NTA JEE Main 12th April 2019 Shift 2 - Mathematics

If $$\alpha$$, $$\beta$$ and $$\gamma$$ are three consecutive terms of a non-constant G.P. Such that the equations $$\alpha x^2 + 2\beta x + \gamma = 0$$ and $$x^2 + x - 1 = 0$$ have a common root, then $$\alpha(\beta + \gamma)$$ is equal to:

Let $$z \in C$$ with Im(z) = 10 and it satisfies $$\frac{2z - n}{2z + n} = 2i - 1$$ for some natural number n. Then

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to

If $$a_1, a_2, a_3, \ldots$$ are in A.P. such that $$a_1 + a_7 + a_{16} = 40$$, then the sum of the first 15 terms of this A.P is:

If $$^{20}C_1 + (2^2)\,^{20}C_2 + (3^2)\,^{20}C_3 + \ldots + (20^2)\,^{20}C_{20} = A(2^\beta)$$, then the ordered pair $$(A, \beta)$$ is equal to

The term independent of x in the expansion of $$\left(\frac{1}{60} - \frac{x^8}{81}\right) \cdot \left(2x^2 - \frac{3}{x^2}\right)^6$$ is equal to

Let S be the set of all $$\alpha \in R$$ such that the equation, $$\cos 2x + \alpha \sin x = 2\alpha - 7$$ has a solution. Then S is equal to:

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the centroid of this triangle is:

A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an equation of the line L is:

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point:

The tangents to the curve $$y = (x - 2)^2 - 1$$ at its points of intersection with the line $$x - y = 3$$, intersect at the point:

An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?

The equation of a common tangent to the curves, $$y^2 = 16x$$ and $$xy = -4$$, is:

$$\lim_{x \to 0} \frac{x + 2\sin x}{\sqrt{x^2 + 2\sin x + 1} - \sqrt{\sin^2 x - x + 1}}$$ is

The Boolean expression $$\sim(p \Rightarrow (\sim q))$$ is equivalent to

The angle of the top of a vertical tower standing on a horizontal plane is observed to be 45° from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be 30°, then the distance (in m) of the foot of the tower from the point A is:

Let A, B and C be sets such that $$\phi \neq A \cap B \subseteq C$$. Then which of the following statements is not true?

A value of $$\theta \in \left(0, \frac{\pi}{3}\right)$$, for which $$\begin{vmatrix} 1 + \cos^2\theta & \sin^2\theta & 4\cos 6\theta \\ \cos^2\theta & 1 + \sin^2\theta & 4\cos 6\theta \\ \cos^2\theta & \sin^2\theta & 1 + 4\cos 6\theta \end{vmatrix} = 0$$, is

If [x] denotes the greatest integer $$\leq x$$, then the system of linear equations $$[\sin\theta]x + [-\cos\theta]y = 0$$, $$[\cot\theta]x + y = 0$$

The derivative of $$\tan^{-1}\left(\frac{\sin x - \cos x}{\sin x + \cos x}\right)$$ with respect to $$\frac{x}{2}$$, where $$x \in \left(0, \frac{\pi}{2}\right)$$, is

Let $$f(x) = 5 - |x - 2|$$ and $$g(x) = |x + 1|$$, $$x \in R$$. If $$f(x)$$ attains maximum value at $$\alpha$$ and $$g(x)$$ attains minimum value at $$\beta$$, then $$\lim_{x \to -\alpha\beta} \frac{(x - 1)(x^2 - 5x + 6)}{x^2 - 6x + 8}$$ is equal to

Let $$\alpha \in \left(0, \frac{\pi}{2}\right)$$, be constant. If the integral $$\int \frac{\tan x + \tan\alpha}{\tan x - \tan\alpha} dx = A(x)\cos 2\alpha + B(x)\sin 2\alpha + C$$, where C is a constant of integration, then the functions A(x) and B(x) are respectively

A value of $$\alpha$$ such that $$\int_\alpha^{\alpha+1} \frac{dx}{(x + \alpha)(x + \alpha + 1)} = \log_e\left(\frac{9}{8}\right)$$ is

If the area (in sq. units) bounded by the parabola $$y^2 = 4\lambda x$$ and the line $$y = \lambda x$$, $$\lambda > 0$$, is $$\frac{1}{9}$$, then $$\lambda$$ is equal to

The general solution of the differential equation $$(y^2 - x^3)dx - xy\,dy = 0$$, $$(x \neq 0)$$ is (where c is a constant of integration)

Let $$\alpha \in R$$ and the three vectors $$\vec{a} = \alpha\hat{i} + \hat{j} + 3\hat{k}$$, $$\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}$$ and $$\vec{c} = \alpha\hat{i} - 2\hat{j} + 3\hat{k}$$. Then the set S = {$$\alpha$$: $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ are coplanar}

A plane which bisects the angle between the two given planes $$2x - y + 2z - 4 = 0$$ and $$x + 2y + 2z - 2 = 0$$, passes through the point

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines $$\vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})$$ and $$\vec{r} = (\hat{i} + \hat{j}) + \mu(-\hat{i} + \hat{j} - 2\hat{k})$$ is

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is:

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $$\frac{4}{5}$$, then the probability that he is unable to solve less than two problems is