NTA JEE Main 9th April 2019 Shift 1 - Mathematics

Let $$p, q \in Q$$. If $$2 - \sqrt{3}$$ is a root of the quadratic equation $$x^2 + px + q = 0$$, then:

All the points in the set $$S = \left\{\frac{\alpha + i}{\alpha - i}, \alpha \in R\right\}$$, $$i = \sqrt{-1}$$ lie on a:

A committee of 11 members is to be formed from 8 males and 5 females. If $$m$$ is the number of ways the committee is formed with at least 6 males and $$n$$ is the number of ways the committee is formed with at least 3 females, then:

Let the sum of the first $$n$$ terms of a non-constant A.P., $$a_1, a_2, a_3, \ldots, a_n$$ be $$50n + \frac{n(n-7)}{2}A$$, where A is a constant. If $$d$$ is the common difference of this A.P., then the ordered pair $$(d, a_{50})$$ is equal to:

If the fourth term in the Binomial expansion of $$\left(\frac{2}{x} + x^{\log_8 x}\right)^6$$, $$(x > 0)$$ is $$20 \times 8^7$$, then a value of $$x$$ is:

The value of $$\cos^2 10° - \cos 10° \cos 50° + \cos^2 50°$$ is:

Let $$S = \{\theta \in [-2\pi, 2\pi] : 2\cos^2\theta + 3\sin\theta = 0\}$$. Then the sum of the elements of S is:

Slope of a line passing through $$P(2, 3)$$ and intersecting the line $$x + y = 7$$ at a distance of 4 units from $$P$$, is:

If a tangent to the circle $$x^2 + y^2 = 1$$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

If one end of a focal chord of the parabola, $$y^2 = 16x$$ is at $$(1, 4)$$, then the length of this focal chord is:

If the line $$y = mx + 7\sqrt{3}$$ is normal to the hyperbola $$\frac{x^2}{24} - \frac{y^2}{18} = 1$$, then a value of $$m$$ is:

For any two statement $$p$$ and $$q$$, the negative of the expression $$p \lor (\sim p \land q)$$ is:

If the standard deviation of the numbers $$-1, 0, 1, k$$ is $$\sqrt{5}$$ where $$k \gt 0$$, then $$k$$ is equal to:

If $$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \cdots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix}$$, then the inverse of $$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$$ is:

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + x + 1 = 0$$. Then for $$y \neq 0$$ in R, $$\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{vmatrix}$$ is equal to:

If the function $$f: R - \{1, -1\} \rightarrow A$$ defined by $$f(x) = \frac{x^2}{1 - x^2}$$, is surjective, then $$A$$ is equal to:

Let $$f(x) = 15 - |x - 10|$$; $$x \in R$$. Then the set of all values of $$x$$, at which the function $$g(x) = f(f(x))$$ is not differentiable, is:

If the function $$f$$ defined on $$\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$$ by $$f(x) = \begin{cases} \frac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$$ is continuous, then $$k$$ is equal to:

Let $$\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)$$, where the function $$f$$ satisfies $$f(x + y) = f(x)f(y)$$ for all natural numbers $$x$$, $$y$$ and $$f(1) = 2$$. Then the natural number 'a' is:

If $$f(x)$$ is a non-zero polynomial of degree four, having local extreme points at $$x = -1, 0, 1$$; then the set $$S = \{x \in R : f(x) = f(0)\}$$ contains exactly:

If the tangent to the curve, $$y = x^3 + ax - b$$ at the point $$(1, -5)$$ is perpendicular to the line, $$-x + y + 4 = 0$$, then which one of the following points lies on the curve?

Let $$S$$ be the set of all values of $$x$$ for which the tangent to the curve $$y = f(x) = x^3 - x^2 - 2x$$ at $$(x, y)$$ is parallel to the line segment joining the points $$(1, f(1))$$ and $$(-1, f(-1))$$, then $$S$$ is equal to:

$$\int \sec^2 x \cdot \cot^{4/3} x \, dx$$ is equal to:

The value of $$\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x} dx$$ is:

The area (in sq. units) of the region $$A = \{(x, y) : x^2 \le y \le x + 2\}$$ is:

The solution of the differential equation $$x\frac{dy}{dx} + 2y = x^2$$, $$(x \neq 0)$$ with $$y(1) = 1$$, is:

Let $$\vec{\alpha} = 3\hat{i} + \hat{j}$$ and $$\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$$. If $$\vec{\beta} = \vec{\beta_1} - \vec{\beta_2}$$, where $$\vec{\beta_1}$$ is parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ is perpendicular to $$\vec{\alpha}$$, then $$\vec{\beta_1} \times \vec{\beta_2}$$ is equal to:

A plane passing through the points $$(0, -1, 0)$$ and $$(0, 0, 1)$$ and making an angle $$\frac{\pi}{4}$$ with the plane $$y - z + 5 = 0$$, also passes through the point:

If the line, $$\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}$$ meets the plane, $$x + 2y + 3z = 15$$ at a point P, then the distance of P from the origin is:

Four persons can hit a target correctly with probabilities $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$ and $$\frac{1}{8}$$ respectively. If all hit at the target independently, then the probability that the target would be hit, is: