Electric field in a certain region is given by $$\vec{E} = \frac{A}{x^2}\hat{i} + \frac{B}{y^3}\hat{j}$$. The SI unit of $$A$$ and $$B$$ are:
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Electric field in a certain region is given by $$\vec{E} = \frac{A}{x^2}\hat{i} + \frac{B}{y^3}\hat{j}$$. The SI unit of $$A$$ and $$B$$ are:
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Match Column-I with Column-II:
Choose the correct answer from the options given below:
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The figure represents the momentum time $$(p - t)$$ curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively?
If $$t_3 - t_2 < t_1$$
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As per the given figure, a small ball $$P$$ slides down the quadrant of a circle and hits the other ball $$Q$$ of equal mass which is initially at rest. Neglecting the effect of friction and assume the collision to be elastic, the velocity of ball $$Q$$ after collision will be: ($$g = 10$$ m s$$^{-2}$$)
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A ball of mass $$200$$ g rests on a vertical post of height $$20$$ m. A bullet of mass $$10$$ g, travelling in horizontal direction, hits the centre of the ball. After collision both travels independently. The ball hits the ground at a distance $$30$$ m and the bullet at a distance of $$120$$ m from the foot of the post. The value of initial velocity of the bullet will be (if $$g = 10$$ m s$$^{-2}$$):
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If the gravitational field in the space is given as $$-\frac{K}{r^2}$$. Taking the reference point to be at $$r = 2$$ cm with gravitational potential $$V = 10$$ J kg$$^{-1}$$. Find the gravitational potentials at $$r = 3$$ cm in SI unit (Given, that $$K = 6$$ J cm kg$$^{-1}$$)
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Choose the correct relationship between Poisson ratio $$\sigma$$, bulk modulus $$(K)$$ and modulus of rigidity $$\eta$$ of a given solid object:
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The height of liquid column raised in a capillary tube of certain radius when dipped in liquid $$A$$ vertically is, $$5$$ cm. If the tube is dipped in a similar manner in another liquid $$B$$ of surface tension and density double the values of liquid $$A$$, the height of liquid column raised in liquid B would be ______ m.
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Heat is given to an ideal gas in an isothermal process.
A. Internal energy of the gas will decrease.
B. Internal energy of the gas will increase.
C. Internal energy of the gas will not change.
D. The gas will do positive work.
E. The gas will do negative work.
Choose the correct answer from the options given below:
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The pressure $$(P)$$ and temperature $$(T)$$ relationship of an ideal gas obeys the equation $$PT^2 =$$ constant. The volume expansion coefficient of the gas will be:
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Two isolated metallic solid spheres of radii $$R$$ and $$2R$$ are charged such that both have same charge density $$\sigma$$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $$\sigma'$$. The ratio $$\frac{\sigma'}{\sigma}$$ is:
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The charge flowing in a conductor changes with time as $$Q(t) = \alpha t - \beta t^2 + \gamma t^3$$, where $$\alpha$$, $$\beta$$ and $$\gamma$$ are constants. Minimum value of current is:
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A massless square loop, of wire resistance $$10$$ $$\Omega$$, supporting a mass of $$1$$ g, hangs vertically with one of its sides in a uniform magnetic field of $$10^3$$ G, directed outwards in the shaded region. A dc voltage $$V$$ is applied to the loop. For what value of $$V$$, the magnetic force will exactly balance the weight of the supporting mass of $$1$$ g?
(If sides of the loop $$= 10$$ cm, $$g = 10$$ m s$$^{-2}$$)
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The magnetic moments associated with two closely wound circular coils $$A$$ and $$B$$ of radius $$r_A = 10$$ cm and $$r_B = 20$$ cm respectively are equal if: (Where $$N_A$$, $$I_A$$ and $$N_B$$, $$I_B$$ are number of turn and current of $$A$$ and $$B$$ respectively)
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In a series $$LR$$ circuit with $$X_L = R$$, power factor is $$P_1$$. If a capacitor of capacitance $$C$$ with $$X_C = X_L$$ is added to the circuit the power factor becomes $$P_2$$. The ratio of $$P_1$$ to $$P_2$$ will be:
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A person has been using spectacles of power $$-1.0$$ diopter for distant vision and a separate reading glass of power $$2.0$$ diopters. What is the least distance of distinct vision for this person:
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A small object at rest, absorbs a light pulse of power $$20$$ mW and duration $$300$$ ns. Assuming speed of light as $$3 \times 10^8$$ m s$$^{-1}$$. The momentum of the object becomes equal to:
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Speed of an electron in Bohr's $$7^{th}$$ orbit for Hydrogen atom is $$3.6 \times 10^6$$ m s$$^{-1}$$. The corresponding speed of the electron in $$3^{rd}$$ orbit, in m s$$^{-1}$$ is:
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The output waveform of the given logical circuit for the following inputs $$A$$ and $$B$$ as shown below, is
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A sinusoidal carrier voltage is amplitude modulated. The resultant amplitude modulated wave has maximum and minimum amplitude of $$120$$ V and $$80$$ V respectively. The amplitude of each side band is:
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A horse rider covers half the distance with $$5$$ m s$$^{-1}$$ speed. The remaining part of the distance was travelled with speed $$10$$ m s$$^{-1}$$ for half the time and with speed $$15$$ m s$$^{-1}$$ for other half of the time. The mean speed of the rider averaged over the whole time of motion is $$\frac{x}{7}$$ m s$$^{-1}$$. The value of $$x$$ is ______.
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A thin uniform rod of length $$2$$ m, cross sectional area $$A$$ and density $$d$$ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $$\omega$$. If value of $$\omega$$ in terms of its rotational kinetic energy $$E$$ is $$\sqrt{\frac{\alpha E}{Ad}}$$, then the value of $$\alpha$$ is ______.
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The general displacement of a simple harmonic oscillator is $$x = A \sin \omega t$$. Let $$T$$ be its time period. The slope of its potential energy $$(U)$$ - time $$(t)$$ curve will be maximum when $$t = \frac{T}{\beta}$$. The value of $$\beta$$ is ______.
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A capacitor of capacitance $$900$$ $$\mu$$F is charged by a $$100$$ V battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capacitor connected to negative plate of the charged capacitor. The loss of energy in this process is measured as $$x \times 10^{-2}$$ J. The value of $$x$$ is ______.
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In the following circuit, the magnitude of current $$I_1$$, is ______ A.
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As per the given figure, if $$\frac{dI}{dt} = -1$$ A s$$^{-1}$$, then the value of $$V_{AB}$$ at this instant will be ______ V.
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In an experiment for estimating the value of focal length of converging mirror, image of an object placed at $$40$$ cm from the pole of the mirror is formed at distance $$120$$ cm from the pole of the mirror. These distances are measured with a modified scale in which there are 20 small divisions in $$1$$ cm. The value of error in measurement of focal length of the mirror is $$\frac{1}{K}$$ cm. The value of $$K$$ is ______.
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In Young's double slit experiment, two slits $$S_1$$ and $$S_2$$ are $$d$$ distance apart and the separation from slits to screen is $$D$$ (as shown in figure). Now if two transparent slabs of equal thickness $$0.1$$ mm but refractive index $$1.51$$ and $$1.55$$ are introduced in the path of beam $$\lambda = 4000$$ $$\text{\AA}$$ from $$S_1$$ and $$S_2$$ respectively. The central bright fringe spot will shift by ______ number of fringes.
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A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of $$24$$ W. The radius of curvature of hemisphere is $$10$$ cm and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is ______ $$\times 10^{-8}$$ N.
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In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by $$0.5$$ mm on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, 4 linear scale divisions are clearly visible while $$46^{th}$$ division the circular scale coincide with the reference line. The diameter of the wire is ______ $$\times 10^{-2}$$ mm.
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Match List - I with List - II
| LIST-I (Atomic number) | LIST-II (Block of periodic table) |
|---|---|
| (A) 37 | I. p-block |
| (B) 78 | II. d-block |
| (C) 52 | III. f-block |
| (D) 65 | IV. s-block |
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For OF$$_2$$ molecule consider the following:
(A) Number of lone pairs on oxygen is 2.
(B) FOF angle is less than 104.5°.
(C) Oxidation state of O is -2.
(D) Molecule is bent 'V' shaped.
(E) Molecular geometry is linear.
Correct options are:
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Match List I with List II
| List I (molecules/ions) | List II (No. of lone pairs of e$$^-$$ on central atom) |
|---|---|
| (A) IF$$_7$$ | I. Three |
| (B) ICl$$_4^-$$ | II. One |
| (C) XeF$$_6$$ | III. Two |
| (D) XeF$$_2$$ | IV. Zero |
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The alkaline earth metal sulphate(s) which are readily soluble in water is/are:
(A) BeSO$$_4$$
(B) MgSO$$_4$$
(C) CaSO$$_4$$
(D) SrSO$$_4$$
(E) BaSO$$_4$$
Choose the correct answer from the options below:
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Lithium aluminium hydride can be prepared from the reaction of
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): In expensive scientific instruments, silica gel is kept in watch-glasses or in semipermeable membrane bags.
Reason (R): Silica gel adsorbs moisture from air via adsorption, thus protects the instrument from water corrosion (rusting) and / or prevents malfunctioning.
In the light of the above statements, choose the correct answer from the options given below:
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What is the correct order of acidity of the protons marked $$A - D$$ in the given compounds?
The major products 'A' and 'B', respectively, are
Formation of photochemical smog involves the following reaction in which A, B and C are respectively.
(i) NO$$_2$$ $$\xrightarrow{h\nu}$$ A + B
(ii) B + O$$_2$$ $$\to$$ C
(iii) A + C $$\to$$ NO$$_2$$ + O$$_2$$
Choose the correct answer from the options below:
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In the extraction of copper, its sulphide ore is heated in a reverberatory furnace after mixing with silica to:
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During the qualitative analysis of SO$$_3^{2-}$$ using dilute H$$_2$$SO$$_4$$, SO$$_2$$ gas is evolved which turns K$$_2$$Cr$$_2$$O$$_7$$ solution (acidified with dilute H$$_2$$SO$$_4$$):
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Which of the following is correct order of ligand field strength?
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To inhibit the growth of tumours, identify the compounds used from the following:
(A) EDTA
(B) Coordination Compounds of Pt
(C) D-Penicillamine
(D) Cis - Platin
Choose the correct answer from the option given below:
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Match List I with List II
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Which of the following compounds would give the following set of qualitative analysis?
(i) Fehling's Test: Positive
(ii) Na fusion extract upon treatment with sodium nitroprusside gives a blood red colour but not prussian blue.
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Benzyl isocyanide can be obtained by:
(A)
(B)
(C)
(D)
Choose the correct answer from the options given below:
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Caprolactam when heated at high temperature in presence of water, gives
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Amongst the following compounds, which one is an antacid?
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In the wet tests for identification of various cations by precipitation, which transition element cation doesn't belong to group IV in qualitative inorganic analysis?
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Ketoses give Seliwanoff's test faster than Aldoses.
Reason (R): Ketoses undergo $$\beta$$-elimination followed by formation of furfural.
In the light of the above statements, choose the correct answer from the options given below:
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The energy of one mole of photons of radiation of frequency $$2 \times 10^{12}$$ Hz in J mol$$^{-1}$$ is ______ . (Nearest integer)
(Given: h $$= 6.626 \times 10^{-34}$$ Js, N$$_A = 6.022 \times 10^{23}$$ mol$$^{-1}$$)
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When 2 litre of ideal gas expands isothermally into vacuum to a total volume of 6 litre, the change in internal energy is ______ J. (Nearest integer)
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600 mL of 0.01 M HCl is mixed with 400 mL of 0.01 M H$$_2$$SO$$_4$$. The pH of the mixture is ______ $$\times 10^{-2}$$. (Nearest integer)
[Given log2 = 0.30, log3 = 0.48, log5 = 0.69, log7 = 0.84, log11 = 1.04]
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A 300 mL bottle of soft drink has 0.2M CO$$_2$$ dissolved in it. Assuming CO$$_2$$ behaves as an ideal gas, the volume of the dissolved CO$$_2$$ at STP is ______ mL. (Nearest integer)
Given: At STP, molar volume of an ideal gas is 22.7 L mol$$^{-1}$$
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A solution containing 2 g of a non-volatile solute in 20 g of water boils at 373.52 K. The molecular mass of the solute is ______ g mol$$^{-1}$$. (Nearest integer)
Given, water boils at 373 K, K$$_b$$ for water $$= 0.52$$ K kg mol$$^{-1}$$
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Some amount of dichloromethane CH$$_2$$Cl$$_2$$ is added to 671.141 mL of chloroform CHCl$$_3$$ to prepare $$2.6 \times 10^{-3}$$ M solution of CH$$_2$$Cl$$_2$$ DCM. The concentration of DCM is ______ ppm (by mass).
Given: Atomic mass: C = 12; H: 1; Cl = 35.5 density of CHCl$$_3$$ = 1.49 g cm$$^{-3}$$
Consider the cell
Pt, H$$_2$$(g, 1 atm) | H$$^+$$ aq, 1M | Fe$$^{3+}$$ aq, Fe$$^{2+}$$ aq | Pt s
When the potential of the cell is 0.712 V at 298 K, the ratio Fe$$^{2+}$$/Fe$$^{3+}$$ is ______ (Nearest integer)
Given: Fe$$^{3+}$$ + e$$^-$$ = Fe$$^{2+}$$, E° Fe$$^{3+}$$, Fe$$^{2+}$$ Pt = 0.771, $$\frac{2.303RT}{F} = 0.06$$ V
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If compound A reacts with B following first order kinetics with rate constant $$2.011 \times 10^{-3}$$ s$$^{-1}$$. The time taken by A (in seconds) to reduce from $$7$$ g to $$2$$ g will be ______. (Nearest Integer)
log5 = 0.698, log7 = 0.845, log2 = 0.301
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The number of electrons involved in the reduction of permanganate to manganese dioxide in acidic medium is ______.
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A trisubstituted compound 'A', C$$_{10}$$H$$_{12}$$O$$_2$$ gives neutral FeCl$$_3$$ test positive. Treatment of compound 'A' with NaOH and CH$$_3$$Br gives C$$_{11}$$H$$_{14}$$O$$_2$$, with hydroiodic acid gives methyl iodide and with hot conc. NaOH gives a compound B, C$$_{10}$$H$$_{12}$$O$$_2$$. Compound 'A' also decolourises alkaline KMnO$$_4$$. The number of $$\pi$$ bond/s present in the compound 'A' is ______.
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If the solution of the equation $$\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1$$, $$x \in (0, \frac{\pi}{2})$$ is $$\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to:
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If $$a_n = \frac{-2}{4n^2 - 16n + 15}$$, then $$a_1 + a_2 + \ldots + a_{25}$$ is equal to:
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If the coefficient of $$x^{15}$$ in the expansion of $$\left(ax^3 + \frac{1}{bx^{\frac{1}{3}}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(ax^{\frac{1}{3}} - \frac{1}{bx^3}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$a, b$$:
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The coefficient of $$x^{301}$$ in $$1 + x^{500} + x \cdot 1 + x^{499} + x^2 \cdot 1 + x^{498} + \ldots + x^{500}$$ is:
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If $$\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$$, then the value of $$a + \frac{1}{a}$$ is:
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Let $$y = x + 2$$, $$4y = 3x + 6$$ and $$3y = 4x + 1$$ be three tangent lines to the circle $$(x - h)^2 + (y - k)^2 = r^2$$. Then $$h + k$$ is equal to:
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If $$P(h, k)$$ be point on the parabola $$x = 4y^2$$, which is nearest to the point $$Q(0, 33)$$, then the distance of $$P$$ from the directrix of the parabola $$y^2 = 4(x + y)$$ is equal to:
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Among the statements:
S1: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r$$
S2: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r \vee q \Rightarrow r$$
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The minimum number of elements that must be added to the relation $$R = \{(a, b), (b, c)\}$$ on the set $$\{a, b, c\}$$ so that it becomes symmetric and transitive is:
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Let $$A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}$$, $$d = |A| \neq 0$$ and $$\left| A - d(\text{Adj } A) \right| = 0$$. Then
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Let the system of linear equations
$$x + y + kz = 2$$
$$2x + 3y - z = 1$$
$$3x + 4y + 2z = k$$
have infinitely many solutions. Then the system
$$(k+1)x + (2k-1)y = 7$$
$$(2k+1)x + (k+5)y = 10$$ has:
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Suppose $$f: R \to (0, \infty)$$ be a differentiable function such that $$5f(x+y) = f(x) \cdot f(y), \forall x, y \in R$$. If $$f(3) = 320$$, then $$\sum_{n=0}^{5} f(n)$$ is equal to:
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The number of points on the curve $$y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x$$ at which the normal lines are parallel to $$x + 90y + 2 = 0$$ is:
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If $$[t]$$ denotes the greatest integer $$\leq 1$$, then the value of $$\frac{3(e-1)}{e} \int_1^2 x^2 e^{[x]+[x^3]} dx$$ is:
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Let the solution curve $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - \frac{3x^5\tan^{-1}x^3}{1+x^{6\cdot 7}} y = 2x \exp\frac{x^3 - \tan^{-1}x^3}{(1+x)^7}$$ pass through the origin. Then $$y(1)$$ is equal to:
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If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are three non-zero vectors and $$\hat{n}$$ is a unit vector perpendicular to $$\vec{c}$$ such that $$\vec{a} = \alpha\vec{b} - \hat{n}$$, $$\alpha \neq 0$$ and $$\vec{b} \cdot \vec{c} = 12$$, then $$\vec{c} \times \vec{a} \times \vec{b}$$ is equal to:
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The line $$l_1$$ passes through the point $$(2, 6, 2)$$ and is perpendicular to the plane $$2x + y - 2z = 10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$$ is:
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Let a unit vector $$\vec{OP}$$ make angle $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $$\beta \in (0, \frac{\pi}{2})$$. $$\vec{OP}$$ is perpendicular to the plane through points $$(1, 2, 3)$$, $$(2, 3, 4)$$ and $$(1, 5, 7)$$, then which one is true?
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If an unbiased die, marked with $$-2, -1, 0, 1, 2, 3$$ on its faces is thrown five times, then the probability that the product of the outcomes is positive, is:
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A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of $$\frac{\pi}{6}$$ with positive direction of y-axis and the area of $$\triangle OAB$$ is $$\frac{98}{3}\sqrt{3}$$, then $$a^2 - b^2$$ is equal to:
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Let $$z = 1 + i$$ and $$z_1 = \frac{1 + i\bar{z}}{\bar{z}(1-z) + \frac{1}{z}}$$. Then $$\frac{12}{\pi} \arg z_1$$ is equal to
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Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to
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$$\sum_{n=0}^{\infty} \frac{n^3((2n)!) + (2n-1)(n!)}{(n!)(2n)!} = ae + \frac{b}{e} + c$$ where $$a, b, c \in \mathbb{Z}$$ and $$e = \sum_{n=0}^{\infty} \frac{1}{n!}$$. Then $$a^2 - b + c$$ is equal to ______.
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The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted, $$a$$ and $$b$$ are respectively mean and variance of remaining 6 observation, then $$a + 3b - 5$$ is equal to ______.
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Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the number of one-one functions $$f: S \to P(S)$$, where $$P(S)$$ denote the power set of $$S$$, such that $$f(n) \subset f(m)$$ where $$n < m$$ is
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Let $$f^1(x) = \frac{3x+2}{2x+3}$$, $$x \in R - \{-\frac{3}{2}\}$$. For $$n \geq 2$$, define $$f^n x = f^1 \circ f^{n-1}(x)$$. If $$f^5 x = \frac{ax+b}{bx+a}$$, $$\gcd(a,b) = 1$$, then $$a + b$$ is equal to ______.
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$$\lim_{x \to 0} \frac{48}{x^4} \int_0^x \frac{t^3}{t^6+1} dt$$ is equal to
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Let $$\alpha$$ be the area of the larger region bounded by the curve $$y^2 = 8x$$ and the lines $$y = x$$ and $$x = 2$$, which lies in the first quadrant. Then the value of $$3\alpha$$ is equal to
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If the equation of the plane passing through the point $$(1, 1, 2)$$ and perpendicular to the line $$x - 3y + 2z - 1 = 0 = 4x - y + z$$ is $$Ax + By + Cz = 1$$, then $$140(C - B + A)$$ is equal to
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If $$\lambda_1 < \lambda_2$$ are two values of $$\lambda$$ such that the angle between the planes $$P_1: \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7$$ and $$P_2: \vec{r} \cdot (\lambda\hat{i} + \hat{j} - 3\hat{k}) = 9$$ is $$\sin^{-1}\frac{2\sqrt{6}}{5}$$, then the square of the length of perpendicular from the point $$(38\lambda_1, 10\lambda_2, 2)$$ to the plane $$P_1$$ is
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