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.
If the orthocentre of the triangle formed by the lines $$y = x + 1$$, $$y = 4x - 8$$ and $$y = mx + c$$ is at $$(3, -1)$$, then $$m - c$$ is :
Login to view the detailed solution.
Let $$\vec{a}$$ and $$\vec{b}$$ be the vectors of the same magnitude such that $$\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|} = \sqrt{2}+1$$. Then $$\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$$ is :
Loading solution...
Let $$A = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}$$ and $$B = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}$$. Then
Loading solution...
.webp)
If the range of the function $$f(x) = \frac{5 - x}{x^2 - 3x + 2}$$, $$x \ne 1, 2$$, is $$(-\infty, \alpha] \cup [\beta, \infty)$$, then $$\alpha^2 + \beta^2$$ is equal to :
Loading solution...
A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$n^2 - m^2$$ is equal to :
Loading solution...
Let a random variable X take values 0, 1, 2, 3 with $$P(X = 0) = P(X = 1) = p$$, $$P(X = 2) = P(X = 3) = q$$ and $$E(X^2) = 2E(X)$$. Then the value of $$8p - 1$$ is :
Loading solution...
.webp)
If the area of the region $$\{(x, y) : 1 + x^2 \le y \le \min\{x + 7, 11 - 3x\}\}$$ is A, then $$3A$$ is equal to
Loading solution...
Let $$f : \mathbf{R} \to \mathbf{R}$$ be a polynomial function of degree four having extreme values at $$x = 4$$ and $$x = 5$$. If $$\lim_{x \to 0} \frac{f(x)}{x^2} = 5$$, then $$f(2)$$ is equal to :
Loading solution...
The number of solutions of the equation $$\cos 2\theta \cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$$ in $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ is :
Loading solution...
.webp)
Let $$a_n$$ be the $$n^{\text{th}}$$ term of an A.P. If $$S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$$, $$a_6 = 7$$ and $$S_7 = 7$$, then $$a_{n}$$ is equal to :
Loading solution...
If the locus of $$z \in \mathbb{C}$$, such that $$\text{Re}\left(\frac{z-1}{2z+i}\right) + \text{Re}\left(\frac{\bar{z}-1}{2\bar{z}-i}\right) = 2$$, is a circle of radius $$r$$ and center $$(a, b)$$ then $$\frac{15ab}{r^2}$$ is equal to :
Loading solution...
Let the length of a latus rectum of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be 10. If its eccentricity is the minimum value of the function $$f(t) = t^2 + t + \frac{11}{12}$$, $$t \in \mathbf{R}$$, then $$a^2 + b^2$$ is equal to :
Loading solution...
.webp)
Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$$, $$y(0) = 1$$. Then $$\int_{-3}^{3} y(x) \, dx$$ is :
Loading solution...
If the equation of the line passing through the point $$\left(0, -\frac{1}{2}, 0\right)$$ and perpendicular to the lines $$\vec{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})$$ and $$\vec{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})$$ is $$\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}$$, then $$a + b + c + d$$ is equal to :
Loading solution...
Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If $$p + q = 126$$, then the eccentricity of the ellipse $$\frac{x^2}{16} + \frac{y^2}{n} = 1$$ is :
Loading solution...
Consider the lines $$L_1 : x - 1 = y - 2 = z$$ and $$L_2 : x - 2 = y = z - 1$$. Let the feet of the perpendiculars from the point $$P(5, 1, -3)$$ on the lines $$L_1$$ and $$L_2$$ be Q and R respectively. If the area of the triangle PQR is A, then $$4A^2$$ is equal to :
Loading solution...
The number of real roots of the equation $$x|x - 2| + 3|x - 3| + 1 = 0$$ is :
Loading solution...
Let $$e_1$$ and $$e_2$$ be the eccentricities of the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{25} = 1$$ and the hyperbola $$\frac{x^2}{16} - \frac{y^2}{b^2} = 1$$, respectively. If $$b < 5$$ and $$e_1 e_2 = 1$$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :
Loading solution...
_1.webp)
Let the system of equations $$x + 5y - z = 1$$, $$4x + 3y - 3z = 7$$, $$24x + y + \lambda z = \mu$$, $$\lambda, \mu \in \mathbf{R}$$, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $$7 \le x + y + z \le 77$$, is
Loading solution...
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :
Loading solution...
If the function $$f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$$ is continuous at $$x = 0$$, then $$f(0)$$ is equal to _____.
Loading solution...
If $$\int \left(\frac{1}{x} + \frac{1}{x^3}\right)\left(\sqrt[23]{3x^{-24} + x^{-26}}\right) dx = -\frac{\alpha}{3(\alpha+1)}(3x^\beta + x^\gamma)^{\frac{\alpha+1}{\alpha}} + C$$, $$x > 0$$, $$(\alpha, \beta, \gamma \in \mathbb{Z})$$, where C is the constant of integration, then $$\alpha + \beta + \gamma$$ is equal to _____.
Loading solution...
For $$t > -1$$, let $$\alpha_t$$ and $$\beta_t$$ be the roots of the equation $$\left((t+2)^{1/7} - 1\right)x^2 + \left((t+2)^{1/6} - 1\right)x + \left((t+2)^{1/21} - 1\right) = 0$$. If $$\lim_{t \to -1^+} \alpha_t = a$$ and $$\lim_{t \to -1^+} \beta_t = b$$, then $$72(a + b)^2$$ is equal to _____.
Loading solution...
Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be $$(-5, 0)$$ and $$5x + 9 = 0$$, respectively. If the product of the focal distances of a point $$\left(\alpha, 2\sqrt{5}\right)$$ on the hyperbola is p, then 4p is equal to _____.
Loading solution...
The sum of the series $$2 \times 1 \times {^{20}C_4} - 3 \times 2 \times {^{20}C_5} + 4 \times 3 \times {^{20}C_6} - 5 \times 4 \times {^{20}C_7} + \ldots + 18 \times 17 \times {^{20}C_{20}}$$, is equal to
Loading solution...
Educational materials for JEE preparation
Share