NTA JEE Main 16th March 2021 Shift 1

If for $$x \in \left(0, \frac{\pi}{2}\right)$$, $$\log_{10} \sin x + \log_{10} \cos x = -1$$ and $$\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10} n - 1)$$, $$n > 0$$, then the value of $$n$$ is equal to:

Let a complex number $$z$$, $$|z| \neq 1$$, satisfy $$\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$$. Then, the largest value of $$|z|$$ is equal to:

If $$n$$ is the number of irrational terms in the expansion of $$\left(3^{1/4} + 5^{1/8}\right)^{60}$$, then $$(n-1)$$ is divisible by:

Let $$[x]$$ denote greatest integer less than or equal to $$x$$. If for $$n \in N$$, $$\left(1 - x + x^3\right)^n = \sum_{j=0}^{3n} a_j x^j$$, then $$\sum_{j=0}^{\left[\frac{3n}{2}\right]} a_{2j} + 4\sum_{j=0}^{\left[\frac{3n-1}{2}\right]} a_{2j+1}$$ is equal to:

The number of roots of the equation, $$(81)^{\sin^2 x} + (81)^{\cos^2 x} = 30$$ in the interval $$[0, \pi]$$ is equal to:

If the three normals drawn to the parabola, $$y^2 = 2x$$ pass through the point $$(a, 0)$$, $$a \neq 0$$, then $$a$$ must be greater than:

The locus of the midpoints of the chord of the circle, $$x^2 + y^2 = 25$$ which is tangent to the hyperbola, $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$ is:

Which of the following Boolean expression is a tautology?

Consider three observations $$a, b$$ and $$c$$ such that $$b = a + c$$. If the standard deviation of $$a+2, c+2$$ is $$d$$, then which of the following is true?

Let $$A = \begin{bmatrix} i & -i \\ -i & i \end{bmatrix}$$, $$i = \sqrt{-1}$$. Then, the system of linear equations $$A^8 \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 64 \end{bmatrix}$$ has: