NTA JEE Main 24th February 2021 Shift 1

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: