NTA JEE Main 15th April 2018 Shift 1 - Mathematics

If $$\lambda \in R$$ is such that the sum of the cubes of the roots of the equation, $$x^2 + (2 - \lambda)x + (10 - \lambda) = 0$$ is minimum, then the magnitude of the difference of the roots of this equation is:

The set of all $$\alpha \in R$$, for which $$w = \frac{1+(1-8\alpha)z}{1-z}$$ is a purely imaginary number, for all $$z \in C$$ satisfying $$|z| = 1$$ and Re(z) $$\neq$$ 1, is:

n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is:

If b is the first term of an infinite G.P whose sum is five, then b lies in the interval:

If $$x_1, x_2, \ldots, x_n$$ and $$\frac{1}{h_1}, \frac{1}{h_2}, \ldots, \frac{1}{h_n}$$ are two A.P.s such that $$x_3 = h_2 = 8$$ and $$x_8 = h_7 = 20$$, then $$x_5 \cdot h_{10}$$ equals:

If n is the degree of the polynomial, $$\left[\frac{1}{\sqrt{5x^3+1} - \sqrt{5x^3-1}}\right]^8 + \left[\frac{1}{\sqrt{5x^3+1} + \sqrt{5x^3-1}}\right]^8$$ and m is the coefficient of $$x^n$$ in it, then the ordered pair (n, m) is equal to:

If $$\tan A$$ and $$\tan B$$ are the roots of the quadratic equation, $$3x^2 - 10x - 25 = 0$$ then the value of $$3\sin^2(A+B) - 10\sin(A+B) \cdot \cos(A+B) - 25\cos^2(A+B)$$ is:

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are $$x + y = 5$$ and $$x = 4$$ respectively. Then area of $$\triangle ABC$$ (in sq. units) is:

A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $$y - 4x + 3 = 0$$, then its radius is equal to:

Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is:

If $$\beta$$ is one of the angles between the normals to the ellipse, $$x^2 + 3y^2 = 9$$ at the points $$(3\cos\theta, \sqrt{3}\sin\theta)$$ and $$(-3\sin\theta, \sqrt{3}\cos\theta)$$; $$\theta \in (0, \frac{\pi}{2})$$; then $$\frac{2\cot\beta}{\sin 2\theta}$$ is equal to:

If the tangents drawn to the hyperbola $$4y^2 = x^2 + 1$$ intersect the co-ordinate axes at the distinct points A and B, then the locus of the mid point of AB is:

If $$(p \wedge \sim q) \wedge (p \wedge r) \rightarrow \sim p \vee q$$ is false, then the truth values of p, q and r are respectively:

The mean of a set of 30 observations is 75. If each observation is multiplied by a nonzero number $$\lambda$$ and then each of them is decreased by 25, their mean remains the same. The $$\lambda$$ is equal to $$\{0\}$$:

An aeroplane flying at a constant speed, parallel to the horizontal ground, $$\sqrt{3}$$ km above it, is observed at an elevation of 60$$^\circ$$ from a point on the ground. If, after five seconds, its elevation from the same point is 30$$^\circ$$, then the speed (in km/hr) of the aeroplane is:

Consider the following two binary relations on the set $$A = \{a, b, c\}$$: $$R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$$ and $$R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$$. Then:

Let A be a matrix such that $$A \cdot \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$$ is a scalar matrix and $$|3A| = 108$$. Then $$A^2$$ equals:

If $$f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$$, then $$\lim_{x \to 0} \frac{f'(x)}{x}$$:

Let S be the set of all real values of k for which the system of linear equations
$$x + y + z = 2$$
$$2x + y - z = 3$$
$$3x + 2y + kz = 4$$
has a unique solution. Then S is:

Let $$S = \{(\lambda, \mu) \in R \times R : f(t) = (|\lambda|e^t - \mu) \cdot \sin(2|t|), t \in R$$, is a differentiable function$$\}$$. Then S is a subset of?

If $$x^2 + y^2 + \sin y = 4$$, then the value of $$\frac{d^2y}{dx^2}$$ at the point (-2, 0) is:

If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm$$^2$$) of this cone is:

If $$f\left(\frac{x-4}{x+2}\right) = 2x + 1$$, $$(x \in R - \{1, -2\})$$, then $$\int f(x)dx$$ is equal to (where C is a constant of integration):

The value of the integral $$\int_{-\pi/2}^{\pi/2} \sin^4 x\left(1 + \log\left(\frac{2+\sin x}{2-\sin x}\right)\right)dx$$ is:

The area (in sq. units) of the region $$\{x \in R : x \geq 0, y \geq 0, y \geq x - 2$$ and $$y \leq \sqrt{x}\}$$, is:

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + 2y = f(x)$$, where $$f(x) = \begin{cases} 1, & x \in [0, 1] \\ 0, & \text{otherwise} \end{cases}$$. If $$y(0) = 0$$, then $$y\left(\frac{3}{2}\right)$$ is:

If $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$ are unit vectors such that $$\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$$, then $$|\vec{a} \times \vec{c}|$$ is equal to:

A variable plane passes through a fixed point (3, 2, 1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz-plane through A, a second plane is drawn parallel zx plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is:

An angle between the plane, $$x + y + z = 5$$ and the line of intersection of the planes, $$3x + 4y + z - 1 = 0$$ and $$5x + 8y + 2z + 14 = 0$$, is:

A box 'A' contains 2 white, 3 red and 2 black balls. Another box 'B' contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is: