NTA JEE Main 7th January 2020 Shift 2 - Mathematics

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 - x - 1 = 0$$. If $$p_k = (\alpha)^k + (\beta)^k$$, $$k \ge 1$$, then which one of the following statements is not true?

If $$\frac{3+i\sin\theta}{4-i\cos\theta}$$, $$\theta \in [0, 2\pi]$$, is a real number, then an argument of $$\sin\theta + i\cos\theta$$ is

Let $$a_1, a_2, a_3, \ldots$$ be a G.P. such that $$a_1 < 0$$, $$a_1 + a_2 = 4$$ and $$a_3 + a_4 = 16$$. If $$\sum_{i=1}^{9} a_i = 4\lambda$$, then $$\lambda$$ is equal to.

If the sum of the first 40 terms of the series, $$3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \ldots$$ is $$(102)m$$, then $$m$$ is equal to

The coefficient of $$x^7$$ in the expression $$(1 + x)^{10} + x(1 + x)^9 + x^2(1 + x)^8 + \ldots + x^{10}$$, is

The number of ordered pairs $$(r, k)$$ for which $$6 \cdot {}^{35}C_r = (k^2 - 3) \cdot {}^{36}C_{r+1}$$, where $$k$$ is an integer is

The locus of the mid-points of the perpendiculars drawn from points on the line $$x = 2y$$, to the line $$x = y$$, is

Let the tangents drawn from the origin to the circle, $$x^2 + y^2 - 8x - 4y + 16 = 0$$ touch it at the points A and B. Then $$(AB)^2$$ is equal to

If $$3x + 4y = 12\sqrt{2}$$ is a tangent to the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{9} = 1$$ for some $$a \in R$$, then the distance between the foci of the ellipse is

Let $$A, B, C$$ and $$D$$ be four non-empty sets. The contrapositive statement of "If $$A \subseteq B$$ and $$B \subseteq D$$, then $$A \subseteq C$$" is

Let $$A = [a_{ij}]$$ and $$B = [b_{ij}]$$ be two $$3 \times 3$$ real matrices such that $$b_{ij} = (3)^{(i+j-2)} a_{ij}$$, where $$i, j = 1, 2, 3$$. If the determinant of B is 81, then determinant of A is

Let $$y = y(x)$$ be a function of $$x$$ satisfying $$y\sqrt{1 - x^2} = k - x\sqrt{1 - y^2}$$ where $$k$$ is a constant and $$y\left(\frac{1}{2}\right) = -\frac{1}{4}$$. Then $$\frac{dy}{dx}$$ at $$x = \frac{1}{2}$$, is equal to

The value of $$c$$, in the Lagrange's mean value theorem for the function $$f(x) = x^3 - 4x^2 + 8x + 11$$, when $$x \in [0, 1]$$, is

Let $$f(x)$$ be a polynomial of degree 5 such that $$x = \pm 1$$ are its critical points. If $$\lim_{x \to 0}\left(2 + \frac{f(x)}{x^3}\right) = 4$$, then which one of the following is not true?

The value of $$\alpha$$ for which $$4\alpha \int_{-1}^{2} e^{-\alpha|x|}dx = 5$$, is

If $$\theta_1$$ and $$\theta_2$$ be respectively the smallest and the largest values of $$\theta$$ in $$(0, 2\pi) - \{\pi\}$$ which satisfy the equation, $$2\cot^2\theta - \frac{5}{\sin\theta} + 4 = 0$$, then $$\int_{\theta_1}^{\theta_2} \cos^2 3\theta \, d\theta$$ is equal to:

The area (in sq. units) of the region $$\{(x, y) \in R^2 | 4x^2 \le y \le 8x + 12\}$$ is

Let $$y = y(x)$$ be the solution curve of the differential equation, $$(y^2 - x)\frac{dy}{dx} = 1$$, satisfying $$y(0) = 1$$. This curve intersects the X-axis at a point whose abscissa is

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three unit vectors such that $$\vec{a} + \vec{b} + \vec{c} = 0$$. If $$\lambda = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$$ and $$\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$$, then the order pair, $$\left(\lambda, \vec{d}\right)$$, is equal to

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $$\frac{1}{4}$$. If the probability that at most two machines will be out of service on the same day is $$\left(\frac{3}{4}\right)^3 k$$, then $$k$$ is equal to

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, $$x$$ and $$y$$ be 10 and 25 respectively, then $$x \cdot y$$ is equal to

Let $$X = \{n \in N : 1 \le n \le 50\}$$. If $$A = \{n \in X : n \text{ is a multiple of } 2\}$$ and $$B = \{n \in X : n \text{ is a multiple of } 7\}$$, then the number of elements in the smallest subset of X, containing both A and B, is

If the system of linear equations,
$$x + y + z = 6$$
$$x + 2y + 3z = 10$$
$$3x + 2y + \lambda z = \mu$$
has more than two solutions, then $$\mu - \lambda^2$$ is equal to

If the function $$f$$ defined on $$\left(-\frac{1}{3}, \frac{1}{3}\right)$$ by $$f(x) = \begin{cases} \frac{1}{x}\log_e\left(\frac{1+3x}{1-2x}\right), & \text{when } x \neq 0 \\ k, & \text{when } x = 0 \end{cases}$$, is continuous, then $$k$$ is equal to

If the foot of the perpendicular drawn from the point $$(1, 0, 3)$$ on a line passing through $$(\alpha, 7, 1)$$ is $$\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)$$, then $$\alpha$$ is equal to