If $$\alpha$$ and $$\beta$$ are the roots of the equation $$375x^2 - 25x - 2 = 0$$, then $$\lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r$$ is equal to:
JEE WhatsApp Group
Sign in
Please select an account to continue using cracku.in
↓ →
JEE WhatsApp Group
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
If $$\alpha$$ and $$\beta$$ are the roots of the equation $$375x^2 - 25x - 2 = 0$$, then $$\lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r$$ is equal to:
Login to view the detailed solution.
The equation $$|z - i| = |z - 1|$$, $$i = \sqrt{-1}$$, represents:
Login to view the detailed solution.
The Number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is:
Login to view the detailed solution.
.webp)
If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is:
Let S$$_n$$ denote the sum of the first n terms of an A.P. If S$$_4$$ = 16 and S$$_6$$ = -48, then S$$_{10}$$ is equal to:
Login to view the detailed solution.
The coefficient of $$x^{18}$$ in the product $$(1 + x)(1 - x)^{10}(1 + x + x^2)^9$$ is
Login to view the detailed solution.
.webp)
The equation $$y=\sin x\sin\left(x+2\right)-\sin^2(x+1)$$ represents a straight line lying in:
Login to view the detailed solution.
The number of solutions of the equation $$1 + \sin^4 x = \cos^2 3x$$, $$x \in \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]$$ is:
Login to view the detailed solution.
If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is:
Login to view the detailed solution.
.webp)
If the normal to the ellipse $$3x^2 + 4y^2 = 12$$ at a point P on it is parallel to the line, $$2x + y = 4$$ and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to:
Login to view the detailed solution.
Let P be the point of intersection of the common tangents to the parabola $$y^2 = 12x$$ and the hyperbola $$8x^2 - y^2 = 8$$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:
Login to view the detailed solution.
If the truth value of the statement $$p \to (\sim q \vee r)$$ is false F, then the truth values of the statements p, q, r are respectively
Login to view the detailed solution.
.webp)
If the data $$x_1, x_2, \ldots, x_{10}$$ is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2000, then the standard deviation of this data is:
Login to view the detailed solution.
If $$B = \begin{pmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{pmatrix}$$ is the inverse of a 3$$\times$$3 matrix A, then the sum of all values of $$\alpha$$ for which det(A) + 1 = 0, is:
Login to view the detailed solution.
If A is a symmetric matrix and B is skew-symmetric matrix such that $$A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix}$$, then AB is equal to:
Login to view the detailed solution.
The value of $$\sin^{-1}\frac{12}{13} - \sin^{-1}\frac{3}{5}$$ is equal to:
Login to view the detailed solution.
For $$x \in R$$, Let [x] denotes the greatest integer $$\leq x$$, then the sum of the series $$\left[-\frac{1}{3}\right] + \left[-\frac{1}{3} - \frac{1}{100}\right] + \left[-\frac{1}{3} - \frac{2}{100}\right] + \ldots + \left[-\frac{1}{3} - \frac{99}{100}\right]$$ is
Login to view the detailed solution.
For $$x \in \left(0, \frac{3}{2}\right)$$, let $$f(x) = \sqrt{x}$$, $$g(x) = \tan x$$ and $$h(x) = \frac{1 - x^2}{1 + x^2}$$. If $$\phi(x) = ((h \circ f) \circ g)(x)$$, then $$\phi\left(\frac{\pi}{3}\right)$$ is equal to:
Login to view the detailed solution.
If $$e^y + xy = e$$, the ordered pair $$\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)$$ at x = 0 is equal to
Login to view the detailed solution.
A 2m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is:
Login to view the detailed solution.
If m is the minimum value of k for which the function $$f(x) = x\sqrt{kx - x^2}$$ is increasing in the interval [0, 3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to:
Login to view the detailed solution.
The integral $$\int \frac{2x^3 - 1}{x^4 + x} dx$$, is equal to
Login to view the detailed solution.
Let $$f: R \to R$$ be a continuous and differentiable function such that $$f(2) = 6$$ and $$f'(2) = \frac{1}{48}$$. If $$\int_6^{f(x)}4t^3dt=\left(x-2\right)g(x)$$, then $$\lim_{x \to 2} g(x)$$ is equal to
Login to view the detailed solution.
If $$\int_0^{\pi/2} \frac{\cot x}{\cot x + \text{cosec} x} dx = m(\pi + n)$$, then mn is equal to
Login to view the detailed solution.
If the area (in sq. units) of the region $$\{(x, y): y^2 \leq 4x, x + y \leq 1, x \geq 0, y \geq 0\}$$ is $$a\sqrt{2} + b$$, then a - b is
Login to view the detailed solution.
Consider the differential equation, $$y^2 dx + \left(x - \frac{1}{y}\right) dy = 0$$. If value of y is 1 when x = 1, then the value of x for which y = 2, is
Login to view the detailed solution.
If the volume of parallelepiped formed by the vectors $$\hat{i} + \lambda\hat{j} + \hat{k}$$, $$\hat{j} + \lambda\hat{k}$$ and $$\lambda\hat{i} + \hat{k}$$ is minimum, then $$\lambda$$ is
Login to view the detailed solution.
Let $$\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$$ be two vectors. If a vector perpendicular to both the vectors $$\vec{a} + \vec{b}$$ and $$\vec{a} - \vec{b}$$ has the magnitude 12 then one such vector is:
Login to view the detailed solution.
If the line $$\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-1}{-1}$$ intersects the plane $$2x + 3y - z + 13 = 0$$ at a point P and the plane $$3x + y + 4z = 16$$ at a point Q, then PQ is equal to
Login to view the detailed solution.
Let a random variable X has a binomial distribution with mean 8 and variance 4. If $$P(X \leq 2) = \frac{k}{2^{16}}$$, then the value of k is equal to
Login to view the detailed solution.
Educational materials for JEE preparation
Share