The sum of all possible values of $$\theta \in [-\pi, 2\pi]$$, for which $$\frac{1 + i\cos\theta}{1 - 2i\cos\theta}$$ is purely imaginary, 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.
The sum of all possible values of $$\theta \in [-\pi, 2\pi]$$, for which $$\frac{1 + i\cos\theta}{1 - 2i\cos\theta}$$ is purely imaginary, is equal to :
Login to view the detailed solution.
The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :
Login to view the detailed solution.
In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $$\frac{70}{3}$$ and the product of the third and fifth terms is 49. Then the sum of the $$4^{th}$$, $$6^{th}$$ and $$8^{th}$$ terms is equal to :
Login to view the detailed solution.
.webp)
If the term independent of $$x$$ in the expansion of $$\left(\sqrt{a}x^2 + \frac{1}{2x^3}\right)^{10}$$ is 105, then $$a^2$$ is equal to :
Login to view the detailed solution.
If the value of $$\frac{3\cos 36° + 5\sin 18°}{5\cos 36° - 3\sin 18°}$$ is $$\frac{a\sqrt{5} - b}{c}$$, where $$a, b, c$$ are natural numbers and $$\gcd(a, c) = 1$$, then $$a + b + c$$ is equal to :
Login to view the detailed solution.
If the image of the point $$(-4, 5)$$ in the line $$x + 2y = 2$$ lies on the circle $$(x + 4)^2 + (y - 3)^2 = r^2$$, then $$r$$ is equal to :
Login to view the detailed solution.
.webp)
If the line segment joining the points $$(5, 2)$$ and $$(2, a)$$ subtends an angle $$\frac{\pi}{4}$$ at the origin, then the absolute value of the product of all possible values of $$a$$ is :
Login to view the detailed solution.
Let $$A = \{2, 3, 6, 8, 9, 11\}$$ and $$B = \{1, 4, 5, 10, 15\}$$. Let $$R$$ be a relation on $$A \times B$$ defined by $$(a, b)R(c, d)$$ if and only if $$3ad - 7bc$$ is an even integer. Then the relation $$R$$ is :
Login to view the detailed solution.
If $$\alpha \neq a, \beta \neq b, \gamma \neq c$$ and $$\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0$$, then $$\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c}$$ is equal to :
Login to view the detailed solution.
.webp)
If the system of equations $$x + 4y - z = \lambda$$, $$7x + 9y + \mu z = -3$$, $$5x + y + 2z = -1$$ has infinitely many solutions, then $$(2\mu + 3\lambda)$$ is equal to :
Login to view the detailed solution.
Let $$f(x) = \begin{cases} -a & \text{if } -a \leq x \leq 0 \\ x + a & \text{if } 0 < x \leq a \end{cases}$$ where $$a > 0$$ and $$g(x) = (f(x) - |f(x)|)/2$$. Then the function $$g : [-a, a] \rightarrow [-a, a]$$ is :
Login to view the detailed solution.
For $$a, b > 0$$, let $$f(x) = \begin{cases} \frac{\tan((a+1)x) + b\tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{ax}\sqrt{x}}, & x > 0 \end{cases}$$ be a continuous function at $$x = 0$$. Then $$\frac{b}{a}$$ is equal to :
Login to view the detailed solution.
.webp)
If the function $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$, $$a > 0$$ has a local maximum at $$x = \alpha$$ and a local minimum at $$x = \alpha^2$$, then $$\alpha$$ and $$\alpha^2$$ are the roots of the equation :
Login to view the detailed solution.
Let $$\int_{\alpha}^{\log_e 4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}$$. Then $$e^{\alpha}$$ and $$e^{-\alpha}$$ are the roots of the equation :
Login to view the detailed solution.
The area of the region in the first quadrant inside the circle $$x^2 + y^2 = 8$$ and outside the parabola $$y^2 = 2x$$ is equal to :
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution curve of the differential equation $$\sec y \frac{dy}{dx} + 2x\sin y = x^3\cos y$$, $$y(1) = 0$$. Then $$y(\sqrt{3})$$ is equal to :
Login to view the detailed solution.
Let $$\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$$, $$\vec{b} = 11\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (-2\vec{a} + 3\vec{b})$$. If $$(2\vec{a} + 3\vec{b}) \cdot \vec{c} = 1670$$, then $$|\vec{c}|^2$$ is equal to :
Login to view the detailed solution.
Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$$ and $$\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}$$ be three vectors. Let $$\vec{r}$$ be a unit vector along $$\vec{b} + \vec{c}$$. If $$\vec{r} \cdot \vec{a} = 3$$, then $$3\lambda$$ is equal to :
Login to view the detailed solution.
If the shortest distance between the lines $$\frac{x - \lambda}{2} = \frac{y - 4}{3} = \frac{z - 3}{4}$$ and $$\frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8}$$ is $$\frac{13}{\sqrt{29}}$$, then a value of $$\lambda$$ is :
Login to view the detailed solution.
There are three bags $$X, Y$$ and $$Z$$. Bag $$X$$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $$Y$$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $$Z$$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :
Login to view the detailed solution.
The number of distinct real roots of the equation $$|x + 1||x + 3| - 4|x + 2| + 5 = 0$$, is _____
Login to view the detailed solution.
An arithmetic progression is written in the following way:
The sum of all the terms of the $$10^{th}$$ row is _____
Login to view the detailed solution.
Let a ray of light passing through the point $$(3, 10)$$ reflects on the line $$2x + y = 6$$ and the reflected ray passes through the point $$(7, 2)$$. If the equation of the incident ray is $$ax + by + 1 = 0$$, then $$a^2 + b^2 + 3ab$$ is equal to _____
Login to view the detailed solution.
Let S be the focus of the hyperbola $$\frac{x^2}{3} - \frac{y^2}{5} = 1$$, on the positive x-axis. Let C be the circle with its centre at $$A(\sqrt{6}, \sqrt{5})$$ and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to _____
Login to view the detailed solution.
If $$\alpha = \lim_{x \to 0^+} \left(\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}}\right)$$ and $$\beta = \lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$$ are the roots of the quadratic equation $$ax^2 + bx - \sqrt{e} = 0$$, then $$12\log_e(a + b)$$ is equal to _____
Login to view the detailed solution.
Let $$a, b, c \in \mathbb{N}$$ and $$a < b < c$$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $$9, 25, a, b, c$$ be $$18, 4$$ and $$\frac{136}{5}$$, respectively. Then $$2a + b - c$$ is equal to _____
Login to view the detailed solution.
Let A be the region enclosed by the parabola $$y^2 = 2x$$ and the line $$x = 24$$. Then the maximum area of the rectangle inscribed in the region A is _____
Login to view the detailed solution.
If $$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} dx = A\left(\frac{\alpha x - 1}{\beta x + 3}\right)^B + C$$, where C is the constant of integration, then the value of $$\alpha + \beta + 20AB$$ is _____
Login to view the detailed solution.
Let $$\alpha|x| = |y|e^{xy - \beta}$$, $$\alpha, \beta \in \mathbb{N}$$ be the solution of the differential equation $$x\,dy - y\,dx + xy(x\,dy + y\,dx) = 0$$, $$y(1) = 2$$. Then $$\alpha + \beta$$ is equal to _____
Login to view the detailed solution.
Let $$P(\alpha, \beta, \gamma)$$ be the image of the point $$Q(1, 6, 4)$$ in the line $$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$$. Then $$2\alpha + \beta + \gamma$$ is equal to _____
Login to view the detailed solution.
Educational materials for JEE preparation
Share