NTA JEE Main 15th April 2018 Shift 2

If $$f(x)$$ is a quadratic expression such that $$f(1) + f(2) = 0$$, and -1 is a root of $$f(x) = 0$$, then the other root of $$f(x) = 0$$ is:

Let $$f(x)$$ be a polynomial of degree 4 having extreme values at $$x = 1$$ and $$x = 2$$. If $$\lim_{x \to 0}\left(\frac{f(x)}{x^2} + 1\right) = 3$$, then $$f(-1)$$ is equal to:

$$\int \frac{2x+5}{\sqrt{7 - 6x - x^2}} dx = A\sqrt{7 - 6x - x^2} + B\sin^{-1}\left(\frac{x+3}{4}\right) + C$$
(where C is a constant of integration), then the ordered pair (A, B) is equal to:

The value of integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x}{1 + \sin x} dx$$ is:

If $$I_1 = \int_0^1 e^{-x} \cos^2 x \, dx$$; $$I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx$$ and $$I_3 = \int_0^1 e^{-x^2} dx$$; then:

The curve satisfying the differential equation, $$(x^2 - y^2)dx + 2xydy = 0$$ and passing through the point (1, 1) is:

If the position vectors of the vertices A, B and C of a $$\triangle$$ABC are respectively $$4\hat{i} + 7\hat{j} + 8\hat{k}$$, $$2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$2\hat{i} + 5\hat{j} + 7\hat{k}$$, then the position vector of the point, where the bisector of $$\angle A$$ meets BC is:

An angle between the lines whose direction cosines are given by the equations, $$l + 3m + 5n = 0$$ and $$5lm - 2mn + 6nl = 0$$, is:

A plane bisects the line segment joining the points (1, 2, 3) and (-3, 4, 5) at right angles. Then this plane also passes through the point:

A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of 'p' is: