NTA JEE Main 9th January 2020 Shift 1

The number of real roots of the equation, $$e^{4x} + e^{3x} - 4e^{2x} + e^x + 1 = 0$$ is:

Let $$z$$ be a complex number such that $$\left|\frac{z-i}{z+2i}\right| = 1$$ and $$|z| = \frac{5}{2}$$. Then, the value of $$|z + 3i|$$ is:

If the number of five digit numbers with distinct digits and 2 at the $$10^{th}$$ place is $$336k$$, then $$k$$ is equal to:

The product $$2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots$$ to $$\infty$$ is equal to:

The value of $$\cos^3\left(\frac{\pi}{8}\right) \cdot \cos\left(\frac{3\pi}{8}\right) + \sin^3\left(\frac{\pi}{8}\right) \cdot \sin\left(\frac{3\pi}{8}\right)$$ is:

A circle touches the y-axis at the point $$(0, 4)$$ and passes through the point $$(2, 0)$$. Which of the following lines is not a tangent to this circle?

If $$e_1$$ and $$e_2$$ are the eccentricities of the ellipse $$\frac{x^2}{18} + \frac{y^2}{4} = 1$$ and the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$ respectively and $$(e_1, e_2)$$ is a point on the ellipse $$15x^2 + 3y^2 = k$$, then the value of $$k$$ is equal to:

Negation of the statement: $$\sqrt{5}$$ is an integer or 5 is irrational is:

Let the observation $$x_i(1 \le i \le 10)$$ satisfy the equations $$\sum_{i=1}^{10}(x_i - 5) = 10$$, $$\sum_{i=1}^{10}(x_i - 5)^2 = 40$$. If $$\mu$$ and $$\lambda$$ are the mean and the variance of the observations, $$x_1 - 3, x_2 - 3, \ldots, x_{10} - 3$$, then the ordered pair $$(\mu, \lambda)$$ is equal to:

If $$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3 \end{bmatrix}$$, $$B = adj \; A$$ and $$C = 3A$$, then $$\frac{|adj \; B|}{|C|}$$ is equal to: