NTA JEE Mains 27th Jan 2024 Shift 2

Let the complex numbers $$\alpha$$ and $$\frac{1}{\bar{\alpha}}$$ lie on the circles $$|z - z_0|^2 = 4$$ and $$|z - z_0|^2 = 16$$ respectively, where $$z_0 = 1 + i$$. Then, the value of $$100|\alpha|^2$$ is _____.

The coefficient of $$x^{2012}$$ in the expansion of $$(1 - x)^{2008}(1 + x + x^2)^{2007}$$ is equal to _____.

If the sum of squares of all real values of $$\alpha$$, for which the lines $$2x - y + 3 = 0$$, $$6x + 3y + 1 = 0$$ and $$\alpha x + 2y - 2 = 0$$ do not form a triangle is p, then the greatest integer less than or equal to p is _____.

Consider a circle $$(x - \alpha)^2 + (y - \beta)^2 = 50$$, where $$\alpha, \beta > 0$$. If the circle touches the line $$y + x = 0$$ at the point P, whose distance from the origin is $$4\sqrt{2}$$, then $$(\alpha + \beta)^2$$ is equal to _____.

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12. If $$\mu$$ and $$\sigma^2$$ denote the mean and variance of the correct observations respectively, then $$15(\mu + \mu^2 + \sigma^2)$$ is equal to _____.

Let A be a $$2 \times 2$$ real matrix and I be the identity matrix of order 2. If the roots of the equation $$|A - xI| = 0$$ be -1 and 3, then the sum of the diagonal elements of the matrix $$A^2$$ is _____.

Let $$f(x) = \int_0^x g(t)\log_e\frac{1-t}{1+t}dt$$, where g is a continuous odd function. If $$\int_{-\pi/2}^{\pi/2} \left(f(x) + \frac{x^2\cos x}{1 + e^x}\right) dx = \left(\frac{\pi}{\alpha}\right)^2 - \alpha$$, then $$\alpha$$ is equal to _____.

If the area of the region $$\{(x, y) : 0 \leq y \leq \min(2x, 6x - x^2)\}$$ is A, then 12A is equal to _____.

If the solution curve, of the differential equation $$\frac{dy}{dx} = \frac{x + y - 2}{x - y}$$ passing through the point $$(2, 1)$$ is $$\tan^{-1}\frac{y-1}{x-1} - \frac{1}{\beta}\log_e\left(\alpha + \left(\frac{y-1}{x-1}\right)^2\right) = \log_e(x-1)$$, then $$5\beta + \alpha$$ is equal to

The lines $$\frac{x-2}{2} = \frac{y}{-2} = \frac{z-7}{16}$$ and $$\frac{x+3}{4} = \frac{y+2}{3} = \frac{z+2}{1}$$ intersect at the point P. If the distance of P from the line $$\frac{x+1}{2} = \frac{y-1}{3} = \frac{z-1}{1}$$ is $$l$$, then $$14l^2$$ is equal to _____.