NTA JEE Main 24th February 2021 Shift 1 - Mathematics

Let $$p$$ and $$q$$ be two positive numbers such that $$p + q = 2$$ and $$p^4 + q^4 = 272$$. Then $$p$$ and $$q$$ are roots of the equation:

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:

If $$e^{\cos^2 x + \cos^4 x + \cos^6 x + \ldots \infty} \log_e 2$$ satisfies the equation $$t^2 - 9t + 8 = 0$$, then the value of $$\frac{2\sin x}{\sin x + \sqrt{3}\cos x}$$, where $$0 < x < \frac{\pi}{2}$$, is equal to

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $$\frac{1}{4}$$. Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?

The value of $$-{}^{15}C_1 + 2 \cdot {}^{15}C_2 - 3 \cdot {}^{15}C_3 + \ldots - 15 \cdot {}^{15}C_{15} + {}^{14}C_1 + {}^{14}C_3 + {}^{14}C_5 + \ldots + {}^{14}C_{11}$$ is equal to

The locus of the mid-point of the line segment joining the focus of the parabola $$y^2 = 4ax$$ to a moving point of the parabola, is another parabola whose directrix is:

The statement among the following that is a tautology is:

Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is:

The system of linear equations
$$3x - 2y - kz = 10$$
$$2x - 4y - 2z = 6$$
$$x + 2y - z = 5m$$
is inconsistent if:

Let $$f: R \to R$$ be defined as $$f(x) = 2x - 1$$ and $$g: R - \{1\} \to R$$. be defined as $$g(x) = \frac{x - \frac{1}{2}}{x - 1}$$. Then the composition function $$f(g(x))$$ is:

If $$f: R \to R$$ is a function defined by $$f(x) = x - 1\cos\frac{2x-1}{2}\pi$$, where $$[\cdot]$$ denotes the greatest integer function, then $$f$$ is:

The function $$f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$$:

If the tangent to the curve $$y = x^3$$ at the point $$P(t, t^3)$$ meets the curve again at $$Q$$, then the ordinate of the point which divides $$PQ$$ internally in the ratio 1 : 2 is:

If $$\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx = a\sin^{-1}\frac{\sin x + \cos x}{b} + c$$, where $$c$$ is a constant of integration, then the ordered pair $$(a, b)$$ is equal to:

$$\lim_{x \to 0} \frac{\int_0^{x^2} \sin\sqrt{ t} \, dt}{x^3}$$ is equal to:

The area (in sq. units) of the part of the circle $$x^2 + y^2 = 36$$, which is outside the parabola $$y^2 = 9x$$, is equal to

The population $$P = P(t)$$ at time $$t$$ of a certain species follows the differential equation $$\frac{dP}{dt} = 0.5P - 450$$. If $$P(0) = 850$$, then the time at which population becomes zero is:

The distance of the point (1, 1, 9) from the point of intersection of the line $$\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$$ and the plane $$x + y + z = 17$$ is:

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes $$3x + y - 2z = 5$$ and $$2x - 5y - z = 7$$, is

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is:

If the least and the largest real values of $$\alpha$$, for which the equation $$z + \alpha|z - 1| + 2i = 0$$ ($$z \in C$$ and $$i = \sqrt{-1}$$) has a solution, are $$p$$ and $$q$$ respectively; then $$4p^2 + q^2$$ is equal to ______.

If one of the diameters of the circle $$x^2 + y^2 - 2x - 6y + 6 = 0$$ is a chord of another circle $$C$$, whose center is at (2, 1), then its radius is ______.

Let $$A = \{n \in N : n \text{ is a 3-digit number}\}$$, $$B = \{9k + 2 : k \in N\}$$ and $$C = \{9k + l : k \in N\}$$ for some $$l$$ ($$0 < l < 9$$). If the sum of all the elements of the set $$A \cap (B \cup C)$$ is $$274 \times 400$$, then $$l$$ is equal to ______

Let $$P = \begin{pmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{pmatrix}$$, where $$\alpha \in R$$. Suppose $$Q = [q_{ij}]$$ is a matrix satisfying $$PQ = kI_3$$ for some non-zero $$k \in R$$. If $$q_{23} = -\frac{k}{8}$$ and $$Q = \frac{k^2}{2}$$, then $$\alpha^2 + k^2$$ is equal to ______.

Let $$M$$ be any $$3 \times 3$$ matrix with entries from the set $$\{0, 1, 2\}$$. The maximum number of such matrices, for which the sum of diagonal elements of $$M^T M$$ is seven, is ______.

$$\lim_{n \to \infty} \tan \sum_{r=1}^{n} \tan^{-1}\frac{1}{1 + r + r^2}$$ is equal to ______.

The minimum value of $$\alpha$$ for which the equation $$\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha$$ has at least one solution in $$\left(0, \frac{\pi}{2}\right)$$ is ______.

If $$\int_{-a}^{a} (|x| + |x - 2|) dx = 22$$, $$a > 2$$ and $$x$$ denotes the greatest integer $$\leq x$$, then $$\int_{a}^{a} (x + |x|) dx$$ is equal to ______

Let three vectors $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be such that $$\vec{c}$$ is coplanar with $$\vec{a}$$ and $$\vec{b}$$, $$\vec{a} \cdot \vec{c} = 7$$ and $$\vec{b}$$ is perpendicular to $$\vec{c}$$, where $$\vec{a} = -\hat{i} + \hat{j} + \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{k}$$, then the value of $$|2\vec{a} + \vec{b} + \vec{c}|^2$$ is ______

Let $$B_i$$ ($$i = 1, 2, 3$$) be three independent events in a sample space. The probability that only $$B_1$$ occur is $$\alpha$$, only $$B_2$$ occurs is $$\beta$$ and only $$B_3$$ occurs is $$\gamma$$. Let $$p$$ be the probability that none of the events $$B_i$$ occurs and these 4 probabilities satisfy the equations $$(\alpha - 2\beta)p = \alpha\beta$$ and $$(\beta - 3\gamma)p = 2\beta\gamma$$ (All the probabilities are assumed to lie in the interval (0, 1)). Then $$\frac{P(B_1)}{P(B_3)}$$ is equal to ______.