If $$\lambda$$ be the ratio of the roots of the quadratic equation in x, $$3m^2x^2 + m(m-4)x + 2 = 0$$, then the least value of m for which $$\lambda + \frac{1}{\lambda} = 1$$, is:
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 $$\lambda$$ be the ratio of the roots of the quadratic equation in x, $$3m^2x^2 + m(m-4)x + 2 = 0$$, then the least value of m for which $$\lambda + \frac{1}{\lambda} = 1$$, is:
Login to view the detailed solution.
If $$\frac{z - \alpha}{z + \alpha}$$ $$(\alpha \in R)$$ is a purely imaginary number and $$|z| = 2$$, then a value of $$\alpha$$ is:
Login to view the detailed solution.
Let $$S = \{1, 2, 3, \ldots, 100\}$$, then number of non-empty subsets A of S such that the product of elements in A is even is:
Login to view the detailed solution.
.webp)
Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by $$n_i$$, the label of the ball drawn from the $$i^{th}$$ box, $$(i = 1, 2, 3)$$. Then, the number of ways in which the balls can be chosen such that $$n_1 < n_2 < n_3$$ is:
Login to view the detailed solution.
Let $$S_k = \frac{1+2+3+\ldots+k}{k}$$. If $$S_1^2 + S_2^2 + \ldots + S_{10}^2 = \frac{5}{12}A$$, then A is equal to:
Login to view the detailed solution.
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P., then the sum of the original three terms of the given G.P. is:
Login to view the detailed solution.
.webp)
A ratio of the 5$$^{th}$$ term from the beginning to the 5$$^{th}$$ term from the end in the binomial expansion of $$\left(2^{1/3} + \frac{1}{2(3)^{1/3}}\right)^{10}$$ is
Login to view the detailed solution.
The maximum value of $$3\cos\theta + 5\sin\left(\theta - \frac{\pi}{6}\right)$$ for any real value of $$\theta$$ is:
Login to view the detailed solution.
If the straight line $$2x - 3y + 17 = 0$$ is perpendicular to the line passing through the points $$(7, 17)$$ and $$(15, \beta)$$, then $$\beta$$ equals:
Login to view the detailed solution.
.webp)
Let $$C_1$$ and $$C_2$$ be the centres of the circles $$x^2 + y^2 - 2x - 2y - 2 = 0$$ and $$x^2 + y^2 - 6x - 6y + 14 = 0$$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $$PC_1QC_2$$ is:
Login to view the detailed solution.
If a variable line $$3x + 4y - \lambda = 0$$ is such that the two circles $$x^2 + y^2 - 2x - 2y + 1 = 0$$ and $$x^2 + y^2 - 18x - 2y + 78 = 0$$ are on its opposite sides, then the set of all values of $$\lambda$$ is the interval:
Login to view the detailed solution.
Let $$P(4, -4)$$ and $$Q(9, 6)$$ be two points on the parabola, $$y^2 = 4x$$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta PXQ$$ is maximum. Then this maximum area (in sq. units) is:
Login to view the detailed solution.
.webp)
If the vertices of a hyperbola be at $$(-2, 0)$$ and $$(2, 0)$$ and one of its foci be at $$(-3, 0)$$, then which one of the following points does not lie on this hyperbola?
Login to view the detailed solution.
$$\lim_{x \to \frac{\pi}{4}} \frac{\cot^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}$$ is
Login to view the detailed solution.
The Boolean expression $$((p \wedge q) \vee (p \vee \sim q)) \wedge (\sim p \wedge \sim q)$$ is equivalent to
Login to view the detailed solution.
If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:
Login to view the detailed solution.
Let $$P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$$ and $$Q = [q_{ij}]$$ be two $$3 \times 3$$ matrices such that $$Q - P^5 = I_3$$. Then $$\frac{q_{21} + q_{31}}{q_{32}}$$ is equal to:
Login to view the detailed solution.
An ordered pair $$(\alpha, \beta)$$ for which the system of linear equations $$(1 + \alpha)x + \beta y + z = 2$$, $$\alpha x + (1 + \beta)y + z = 3$$, $$\alpha x + \beta y + 2z = 2$$ has a unique solution, is:
Login to view the detailed solution.
Considering only the principal values of inverse functions, the set $$A = \{x \ge 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$$
Let S be the set of all points in $$(-\pi, \pi)$$ at which the function, $$f(x) = \min\{\sin x, \cos x\}$$ is not differentiable. Then S is a subset of which of the following?
Login to view the detailed solution.
For $$x \gt 1$$, if $$(2x)^{2y} = 4e^{2x-2y}$$, then $$(1 + \log_e 2x)^2 \frac{dy}{dx}$$ is equal to
Login to view the detailed solution.
The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, $$y = 12 - x^2$$ such that the rectangle lies inside the parabola, is:
Login to view the detailed solution.
The integral $$\int \cos(\ln x) dx$$, is equal to
Login to view the detailed solution.
Let f and g be continuous functions on [0, a] such that $$f(x) = f(a-x)$$ and $$g(x) + g(a-x) = 4$$, then $$\int_0^a f(x)g(x)dx$$ is equal to
Login to view the detailed solution.
The area (in sq. units) of the region bounded by the parabola, $$y = x^2 + 2$$ and the lines, $$y = x + 1$$, $$x = 0$$ and $$x = 3$$, is
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation, $$x\frac{dy}{dx} + y = x\log_e x$$, $$(x > 1)$$. If $$2y(2) = \log_e 4 - 1$$, then $$y(e)$$ is equal to
Login to view the detailed solution.
The sum of the distinct real values of $$\mu$$ for which the vectors $$\mu\hat{i} + \hat{j} + \hat{k}$$, $$\hat{i} + \mu\hat{j} + \hat{k}$$, $$\hat{i} + \hat{j} + \mu\hat{k}$$ are co-planar, is
A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(-1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is
Login to view the detailed solution.
The perpendicular distance from the origin to the plane containing the two lines, $$\frac{x+2}{3} = \frac{y-2}{5} = \frac{z+5}{7}$$ and $$\frac{x-1}{1} = \frac{y-4}{4} = \frac{z+4}{7}$$, is
Login to view the detailed solution.
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to:
Login to view the detailed solution.
Educational materials for JEE preparation
Share