NTA JEE Main 27th June 2022 Shift 1

The number of distinct real roots of $$x^4 - 4x + 1 = 0$$ is

The lengths of the sides of a triangle are $$10 + x^2, 10 + x^2$$ and $$20 - 2x^2$$. If for $$x = k$$, the area of the triangle is maximum, then $$3k^2$$ is equal to

$$\int \frac{(x^2+1)e^x}{(x+1)^2} dx = f(x)e^x + C$$, where $$C$$ is a constant, then $$\frac{d^3f}{dx^3}$$ at $$x = 1$$ is equal to

The value of the integral $$\int_{-2}^{2} \frac{|x^3+x|}{(e^{x|x|}+1)} dx$$ is equal to

Let $$\frac{dy}{dx} = \frac{ax - by + a}{bx + cy + a}$$, where $$a, b, c$$ are constants, represent a circle passing through the point $$(2, 5)$$. Then the shortest distance of the point $$(11, 6)$$ from this circle is

If $$\frac{dy}{dx} + \frac{2^x y(2y-1)}{2^x - 1} = 0, x, y > 0, y(1) = 1$$, then $$y(2)$$ is equal to

Let $$\vec{a} = \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{c} = 2\hat{i} - 3\hat{j} + 2\hat{k}$$. Then the number of vectors $$\vec{b}$$ such that $$\vec{b} \times \vec{c} = \vec{a}$$ and $$|\vec{b}| \in \{1, 2, \ldots, 10\}$$ is

If two straight lines whose direction cosines are given by the relations $$l + m - n = 0, 3l^2 + m^2 + cnl = 0$$ are parallel, then the positive value of $$c$$ is

Five numbers $$x_1, x_2, x_3, x_4, x_5$$ are randomly selected from the numbers $$1, 2, 3, \ldots, 18$$ and are arranged in the increasing order $$(x_1 < x_2 < x_1 < x_4 < x_2)$$. The probability that $$x_2 = 7$$ and $$x_4 = 11$$ is

Let $$X$$ be a random variable having binomial distribution $$B(7, p)$$. If $$P(X = 3) = 5P(X = 4)$$, then the sum of the mean and the variance of $$X$$ is