NTA JEE Main 11th January 2019 Shift 1

The straight line $$x + 2y = 1$$ meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:

If tangents are drawn to the ellipse $$x^2 + 2y^2 = 2$$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve:

Equation of a common tangent to the parabola $$y^2 = 4x$$ and the hyperbola $$xy = 2$$ is:

Let $$[x]$$ denote the greatest integer less than or equal to x. Then: $$\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}$$

If q is false and $$p \wedge q \leftrightarrow r$$ is true, then which one of the following statements is a tautology?

The outcome of each of 30 items was observed; 10 items gave an outcome $$\frac{1}{2} - d$$ each, 10 items gave outcome $$\frac{1}{2}$$ each and the remaining 10 items gave outcome $$\frac{1}{2} + d$$ each. If the variance of this outcome data is $$\frac{4}{3}$$ then $$|d|$$ equals:

Let $$A = \begin{pmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{pmatrix}$$. If $$AA^T = I_3$$, then $$|p|$$ is:

If the system of linear equations $$2x + 2y + 3z = a$$, $$3x - y + 5z = b$$, $$x - 3y + 2z = c$$ where $$a, b, c$$ are non-zero real numbers, has more than one solution, then

Let $$f : R \to R$$ be defined by $$f(x) = \frac{x}{1+x^2}$$, $$x \in R$$. Then the range of $$f$$ is

Let $$f(x) = \begin{cases} -1, & -2 \le x \lt 0 \\ x^2 - 1, & 0 \le x \le 2 \end{cases}$$ and $$g(x) = |\eta(x)| + f(|x|)$$. Then, in the interval $$(-2, 2)$$, $$g$$ is: