NTA JEE Main 9th January 2019 Shift 2 - Mathematics

The number of all possible positive integral values of $$\alpha$$ for which the roots of the quadratic equation $$6x^2 - 11x + \alpha = 0$$ are rational numbers is:

If both the roots of the quadratic equation $$x^2 - mx + 4 = 0$$ are real and distinct and they lie in the interval (1, 5), then $$m$$ lies in the interval:
Note: In the actual JEE paper interval was $$[ 1, 5 ]$$

Let $$z_0$$ be a root of quadratic equation, $$x^2 + x + 1 = 0$$. If $$z = 3 + 6iz_0^{81} - 3iz_0^{93}$$, then $$\arg(z)$$ is equal to:

The number of natural numbers less than 7000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to:

The sum of the following series $$1 + 6 + \frac{9(1^2 + 2^2 + 3^2)}{7} + \frac{12(1^2 + 2^2 + 3^2 + 4^2)}{9} + \frac{15(1^2 + 2^2 + \ldots + 5^2)}{11} + \ldots$$ up to 15 terms, is:

Let $$a$$, $$b$$ and $$c$$ be the 7$$^{th}$$, 11$$^{th}$$ and 13$$^{th}$$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $$\frac{a}{c}$$ is equal to:

The coefficient of $$t^4$$ in the expansion of $$\left(\frac{1 - t^6}{1 - t}\right)^3$$ is:

If $$0 \leq x < \frac{\pi}{2}$$, then the number of values of $$x$$ for which $$\sin x - \sin 2x + \sin 3x = 0$$, is:

Let $$S$$ be the set of all triangles in the $$xy$$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $$S$$ has area 50 sq. units, then the number of elements in the set $$S$$ is:

Let the equations of two sides of a triangle be $$3x - 2y + 6 = 0$$ and $$4x + 5y - 20 = 0$$. If the orthocenter of this triangle is at $$(1, 1)$$ then the equation of its third side is:

If the circles $$x^2 + y^2 - 16x - 20y + 164 = r^2$$ and $$(x-4)^2 + (y-7)^2 = 36$$ intersect at two distinct points, then:

Let $$A(4, -4)$$ and $$B(9, 6)$$ be points on the parabola, $$y^2 = 4x$$. Let $$C$$ be chosen on the arc AOB of the parabola, where $$O$$ is the origin, such that the area of $$\triangle ACB$$ is maximum. Then, the area (in sq. units) of $$\triangle ACB$$, is:

A hyperbola has its centre at the origin, passes through the point $$(4, 2)$$ and has transverse axis of length 4 along the $$x$$-axis. Then the eccentricity of the hyperbola is:

For each $$x \in R$$, let $$[x]$$ be the greatest integer less than or equal to $$x$$. Then $$\lim_{x \to 0^{-}} \frac{x([x] + |x|)\sin[x]}{|x|}$$ is equal to:

The logical statement $$[\sim(\sim p \vee q) \vee (p \wedge r)] \wedge (\sim q \wedge r)$$ is equivalent to:

A data consists of $$n$$ observations: $$x_1, x_2, \ldots, x_n$$. If $$\sum_{i=1}^{n}(x_i + 1)^2 = 9n$$ and $$\sum_{i=1}^{n}(x_i - 1)^2 = 5n$$, then the standard deviation of this data is:

If $$A = \begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}$$, then $$A$$ is:

If the system of linear equations $$x - 4y + 7z = g$$; $$3y - 5z = h$$; $$-2x + 5y - 9z = k$$ is consistent, then:

If $$x = \sin^{-1}(\sin 10)$$ and $$y = \cos^{-1}(\cos 10)$$, then $$y - x$$ is equal to:

Let $$f: [0,1] \to R$$ be such that $$f(xy) = f(x) \cdot f(y)$$, for all $$x, y \in [0,1]$$, and $$f(0) \neq 0$$. If $$y = y(x)$$ satisfies the differential equation, $$\frac{dy}{dx} = f(x)$$ with $$y(0) = 1$$, then $$y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$$ is equal to:

Let $$A = \{x \in R : x$$ is not a positive integer$$\}$$. Define a function $$f: A \to R$$ as $$f(x) = \frac{2x}{x-1}$$, then $$f$$ is:

Let $$f$$ be a differentiable function from $$R$$ to $$R$$ such that $$|f(x) - f(y)| \leq 2|x - y|^{3/2}$$, for all $$x, y \in R$$. If $$f(0) = 1$$ then $$\int_0^1 f^2(x)dx$$ is equal to:

If $$x = 3\tan t$$ and $$y = 3\sec t$$, then the value of $$\frac{d^2y}{dx^2}$$ at $$t = \frac{\pi}{4}$$, is:

If $$f(x) = \int \frac{(5x^8 + 7x^6)}{(x^2 + 1 + 2x^7)^2}dx$$, $$(x \geq 0)$$, and $$f(0) = 0$$, then the value of $$f(1)$$ is:

If $$\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}} d\theta = 1 - \frac{1}{\sqrt{2}}$$, $$(k \gt 0)$$, then the value of $$k$$ is:

The area of the region $$A = \{(x, y) : 0 \leq y \leq x|x| + 1$$ and $$-1 \leq x \leq 1\}$$ in sq. units, is:

Let $$\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$$, $$\vec{b} = b_1\hat{i} + b_2\hat{j} + \sqrt{2}\hat{k}$$ and $$\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}$$ be three vectors such that the projection vector of $$\vec{b}$$ on $$\vec{a}$$ is $$|\vec{a}|$$. If $$\vec{a} + \vec{b}$$ is perpendicular to $$\vec{c}$$, then $$|\vec{b}|$$ is equal to:

If the lines $$x = ay + b$$, $$z = cy + d$$ and $$x = a'z + b'$$, $$y = c'z + d'$$ are perpendicular, then:

The equation of the plane containing the straight line $$\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$$ and perpendicular to the plane containing the straight lines $$\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$$ and $$\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$$ is:

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is: