NTA JEE Main 9th January 2019 Shift 1 - Mathematics

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + 2x + 2 = 0$$, then $$\alpha^{15} + \beta^{15}$$ is equal to:

Let $$A = \{\theta \in (-\frac{\pi}{2}, \pi): \frac{3 + 2i\sin\theta}{1 - 2i\sin\theta}$$ is purely imaginary$$\}$$. Then the sum of the elements in $$A$$ is:

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is:

If $$a$$, $$b$$ and $$c$$ be three distinct real numbers in G.P. and $$a + b + c = xb$$, then $$x$$ cannot be:

Let $$a_1, a_2, \ldots, a_{30}$$ be an A.P., $$S = \sum_{i=1}^{30} a_i$$ and $$T = \sum_{i=1}^{15} a_{(2i-1)}$$. If $$a_5 = 27$$ and $$S - 2T = 75$$, then $$a_{10}$$ is equal to:

If the fractional part of the number $$\frac{2^{403}}{15}$$ is $$\frac{k}{15}$$, then $$k$$ is equal to:

For any $$\theta \in \frac{\pi}{4}, \frac{\pi}{2}$$, the expression $$3\sin\theta - \cos\theta^4 + 6\sin\theta + \cos\theta^2 + 4\sin^6\theta$$ equals:

Consider the set of all lines $$px + qy + r = 0$$ such that $$3p + 2q + 4r = 0$$. Which one of the following statements is true?

Three circles of radii $$a$$, $$b$$, $$c$$ ($$a < b < c$$) touch each other externally. If they have $$x$$-axis as a common tangent, then:

Axis of a parabola lies along $$x$$-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive $$x$$-axis then which of following points does not lie on it?

Equation of a common tangent to the circle, $$x^2 + y^2 - 6x = 0$$ and the parabola, $$y^2 = 4x$$ is:

Let $$0 < \theta < \frac{\pi}{2}$$. If the eccentricity of the hyperbola $$\frac{x^2}{\cos^2\theta} - \frac{y^2}{\sin^2\theta} = 1$$ is greater than 2, then the length of its latus rectum lies in the interval:

The value of $$\lim_{y \to 0} \frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt{2}}{y^4}$$ is:

If the Boolean expression $$p \oplus q \wedge \sim p \odot q$$ is equivalent to $$p \wedge q$$, where $$\oplus$$, $$\odot \in \{\wedge, \vee\}$$, then the ordered pair $$(\oplus, \odot)$$ is:

5 students of a class have an average height 150 cm and variance 18 cm$$^2$$. A new student, whose height is 156 cm, joined them. The variance in cm$$^2$$ of the height of these six students is:

If $$A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$, then the matrix $$A^{-50}$$ when $$\theta = \frac{\pi}{12}$$, is equal to:

The system of linear equations
$$x + y + z = 2$$
$$2x + 3y + 2z = 5$$
$$2x + 3y + (a^2 - 1)z = a + 1$$

If $$\cos^{-1}\left(\frac{2}{3x}\right) + \cos^{-1}\left(\frac{3}{4x}\right) = \frac{\pi}{2}$$, ($$x > \frac{3}{4}$$), then $$x$$ is equal to:

For $$x \in R - \{0, 1\}$$, let $$f_1(x) = \frac{1}{x}$$, $$f_2(x) = 1 - x$$ and $$f_3(x) = \frac{1}{1-x}$$ be three given functions. If a function, $$J(x)$$ satisfies $$(f_2 \circ J \circ f_1)(x) = f_3(x)$$ then $$J(x)$$ is equal to:

Let $$f: R \to R$$ be a function defined as
$$f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1 < x < 3 \\ b + 5x, & \text{if } 3 \leq x < 5 \\ 30, & \text{if } x \geq 5 \end{cases}$$

Then $$f$$ is:

The maximum volume in cu.m of the right circular cone having slant height 3 m is:

If $$\theta$$ denotes the acute angle between the curves, $$y = 10 - x^2$$ and $$y = 2 + x^2$$ at a point of their intersection, then $$\tan\theta$$ is equal to:

For $$x^2 \neq n\pi + 1$$, $$n \in N$$ (the set of natural numbers), the integral $$\int x\sqrt{\frac{2\sin(x^2-1) - \sin 2(x^2-1)}{2\sin(x^2-1) + \sin 2(x^2-1)}} \; dx$$ is equal to (where c is a constant of integration):

The value of $$\int_0^{\pi} \cos x^3 \; dx$$ is:

The area (in sq. units) bounded by the parabola $$y = x^2 - 1$$, the tangent at the point (2, 3) to it and the $$y$$-axis is:

If $$y = y(x)$$ is the solution of the differential equation, $$x\frac{dy}{dx} + 2y = x^2$$ satisfying $$y(1) = 1$$, then $$y\left(\frac{1}{2}\right)$$ is equal to:

Let $$\vec{a} = \hat{i} - \hat{j}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$\vec{a} \times \vec{c} + \vec{b} = \vec{0}$$ and $$\vec{a} \cdot \vec{c} = 4$$, then $$|\vec{c}|^2$$ is equal to:

The plane through the intersection of the planes $$x + y + z = 1$$ and $$2x + 3y - z + 4 = 0$$ and parallel to $$y$$-axis also passes through the point:

The equation of the line passing through $$(-4, 3, 1)$$, parallel to the plane $$x + 2y - z - 5 = 0$$ and intersecting the line $$\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z-2}{-1}$$ is:

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then $$P(X = 1) + P(X = 2)$$ equals: