JEE Main 2021 (18 March Shift 1)

Let $$z_1, z_2$$ be the roots of the equation $$z^2 + az + 12 = 0$$ and $$z_1, z_2$$ form an equilateral triangle with origin. Then, the value of $$|a|$$ is ________.

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is ________.

The number of solutions of the equation $$|\cot x| = \cot x + \frac{1}{\sin x}$$ in the interval $$[0, 2\pi]$$ is ________.

A square $$ABCD$$ has all its vertices on the curve $$x^2 y^2 = 1$$. The midpoints of its sides also lie on the same curve. Then, the square of area of $$ABCD$$ is ________.

The missing value in the following figure is ________.

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is ________.

If $$f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} dx$$, $$(x \geq 0)$$, $$f(0) = 0$$ and $$f(1) = \frac{1}{K}$$, then the value of $$K$$ is ________.

Let $$f(x)$$ and $$g(x)$$ be two functions satisfying $$f(x^2) + g(4-x) = 4x^3$$ and $$g(4-x) + g(x) = 0$$, then the value of $$\int_{-4}^{4} f(x^2) dx$$ is ________.

Let the plane $$ax + by + cz + d = 0$$ bisect the line joining the points $$(4, -3, 1)$$ and $$(2, 3, -5)$$ at the right angles. If $$a, b, c, d$$ are integers, then the minimum value of $$(a^2 + b^2 + c^2 + d^2)$$ is ________.

The equation of the planes parallel to the plane $$x - 2y + 2z - 3 = 0$$ which are at unit distance from the point $$(1, 2, 3)$$ is $$ax + by + cz + d = 0$$. If $$(b-d) = K(c-a)$$, then the positive value of $$K$$ is ________.