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 $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + (2i - 1) = 0$$. Then, the value of $$|\alpha^8 + \beta^8|$$ is equal to
Login to view the detailed solution.
Let $$\{a_n\}_{n=0}^{\infty}$$ be a sequence such that $$a_0 = a_1 = 0$$ and $$a_{n+2} = 2a_{n+1} - a_n + 1$$ for all $$n \geq 0$$. Then, $$\sum_{n=2}^{\infty} \frac{a_n}{7^n}$$ is equal to
Login to view the detailed solution.
If the constant term in the expansion of $$\left(3x^3 - 2x^2 + \frac{5}{x^5}\right)^{10}$$ is $$2^k \cdot l$$, where $$l$$ is an odd integer, then the value of $$k$$ is equal to
Login to view the detailed solution.
.webp)
The distance between the two points $$A$$ and $$A'$$ which lie on $$y = 2$$ such that both the line segments $$AB$$ and $$A'B$$ (where $$B$$ is the point $$(2, 3)$$) subtend angle $$\frac{\pi}{4}$$ at the origin, is equal to
Login to view the detailed solution.
Let the tangent to the circle $$C_1 : x^2 + y^2 = 2$$ at the point $$M(-1, 1)$$ intersect the circle $$C_2 : (x-3)^2 + (y-2)^2 = 5$$, at two distinct points $$A$$ and $$B$$. If the tangents to $$C_2$$ at the points $$A$$ and $$B$$ intersect at $$N$$, then the area of the triangle $$ANB$$ is equal to
Let $$PQ$$ be a focal chord of the parabola $$y^2 = 4x$$ such that it subtends an angle of $$\frac{\pi}{2}$$ at the point $$(3, 0)$$. Let the line segment $$PQ$$ be also a focal chord of the ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a^2 > b^2$$. If $$e$$ is the eccentricity of the ellipse $$E$$, then the value of $$\frac{1}{e^2}$$ is equal to
Login to view the detailed solution.
.webp)
Let $$\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$$ be such that $$(p \wedge q)\Delta((p \vee q) \Rightarrow q)$$ is a tautology. Then $$\Delta$$ is equal to
Login to view the detailed solution.
Let the mean and the variance of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ be $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively. If the mean and variance of the first 4 observations are $$\frac{7}{2}$$ and $$a$$ respectively, then $$(4a + x_5)$$ is equal to
Login to view the detailed solution.
Let a set $$A = A_1 \cup A_2 \cup \ldots \cup A_k$$, where $$A_i \cap A_j = \phi$$ for $$i \neq j$$; $$1 \leq i, j \leq k$$. Define the relation $$R$$ from $$A$$ to $$A$$ by $$R = \{(x,y) : y \in A_i$$ if and only if $$x \in A_i, 1 \leq i \leq k\}$$. Then, $$R$$ is:
Login to view the detailed solution.
.webp)
The probability that a randomly chosen $$2 \times 2$$ matrix with all the entries from the set of first 10 primes, is singular, is equal to
Login to view the detailed solution.
Let $$A = [a_{ij}]$$ be a square matrix of order 3 such that $$a_{ij} = 2^{j-i}$$, for all $$i, j = 1, 2, 3$$. Then, the matrix $$A^2 + A^3 + \ldots + A^{10}$$ is equal to
If the system of linear equations
$$2x + y - z = 7$$
$$x - 3y + 2z = 1$$
$$x + 4y + \delta z = k$$, where $$\delta, k \in R$$
has infinitely many solutions, then $$\delta + k$$ is equal to
Login to view the detailed solution.
.webp)
The domain of the function $$\cos^{-1}\left(\frac{2\sin^{-1}\left(\frac{1}{4x^2-1}\right)}{\pi}\right)$$ is
Let $$f : R \to R$$ be a function defined by:
$$f(x) = \begin{cases} \max\{t^3 - 3t\}; & t \leq x, \quad x \leq 2 \\ x^2 + 2x - 6; & 2 < x < 3 \\ [x-3] + 9; & 3 \leq x \leq 5 \\ 2x + 1; & x > 5 \end{cases}$$
Where $$[t]$$ is the greatest integer less than or equal to $$t$$. Let $$m$$ be the number of points where $$f$$ is not differentiable and $$I = \int_{-2}^{2} f(x) dx$$. Then the ordered pair $$(m, I)$$ is equal to
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is
Login to view the detailed solution.
$$\int_0^5 \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right)dx$$, where $$[t]$$ denotes greatest integer less than or equal to $$t$$, is equal to
Login to view the detailed solution.
The area enclosed by $$y^2 = 8x$$ and $$y = \sqrt{2}x$$ that lies outside the triangle formed by $$y = \sqrt{2}x$$, $$x = 1$$, $$y = 2\sqrt{2}$$, is equal to
Let the solution curve of the differential equation $$x\frac{dy}{dx} - y = \sqrt{y^2 + 16x^2}$$, $$y(1) = 3$$ be $$y = y(x)$$. Then $$y(2)$$ is equal to
Login to view the detailed solution.
_1.webp)
Let $$\vec{a} = \alpha \hat{i} + 3\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} - \beta \hat{j} + 4\hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 2\hat{k}$$ where $$\alpha, \beta \in R$$. If the projection of $$\vec{a}$$ on $$\vec{c}$$ is $$\frac{10}{3}$$ and $$\vec{b} \times \vec{c} = -6\hat{i} + 10\hat{j} + 7\hat{k}$$, then the value of $$\alpha + \beta$$ equal to
Login to view the detailed solution.
If the mirror image of the point $$(2, 4, 7)$$ in the plane $$3x - y + 4z = 2$$ is $$(a, b, c)$$, the $$2a + b + 2c$$ is equal to
Login to view the detailed solution.
Let $$S = \{z \in C : |z - 2| \leq 1, z(1+i) + \bar{z}(1-i) \leq 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at $$z_1 \in S$$ and $$z_2 \in S$$. If $$5(|z_1|^2 + |z_2|^2) = \alpha + \beta\sqrt{5}$$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha + \beta$$ is equal to ______.
Let $$b_1b_2b_3b_4$$ be a 4-element permutation with $$b_i \in \{1, 2, 3, \ldots, 100\}$$ for $$1 \leq i \leq 4$$ and $$b_i \neq b_j$$ for $$i \neq j$$, such that either $$b_1, b_2, b_3$$ are consecutive integers or $$b_2, b_3, b_4$$ are consecutive integers. Then the number of such permutations $$b_1b_2b_3b_4$$ is equal to ______.
Login to view the detailed solution.
The number of elements in the set $$S = \{\theta \in [-4\pi, 4\pi] : 3\cos^2 2\theta + 6\cos 2\theta - 10\cos^2\theta + 5 = 0\}$$ is ______.
Login to view the detailed solution.
The number of solutions of the equation $$2\theta - \cos^2\theta + \sqrt{2} = 0$$ in $$R$$ is equal to ______.
Let $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$, $$a > 0$$, $$b > 0$$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $$4(2\sqrt{2} + \sqrt{14})$$. If the eccentricity $$H$$ is $$\frac{\sqrt{11}}{2}$$, then value of $$a^2 + b^2$$ is equal to ______.
Login to view the detailed solution.
$$50\tan\left(3\tan^{-1}\left(\frac{1}{2}\right) + 2\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right) + 4\sqrt{2}\tan\left(\frac{1}{2}\tan^{-1}(2\sqrt{2})\right)$$ is equal to ______.
Let $$c, k \in R$$. If $$f(x) = (c+1)x^2 + (1-c^2)x + 2k$$ and $$f(x+y) = f(x) + f(y) - xy$$, for all $$x, y \in R$$, then the value of $$|2(f(1) + f(2) + f(3) + \ldots + f(20))|$$ is equal to ______.
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{\sqrt{2}y}{2\cos^4x - \cos 2x} = xe^{\tan^{-1}(\sqrt{2}\cot 2x)}$$, $$0 < x < \frac{\pi}{2}$$ with $$y\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32}$$. If $$y\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$$, then the value of $$3\alpha^2$$ is equal to ______.
Let $$d$$ be the distance between the foot of perpendiculars of the points $$P(1, 2, -1)$$ and $$Q(2, -1, 3)$$ on the plane $$-x + y + z = 1$$. Then $$d^2$$ is equal to ______.
Login to view the detailed solution.
Let $$P_1 : \vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 4$$ be a plane. Let $$P_2$$ be another plane which passes through the points $$(2, -3, 2)$$, $$(2, -2, -3)$$ and $$(1, -4, 2)$$. If the direction ratios of the line of intersection of $$P_1$$ and $$P_2$$ be $$16, \alpha, \beta$$, then the value of $$\alpha + \beta$$ is equal to ______.
Login to view the detailed solution.
Educational materials for JEE preparation
Share
