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.
Let the ellipse $$E:\frac{x^{2}}{144}+\frac{y^{2}}{169}=1$$ and the hyperbola $$H:\frac{x^{2}}{16}-\frac{y^{2}}{\lambda^{2}}=-1$$ have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H , then the value of 24(e+ L) is:
Login to view the detailed solution.
Given below are two statements :
Statement I : The function $$f:R\rightarrow R $$ defined by $$f(x)=\frac{x}{1+\mid x\mid}$$ is one-one.
Statement II : The function $$f:R\rightarrow R $$ defined by $$f(x)=\frac{x^{2}+4x-30}{x^{2}-8x+18}$$ is many-one.
In the light of the above statements, choose the correct answer from the options given below :
Login to view the detailed solution.
The sum of the coefficients of $$x^{499} \text{ and }x^{500} \text{ in } (1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+....+x^{1000} \text{ is: }$$
Login to view the detailed solution.
.webp)
Let P be a point in the plane of the vectors $$ \overrightarrow{AB}=3\widehat{i} + \widehat{j}-\widehat{k} \text{ and }\overrightarrow{AC}=\widehat{i}-\widehat{j}+3\widehat{k}$$ such that P is equidistant from the Lines AB and AC. If $$ \mid \overrightarrow{AP} \mid=\frac{\sqrt{5}}{2} $$ then the area of the triangle ABP is:
Login to view the detailed solution.
The sum of all the elements in the range of $$f(x) =Sgn(\sin x) + Sgn(\cos x) +Sgn(\tan x) +Sg n(\cot x)$$, $$x \neq \frac{n\pi}{2}, n\epsilon Z, \text{ where } Sgn(t)=\begin{cases}1, & \text{ if } t>0\\-1 & \text{ if } t<0\end{cases} ,is:$$
Login to view the detailed solution.
Let $$P_1 : y=4x^2 \text{ and } P_2 : y=x^2 + 27$$ be two parabolas. If the area of the bounded region enclosed between$$P_1$$ and $$P_2$$ is six times the area of the bounded region enclosed between the line $$y = c\alpha x, \alpha > 0 \text{ and } P_1,$$ then $$\alpha$$ is equal to:
Login to view the detailed solution.
.webp)
Let the circle $$x^{2}+y^{2}=4$$ interesect x-axis at the points A(a,0), a > 0 and B(b, 0). let $$P(2 \cos \alpha, 2 \sin \alpha),0 \lt \alpha \lt \frac{\pi}{2} \text{and } Q(2\cos \beta, 2\sin \beta)$$ be two points such that $$( \alpha - \beta) =\frac {\pi}{2}$$. Then the point of intersection of AQ and BP lies on:
Login to view the detailed solution.
Let $$f(x)=\int\frac{dx}{x^{\left(\frac{2}{3}\right)}+2x^{\left(\frac{1}{2}\right)}} $$ be such that $$f(0)=-26+24\log_{e}{(2)}. \text { If } f(1)=a+b \log_{e}{(3)}, \text{ where } a,b \in Z$$, then a+b is equal to:
Login to view the detailed solution.
$$\dfrac{6}{3^{26}}+\dfrac{10.1}{3^{25}}+\dfrac{10.2}{3^{24}}+\dfrac{10.2^{2}}{3^{23}}+...+\dfrac{10.2^{24}}{3}$$ is equal to :
Login to view the detailed solution.
.webp)
Let $$[\cdot]$$ denote the greatest integer function. Then $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$$ is equal to:
Login to view the detailed solution.
Considering the principal values of inverse trigonometric functions, the value of the expression $$ \tan\left( 2\sin^{-1} \left( \frac{2}{\sqrt{13}}-2\cos ^{-1}\left( \frac{3}{\sqrt{10}}\right)\right)\right) $$
is equal to:
Login to view the detailed solution.
Let $$f(x) = \lim_{\theta \to 0}\left(\frac{\cos\pi x - x^{\frac{2}{\theta}} \sin(x - 1)}{1 + x^{\left(\frac{2}{\theta}\right)} (x - 1)}\right), \quad x \in \mathbb{R}$$. Consider the following two statements :
(I) $$f(x)$$ is discontinuous at $$x=1$$.
(II) $$f(x)$$ is continuous at $$x= - 1$$.
Then,
Login to view the detailed solution.
.webp)
The probability distribution of a random variable X is given below:
If $$ E(X)=\frac{263}{15} $$. then $$ P(X<20)$$ is equal to:
Login to view the detailed solution.
Let $$A = \{ z \in \mathbb {C} : |z - 2| \le 4 \}\quad$$ and $$\quad B = \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 5 \}.$$ Then the maximum of $$\left\{ |z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B \right\}text{ is:}$$
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation $$x\frac{dy}{dx} - y = x^2 \cot x, \quad x \in (0, \pi).$$ If $$y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$$, then $$6y\left(\frac{\pi}{6}\right) - 8y\left(\frac{\pi}{4}\right)$$ is equal to :
Login to view the detailed solution.
Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line $$ \frac{x-1}{1} = \frac{y}{2} = \frac{z-1}{1}$$ Then the distance of Q from the line $$ \frac{x-9}{3} = \frac{y-9}{2} = \frac{z-5}{-2} $$ is
Login to view the detailed solution.
Let A be the focus of the parabolay $$y^{2}=8x$$. Let the line $$y= mx +c$$ intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $$\left(\frac {7}{3},\frac{4}{3}\right)$$, then $$ (BC)^{2}$$ is equal to:
Login to view the detailed solution.
Let the arithmetic mean of $$\dfrac{1}{a}$$ and $$\dfrac{1}{b}$$ be $$\dfrac{5}{16}$$, $$\text{a > 2}$$. If $$\alpha$$ is such that $$ a,\alpha,b $$ are in A.P., then the equation $$\alpha x^{2}-ax+2(\alpha-2b)=0$$ has:
Login to view the detailed solution.
An ellipse has its center at (1, - 2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is:
Login to view the detailed solution.
Given below ar e two statements :
Statement I: $$ 25^{13}+20^{13}+8^{13}+3^{13} $$ is divisible by 7.
Statement II: The integral part of $$(7 + 4\sqrt{3})^{25}$$ is an odd number.
ln the light of the above statements , choose the correct answer from the options given be low :
Login to view the detailed solution.
If $$\sum_{r=1}^{25}\left( \frac{r}{r^{4}+r^{2}+1} \right)=\frac{p}{q},$$ where p and q are positive integers such that gcd(p,q)=1, then p+q is equal to ___________
Login to view the detailed solution.
Three persons enter in a lift at the ground floor. The lift will go upto $$10^{th}$$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to________
Login to view the detailed solution.
Let $$A=\begin{bmatrix}3 & -4 \\1 & -1 \end{bmatrix}$$ and B be two matrices such that $$A^{100}=100B+I$$. Then the sum of all the elements of $$B^{100}$$ is_______
Login to view the detailed solution.
Let $$ f$$ be a differentiable function satisfying $$f(x)=1-2x+\int_{0}^{x} e^{(x-t)}f(t)dt, x \in \mathbb{R}$$ and let $$g(x)=\int_{0}^{x} (f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x \in \mathbb{R}.$$ If p and q are respectively the points of local minima and local maxima of g, then the value of $$\mid p+q \mid $$ is equal to _________
Login to view the detailed solution.
If the distance of the point $$P(43, \alpha, \beta), \beta<0,$$ from the line $$\overrightarrow{r} = 4\widehat{i}-\widehat{k}+\mu(2\widehat{i}+3\widehat{k}), \mu \in \mathbb{R}$$ along a line with direction ratios 3, -1, 0 is $$13\sqrt{10},$$ then $$ \alpha ^{2}+ \beta^{2}$$ is equal to______
Login to view the detailed solution.
Educational materials for JEE preparation
Share