NTA JEE Main 10th January 2019 Shift 2 - Mathematics

The value of $$\lambda$$ such that sum of the squares of the roots of the quadratic equation, $$x^2 + (3-\lambda)x + 2 = \lambda$$ has the least value is:

Let $$z = \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^5 + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^5$$. If $$R(z)$$ and $$I(z)$$ respectively denote the real and imaginary parts of $$z$$, then:

If $$\sum_{r=0}^{25} \{^{50}C_r \cdot ^{50-r}C_{25-r}\} = K \cdot ^{50}C_{25}$$, then $$K$$ is equal to:

The positive value of $$\lambda$$ for which the co-efficient of $$x^2$$ in the expansion $$x^2\left(\sqrt{x} + \frac{\lambda}{x^2}\right)^{10}$$ is 720, is:

The value of $$\cos\frac{\pi}{2^2} \cdot \cos\frac{\pi}{2^3} \cdots \cos\frac{\pi}{2^{10}} \cdot \sin\frac{\pi}{2^{10}}$$ is:

Two vertices of a triangle are $$(0, 2)$$ and $$(4, 3)$$. If its orthocenter is at the origin, then its third vertex lies in which quadrant?

Two sides of a parallelogram are along the lines, $$x + y = 3$$ and $$x - y + 3 = 0$$. If its diagonals intersect at $$(2, 4)$$, then one of its vertex is:

If the area of an equilateral triangle inscribed in the circle $$x^2 + y^2 + 10x + 12y + c = 0$$ is $$27\sqrt{3}$$ sq. units, then $$c$$ is equal to:

The length of the chord of the parabola $$x^2 = 4y$$ having equation $$x - \sqrt{2}y + 4\sqrt{2} = 0$$ is:

Let $$S = \left\{(x, y) \in R^2 : \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\right\}$$, where $$r \neq \pm 1$$. Then $$S$$ represents:

Consider the following three statements:
P: 5 is a prime number
Q: 7 is a factor of 192
R: LCM of 5 and 7 is 35
Then the truth value of which one of the following statements is true?

If the mean and standard deviation of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ are 10 and 3, respectively, then the variance of 6 observations $$x_1, x_2, \ldots, x_5$$ and $$-50$$ is equal to:

With the usual notation in $$\triangle ABC$$, if $$\angle A + \angle B = 120^{\circ}$$, $$a = \sqrt{3} + 1$$ units and $$b = \sqrt{3} - 1$$ units, then the ratio $$\angle A : \angle B$$ is:

Let $$A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$$, where $$b \gt 0$$. Then the minimum value of $$\frac{\det(A)}{b}$$ is:

The number of values of $$\theta \in (0, \pi)$$ for which the system of linear equations
$$x + 3y + 7z = 0$$
$$-x + 4y + 7z = 0$$
$$(\sin 3\theta)x + (\cos 2\theta)y + 2z = 0$$
has a non-trivial solution, is:

Let $$a_1, a_2, a_3, \ldots, a_{10}$$ be in G.P. with $$a_i > 0$$ for $$i = 1, 2, \ldots, 10$$ and $$S$$ be the set of pairs $$(r, k)$$, $$r, k \in N$$ (the set of natural numbers) for which $$\begin{vmatrix} \log_e a_1^r a_2^k & \log_e a_2^r a_3^k & \log_e a_3^r a_4^k \\ \log_e a_4^r a_5^k & \log_e a_5^r a_6^k & \log_e a_6^r a_7^k \\ \log_e a_7^r a_8^k & \log_e a_8^r a_9^k & \log_e a_9^r a_{10}^k \end{vmatrix} = 0$$. Then the number of elements in $$S$$, is:

The value of $$\cot\left(\sum_{n=1}^{19} \cot^{-1}\left(1 + \sum_{p=1}^{n} 2p\right)\right)$$ is:

Let $$N$$ be the set of natural numbers and two functions $$f$$ and $$g$$ be defined as $$f, g: N \to N$$ such that $$f(n) = \begin{cases} \frac{n+1}{2}, & \text{if n is odd} \\ \frac{n}{2}, & \text{if n is even} \end{cases}$$ and $$g(n) = n - (-1)^n$$. Then $$fog$$ is:

Let $$f: (-1, 1) \to R$$ be a function defined by $$f(x) = \max\left\{-|x|, -\sqrt{1-x^2}\right\}$$. If $$K$$ be the set of all points at which $$f$$ is not differentiable, then $$K$$ has exactly:

A helicopter is flying along the curve given by $$y - x^{3/2} = 7$$, $$(x \geq 0)$$. A soldier positioned at the point $$\left(\frac{1}{2}, 7\right)$$, who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:

The tangent to the curve, $$y = xe^{x^2}$$ passing through the point $$(1, e)$$ also passes through the point:

If $$\int x^5 e^{-4x^3}dx = \frac{1}{48}e^{-4x^3}f(x) + C$$, where $$C$$ is a constant of integration, then $$f(x)$$ is equal to:

The value of $$\int_{-\pi/2}^{\pi/2} \frac{dx}{[x] + [\sin x] + 4}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is:

If $$\int_0^x f(t)dt = x^2 + \int_x^1 t^2 f(t)dt$$, then $$f'\left(\frac{1}{2}\right)$$ is:

A curve amongst the family of curves represented by the differential equation, $$(x^2 - y^2)dx + 2xy \; dy = 0$$ which passes through $$(1, 1)$$, is:

Let $$f(x)$$ be a differentiable function such that $$f'(x) = 7 - \frac{3}{4}\frac{f(x)}{x}$$, $$(x > 0)$$ and $$f(1) \neq 4$$. Then $$\lim_{x \to 0^+} xf\left(\frac{1}{x}\right)$$:

Let $$\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$$ and $$\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$$, be two given vectors where vectors $$\vec{a}$$ and $$\vec{b}$$ are non-collinear. The value of $$\lambda$$ for which vectors $$\vec{\alpha}$$ and $$\vec{\beta}$$ are collinear, is:

The plane which bisects the line segment joining the points $$(-3, -3, 4)$$ and $$(3, 7, 6)$$ at right angles, passes through which one of the following points?

On which of the following lines lies the point of intersection of the line, $$\frac{x-4}{2} = \frac{y-5}{2} = \frac{z-3}{1}$$ and the plane, $$x + y + z = 2$$?

If the probability of hitting a target by a shooter, in any shot is $$\frac{1}{3}$$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $$\frac{5}{6}$$, is: