NTA JEE Main 11th January 2019 Shift 1 - Mathematics

If one real root of the quadratic equation $$81x^2 + kx + 256 = 0$$ is cube of the other root, then a value of k is:

Let $$\left(-2 - \frac{1}{3}i\right)^3 = \frac{x+iy}{27}$$ $$(i = \sqrt{-1})$$, where $$x$$ and $$y$$ are real numbers then $$y - x$$ equals

Let $$a_1, a_2, \ldots, a_{10}$$ be a G.P. If $$\frac{a_3}{a_1} = 25$$, then $$\frac{a_9}{a_5}$$ equals:

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $$\frac{27}{19}$$. Then the common ratio of this series is:

The sum of the real values of $$x$$ for which the middle term in the binomial expansion of $$\left(\frac{x^3}{3} + \frac{3}{x}\right)^8$$ equals 5670 is:

The value of $$r$$ for which $${}^{20}C_r \cdot {}^{20}C_0 + {}^{20}C_{r-1} \cdot {}^{20}C_1 + {}^{20}C_{r-2} \cdot {}^{20}C_2 + \ldots + {}^{20}C_0 \cdot {}^{20}C_r$$ is maximum, is:

Let $$f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$$ for $$k = 1, 2, 3, \ldots$$. Then for all $$x \in R$$, the value of $$f_4(x) - f_6(x)$$ is equal to:

In a triangle, the sum of lengths of two sides is $$x$$ and the product of the lengths of the same two sides is $$y$$. If $$x^2 - c^2 = y$$, where $$c$$ is the length of the third side of the triangle, then the circumradius of the triangle is

A square is inscribed in the circle $$x^2 + y^2 - 6x + 8y - 103 = 0$$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

Two circles with equal radii are intersecting at the points (0,1) and (0,-1). The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

The straight line $$x + 2y = 1$$ meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:

If tangents are drawn to the ellipse $$x^2 + 2y^2 = 2$$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve:

Equation of a common tangent to the parabola $$y^2 = 4x$$ and the hyperbola $$xy = 2$$ is:

Let $$[x]$$ denote the greatest integer less than or equal to x. Then: $$\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}$$

If q is false and $$p \wedge q \leftrightarrow r$$ is true, then which one of the following statements is a tautology?

The outcome of each of 30 items was observed; 10 items gave an outcome $$\frac{1}{2} - d$$ each, 10 items gave outcome $$\frac{1}{2}$$ each and the remaining 10 items gave outcome $$\frac{1}{2} + d$$ each. If the variance of this outcome data is $$\frac{4}{3}$$ then $$|d|$$ equals:

Let $$A = \begin{pmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{pmatrix}$$. If $$AA^T = I_3$$, then $$|p|$$ is:

If the system of linear equations $$2x + 2y + 3z = a$$, $$3x - y + 5z = b$$, $$x - 3y + 2z = c$$ where $$a, b, c$$ are non-zero real numbers, has more than one solution, then

Let $$f : R \to R$$ be defined by $$f(x) = \frac{x}{1+x^2}$$, $$x \in R$$. Then the range of $$f$$ is

Let $$f(x) = \begin{cases} -1, & -2 \le x \lt 0 \\ x^2 - 1, & 0 \le x \le 2 \end{cases}$$ and $$g(x) = |\eta(x)| + f(|x|)$$. Then, in the interval $$(-2, 2)$$, $$g$$ is:

If $$x \log_e(\log_e x) - x^2 + y^2 = 4$$ $$(y > 0)$$, then $$\frac{dy}{dx}$$ at $$x = e$$ is equal to:

The maximum value of the function $$f(x) = 3x^3 - 18x^2 + 27x - 40$$ on the set $$S = \{x \in R : x^2 + 30 \le 11x\}$$ is:

If $$\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)\left(\sqrt{1-x^2}\right)^m + C$$, for a suitable chosen integer m and a function $$A(x)$$, where C is a constant of integration, then $$(A(x))^m$$ equals:

The value of the integral $$\int_{-2}^{2} \frac{\sin^2 x}{[\frac{x}{\pi}] + \frac{1}{2}} dx$$ (where $$[x]$$ denotes the greatest integer less than or equal to x) is

The area (in sq. units) of the region bounded by the curve $$x^2 = 4y$$ and the straight line $$x = 4y - 2$$ is:

If $$y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \left(\frac{2x+1}{x}\right)y = e^{-2x}$$, $$x \gt 0$$, where $$y(1) = \frac{1}{2}e^{-2}$$, then:

Let $$\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$$, $$\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}$$ and $$\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}$$ be coplanar vectors. Then the non-zero vector $$\vec{a} \times \vec{c}$$ is:

The plane containing the line $$\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z-1}{3}$$ and also containing its projection on the plane $$2x + 3y - z = 5$$, contains which one of the following points?

The direction ratios of normal to the plane through the points (0,-1,0) and (0,0,1) and making an angle $$\frac{\pi}{4}$$ with the plane $$y - z + 5 = 0$$ are: 2,-1,1; $$2, \sqrt{2} - \sqrt{2}$$; $$\sqrt{2}, 1, -1$$; $$2\sqrt{3}, 1, -1$$

Two integers are selected at random from the set $$\{1, 2, \ldots, 11\}$$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is: