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 $$x_{1},x_{2},...x_{10}$$ be ten observations such that $$\sum_{i=1}^{10}(x_{i}-2)=30,\sum_{i=1}^{10}(x_{i}-\beta)^{2}=98,\beta > 2$$, and their variance is $$\frac{4}{5}$$. If $$\mu$$ and $$\sigma^{2}$$ are respectively the mean and the variance of $$2(x_{1}-1)+4\beta, 2(x_{2}-1)+4\beta,....,2(x_{10}-1)+4\beta$$, then $$\frac{\beta \mu}{\sigma^{2}}$$ is equal to :
Login to view the detailed solution.
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $$11^{th}$$ term is :
Login to view the detailed solution.
The number of solutions of the equation $$\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0$$ is:
Login to view the detailed solution.
.webp)
Define a relation R on the interval $$[0,\frac{\pi}{2})$$ by $$xRy$$ if and only if $$\sec^{2} x-\tan^{2} y=1$$. Then R is :
Login to view the detailed solution.
Two parabolas have the same focus (4,3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then $$(AB)^{2}$$ is equal to :
Login to view the detailed solution.
Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set $$P$$ is :
Login to view the detailed solution.
Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k}$$ and $$\overrightarrow{b}=2\hat{i}+7\hat{j}+3\hat{k}$$. Let $$L_{1}:\overrightarrow{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \overrightarrow{a},\lambda \in R$$. and $$L_{2}: \overrightarrow{r}=(\hat{j}+\hat{k})+\mu \overrightarrow{b}, \mu \in R$$ be two lines. If the line $$L_{3}$$ passes through the point of intersection of $$L_{1}$$ and $$L_{2}$$, and is parallel to $$\overrightarrow{a}+\overrightarrow{b}$$, then $$L_{3}$$ passes through the point :
Login to view the detailed solution.
Let $$\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}, \overrightarrow{c}=3\hat{i}-5\hat{j}+\hat{k}$$ and $$\overrightarrow{c}$$ be a vector such that $$\overrightarrow{c} \times \overrightarrow{c} = \overrightarrow{c} \times \overrightarrow{b}$$ and $$(\overrightarrow{a}+\overrightarrow{c}).(\overrightarrow{b}.\overrightarrow{c})=168$$. Then the maximum value of $$|\overrightarrow{c}|^{2}$$ is :
Login to view the detailed solution.
The integral $$80\int_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta + \cos \theta}{9+16\sin 2\theta}\right)d\theta$$ is equaol to :
Login to view the detailed solution.
Let the ellipse $$E_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a \gt b$$ and $$E_{2}:\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1,A \lt B$$ have same eccentricity $$\frac{1}{\sqrt{3}}$$. Let the product of their lengths of latus rectums be $$\frac{32}{\sqrt{3}}$$, and the distance between the foci of $$E_{1}$$ be 4. If $$E_{1}$$ and $$E_{2}$$ meet at $$A,B,C$$ and $$D,$$ then the area of the quadrilateral $$ABCD$$ equals:
Login to view the detailed solution.
Let $$A = [a_{ij}] = \begin{bmatrix}\log_{5}{128} & \log_{4}5 \\\log_{5}8 & \log_{4}25 \end{bmatrix}$$. If $$A_{ij}$$ is the cofactor of $$a_{ij},C_{jk} = \sum_{k=1}^{2}a_{ik}A_{ik},1 \leq i,j \leq 2$$,and $$C = [C_{ij}],$$ then $$8|C|$$ is equal to :
Login to view the detailed solution.
Let $$|z_{1}-8-2i| \leq 1$$ and $$|z_{2}-2+6i| \leq 2,z_{1},z_{2} \in C$$. Then the minimum value of $$|z_{1}-z_{2}|$$ is :
Login to view the detailed solution.
Let $$L_{1}: \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2}$$ and $$L_{2}: \frac{x+1}{-1}=\frac{y-2}{2}=\frac{z}{1}$$ be two lines. Let $$L_{3}$$ be a line passing through the point $$(\alpha ,\beta ,\gamma)$$ and be perpendicular to both $$L_{1}$$ and $$L_{2}$$. If $$L_{3}$$ intersects $$L_{1}$$, then $$|5\alpha -11\beta -8\gamma|$$ equals:
Login to view the detailed solution.
Let M and m respectively be the maximum and the minimum value of
$$f(x) =\begin{vmatrix}\mathbf{1+\sin^{2}x} & \mathbf{\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{1+\cos^{2}x} & \mathbf{4\sin 4x} \\\mathbf{\sin^{2}x} &\mathbf{\cos^{2}x} & \mathbf{1+4\sin 4x}\end{vmatrix}$$, $$x \in R$$ then $$M^{4}-m^{4}$$ is equal to :
Login to view the detailed solution.
Let $$ABCD$$ be a triangle formed by the lines $$7x − 6y + 3 = 0, x + 2y − 31 = 0$$ and $$9x − 2y − 19 = 0.$$ Let the point $$(h,k)$$ be the image of the centroid of $$\triangle ABC$$ in the line $$3x + 6y − 53 = 0.$$ Then $$h^{2}+k^{2}+hk$$ is equal to :
Login to view the detailed solution.
The value of $$\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{(k+3)!}\right)$$ is:
Login to view the detailed solution.
The least value of n for which the number of integral terms in the Binomial expansion of $$(\sqrt[3]{7}+\sqrt[12]{11})^{n}$$ is 183, is :
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation $$\cos x(\log_{e}(\cos x))^{2}dy + (\sin x-3y\sin x\log_{e}(\cos x))dx=0,x \in (0,\frac{\pi}{2})$$. if $$y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_{e}2}$$, then $$y\left(\frac{\pi}{3}\right)$$ is equal to :
Login to view the detailed solution.
Let the line $$x + y = 1$$ meet the circle $$x^{2}+y^{2}=4$$ at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at $$C$$ and $$D$$, then the area of the quadrilateral ADBC is equal to :
Login to view the detailed solution.
Let the area of the region $$\left\{(x,y): 2y \leq x^{2}+3,y+|x| \leq 3,y \geq |x-1|\right\}$$ be A.Then 6 A is equal to :
Login to view the detailed solution.
Let $$S = \left\{x : \cos^{-1} x = \pi + \sin^{-1} x+\sin^{-1}(2x+1)\right\}$$. Then $$\sum_{x \in S}^{}(2x-1)^{2}$$ is equal to_________.
Login to view the detailed solution.
Let $$F : \left(0,\infty\right)\rightarrow R$$ be a twice differentiable function. If for some $$a \neq 0,\int_{0}^{1}f(\lambda x)d\lambda = af(x),f(1)=1$$ and $$f(16)=\frac{1}{8}$$, then $$16-f'\left(\frac{1}{16}\right)$$ is equal to _______.
Login to view the detailed solution.
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.
Login to view the detailed solution.
Let $$S = \left\{m \in Z : A^{m^{2}}+A^{m} = 3I - A^{-6}\right\}$$, where $$ A =\begin{bmatrix}2 & -1 \\1 & 0 \end{bmatrix}$$. Then n(S) is equal to ______.
Login to view the detailed solution.
Let [t] be the greatest integer less than or equal to t. Then the least value of $$p \in N$$ for which $$\lim_{x\rightarrow 0^{+}}\left(x([\frac{1}{x}]+[\frac{2}{x}]+...+[\frac{p}{x}])-x^{2}([\frac{1}{x^{2}}]+[\frac{2^{2}}{x^{2}}]+...+[\frac{9^{2}}{x^{2}}])\right) \geq 1$$ is equal to_______.
Login to view the detailed solution.
An electric dipole of mass m, charge q, and length $$l$$ is placed in a uniform electric field $$\overrightarrow{E} = E_{\circ}\hat{i}$$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be:
Login to view the detailed solution.
A coil of area A and N turns is rotating with angular velocity $$\omega$$ in a uniform magnetic field $$\overrightarrow{B}$$ about an axis perpendicular to $$\overrightarrow{B}$$. Magnetic flux $$\varphi$$ and induced emf ε across it, at an instant when $$\overrightarrow{B}$$ is parallel to the plane of coil, are :
Login to view the detailed solution.
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.
Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor $$\left(R/\sqrt{R^{2}+\omega^{2}L^{2}}\right)$$, where ω frequency of the supply across resistor R and inductor L. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.
In the light of the above statements, choose the most appropriate answer from the options given below :
Login to view the detailed solution.
As shown below, bob A of a pendulum having massless string of length 'R' is released from $$60^{\circ}$$ to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.)
Login to view the detailed solution.
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Electromagnetic waves carry energy but not momentum. Reason (R): Mass of a photon is zero. In the light of the above statements, choose the most appropriate answer from the options given below :
Login to view the detailed solution.
Two projectiles are fired with same initial speed from same point on ground at angles of $$(45^{\circ}-\alpha)$$ and$$ (45^{\circ}+\alpha)$$, respectively, with the horizontal direction. The ratio of their maximum heights attained is :
Login to view the detailed solution.
If $$\lambda$$ and $$K$$ are de Broglie wavelength and kinetic energy, respectively, of a particle with constant mass. The correct graphical representation for the particle will be
Login to view the detailed solution.
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance. Reason (R) : A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below :
Login to view the detailed solution.
Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be
Login to view the detailed solution.
At the interface between two materials having refractive indices $$n_{1}$$ and $$n_{2}$$, the critical angle for reflection of an em wave is $$\theta_{1C}$$. The $$n_{2}$$ material is replaced by another material having refractive index $$n_{3}$$ such that the critical angle at the interface between $$n_{1}$$ and $$n_{3}$$ materials is $$\theta_{2C}$$. If $$n_{3} > n_{2} > n_{1};\frac{n_{2}}{n_{3}}=\frac{2}{5}$$ and $$\sin \theta_{2C}-\sin \theta_{1C}=\frac{1}{2}$$, then $$\theta_{1C}$$ is
Login to view the detailed solution.
Let u and $$\upsilon$$ be the distances of the object and the image from a lens of focal length $$f$$. The correct graphical representation of u and $$\upsilon$$ for a convex lens when $$|u| > f$$, is
Login to view the detailed solution.
Choose the correct answer from the options given below :
Login to view the detailed solution.
For the circuit shown above, equivalent GATE is :
Login to view the detailed solution.
The expression given below shows the variation of velocity $$(\upsilon)$$ with time $$(t),\upsilon = At^{2}+\frac{Bt}{C+t}$$. The dimension of ABC is :
Login to view the detailed solution.
The workdone in an adiabatic change in an ideal gas depends upon only :
Login to view the detailed solution.
The fractional compression $$(\frac{\Delta V}{V})$$ of water at the depth of 2.5 km below the sea level is ______%.Given, the Bulk modulus of water $$= 2\times 10^{9}Nm^{-2}$$, density of water $$= 10^{3}kgm^{-3}$$, acceleration due to gravity $$= g =10 m s^{-2}$$.
Login to view the detailed solution.
The pair of physical quantities not having same dimensions is :
Login to view the detailed solution.
Consider $$I_{1}$$ and $$I_{2}$$ are the currents flowing simultaneously in two nearby coils 1 & 2, respectively. If $$L_{1}$$ = self inductance of coil 1, $$M_{12}$$ = mutual inductance of coil 1 with respect to coil 2, then the value of induced emf in coil 1 will be Options
Login to view the detailed solution.
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below :
Login to view the detailed solution.
A body of mass 'm' connected to a massless and unstretchable string goes in verticle circle of radius 'R' under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $$n\sqrt{gR}$$, where, $$n /geq$$, then ratio of kinetic energy of the body at bottom to that at top of the circle is
Login to view the detailed solution.
In a hydraulic lift, the surface area of the input piston is $$6 cm^{2}$$ and that of the output piston is $$1500 cm^{2}$$. If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is _______ kJ.
Login to view the detailed solution.
The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of $$\overrightarrow{F} = \hat{i} - \hat{j} + \hat{k}$$ acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is_________.
Login to view the detailed solution.
Two light beams fall on a transparent material block at point 1 and 2 with angle $$\theta_{1}$$ and $$\theta_{2}$$, respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2, $$d = 4\sqrt{3} cm$$ and $$\theta_{1} = \theta_{2} = \cos^{-1}(\frac{n_{2}}{2n_{1}})$$, where refractive index of the block $$n_{2} > $$ refractive index of the outside medium $$n_{1}$$, then the thickness of the block is ________cm.
Login to view the detailed solution.
A container of fixed volume contains a gas at $$27^{\circ}C$$. To double the pressure of the gas, the temperature of gas should be raised to ______ $$^{\circ}C$$.
Login to view the detailed solution.
The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the river bank, is _______ cm. (Take g = $$10 m/s^{2}$$)
Login to view the detailed solution.
Choose the correct answer from the options given below :
500J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm. The final temperature and the change in internal energy respectively are: Given : $$R = 8.3 J K^{-1}mol^{-1}$$
At temperature $$T$$, compound $$AB_{2(g)}$$ dissociates as $$AB_{2} \rightleftharpoons AB_{(g)}+\frac{1}{2}B_{2(g)}$$ having degree of dissociation $$x$$ (small compared to unity). The correct expression for $$x$$ in terms of $$K_{p}$$ and p is
Login to view the detailed solution.
An element 'E'has the ionisation enthalpy value of $$374 kJ mol^{-1}$$.'E'reacts with elements A, B, C and D with electron gain enthalpy values of −328, −349, −325 and $$-295 kJ mol^{-1}$$, respectively. The correct order of the products EA, EB, EC and ED in terms of ionic character is :
Total number of nucleophiles from the following is :
$$NH_{3},PhSH,(H_{3}C)_{2}S,H_{2}C=CH_{2},\ominus\\O H,H_{3}O^{\oplus},(CH_{3})_{2}CO,\rightleftharpoons NCH_{3}$$
Login to view the detailed solution.
The steam volatile compounds among the following are :
(A)
(B)
(C)
(D)
Choose the correct answer from the options given below :
Given below are two statements : Statement (I): The radii of isoelectronic species increases in the order. $$Mg^{2+} < Na^{+} < F^{-} < O^{2-}$$ Statement (II): The magnitude of electron gain enthalpy of halogen decreases in the order. Cl > F > Br > I In the light of the above statements, choose the most appropriate answer from the options given below :
Match List - I with List - II.
choose the correct answer from the options given below :
The molar conductivity of a weak electrolyte when plotted against the square root of its concentration, which of the following is expected to be observed ?
Login to view the detailed solution.
The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.
Login to view the detailed solution.
The product (P) formed in the following reaction is :
If $$a_{\circ}$$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength $$(\lambda)$$ of the electron present in the second orbit of hydrogen atom? [n : any integer]
Login to view the detailed solution.
Match List - I with List - II.
Choose the correct answer from the options given below :
The reaction $$A_{2}+B_{2}\rightarrow 2AB$$ follows the mechanism
$$A_{2}k_{1}A+A(fast)\\ \text{ }{_{k_{-1}}}\\ \\A+B_{2}\xrightarrow{k_{2}}AB+B(slow)\\A+B \rightarrow AB(fast)$$
The overall order of the reaction is :
Login to view the detailed solution.
1.24 g of $$AX_{2}$$ (molar mass 124 g $$mol^{-1}$$) is dissolved in 1 kg of water to form a solution with boiling point of $$100.0156^{\circ}C$$, while $$25.4g^{\circ}$$ of $$AY_{2}$$ (molar mass 250 g $$mol^{-1}$$) in 2 kg of water constitutes a solution with a boiling point of $$100.0260^{\circ}C.K_{b}(H_{2}O)=0.52 K$$ kg $$mol^{-1}$$ Which of the following is correct ?
Login to view the detailed solution.
Choose the correct statements. (A) Weight of a substance is the amount of matter present in it. (B) Mass is the force exerted by gravity on an object. (C) Volume is the amount of space occupied by a substance. (D) Temperatures below $$O^{\circ}C$$ are possible in Celsius scale, but in Kelvin scale negative temperature is not possible. (E) Precision refers to the closeness of various measurements for the same quantity. Choose the correct answer from the options given below :
Login to view the detailed solution.
The correct option with order of melting points of the pairs (Mn, Fe), (Tc, Ru) and (Re, Os) is :
For a $$Mg|Mg^{2+}(aq)||Ag^{+}(aq)|Ag$$ the correct Nernst Equation is :
Login to view the detailed solution.
In The following substitution reaction :
product 'P'formed is :
The correct increasing order of stability of the complexes based on $$\Delta_{\circ}$$ value is: I.$$[Mn(CN)_{6}]^{3}$$ II.$$[Co(CN)_{6}]^{4-}$$ III.$$[Fe(CN)_{6}]^{4-}$$ IV.$$[Fe(CN)_{6}]^{3-}$$
The molar mass of the water insoluble product formed from the fusion of chromite ore $$(FeCr_{2}O_{4})$$ with $$Na_{2}CO_{3}$$ in presence of $$O_{2}$$ is_______$$gmol^{-1}$$.
Login to view the detailed solution.
Given below are some nitrogen containing compounds
Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume _______ mg of HCl. (Given molar mass in $$gmol^{-1}$$ C : 12, H : 1, O : 16, Cl : 35.5)
The sum of sigma $$(\sigma)$$ and pi$$(\pi)$$ bonds in Hex-1,3-dien-5-yne is _______.
Login to view the detailed solution.
0.1 mole of compound ' S ' will weigh _______ g. (Given molar mass in $$gmol^{-1}$$ C : 12, H : 1, O : 16)
If $$A_{2}B$$ is 30% ionised in an aqueous solution, then the value of van't Hoff factor (i) is ______ $$\times 10^{-1}$$
Login to view the detailed solution.
Educational materials for JEE preparation
Share