The integer $$k$$, for which the inequality $$x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$$ is valid for every $$x$$ in $$R$$ is:
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
Sign in
Please select an account to continue using cracku.in
↓ →
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
The integer $$k$$, for which the inequality $$x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$$ is valid for every $$x$$ in $$R$$ is:
Login to view the detailed solution.
Let the lines $$(2 - i)z = (2 + i)\bar{z}$$ and $$(2 + i)z + (i - 2)\bar{z} - 4i = 0$$, (here $$i^2 = -1$$) be normal to a circle $$C$$. If the line $$iz + \bar{z} + 1 + i = 0$$ is tangent to this circle $$C$$, then its radius is:
Loading solution...
The total number of positive integral solutions $$(x, y, z)$$ such that $$xyz = 24$$ is:
Loading solution...
.webp)
If $$0 < \theta, \phi < \frac{\pi}{2}$$, $$x = \sum_{n=0}^{\infty} \cos^{2n}\theta$$, $$y = \sum_{n=0}^{\infty} \sin^{2n}\phi$$ and $$z = \sum_{n=0}^{\infty} \cos^{2n}\theta \cdot \sin^{2n}\phi$$ then:
Loading solution...
All possible values of $$\theta \in [0, 2\pi]$$ for which $$\sin 2\theta + \tan 2\theta > 0$$ lie in:
Loading solution...
The image of the point (3, 5) in the line $$x - y + 1 = 0$$, lies on:
Loading solution...
.webp)
A tangent is drawn to the parabola $$y^2 = 6x$$ which is perpendicular to the line $$2x + y = 1$$. Which of the following points does NOT lie on it?
Loading solution...
If the curves, $$\frac{x^2}{a} + \frac{y^2}{b} = 1$$ and $$\frac{x^2}{c} + \frac{y^2}{d} = 1$$ intersect each other at an angle of 90°, then which of the following relations is TRUE?
Loading solution...
$$\lim_{n \to \infty} \left(1 + \frac{1 + \frac{1}{2} + \ldots + \frac{1}{n}}{n^2}\right)^n$$ is equal to
Loading solution...
.webp)
The statement $$A \to (B \to A)$$ is equivalent to:
Loading solution...
A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30° (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45°. Then the time taken (in seconds) by the boat from B to reach the base of the tower is:
Loading solution...
Let $$f, g : N \to N$$ such that $$f(n + 1) = f(n) + f(1)$$ $$\forall n \in N$$ and $$g$$ be any arbitrary function. Which of the following statements is NOT true?
Loading solution...
.webp)
If Rolle's theorem holds for the function $$f(x) = x^3 - ax^2 + bx - 4$$, $$x \in [1, 2]$$ with $$f'\left(\frac{4}{3}\right) = 0$$, then ordered pair $$(a, b)$$ is equal to:
Loading solution...
The value of the integral $$\int \frac{\sin\theta \cdot \sin 2\theta (\sin^6\theta + \sin^4\theta + \sin^2\theta)\sqrt{2\sin^4\theta + 3\sin^2\theta + 6}}{1 - \cos 2\theta} d\theta$$ is (where $$c$$ is a constant of integration)
Loading solution...
The value of $$\int_{-1}^{1} x^2 e^{[x^3]} dx$$, where $$[t]$$ denotes the greatest integer $$\leq t$$, is:
Loading solution...
If a curve passes through the origin and the slope of the tangent to it at any point $$(x, y)$$ is $$\frac{x^2 - 4x + y + 8}{x - 2}$$, then this curve also passes through the point:
Loading solution...
Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations $$l + m - n = 0$$ and $$l^2 + m^2 - n^2 = 0$$. Then the value of $$\sin^4\alpha + \cos^4\alpha$$ is:
Loading solution...
The equation of the line through the point (0, 1, 2) and perpendicular to the line $$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{-2}$$ is:
Loading solution...
_1.webp)
The coefficients $$a, b$$ and $$c$$ of the quadratic equation, $$ax^2 + bx + c = 0$$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
Loading solution...
When a missile is fired from a ship, the probability that it is intercepted is $$\frac{1}{3}$$ and the probability that the missile hits the target, given that it is not intercepted, is $$\frac{3}{4}$$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
Loading solution...
The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is ______
Loading solution...
Let $$A_1, A_2, A_3, \ldots$$ be squares such that for each $$n \geq 1$$, the length of the side of $$A_n$$ equals the length of diagonal of $$A_{n+1}$$. If the length of $$A_1$$ is 12 cm, then the smallest value of $$n$$ for which area of $$A_n$$ is less than one, is ______
Loading solution...
The locus of the point of intersection of the lines $$\left(\sqrt{3}\right)kx + ky - 4\sqrt{3} = 0$$ and $$\sqrt{3}x - y - 4\left(\sqrt{3}\right)k = 0$$ is a conic, whose eccentricity is ______
Loading solution...
If $$A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$$ and $$(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$, then $$13(a^2 + b^2)$$ is equal to ______.
Loading solution...
Let $$A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$$, where $$x, y$$ and $$z$$ are real numbers such that $$x + y + z > 0$$ and $$xyz = 2$$. If $$A^2 = I_3$$, then the value of $$x^3 + y^3 + z^3$$ is ______
Loading solution...
If the system of equations
$$kx + y + 2z = 1$$
$$3x - y - 2z = 2$$
$$-2x - 2y - 4z = 3$$
has infinitely many solutions, then $$k$$ is equal to ______.
Loading solution...
The number of points, at which the function $$f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2|$$, $$x \in R$$ is not differentiable, is ______
Loading solution...
Let $$f(x)$$ be a polynomial of degree 6 in $$x$$, in which the coefficient of $$x^6$$ is unity and it has extrema at $$x = -1$$ and $$x = 1$$. If $$\lim_{x \to 0} \frac{f(x)}{x^3} = 1$$, then $$5 \cdot f(2)$$ is equal to ______
Loading solution...
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $$A$$. Then $$A^4$$ is equal to ______
Loading solution...
Let $$\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$$, $$\vec{b} = \hat{i} - \hat{j}$$ and $$\vec{c} = \hat{i} - \hat{j} - \hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$\vec{r} \cdot \vec{a}$$ is equal to ______
Loading solution...
Educational materials for JEE preparation
Share