A person moved from $$A$$ to $$B$$ on a circular path as shown in figure. If the distance travelled by him is $$60 \text{ m}$$, then the magnitude of displacement would be: (Given $$\cos 135° = -0.7$$)
JEE WhatsApp Group
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.
A person moved from $$A$$ to $$B$$ on a circular path as shown in figure. If the distance travelled by him is $$60 \text{ m}$$, then the magnitude of displacement would be: (Given $$\cos 135° = -0.7$$)
Login to view the detailed solution.
If momentum $$P$$, area $$A$$ and time $$T$$ are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is
Login to view the detailed solution.
Which of the following physical quantities have the same dimensions?
Login to view the detailed solution.
.webp)
A body of mass $$0.5 \text{ kg}$$ travels on a straight line path with velocity $$v = (3x^2 + 4) \text{ m s}^{-1}$$. The net work done by the force during its displacement from $$x = 0$$ to $$x = 2 \text{ m}$$ is
Login to view the detailed solution.
A solid cylinder and a solid sphere, having same mass $$M$$ and radius $$R$$, roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be
Login to view the detailed solution.
Three identical particles $$A$$, $$B$$ and $$C$$ of mass $$100 \text{ kg}$$ each are placed in a straight line with $$AB = BC = 13 \text{ m}$$. The gravitational force on a fourth particle $$P$$ of the same mass is $$F$$, when placed at a distance $$13 \text{ m}$$ from the particle $$B$$ on the perpendicular bisector of the line $$AC$$. The value of $$F$$ will be approximately
Login to view the detailed solution.
A certain amount of gas of volume $$V$$ at $$27°C$$ temperature and pressure $$2 \times 10^7 \text{ N m}^{-2}$$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use $$\gamma = 1.5$$)
Login to view the detailed solution.
Following statements are given:
(1) The average kinetic energy of a gas molecule decreases when the temperature is reduced.
(2) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature.
(3) The average kinetic energy of a gas molecule decreases with increase in volume.
(4) Pressure of a gas increases with increase in temperature at constant volume.
(5) The volume of gas decreases with increase in temperature.
Choose the correct answer from the options given below:
Login to view the detailed solution.
In figure (A), mass $$2m$$ is fixed on mass $$m$$ which is attached to two springs of spring constant $$k$$. In figure (B), mass $$m$$ is attached to two springs of spring constant $$k$$ and $$2k$$. If mass $$m$$ in (A) and (B) are displaced by distance $$x$$ horizontally and then released, then time period $$T_1$$ and $$T_2$$ corresponding to (A) and (B) respectively follow the relation.
Login to view the detailed solution.
A condenser of $$2 \mu F$$ capacitance is charged steadily from $$0$$ to $$5 \text{ C}$$. Which of the following graph represents correctly the variation of potential difference $$V$$ across its plates with respect to the charge $$Q$$ on the condenser?
Login to view the detailed solution.
Two charged particles, having same kinetic energy, are allowed to pass through a uniform magnetic field perpendicular to the direction of motion. If the ratio of radii of their circular paths is $$6:5$$ and their respective masses ratio is $$9:4$$. Then, the ratio of their charges will be
Login to view the detailed solution.
The magnetic moment of an electron ($$e$$) revolving in an orbit around nucleus with an orbital angular momentum is given by
Login to view the detailed solution.
A small square loop of wire of side $$l$$ is placed inside a large square loop of wire $$L$$ ($$L \gg l$$). Both loops are coplanar and their centres coincide at point $$O$$ as shown in figure. The mutual inductance of the system is
Login to view the detailed solution.
To increase the resonant frequency in series LCR circuit,
Login to view the detailed solution.
The RMS value of conduction current in a parallel plate capacitor is $$6.9 \mu A$$. The capacity of this capacitor, if it is connected to $$230 \text{ V}$$ AC supply with an angular frequency of $$600 \text{ rad s}^{-1}$$, will be
Login to view the detailed solution.
Which of the following statement is correct?
Login to view the detailed solution.
Time taken by light to travel in two different materials $$A$$ and $$B$$ of refractive indices $$\mu_A$$ and $$\mu_B$$ of same thickness is $$t_1$$ and $$t_2$$ respectively. If $$t_2 - t_1 = 5 \times 10^{-10} \text{ s}$$ and the ratio of $$\mu_A$$ to $$\mu_B$$ is $$1:2$$. Then the thickness of material, in meter is: (Given $$v_A$$ and $$v_B$$ are velocities of light in $$A$$ and $$B$$ materials respectively).
Login to view the detailed solution.
A metal exposed to light of wavelength $$800 \text{ nm}$$ and emits photoelectrons with a certain kinetic energy. The maximum kinetic energy of photo-electron doubles when light of wavelength $$500 \text{ nm}$$ is used. The work function of the metal is (Take $$hc = 1230 \text{ eV nm}$$).
Login to view the detailed solution.
The momentum of an electron revolving in $$n^{th}$$ orbit is given by: (Symbols have their usual meanings)
Login to view the detailed solution.
In the circuit, the logical value of $$A = 1$$ or $$B = 1$$ when potential at $$A$$ or $$B$$ is $$5 \text{ V}$$ and the logical value of $$A = 0$$ or $$B = 0$$ when potential at $$A$$ or $$B$$ is $$0 \text{ V}$$.
The truth table of the given circuit will be:
Login to view the detailed solution.
A car is moving with speed of $$150 \text{ km h}^{-1}$$ and after applying the brake it will move $$27 \text{ m}$$ before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling ______ m distance.
Login to view the detailed solution.
Four forces are acting at a point $$P$$ in equilibrium as shown in figure. The ratio of force $$F_1$$ to $$F_2$$ is $$1:x$$ where $$x =$$ ______.
Login to view the detailed solution.
A wire of length $$L$$ and radius $$r$$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $$F$$, its length increases by $$5 \text{ cm}$$. Another wire of the same material of length $$4L$$ and radius $$4r$$ is pulled by a force $$4F$$ under same conditions. The increase in length of this wire is ______ cm.
Login to view the detailed solution.
A unit scale is to be prepared whose length does not change with temperature and remains $$20 \text{ cm}$$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $$40 \text{ cm}$$ and length of iron will be ______ cm.
$$\alpha_{\text{iron}} = 1.2 \times 10^{-5} \text{ K}^{-1}$$ and $$\alpha_{\text{brass}} = 1.8 \times 10^{-5} \text{ K}^{-1}$$.
Login to view the detailed solution.
An observer is riding on a bicycle and moving towards a hill at $$18 \text{ km h}^{-1}$$. He hears a sound from a source at some distance behind him directly as well as after its reflection from the hill. If the original frequency of the sound as emitted by source is $$640 \text{ Hz}$$ and velocity of the sound in air is $$320 \text{ m s}^{-1}$$, the beat frequency between the two sounds heard by observer will be ______ Hz.
Login to view the detailed solution.
The volume charge density of a sphere of radius $$6 \text{ m}$$ is $$2 \mu C \text{ cm}^{-3}$$. The number of lines of force per unit surface area coming out from the surface of the sphere is ______ $$\times 10^{10} \text{ N C}^{-1}$$.
[Given: Permittivity of vacuum $$\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}$$]
Login to view the detailed solution.
In the given figure, the value of $$V_0$$ will be ______ V.
Login to view the detailed solution.
Eight copper wires of length $$l$$ and diameter $$d$$ are joined in parallel to form a single composite conductor of resistance $$R$$. If a single copper wire of length $$2l$$ has the same resistance $$R$$ then its diameter will be ______ $$d$$.
Login to view the detailed solution.
The energy band gap of semiconducting material to produce violet (wavelength $$= 4000$$ $$\mathring{A}$$) LED is ______ eV. (Round off to the nearest integer).
Login to view the detailed solution.
The required height of a TV tower which can cover the population of $$6.03$$ lakh is $$h$$. If the average population density is $$100$$ per square km and the radius of earth is $$6400 \text{ km}$$, then the value of $$h$$ will be ______ m.
Login to view the detailed solution.
$$SO_2Cl_2$$ on reaction with excess of water results into acidic mixture
$$SO_2Cl_2 + 2H_2O \rightarrow H_2SO_4 + 2HCl$$
16 moles of $$NaOH$$ is required for the complete neutralisation of the resultant acidic mixture. The number of moles of $$SO_2Cl_2$$ used is
Login to view the detailed solution.
Which of the following sets of quantum numbers is not allowed?
Login to view the detailed solution.
The IUPAC nomenclature of an element with electronic configuration $$[Rn] 5f^{14} 6d^1 7s^2$$ is
Login to view the detailed solution.
$$20 \text{ mL}$$ of $$0.1 \text{ M } NH_4OH$$ is mixed with $$40 \text{ mL}$$ of $$0.05 \text{ M HCl}$$. The pH of the mixture is nearest to:
(Given: $$K_b(NH_4OH) = 1 \times 10^{-5}$$, $$\log 2 = 0.30$$, $$\log 3 = 0.48$$, $$\log 5 = 0.69$$, $$\log 7 = 0.84$$, $$\log 11 = 1.04$$)
Login to view the detailed solution.
The reaction of $$H_2O_2$$ with potassium permanganate in acidic medium leads to the formation of mainly
Login to view the detailed solution.
Choose the correct order of density of the alkali metals
Login to view the detailed solution.
Match List - I with List - II
Choose the correct answer from the options given below
Login to view the detailed solution.
The geometry around boron in the product 'B' formed from the following reaction is
$$BF_3 + NaH \xrightarrow{450K} A + NaF$$
$$A + NMe_3 \rightarrow B$$
A compound 'A' on reaction with 'X' and 'Y' produces the same major product but different by-products 'a' and 'b'. Oxidation of 'a' gives a substance produced by ants.
'X' and 'Y' respectively are
Login to view the detailed solution.
The photochemical smog does not generally contain
Login to view the detailed solution.
The depression in freezing point observed for a formic acid solution of concentration $$0.5 \text{ mL L}^{-1}$$ is $$0.0405°C$$. Density of formic acid is $$1.05 \text{ g mL}^{-1}$$. The Van't Hoff factor of the formic acid solution is nearly: (Given for water $$K_f = 1.86 \text{ K kg mol}^{-1}$$)
Login to view the detailed solution.
The compound(s) that is(are) removed as slag during the extraction of copper is:
(1) CaO
(2) FeO
(3) Al₂O₃
(4) ZnO
(5) NiO
Choose the correct answer from the options given below
Login to view the detailed solution.
The interhalogen compound formed from the reaction of bromine with excess of fluorine is a
Login to view the detailed solution.
Most stable product of the following reaction is
Login to view the detailed solution.
Given below are two statements:
Statement I: On heating with $$KHSO_4$$, glycerol is dehydrated and acrolein is formed.
Statement II: Acrolein has fruity odour and can be used to test glycerol's presence.
Choose the correct option.
Login to view the detailed solution.
Which one of the following reactions does not represent correct combination of substrate and product under the given conditions?
Login to view the detailed solution.
An organic compound 'A' on reaction with $$NH_3$$ followed by heating gives compound B. Which on further strong heating gives compound C ($$C_8H_5NO_2$$). Compound C on sequential reaction with ethanolic KOH, alkyl chloride and hydrolysis with alkali gives a primary amine. The compound A is
Login to view the detailed solution.
Melamine polymer is formed by the condensation of
Login to view the detailed solution.
Drugs used to bind to receptors, inhibiting its natural function and blocking a message are called
Login to view the detailed solution.
During the denaturation of proteins, which of these structures will remain intact?
Login to view the detailed solution.
Among the following species $$N_2, N_2^+, N_2^-, N_2^{2-}, O_2, O_2^+, O_2^-, O_2^{2-}$$, the number of species showing diamagnetism is ______.
Login to view the detailed solution.
The pressure of a moist gas at $$27°C$$ is $$4 \text{ atm}$$. The volume of the container is doubled at the same temperature. The new pressure of the moist gas is ______ $$\times 10^{-1}$$ atm. (Nearest integer)
(Given: The vapour pressure of water at $$27°C$$ is $$0.4 \text{ atm}$$)
Login to view the detailed solution.
The enthalpy of combustion of propane, graphite and dihydrogen at $$298 \text{ K}$$ are: $$-2220.0 \text{ kJ mol}^{-1}$$, $$-393.5 \text{ kJ mol}^{-1}$$ and $$-285.8 \text{ kJ mol}^{-1}$$ respectively. The magnitude of enthalpy of formation of propane ($$C_3H_8$$) is ______ $$\text{kJ mol}^{-1}$$. (Nearest integer)
Login to view the detailed solution.
While estimating the nitrogen present in an organic compound by Kjeldahl's method, the ammonia evolved from $$0.25 \text{ g}$$ of the compound neutralized $$2.5 \text{ mL}$$ of $$2M \text{ } H_2SO_4$$. The percentage of nitrogen present in organic compound is ______.
Login to view the detailed solution.
The number of $$sp^3$$ hybridised carbons in an acyclic neutral compound with molecular formula $$C_4H_5N$$ is ______.
Login to view the detailed solution.
The cell potential for $$Zn | Zn^{2+}(aq) || Sn^{x+} | Sn$$ is $$0.801 \text{ V}$$ at $$298 \text{ K}$$. The reaction quotient for the above reaction is $$10^{-2}$$. The number of electrons involved in the given electrochemical cell reaction is ______.
(Given $$E^0_{Zn^{2+}|Zn} = -0.763 \text{ V}$$, $$E^0_{Sn^{x+}|Sn} = +0.008 \text{ V}$$ and $$\dfrac{2.303RT}{F} = 0.06 \text{ V}$$)
Login to view the detailed solution.
The half life for the decomposition of gaseous compound A is $$240 \text{ s}$$ when the gaseous pressure was $$500 \text{ Torr}$$ initially. When the pressure was $$250 \text{ Torr}$$, the half life was found to be $$4.0 \text{ min}$$. The order of the reaction is ______ (Nearest integer).
Login to view the detailed solution.
Among $$Co^{3+}$$, $$Ti^{2+}$$, $$V^{2+}$$ and $$Cr^{2+}$$ ions, one if used as a reagent cannot liberate $$H_2$$ from dilute mineral acid solution, its spin-only magnetic moment in gaseous state is ______ B.M. (Nearest integer)
Login to view the detailed solution.
Consider the following metal complexes:
$$[Co(NH_3)]^{3+}$$
$$[CoCl(NH_3)_5]^{2+}$$
$$[Co(CN)_6]^{3-}$$
$$[Co(NH_3)_5(H_2O)]^{3+}$$
The spin-only magnetic moment value of the complex that absorbs light with shortest wavelength is ______ B.M. (Nearest integer)
Login to view the detailed solution.
In the given reaction
(Where Et is $$-C_2H_5$$)
The number of chiral carbon(s) in product A is ______.
Login to view the detailed solution.
If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^4 + x^3 + x^2 + x + 1 = 0$$, then $$\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$$ is equal to
Login to view the detailed solution.
For $$n \in \mathbb{N}$$, let $$S_n = \{z \in \mathbb{C} : |z - 3 + 2i| = \dfrac{n}{4}\}$$ and $$T_n = \{z \in \mathbb{C} : |z - 2 + 3i| = \dfrac{1}{n}\}$$. Then the number of elements in the set $$\{n \in \mathbb{N} : S_n \cap T_n = \phi\}$$ is
Login to view the detailed solution.
The number of solutions of $$\cos x = |\sin x|$$, such that $$-4\pi \le x \le 4\pi$$ is
Login to view the detailed solution.
A line, with the slope greater than one, passes through the point $$A(4, 3)$$ and intersects the line $$x - y - 2 = 0$$ at the point $$B$$. If the length of the line segment $$AB$$ is $$\dfrac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line
Login to view the detailed solution.
Let the locus of the centre $$(\alpha, \beta)$$, $$\beta > 0$$, of the circle which touches the circle $$x^2 + (y - 1)^2 = 1$$ externally and also touches the $$x$$-axis be $$L$$. Then the area bounded by $$L$$ and the line $$y = 4$$ is
Login to view the detailed solution.
If $$\lim_{n \to \infty} \left(\sqrt{n^2 - n - 1} + n\alpha + \beta\right) = 0$$ then $$8(\alpha + \beta)$$ is equal to
Login to view the detailed solution.
Which of the following statements is a tautology?
Login to view the detailed solution.
A tower $$PQ$$ stands on a horizontal ground with base $$Q$$ on the ground. The point $$R$$ divides the tower in two parts such that $$QR = 15 \text{ m}$$. If from a point $$A$$ on the ground the angle of elevation of $$R$$ is $$60°$$ and the part $$PR$$ of the tower subtends an angle of $$15°$$ at $$A$$, then the height of the tower is
Login to view the detailed solution.
The number of $$\theta \in [0, 4\pi]$$ for which the system of linear equations
$$3(\sin 3\theta) x - y + z = 2$$
$$3(\cos 2\theta) x + 4y + 3z = 3$$
$$6x + 7y + 7z = 9$$
has no solution is
Login to view the detailed solution.
The total number of functions, $$f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$$ such that $$f(1) + f(2) = f(3)$$, is equal to
Login to view the detailed solution.
If the absolute maximum value of the function $$f(x) = (x^2 - 2x + 7)e^{(4x^3 - 12x^2 - 180x + 31)}$$ in the interval $$[-3, 0]$$ is $$f(\alpha)$$, then
Login to view the detailed solution.
The curve $$y(x) = ax^3 + bx^2 + cx + 5$$ touches the $$x$$-axis at the point $$P(-2, 0)$$ and cuts the $$y$$-axis at the point $$Q$$ where $$y'$$ is equal to $$3$$. Then the local maximum value of $$y(x)$$ is
Login to view the detailed solution.
For any real number $$x$$, let $$[x]$$ denote the largest integer less than or equal to $$x$$. Let $$f$$ be a real-valued function defined on the interval $$[-10, 10]$$ by
$$f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$$
Then, the value of $$\dfrac{\pi^2}{10} \displaystyle\int_{-10}^{10} f(x) \cos \pi x \, dx$$ is
Login to view the detailed solution.
The area of the region given by $$A = \{(x, y) : x^2 \le y \le \min\{x + 2, 4 - 3x\}\}$$ is
Login to view the detailed solution.
The slope of the tangent to a curve $$C: y = y(x)$$ at any point $$[x, y)$$ on it is $$\dfrac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}$$. If $$C$$ passes through the points $$\left(0, \dfrac{1}{2} + \dfrac{\pi}{2\sqrt{2}}\right)$$ and $$\left(\alpha, \dfrac{1}{2}e^{2\alpha}\right)$$ then $$e^{\alpha}$$ is equal to
Login to view the detailed solution.
The general solution of the differential equation $$(x - y^2)dx + y(5x + y^2)dy = 0$$ is
Login to view the detailed solution.
Let $$ABC$$ be a triangle such that $$\vec{BC} = \vec{a}$$, $$\vec{CA} = \vec{b}$$, $$\vec{AB} = \vec{c}$$, $$|\vec{a}| = 6\sqrt{2}$$, $$|\vec{b}| = 2\sqrt{3}$$ and $$\vec{b} \cdot \vec{c} = 12$$. Consider the statements:
$$S_1: |\vec{a} \times (\vec{b} + \vec{c})| \times |\vec{b} - \vec{c}| = 6(2\sqrt{2} - 1)$$
$$S_2: \angle ABC = \cos^{-1}\sqrt{\dfrac{2}{3}}$$
Then
Login to view the detailed solution.
Let $$P$$ be the plane containing the straight line $$\dfrac{x - 3}{9} = \dfrac{y + 4}{-1} = \dfrac{z - 7}{-5}$$ and perpendicular to the plane containing the straight lines $$\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z}{5}$$ and $$\dfrac{x}{3} = \dfrac{y}{7} = \dfrac{z}{8}$$. If $$d$$ is the distance of $$P$$ from the point $$(2, -5, 11)$$, then $$d^2$$ is equal to
Login to view the detailed solution.
If the sum and the product of mean and variance of a binomial distribution are $$24$$ and $$128$$ respectively, then the probability of one or two successes is
Login to view the detailed solution.
If the numbers appeared on the two throws of a fair six faced die are $$\alpha$$ and $$\beta$$, then the probability that $$x^2 + \alpha x + \beta > 0$$, for all $$x \in \mathbb{R}$$, is
Login to view the detailed solution.
Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^2 - 8ax + 2a = 0$$ and $$q$$ and $$s$$ are the roots of the equation $$x^2 + 12bx + 6b = 0$$, such that $$\dfrac{1}{p}, \dfrac{1}{q}, \dfrac{1}{r}, \dfrac{1}{s}$$ are in A.P., then $$a^{-1} - b^{-1}$$ is equal to ______.
Login to view the detailed solution.
The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.
Login to view the detailed solution.
Let $$a_1 = b_1 = 1$$, $$a_n = a_{n-1} + 2$$ and $$b_n = a_n + b_{n-1}$$ for every natural number $$n \ge 2$$. Then $$\displaystyle\sum_{n=1}^{15} a_n \cdot b_n$$ is equal to ______.
Login to view the detailed solution.
If the maximum value of the term independent of $$t$$ in the expansion of $$\left(t^2 x^{1/5} + \dfrac{(1-x)^{1/10}}{t}\right)^{15}$$, $$x \ge 0$$, is $$K$$, then $$8K$$ is equal to ______.
Login to view the detailed solution.
The sum of diameters of the circles that touch (i) the parabola $$75x^2 = 64(5y - 3)$$ at the point $$\left(\dfrac{8}{5}, \dfrac{6}{5}\right)$$ and (ii) the $$y$$-axis, is equal to ______.
Login to view the detailed solution.
Let the equation of two diameters of a circle $$x^2 + y^2 - 2x + 2fy + 1 = 0$$ be $$2px - y = 1$$ and $$2x + py = 4p$$. Then the slope $$m \in (0, \infty)$$ of the tangent to the hyperbola $$3x^2 - y^2 = 3$$ passing through the centre of the circle is equal to ______.
Login to view the detailed solution.
Let $$A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$$ and $$B = A - I$$. If $$\omega = \dfrac{\sqrt{3}i - 1}{2}$$, then the number of elements in the set $$\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$$ is equal to ______.
Login to view the detailed solution.
Let $$f(x) = \begin{cases} \{4x^2 - 8x + 5\}, & \text{if } 8x^2 - 6x + 1 \ge 0 \\ [4x^2 - 8x + 5], & \text{if } 8x^2 - 6x + 1 < 0 \end{cases}$$, where $$[\alpha]$$ denotes the greatest integer less than or equal to $$\alpha$$ . Then the number of points in $$\mathbb{R}$$ where $$f$$ is not differentiable is ______.
Login to view the detailed solution.
If $$\displaystyle\lim_{n \to \infty} \dfrac{(n+1)^{k-1}}{n^{k+1}} \left[(nk+1) + (nk+2) + \ldots + (nk+n)\right] = 33 \cdot \lim_{n \to \infty} \dfrac{1}{n^{k+1}} \left(1^k + 2^k + 3^k + \ldots + n^k\right)$$, then the integral value of $$k$$ is equal to ______.
Login to view the detailed solution.
The line of shortest distance between the lines $$\dfrac{x-2}{0} = \dfrac{y-1}{1} = \dfrac{z}{1}$$ and $$\dfrac{x-3}{2} = \dfrac{y-5}{2} = \dfrac{z-1}{1}$$ makes an angle of $$\sin^{-1}\sqrt{\dfrac{2}{27}}$$ with the plane $$P: ax - y - z = 0$$, $$a > 0$$. If the image of the point $$(1, 1, -5)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta - \gamma$$ is equal to ______.
Login to view the detailed solution.
Educational materials for JEE preparation
Share