NTA JEE Main 27th August 2021 Shift 1

If the matrix $$A = \begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$$ satisfies $$A(A^3 + 3I) = 2I$$, then the value of $$K$$ is

If $$(\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a$$; $$0 < x < 1$$, $$a \neq 0$$, then the value of $$2x^2 - 1$$ is

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

If $$U_n = \left(1 + \frac{1}{n^2}\right)\left(1 + \frac{2^2}{n^2}\right)^2 \cdots \left(1 + \frac{n^2}{n^2}\right)^n$$, then $$\lim_{n \to \infty} (U_n)^{\frac{-4}{n^2}}$$ is equal to

$$\int_6^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e(x^2 - 44x + 484)} dx$$ is equal to

Let us consider a curve, $$y = f(x)$$ passing through the point $$(-2, 2)$$ and the slope of the tangent to the curve at any point $$(x, f(x))$$ is given by $$f(x) + xf'(x) = x^2$$. Then

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 2(y + 2\sin x - 5)x - 2\cos x$$ such that $$y(0) = 7$$. Then $$y(\pi)$$ is equal to

The distance of the point $$(1, -2, 3)$$ from the plane $$x - y + z = 5$$ measured parallel to a line, whose direction ratios are $$2, 3, -6$$, is

Equation of a plane at a distance $$\sqrt{\frac{2}{21}}$$ units from the origin, which contains the line of intersection of the planes $$x - y - z - 1 = 0$$ and $$2x + y - 3z + 4 = 0$$, is

When a certain biased die is rolled, a particular face occurs with probability $$\frac{1}{6} - x$$ and its opposite face occurs with probability $$\frac{1}{6} + x$$. All other faces occur with probability $$\frac{1}{6}$$. Note that opposite faces sum to 7 in any die. If $$0 < x < \frac{1}{6}$$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is $$\frac{13}{96}$$, then the value of $$x$$ is