NTA JEE Main 2nd September 2020 Shift 1

Let $$S$$ be the set of all $$\lambda \in R$$ for which the system of linear equations
$$2x - y + 2z = 2$$
$$x - 2y + \lambda z = -4$$
$$x + \lambda y + z = 4$$
has no solution. Then the set $$S$$:

The domain of the function $$f(x) = \sin^{-1}\left(\frac{|x|+5}{x^2+1}\right)$$ is $$(-\infty, -a] \cup [a, \infty)$$, then $$a$$ is equal to:

If a function $$f(x)$$ defined by $$f(x) = \begin{cases} ae^x + be^{-x}, & -1 \le x < 1 \\ cx^2, & 1 \le x \le 3 \\ ax^2 + 2cx, & 3 < x \le 4 \end{cases}$$ be continuous for some $$a, b, c \in R$$ and $$f'(0) + f'(2) = e$$, then the value of $$a$$ is:

If the tangent to the curve $$y = x + \sin y$$ at a point $$(a, b)$$ is parallel to the line joining $$(0, \frac{3}{2})$$ and $$(\frac{1}{2}, 2)$$, then:

If $$p(x)$$ be a polynomial of degree three that has a local maximum value 8 at $$x = 1$$ and a local minimum value 4 at $$x = 2$$ then $$p(0)$$ is equal to:

Let $$P(h, k)$$ be a point on the curve $$y = x^2 + 7x + 2$$, nearest to the line, $$y = 3x - 3$$. Then the equation of the normal to the curve at $$P$$ is:

Area (in sq. units) of the region outside $$\frac{|x|}{2} + \frac{|y|}{3} = 1$$ and inside the ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ is:

Let $$y = y(x)$$ be the solution of the differential equation, $$\frac{2+\sin x}{y+1} \cdot \frac{dy}{dx} = -\cos x$$, $$y > 0$$, $$y(0) = 1$$. If $$y(\pi) = a$$ and $$\frac{dy}{dx}$$ at $$x = \pi$$ is $$b$$, then the ordered pair $$(a, b)$$ is equal to:

The plane passing through the points $$(1, 2, 1)$$, $$(2, 1, 2)$$ and parallel to the line, $$2x = 3y$$, $$z = 1$$ also passes through the point:

Box 1 contains 30 cards numbered 1 to 30 and Box 2 contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box 1 is: