NTA JEE Main 12th April 2023 Shift 1

Let the digits $$a$$, $$b$$, $$c$$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Two circles in the first quadrant of radii $$r_1$$ and $$r_2$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x + y = 2$$. Then $$r_1^2 + r_2^2 - r_1 r_2$$ is equal to _____.

Let the positive numbers $$a_1, a_2, a_3, a_4$$ and $$a_5$$ be in a G.P. Let their mean and variance be $$\frac{31}{10}$$ and $$\frac{m}{n}$$ respectively, where $$m$$ and $$n$$ are co-prime. If the mean of their reciprocals is $$\frac{31}{40}$$ and $$a_3 + a_4 + a_5 = 14$$, then $$m + n$$ is equal to _____.

The number of relations, on the set $$\{1, 2, 3\}$$ containing $$(1, 2)$$ and $$(2, 3)$$ which are reflexive and transitive but not symmetric, is _____.

Let $$D_k = \begin{vmatrix} 1 & 2k & 2k-1 \\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2 \end{vmatrix}$$. If $$\sum_{k=1}^{n} D_k = 96$$, then $$n$$ is equal to _____.

Let $$[x]$$ be the greatest integer $$\leq x$$. Then the number of points in the interval $$(-2, 1)$$ where the function $$f(x) = |[x]| + \sqrt{x - [x]}$$ is discontinuous, is _____.

Let $$I(x) = \int \sqrt{\frac{x+7}{x}} \ dx$$ and $$I(9) = 12 + 7\log_e 7$$. If $$I(1) = \alpha + 7\log_e\left(1 + 2\sqrt{2}\right)$$, then $$\alpha^4$$ is equal to _____.

If $$\int_{-0.15}^{0.15} |100x^2 - 1| \ dx = \frac{k}{3000}$$, then $$k$$ is equal to _____.

Let the plane $$x + 3y - 2z + 6 = 0$$ meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle $$ABC$$ is $$\left(\alpha, \beta, \frac{6}{7}\right)$$, then $$98(\alpha + \beta)^2$$ is equal to _____.

A fair $$n$$ ($$n > 1$$) faces die is rolled repeatedly until a number less than $$n$$ appears. If the mean of the number of tosses required is $$\frac{n}{9}$$, then $$n$$ is equal to _____.