NTA JEE Main 12th January 2019 Shift 1 - Mathematics

If $$\lambda$$ be the ratio of the roots of the quadratic equation in x, $$3m^2x^2 + m(m-4)x + 2 = 0$$, then the least value of m for which $$\lambda + \frac{1}{\lambda} = 1$$, is:

If $$\frac{z - \alpha}{z + \alpha}$$ $$(\alpha \in R)$$ is a purely imaginary number and $$|z| = 2$$, then a value of $$\alpha$$ is:

Let $$S = \{1, 2, 3, \ldots, 100\}$$, then number of non-empty subsets A of S such that the product of elements in A is even is:

Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by $$n_i$$, the label of the ball drawn from the $$i^{th}$$ box, $$(i = 1, 2, 3)$$. Then, the number of ways in which the balls can be chosen such that $$n_1 < n_2 < n_3$$ is:

Let $$S_k = \frac{1+2+3+\ldots+k}{k}$$. If $$S_1^2 + S_2^2 + \ldots + S_{10}^2 = \frac{5}{12}A$$, then A is equal to:

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P., then the sum of the original three terms of the given G.P. is:

A ratio of the 5$$^{th}$$ term from the beginning to the 5$$^{th}$$ term from the end in the binomial expansion of $$\left(2^{1/3} + \frac{1}{2(3)^{1/3}}\right)^{10}$$ is

The maximum value of $$3\cos\theta + 5\sin\left(\theta - \frac{\pi}{6}\right)$$ for any real value of $$\theta$$ is:

If the straight line $$2x - 3y + 17 = 0$$ is perpendicular to the line passing through the points $$(7, 17)$$ and $$(15, \beta)$$, then $$\beta$$ equals:

Let $$C_1$$ and $$C_2$$ be the centres of the circles $$x^2 + y^2 - 2x - 2y - 2 = 0$$ and $$x^2 + y^2 - 6x - 6y + 14 = 0$$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $$PC_1QC_2$$ is:

If a variable line $$3x + 4y - \lambda = 0$$ is such that the two circles $$x^2 + y^2 - 2x - 2y + 1 = 0$$ and $$x^2 + y^2 - 18x - 2y + 78 = 0$$ are on its opposite sides, then the set of all values of $$\lambda$$ is the interval:

Let $$P(4, -4)$$ and $$Q(9, 6)$$ be two points on the parabola, $$y^2 = 4x$$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta PXQ$$ is maximum. Then this maximum area (in sq. units) is:

If the vertices of a hyperbola be at $$(-2, 0)$$ and $$(2, 0)$$ and one of its foci be at $$(-3, 0)$$, then which one of the following points does not lie on this hyperbola?

$$\lim_{x \to \frac{\pi}{4}} \frac{\cot^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}$$ is

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

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:

Let $$P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$$ and $$Q = [q_{ij}]$$ be two $$3 \times 3$$ matrices such that $$Q - P^5 = I_3$$. Then $$\frac{q_{21} + q_{31}}{q_{32}}$$ is equal to:

An ordered pair $$(\alpha, \beta)$$ for which the system of linear equations $$(1 + \alpha)x + \beta y + z = 2$$, $$\alpha x + (1 + \beta)y + z = 3$$, $$\alpha x + \beta y + 2z = 2$$ has a unique solution, is:

Considering only the principal values of inverse functions, the set $$A = \{x \ge 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$$

Let S be the set of all points in $$(-\pi, \pi)$$ at which the function, $$f(x) = \min\{\sin x, \cos x\}$$ is not differentiable. Then S is a subset of which of the following?

For $$x \gt 1$$, if $$(2x)^{2y} = 4e^{2x-2y}$$, then $$(1 + \log_e 2x)^2 \frac{dy}{dx}$$ is equal to

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, $$y = 12 - x^2$$ such that the rectangle lies inside the parabola, is:

The integral $$\int \cos(\ln x) dx$$, is equal to

Let f and g be continuous functions on [0, a] such that $$f(x) = f(a-x)$$ and $$g(x) + g(a-x) = 4$$, then $$\int_0^a f(x)g(x)dx$$ is equal to

The area (in sq. units) of the region bounded by the parabola, $$y = x^2 + 2$$ and the lines, $$y = x + 1$$, $$x = 0$$ and $$x = 3$$, is

Let $$y = y(x)$$ be the solution of the differential equation, $$x\frac{dy}{dx} + y = x\log_e x$$, $$(x > 1)$$. If $$2y(2) = \log_e 4 - 1$$, then $$y(e)$$ is equal to

The sum of the distinct real values of $$\mu$$ for which the vectors $$\mu\hat{i} + \hat{j} + \hat{k}$$, $$\hat{i} + \mu\hat{j} + \hat{k}$$, $$\hat{i} + \hat{j} + \mu\hat{k}$$ are co-planar, is

A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(-1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is

The perpendicular distance from the origin to the plane containing the two lines, $$\frac{x+2}{3} = \frac{y-2}{5} = \frac{z+5}{7}$$ and $$\frac{x-1}{1} = \frac{y-4}{4} = \frac{z+4}{7}$$, is

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to: