NTA JEE Main 9th January 2020 Shift 1

If for some $$\alpha$$ and $$\beta$$ in $$R$$, the intersection of the following three planes
$$x + 4y - 2z = 1$$
$$x + 7y - 5z = \beta$$
$$x + 5y + \alpha z = 5$$
is a line in $$R^3$$, then $$\alpha + \beta$$ is equal to:

If $$f(x) = \begin{cases} \frac{\sin(a+2)x + \sin x}{x} & ; x < 0 \\ b & ; x = 0 \\ \frac{(x+3x^2)^{1/3} - x^{1/3}}{x^{1/3}} & ; x > 0 \end{cases}$$ is continuous at $$x = 0$$, then $$a + 2b$$ is equal to:

Let $$f$$ be any function continuous on $$[a, b]$$ and twice differentiable on $$(a, b)$$. If all $$x \in (a, b)$$, $$f'(x) > 0$$ and $$f''(x) < 0$$, then for any $$c \in (a, b)$$, $$\frac{f(c) - f(a)}{f(b) - f(c)}$$ is:

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50 \; cm^3/min$$. When the thickness of ice is 5 cm, then the rate (in cm/min) at which the thickness of ice decreases, is:

The integral $$\int \frac{dx}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}}$$ is equal to: (where $$C$$ is a constant of integration)

If for all real triplets $$(a, b, c)$$, $$f(x) = a + bx + cx^2$$; then $$\int_0^1 f(x) \; dx$$ is equal to:

The value of $$\int_0^{2\pi} \frac{x \sin^8 x}{\sin^8 x + \cos^8 x} dx$$ is equal to:

If $$f'(x) = \tan^{-1}(\sec x + \tan x)$$, $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$ and $$f(0) = 0$$, then $$f(1)$$ is equal to:

Let $$D$$ be the centroid of the triangle with vertices $$(3, -1)$$, $$(1, 3)$$ and $$(2, 4)$$. Let P be the point of intersection of the lines $$x + 3y - 1 = 0$$ and $$3x - y + 1 = 0$$. Then, the line passing through the points $$D$$ and $$P$$ also passes through the point:

In a box, there are 20 cards, out of which 10 are labelled as $$A$$ and the remaining 10 are labelled as $$B$$. Cards are drawn at random, one after the other and with replacement, till a second $$A$$ card is obtained. The probability that the second $$A$$ card appears before the third $$B$$ card is: