NTA JEE Main 10th January 2019 Shift 1

If the parabolas $$y^2 = 4b(x-c)$$ and $$y^2 = 8ax$$ have a common normal, then which one of the following is a valid choice for the ordered triad $$(a, b, c)$$:

The equation of a tangent to the hyperbola, $$4x^2 - 5y^2 = 20$$, parallel to the line $$x - y = 2$$, is:

For each $$t \in R$$, let $$[t]$$ be the greatest integer less than or equal to $$t$$. Then, $$\lim_{x \to 1^+} \frac{(1-|x|+\sin|1-x|)\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}$$

Consider the statement: "$$P(n): n^2 - n + 41$$ is prime". Then which one of the following is true?

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is:

Consider a triangular plot $$ABC$$ with sides $$AB = 7$$ m, $$BC = 5$$ m and $$CA = 6$$ m. A vertical lamp-post at the mid-point $$D$$ of $$AC$$ subtends an angle 30$$^{\circ}$$ at $$B$$. The height (in m) of the lamp-post is:

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

If the system of equations $$x + y + z = 5$$, $$x + 2y + 3z = 9$$, $$x + 3y + \alpha z = \beta$$ has infinitely many solutions, then $$\beta - \alpha$$ equals:

Let $$d \in R$$, and $$A = \begin{bmatrix} -2 & 4+d & (\sin\theta)-2 \\ 1 & (\sin\theta)+2 & d \\ 5 & (2\sin\theta)-d & (-\sin\theta)+2+2d \end{bmatrix}$$, $$\theta \in [0, 2\pi]$$. If the minimum value of $$\det(A)$$ is 8, then a value of $$d$$ is:

Let $$f(x) = \begin{cases} \max(|x|, x^2), & |x| \leq 2 \\ 8-2|x|, & 2 < |x| \leq 4 \end{cases}$$. Let $$S$$ be the set of points in the interval $$(-4, 4)$$ at which $$f$$ is not differentiable. Then $$S$$: