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 $$p, q \in \mathbb{R}$$ and $$(1 - \sqrt{3}i)^{200} = 2^{199}(p + iq)$$, $$i = \sqrt{-1}$$. Then $$p + q + q^2$$ and $$p - q + q^2$$ are roots of the equation.
Login to view the detailed solution.
For three positive integers $$p, q, r$$, $$x^{pq^2} = y^{qr} = z^{p^2r}$$ and $$r = pq + 1$$ such that $$3, 3\log_y x, 3\log_z y, 7\log_x z$$ are in A.P. with common difference $$\frac{1}{2}$$. The $$r - p - q$$ is equal to
Login to view the detailed solution.
The value of $$\sum_{r=0}^{22} {}^{22}C_r \cdot {}^{23}C_r$$ is
Login to view the detailed solution.
.webp)
Let a tangent to the curve $$y^2 = 24x$$ meet the curve $$xy = 2$$ at the points $$A$$ and $$B$$. Then the mid-points of such line segments $$AB$$ lie on a parabola with the
Login to view the detailed solution.
$$\lim_{t \to 0} \left(1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + 3^{\frac{1}{\sin^2 t}} + \cdots + n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}$$ is equal to
Login to view the detailed solution.
The compound statement $$(\sim(P \wedge Q)) \vee ((\sim P) \wedge Q) \Rightarrow ((\sim P) \wedge (\sim Q))$$ is equivalent to
.webp)
The relation $$R = \{(a, b): \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$$ is:
Login to view the detailed solution.
If $$A$$ and $$B$$ are two non-zero $$n \times n$$ matrices such that $$A^2 + B = A^2B$$, then
Let $$N$$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$$x + y + z = 1$$
$$2x + Ny + 2z = 2$$
$$3x + 3y + Nz = 3$$
has unique solution is $$\frac{k}{6}$$, then the sum of value of $$k$$ and all possible values of $$N$$ is
Login to view the detailed solution.
.webp)
Let $$\alpha$$ be a root of the equation $$(a - c)x^2 + (b - a)x + (c - b) = 0$$, where $$a, b, c$$ are distinct real numbers such that the matrix $$\begin{pmatrix} \alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c \end{pmatrix}$$ is singular. Then the value of $$\frac{(a - c)^2}{(b - a)(c - b)} + \frac{(b - a)^2}{(a - c)(c - b)} + \frac{(c - b)^2}{(a - c)(b - a)}$$ is
Login to view the detailed solution.
$$\tan^{-1}\frac{1+\sqrt{3}}{3+\sqrt{3}} + \sec^{-1}\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}} = $$
Login to view the detailed solution.
The equation $$x^2 - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has:
Login to view the detailed solution.
.webp)
Let $$f(x) = \begin{cases} x^2\sin\frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$$, then at $$x = 0$$
Login to view the detailed solution.
The area enclosed between the curves $$y^2 + 4x = 4$$ and $$y - 2x = 2$$ is
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation $$x^3 dy + (xy - 1) dx = 0$$, $$x > 0$$, $$y\left(\frac{1}{2}\right) = 3 - e$$. Then $$y(1)$$ is equal to
Login to view the detailed solution.
Let $$\vec{u} = \hat{i} - \hat{j} - 2\hat{k}$$, $$\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$$, $$\vec{v} \cdot \vec{w} = 2$$ and $$\vec{v} \times \vec{w} = \vec{u} + \lambda\vec{v}$$, then $$\vec{u} \cdot \vec{w}$$ is equal to
Login to view the detailed solution.
Let $$PQR$$ be a triangle. The points $$A$$, $$B$$ and $$C$$ are on the sides $$QR$$, $$RP$$ and $$PQ$$ respectively such that $$\frac{QA}{AR} = \frac{RB}{BP} = \frac{PC}{CQ} = \frac{1}{2}$$. Then $$\frac{\text{Area}\triangle PQR}{\text{Area}\triangle ABC}$$ is equal to
Login to view the detailed solution.
The distance of the point $$(7, -3, -4)$$ from the plane containing the points $$(2, -3, 1)$$, $$(-1, 1, -2)$$ and $$(3, -4, 2)$$ is equal to:
Login to view the detailed solution.
_1.webp)
The distance of the point $$(-1, 9, -16)$$ from the plane $$2x + 3y - z = 5$$ measure parallel to the line $$\frac{x+4}{3} = \frac{2-y}{4} = \frac{z-3}{12}$$ is
Login to view the detailed solution.
Let $$\Omega$$ be the sample space and $$A \subseteq \Omega$$ be an event. Given below are two statements:
(S1): If $$P(A) = 0$$, then $$A = \phi$$
(S2): If $$P(A) = 1$$, then $$A = \Omega$$
Then
Login to view the detailed solution.
Let $$\lambda \in \mathbb{R}$$ and let the equation $$E$$ be $$|x|^2 - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set $$S = \{x + \lambda : x \text{ is an integer solution of } E\}$$ is _____
Login to view the detailed solution.
A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is
Login to view the detailed solution.
The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is _____
Login to view the detailed solution.
The 4$$^{th}$$ term of GP is 500 and its common ratio is $$\frac{1}{m}$$, $$m \in \mathbb{N}$$. Let $$S_n$$ denote the sum of the first $$n$$ terms of this GP. If $$S_6 > S_5 + 1$$ and $$S_7 < S_6 + \frac{1}{2}$$, then the number of possible values of $$m$$ is _____
Login to view the detailed solution.
Suppose $$\sum_{r=0}^{2023} r^2 \cdot {}^{2023}C_r = 2023 \times \alpha \times 2^{2022}$$, then the value of $$\alpha$$ is
Login to view the detailed solution.
Let a tangent to the curve $$9x^2 + 16y^2 = 144$$ intersect the coordinate axes at the points $$A$$ and $$B$$. Then, the minimum length of the line segment $$AB$$ is _____
Login to view the detailed solution.
Let $$C$$ be the largest circle centred at $$(2, 0)$$ and inscribed in the ellipse $$\frac{x^2}{36} + \frac{y^2}{16} = 1$$. If $$(1, \alpha)$$ lies on $$C$$, then $$10\alpha^2$$ is equal to _____
Login to view the detailed solution.
The value of $$\frac{8}{\pi} \int_0^{\pi/2} \frac{\cos x^{2023}}{\sin x^{2023} + \cos x^{2023}} dx$$ is _____.
Login to view the detailed solution.
The value of $$12 \int_0^3 |x^2 - 3x + 2| dx$$ is _____
The shortest distance between the lines $$\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2}$$ and $$\frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0}$$ is equal to _____
Login to view the detailed solution.
Educational materials for JEE preparation
Share
