2a + 3b$$ is a multiple of $$5, a, b \in \mathbb{N}$$. Then $$R$$ is

The set of all values of $$t \in \mathbb{R}$$, for which the matrix $$\begin{bmatrix} e^t & e^{-t}(\sin t - 2\cos t) & e^{-t}(-2\sin t - \cos t) \\ e^t & e^{-t}(2\sin t + \cos t) & e^{-t}(\sin t - 2\cos t) \\ e^t & e^{-t}\cos t & e^{-t}\sin t \end{bmatrix}$$ is invertible, is

Let $$f$$ and $$g$$ be twice differentiable functions on $$R$$ such that
$$f''(x) = g''(x) + 6x$$
$$f'(1) = 4g'(1) - 3 = 9$$
$$f(2) = 3, g(2) = 12$$
Then which of the following is NOT true?

The value of the integral $$\int_1^2 \left(\frac{t^4+1}{t^6+1}\right) dt$$ is: