Let $$S = \{z = x + iy: \frac{2z - 3i}{4z + 2i} \text{ is a real number}\}$$. Then which of the following is NOT correct?
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.
Let $$S = \{z = x + iy: \frac{2z - 3i}{4z + 2i} \text{ is a real number}\}$$. Then which of the following is NOT correct?
Login to view the detailed solution.
Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
Login to view the detailed solution.
If $$S_n = 4 + 11 + 21 + 34 + 50 + \ldots$$ to $$n$$ terms, then $$\frac{1}{60}\left(S_{29}-S_9\right)$$ is equal to
Login to view the detailed solution.
.webp)
Let the number $$(22)^{2022} + (2022)^{22}$$ leave the remainder $$\alpha$$ when divided by 3 and $$\beta$$ when divided by 7. Then $$(\alpha^2 + \beta^2)$$ is equal to
Login to view the detailed solution.
If the coefficients of $$x$$ and $$x^2$$ in $$(1 + x)^p(1 - x)^q$$ are 4 and -5 respectively, then $$2p + 3q$$ is equal to
Login to view the detailed solution.
Let $$S = \{x \in [-\frac{\pi}{2}, \frac{\pi}{2}]: 9^{1-\tan^2 x} + 9^{\tan^2 x} = 10\}$$ and $$\beta = \sum_{x \in S} \tan^2 \frac{x}{3}$$, then $$\frac{1}{6}(\beta - 14)^2$$ is equal to
Login to view the detailed solution.
.webp)
Let $$A$$ be the point (1, 2) and $$B$$ be any point on the curve $$x^2 + y^2 = 16$$. If the centre of the locus of the point $$P$$, which divides the line segment AB in the ratio 3:2 is the point $$C(\alpha, \beta)$$, then the length of the line segment $$AC$$ is
Login to view the detailed solution.
Let a circle of radius 4 be concentric to the ellipse $$15x^2 + 19y^2 = 285$$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
Login to view the detailed solution.
The statement $$\sim p \vee \sim p \wedge q$$ is equivalent to
.webp)
Let $$\mu$$ be the mean and $$\sigma$$ be the standard deviation of the distribution
| $$X_i$$ | 0 | 1 | 2 | 3 | 4 | 5 |
| $$f_i$$ | $$k+2$$ | $$2k$$ | $$k^2-1$$ | $$k^2-1$$ | $$k^2+1$$ | $$k-3$$ |
where $$\Sigma f_i = 62$$. If $$[x]$$ denotes the greatest integer $$\leq x$$, then $$[\mu^2 + \sigma^2]$$ is equal to
Login to view the detailed solution.
Let $$A = \{2, 3, 4\}$$ and $$B = \{8, 9, 12\}$$. Then the number of elements in the relation $$R = \{(a_1, b_1, a_2, b_2) \in A \times B, A \times B: a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\}$$ is
Login to view the detailed solution.
If $$A = \frac{1}{5!6!7!} \begin{vmatrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{vmatrix}$$, then adj $$2A$$ is equal to
Login to view the detailed solution.
.webp)
Let $$g(x) = f(x) + f(1-x)$$ and $$f''(x) > 0$$, $$x \in (0, 1)$$. If $$g$$ is decreasing in the interval $$(0, \alpha)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to
Login to view the detailed solution.
For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int \frac{x^{2x}}{e} + \frac{e^{2x}}{x} \log_e x \, dx = \frac{1}{\alpha e} x^{\beta x} - \frac{1}{\gamma x} e^{\delta x} + C$$, where $$e = \sum_{n=0}^\infty \frac{1}{n!}$$ and C is constant of integration, then $$\alpha + 2\beta + 3\gamma - 4\delta$$ is equal to
Login to view the detailed solution.
Let f be a continuous function satisfying $$\int_0^{t^2} f(x) + x^2 dx = \frac{4}{3}t^3$$, $$\forall t > 0$$. Then $$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to
Login to view the detailed solution.
Let $$\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} + 5\hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d} = 12$$. Then $$(-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$$ is equal to
Login to view the detailed solution.
If the points $$P$$ and $$Q$$ are respectively the circumcenter and the orthocentre of a $$\triangle ABC$$, then $$\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}$$ is equal to
Login to view the detailed solution.
Let the image of the point P(1, 2, 6) in the plane passing through the points A(1, 2, 0) and B(1, 4, 1) C(0, 5, 1) be $$Q(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ equal to
Login to view the detailed solution.
Let the line $$\frac{x}{1} = \frac{6-y}{2} = \frac{z+8}{5}$$ intersect the lines $$\frac{x-5}{4} = \frac{y-7}{3} = \frac{z+2}{1}$$ and $$\frac{x+3}{6} = \frac{3-y}{3} = \frac{z-6}{1}$$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane $$2x - 2y + z = 14$$ is
Login to view the detailed solution.
Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $$\frac{k}{2^{15}}$$, then k is equal to
Login to view the detailed solution.
The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _______.
Login to view the detailed solution.
Suppose $$a_1, a_2, 2, a_3, a_4$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$$, then $$a_4$$ is equal to _______.
Login to view the detailed solution.
Let the equations of two adjacent sides of a parallelogram $$ABCD$$ be $$2x - 3y = -23$$ and $$5x + 4y = 23$$. If the equation of its one diagonal $$AC$$ is $$3x + 7y = 23$$ and the distance of $$A$$ from the other diagonal is $$d$$, then $$50d^2$$ is equal to _______.
Login to view the detailed solution.
Let $$S$$ be the set of values of $$\lambda$$, for which the system of equations $$6\lambda x - 3y + 3z = 4\lambda^2$$, $$2x + 6\lambda y + 4z = 1$$ and $$3x + 2y + 3\lambda z = \lambda$$ has no solution. Then $$12\sum_{\lambda \in S} \lambda$$ is equal to _______.
Login to view the detailed solution.
If the domain of the function $$f(x) = \sec^{-1}\left(\frac{2x}{5x+3}\right)$$ is $$[\alpha, \beta) \cup (\gamma, \delta]$$, then $$3\alpha + 10\beta + \gamma + 21\delta$$ is equal to _______.
Login to view the detailed solution.
In the figure, $$\theta_1 + \theta_2 = \frac{\pi}{2}$$ and $$\sqrt{3}BE = 4AB$$. If the area of $$\triangle CAB$$ is $$2\sqrt{3} - 3$$ unit$$^2$$, when $$\frac{\theta_2}{\theta_1}$$ is the largest, then the perimeter (in unit) of $$\triangle CED$$ is equal to _______.
Login to view the detailed solution.
Let the quadratic curve passing through the point (-1, 0) and touching the line $$y = x$$ at (1, 1) be $$y = f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha + 1)$$ in the first quadrant is _______.
Login to view the detailed solution.
If the area of the region $$\{(x, y): |x^2 - 2| \leq y \leq x\}$$ is A, then $$6A + 16\sqrt{2}$$ is equal to _______.
Login to view the detailed solution.
Let the tangent at any point P on a curve passing through the points (1, 1) and ($$\frac{1}{10}$$, 100), intersect positive x-axis and y-axis at the points A and B respectively. If PA : PB = 1 : k and $$y = y(x)$$ is the solution of the differential equation $$e^{\frac{dy}{dx}} = kx + \frac{k}{2}$$, $$y(0) = k$$, then $$4y(1) - 5\log_e 3$$ is equal to _______.
Login to view the detailed solution.
Let the foot of perpendicular from the point A(4, 3, 1) on the plane P: $$x - y + 2z + 3 = 0$$ be N. If $$B(5, \alpha, \beta)$$, $$\alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3\sqrt{2}$$, then $$\alpha^2 + \beta^2 + \alpha\beta$$ is equal to _______.
Login to view the detailed solution.
Educational materials for JEE preparation
Share