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.
Let $$n$$ denote the number of solutions of the equation $$z^2 + 3\bar{z} = 0$$, where $$z$$ is a complex number. Then the value of $$\sum_{k=0}^{\infty} \frac{1}{n^k}$$ is equal to
Login to view the detailed solution.
Let $$S_n$$ denote the sum of first $$n$$-terms of an arithmetic progression. If $$S_{10} = 530$$, $$S_5 = 140$$, then $$S_{20} - S_6$$ is equal to:
Loading solution...
The number of solutions of $$\sin^7 x + \cos^7 x = 1$$, $$x \in [0, 4\pi]$$ is equal to
Loading solution...
.webp)
Let the circle $$S : 36x^2 + 36y^2 - 108x + 120y + C = 0$$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $$x - 2y = 4$$ and $$2x - y = 5$$ lies inside the circle $$S$$, then:
Loading solution...
Let $$E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$. Let $$E_2$$ be another ellipse such that it touches the end points of major axis of $$E_1$$ and the foci of $$E_2$$ are the end points of minor axis of $$E_1$$. If $$E_1$$ and $$E_2$$ have same eccentricities, then its value is:
Loading solution...
Let a line $$L : 2x + y = k$$, $$k > 0$$ be a tangent to the hyperbola $$x^2 - y^2 = 3$$. If $$L$$ is also a tangent to the parabola $$y^2 = \alpha x$$, then $$\alpha$$ is equal to:
Loading solution...
.webp)
Which of the following Boolean expressions is not a tautology?
Loading solution...
Let $$A = [a_{ij}]$$ be a real matrix of order $$3 \times 3$$, such that $$a_{i1} + a_{i2} + a_{i3} = 1$$, for $$i = 1, 2, 3$$. Then, the sum of all entries of the matrix $$A^3$$ is equal to:
Loading solution...
The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x + y + z = 6$$, $$3x + 5y + 5z = 26$$ and $$x + 2y + \lambda z = \mu$$ has no solution, are:
Loading solution...
.webp)
Let $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then, the values of $$x \in R$$ satisfying the equation $$[e^x]^2 + [e^x + 1] - 3 = 0$$ lie in the interval:
Loading solution...
If the domain of the function $$f(x) = \frac{\cos^{-1}\sqrt{x^2 - x + 1}}{\sqrt{\sin^{-1}\left(\frac{2x-1}{2}\right)}}$$ is the interval $$(\alpha, \beta]$$, then $$\alpha + \beta$$ is equal to:
Loading solution...
Let $$f : R \to R$$ be defined as $$f(x) = \begin{cases} \frac{x^3}{(1-\cos 2x)^2} \log_e\left(\frac{1+2xe^{-2x}}{(1-xe^{-x})^2}\right), & x \neq 0 \\ \alpha, & x = 0 \end{cases}$$
If $$f$$ is continuous at $$x = 0$$, then $$\alpha$$ is equal to:
Loading solution...
.webp)
Let $$f : R \to R$$ be defined as $$f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x > 0 \\ 3xe^x, & x \le 0 \end{cases}$$
Then $$f$$ is increasing function in the interval
Loading solution...
If $$\int_0^{100\pi} \frac{\sin^2 x}{e^{\left(\frac{x}{\pi} - \left[\frac{x}{\pi}\right]\right)}} dx = \frac{\alpha\pi^3}{1+4\pi^2}$$, $$\alpha \in R$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$, then the value of $$\alpha$$ is:
Loading solution...
Let $$y = y(x)$$ be the solution of the differential equation $$\cosec^2 x \, dy + 2dx = (1 + y\cos 2x) \cosec^2 x \, dx$$, with $$y\left(\frac{\pi}{4}\right) = 0$$. Then, the value of $$(y(0) + 1)^2$$ is equal to:
Loading solution...
Let a vector $$\vec{a}$$ be coplanar with vectors $$\vec{b} = 2\hat{i} + \hat{j} + \hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + \hat{k}$$. If $$\vec{a}$$ is perpendicular to $$\vec{d} = 3\hat{i} + 2\hat{j} + 6\hat{k}$$, and $$|\vec{a}| = \sqrt{10}$$. Then a possible value of $$[\vec{a} \ \vec{b} \ \vec{c}] + [\vec{a} \ \vec{b} \ \vec{d}] + [\vec{a} \ \vec{c} \ \vec{d}]$$ is equal to:
Loading solution...
Let three vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be such that $$\vec{a} \times \vec{b} = \vec{c}$$, $$\vec{b} \times \vec{c} = \vec{a}$$ and $$|\vec{a}| = 2$$. Then which one of the following is not true?
Loading solution...
Let $$L$$ be the line of intersection of planes $$\vec{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 2$$ and $$\vec{r} \cdot (2\hat{i} + \hat{j} - \hat{k}) = 2$$. If $$P(\alpha, \beta, \gamma)$$ is the foot of perpendicular on $$L$$ from the point $$(1, 2, 0)$$, then the value of $$35(\alpha + \beta + \gamma)$$ is equal to:
Loading solution...
_1.webp)
If the shortest distance between the straight lines $$3(x-1) = 6(y-2) = 2(z-1)$$ and $$4(x-2) = 2(y-\lambda) = (z-3)$$, $$\lambda \in R$$ is $$\frac{1}{\sqrt{38}}$$, then the integral value of $$\lambda$$ is equal to:
Loading solution...
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $$2 \times 2$$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:
Loading solution...
If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to ___.
Loading solution...
The sum of all the elements in the set $$\{n \in \{1, 2, \ldots, 100\} | \text{H.C.F. of } n \text{ and } 2040 \text{ is } 1\}$$ is equal to ___.
Loading solution...
If the constant term, in binomial expansion of $$\left(2x^r + \frac{1}{x^2}\right)^{10}$$ is 180, then $$r$$ is equal to ___.
Loading solution...
The number of elements in the set $$\{n \in \{1, 2, 3, \ldots, 100\} | (11)^n > (10)^n + (9)^n\}$$ is ___.
Loading solution...
Consider the following frequency distribution:
If mean = $$\frac{309}{22}$$ and median = 14, then the value $$(a-b)^2$$ is equal to ___.
Loading solution...
Let $$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$. Then the number of $$3 \times 3$$ matrices $$B$$ with entries from the set $$\{1, 2, 3, 4, 5\}$$ and satisfying $$AB = BA$$ is ___.
Loading solution...
Let $$A = \{0, 1, 2, 3, 4, 5, 6, 7\}$$. Then the number of bijective functions $$f : A \to A$$ such that $$f(1) + f(2) = 3 - f(3)$$ is equal to ___.
Loading solution...
Let $$f : R \to R$$ be a function defined as $$f(x) = \begin{cases} 3\left(1 - \frac{|x|}{2}\right) & \text{if } |x| \le 2 \\ 0 & \text{if } |x| > 2 \end{cases}$$
Let $$g : R \to R$$ be given by $$g(x) = f(x+2) - f(x-2)$$. If $$n$$ and $$m$$ denote the number of points in $$R$$ where $$g$$ is not continuous and not differentiable, respectively, then $$n + m$$ is equal to ___.
Loading solution...
The area (in sq. units) of the region bounded by the curves $$x^2 + 2y - 1 = 0$$, $$y^2 + 4x - 4 = 0$$ and $$y^2 - 4x - 4 = 0$$ in the upper half plane is ___.
Loading solution...
Let $$y = y(x)$$ be the solution of the differential equation $$\left((x+2)e^{\left(\frac{y+1}{x+2}\right)} + (y+1)\right)dx = (x+2)dy$$, $$y(1) = 1$$. If the domain of $$y = y(x)$$ is an open interval $$(\alpha, \beta)$$, then $$|\alpha + \beta|$$ is equal to ___.
Loading solution...
Educational materials for JEE preparation
Share