NTA JEE Main 25th July 2021 Shift 2 - Mathematics

The number of real solutions of the equation, $$x^2 - |x| - 12 = 0$$ is:

The sum of all those terms which are rational numbers in the expansion of $$\left(2^{\frac{1}{3}} + 3^{\frac{1}{4}}\right)^{12}$$ is:

If the greatest value of the term independent of $$x$$ in the expansion of $$\left(x \sin \alpha + a\frac{\cos \alpha}{x}\right)^{10}$$ is $$\frac{10!}{(5!)^2}$$, then the value of $$a$$ is equal to:

The lowest integer which is greater than $$\left(1 + \frac{1}{10^{100}}\right)^{10^{100}}$$ is

If $$^nP_r = ^nP_{r+1}$$ and $$^nC_r = ^nC_{r-1}$$, then the value of $$r$$ is equal to:

The value of $$\cot \frac{\pi}{24}$$ is:

The number of distinct real roots of $$\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$$ in the interval $$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$$ is:

Let the equation of the pair of lines, $$y = px$$ and $$y = qx$$, can be written as $$(y - px)(y - qx) = 0$$. Then the equation of the pair of the angle bisectors of the lines $$x^2 - 4xy - 5y^2 = 0$$ is:

If a tangent to the ellipse $$x^2 + 4y^2 = 4$$ meets the tangents at the extremities of its major axis at $$B$$ and $$C$$, then the circle with $$BC$$ as diameter passes through the point.

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $$\sqrt{13.44}$$, then the standard deviation of the second sample is:

If $$P = \begin{bmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{bmatrix}$$, then $$P^{50}$$ is:

If $$[x]$$ be the greatest integer less than or equal to $$x$$, then $$\sum_{n=8}^{100} \left[\frac{(-1)^n n}{2}\right]$$ is equal to:

Consider function $$f : A \rightarrow B$$ and $$g : B \rightarrow C$$ $$(A, B, C \subseteq R)$$ such that $$(gof)^{-1}$$ exists, then:

If $$f(x) = \begin{cases} \int_0^x (5 + |1 - t|) \, dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}$$, then

The value of the integral $$\int_{-1}^{1} \log\left(x + \sqrt{x^2 + 1}\right) dx$$ is:

Let $$y = y(x)$$ be the solution of the differential equation $$x \, dy = (y + x^3 \cos x) \, dx$$ with $$y(\pi) = 0$$, then $$y\left(\frac{\pi}{2}\right)$$ is equal to:

Let $$a, b$$ and $$c$$ be distinct positive numbers. If the vectors $$a\hat{i} + a\hat{j} + c\hat{k}$$, $$\hat{i} + \hat{k}$$ and $$c\hat{i} + c\hat{j} + b\hat{k}$$ are co-planar, then $$c$$ is equal to:

If $$|\vec{a}| = 2$$, $$|\vec{b}| = 5$$ and $$|\vec{a} \times \vec{b}| = 8$$, then $$|\vec{a} \cdot \vec{b}|$$ is equal to:

Let $$X$$ be a random variable such that the probability function of a distribution is given by $$P(X = 0) = \frac{1}{2}$$, $$P(X = j) = \frac{1}{3^j}$$ $$(j = 1, 2, 3, \ldots, \infty)$$. Then the mean of the distribution and $$P(X$$ is positive and even) respectively, are:

If $$a + b + c = 1$$, $$ab + bc + ca = 2$$ and $$abc = 3$$, then the value of $$a^4 + b^4 + c^4$$ is equal to:

The equation of a circle is $$\text{Re}(z^2) + 2(\text{Im}(z))^2 + 2\text{Re}(z) = 0$$, where $$z = x + iy$$. A line which passes through the centre of the given circle and the vertex of the parabola, $$x^2 - 6x - y + 13 = 0$$, has $$y$$-intercept equal to _________.

Let $$n \in \mathbf{N}$$ and $$[x]$$ denote the greatest integer less than or equal to $$x$$. If the sum of $$(n + 1)$$ terms of $$^nC_0, 3 \cdot ^nC_1, 5 \cdot ^nC_2, 7 \cdot ^nC_3, \ldots$$ is equal to $$2^{100} \cdot 101$$, then $$2\left[\frac{n-1}{2}\right]$$ is equal to

If the co-efficient of $$x^7$$ and $$x^8$$ in the expansion of $$\left(2 + \frac{x}{3}\right)^n$$ are equal, then the value of $$n$$ is equal to:

Consider the function $$f(x) = \frac{P(x)}{\sin(x - 2)}$$, $$x \neq 2$$, and $$f(x) = 7$$, $$x = 2$$ where $$P(x)$$ is a polynomial such that $$P''(x)$$ is always a constant and $$P(3) = 9$$. If $$f(x)$$ is continuous at $$x = 2$$, then $$P(5)$$ is equal to _________.

If a rectangle is inscribed in an equilateral triangle of side length $$2\sqrt{2}$$ as shown in the figure, then the square of the largest area of such a rectangle is _________.

Let a curve $$y = f(x)$$ pass through the point $$\left(2, (\log_e 2)^2\right)$$ and have slope $$\frac{2y}{x \log_e x}$$ for all positive real values of $$x$$. Then the value of $$f(e)$$ is equal to _________.

If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors and $$\left(\vec{a} + 3\vec{b}\right)$$ is perpendicular to $$\left(7\vec{a} - 5\vec{b}\right)$$ and $$\left(\vec{a} - 4\vec{b}\right)$$ is perpendicular to $$\left(7\vec{a} - 2\vec{b}\right)$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ (in degrees) is _________.

If the lines $$\frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$$ and $$\frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1}$$ are co-planar, then the value of $$k$$ is _________.

A fair coin is tossed $$n-$$ times such that the probability of getting at least one head is at least 0.9. Then the minimum value of $$n$$ is _________.