The velocity-displacement graph of a particle is shown in the figure.
The acceleration-displacement graph of the same particle is represented by :
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The velocity-displacement graph of a particle is shown in the figure.
The acceleration-displacement graph of the same particle is represented by :
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If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately :
(Take : $$g = 10$$ ms$$^{-2}$$, the radius of earth, $$R = 6400 \times 10^3$$ m, Take $$\pi = 3.14$$)
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A particle of mass $$m$$ moves in a circular orbit under the central potential field, $$U(r) = \frac{C}{r}$$, where $$C$$ is a positive constant. The correct radius - velocity graph of the particle's motion is :
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An object of mass $$m_1$$ collides with another object of mass $$m_2$$, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses $$m_2 : m_1$$ is :
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Consider a uniform wire of mass $$M$$ and length $$L$$. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :
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A solid cylinder of mass $$m$$ is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:
[The coefficient of static friction, $$\mu_s$$, is 0.4]
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The angular momentum of a planet of mass $$M$$ moving around the sun in an elliptical orbit is $$\vec{L}$$. The magnitude of the areal velocity of the planet is :
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For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $$\gamma$$ is the ratio of specific heats):
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An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $$S_1$$ and that of the other part is $$S_2$$. Given that $$S_1 > S_2$$. If the piston is removed then the total entropy of the system will be:
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Consider a sample of oxygen behaving like an ideal gas. At 300 K, the ratio of root-mean-square (RMS) velocity to the average velocity of the gas molecule would be :
(Molecular weight of oxygen is 32 g mol$$^{-1}$$; $$R = 8.3$$ J K$$^{-1}$$ mol$$^{-1}$$)
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The function of time representing a simple harmonic motion with a period of $$\frac{\pi}{\omega}$$ is :
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Which of the following statements are correct?
(A) Electric monopoles do not exist whereas magnetic monopoles exist.
(B) Magnetic field lines due to a solenoid at its ends and outside cannot be completely straight and confined.
(C) Magnetic field lines are completely confined within a toroid.
(D) Magnetic field lines inside a bar magnet are not parallel.
(E) $$\chi = -1$$ is the condition for a perfect diamagnetic material, where $$\chi$$ is its magnetic susceptibility.
Choose the correct answer from the options given below :
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A proton and an $$\alpha$$-particle, having kinetic energies $$K_p$$ and $$K_\alpha$$, respectively, enter into a magnetic field at right angles. The ratio of the radii of the trajectory of proton to that of $$\alpha$$-particle is 2 : 1. The ratio of $$K_p : K_\alpha$$ is:
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In a series LCR circuit, the inductive reactance $$(X_L)$$ is 10 $$\Omega$$ and the capacitive reactance $$(X_C)$$ is 4 $$\Omega$$. The resistance $$(R)$$ in the circuit is 6 $$\Omega$$. The power factor of the circuit is :
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The time taken for the magnetic energy to reach 25% of its maximum value, when a solenoid of resistance $$R$$, inductance $$L$$ is connected to a battery, is :
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A plane electromagnetic wave propagating along y-direction can have the following pair of electric field $$(\vec{E})$$ and magnetic field $$(\vec{B})$$ components.
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Three rays of light, namely red (R), green (G) and blue (B) are incident on the face PQ of a right angled prism PQR as shown in figure.
The refractive indices of the material of the prism for red, green and blue wavelength are 1.27, 1.42 and 1.49 respectively. The colour of the ray(s) emerging out of the face PR is :
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The speed of electrons in a scanning electron microscope is $$1 \times 10^7$$ ms$$^{-1}$$. If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of:
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The decay of a proton to neutron is :
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The correct relation between $$\alpha$$ (ratio of collector current to emitter current) and $$\beta$$ (ratio of collector current to base current) of a transistor is :
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The radius of a sphere is measured to be $$(7.50 \pm 0.85)$$ cm. Suppose the percentage error in its volume is $$x$$. The value of $$x$$, to the nearest integer x, is ___.
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The projectile motion of a particle of mass 5 g is shown in the figure.
The initial velocity of the particle is $$5\sqrt{2}$$ ms$$^{-1}$$ and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points A and B is $$x \times 10^{-2}$$ kgms$$^{-1}$$. The value of $$x$$, to the nearest integer, is ___.
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A ball of mass 4 kg, moving with a velocity of 10 m s$$^{-1}$$, collides with a spring of length 8 m and force constant 100 N m$$^{-1}$$. The length of the compressed spring is $$x$$ m. The value of $$x$$, to the nearest integer, is ___.
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Consider a water tank as shown in the figure. It's cross-sectional area is 0.4 m$$^2$$. The tank has an opening B near the bottom whose cross-section area is 1 cm$$^2$$. A load of 24 kg is applied on the water at the top when the height of the water level is 40 cm above the bottom, the velocity of water coming out the opening B is v m s$$^{-1}$$. The value of $$v$$, to the nearest integer, is ___. [Take the value of $$g$$ to be 10 m s$$^{-2}$$]
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A galaxy is moving away from the earth at a speed of 286 km s$$^{-1}$$. The shift in the wavelength of a red line at 630 nm is $$x \times 10^{-10}$$ m. The value of $$x$$, to the nearest integer, is ___.
[Take the value of the speed of the light $$c$$, as $$3 \times 10^8$$ m s$$^{-1}$$]
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An infinite number of point charges, each carrying $$1 \mu C$$ charge, are placed along the y-axis at $$y = 1$$ m, 2 m, 4 m, 8 m .....
The total force on a 1 C point charge, placed at the origin, is $$x \times 10^3$$ N. The value of $$x$$, to the nearest integer, is ___.
[Take $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$$ N m$$^2$$ C$$^{-2}$$]
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Consider a 72 cm long wire AB as shown in the figure. The galvanometer jockey is placed at P on AB at a distance $$x$$ cm from A. The galvanometer shows zero deflection.
The value of $$x$$, to the nearest integer, is ___.
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Two wires of same length and thickness having specific resistances 6 $$\Omega$$ cm and 3 $$\Omega$$ cm respectively are connected in parallel. The effective resistivity is $$\rho$$ $$\Omega$$ cm. The value of $$\rho$$ to the nearest integer, is ___.
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The typical output characteristics curve for a transistor working in the common-emitter configuration is shown in the figure.
The estimated current gain from the figure is ___.
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A TV transmission tower antenna is at a height of 20 m. Suppose that the receiving antenna is at (i) ground level (ii) a height of 5 m. The increase in antenna range in case (ii) relative to case (i) is $$n$$%. The value of $$n$$, to the nearest integer, is ___.
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Given below are two statements:
Statement I : Bohr's theory accounts for the stability and line spectrum of Li$$^+$$ ion.
Statement II : Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field.
In the light of the above statements, choose the most appropriate answer from the options given below:
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The first ionization energy of magnesium is smaller, as compared to that of elements X and Y, but higher than that of Z. The elements X, Y and Z, respectively, are :
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The oxidation states of nitrogen in NO, NO$$_2$$, N$$_2$$O and NO$$_3^-$$ are in the order of :
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In basic medium, H$$_2$$O$$_2$$ exhibits which of the following reactions?
(A) Mn$$^{2+} \rightarrow$$ Mn$$^{4+}$$
(B) I$$_2 \rightarrow$$ I$$^-$$
(C) PbS $$\rightarrow$$ PbSO$$_4$$
Choose the most appropriate answer from the options below :
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Match list-I with list-II :
| (a) | Be | (i) | Treatment of cancer |
| (b) | Mg | (ii) | Extraction of metals |
| (c) | Ca | (iii) | Incendiary bombs and signals |
| (d) | Ra | (iv) | Windows of X-ray tubes |
| (v) | Bearings for motor engines |
Choose the most appropriate answer, the option given below :
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Given below are two statements :
Statement I : C$$_2$$H$$_5$$OH and AgCN both can generate nucleophile.
Statement II : KCN and AgCN both will generate nitrile nucleophile with all reaction conditions.
Choose the most appropriate option :
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In the following molecules,
Hybridisation of carbon a, b and c respectively are :
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Consider the given reaction, percentage yield of :
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Given below are two statements :
Statement I : Non-biodegradable wastes are generated by the thermal power plants.
Statement II : Bio-degradable detergents leads to eutrophication.
In the light of the above statements, choose the most appropriate answer from the option given below:
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The oxide that shows magnetic property is:
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A hard substance melts at high temperature and is an insulator in both solid and in molten state. This solid is most likely to be a / an :
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The charges on the colloidal CdS sol and TiO$$_2$$ sol are, respectively :
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Match list-I with list-II :
| List-I | List-II | |
|---|---|---|
| (a) Mercury | (i) | Vapour phase refining |
| (b) Copper | (ii) | Distillation refining |
| (c) Silicon | (iii) | Electrolytic refining |
| (d) Nickel | (iv) | Zone refining |
Choose the most appropriate answer from the option given below:
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The secondary valency and the number of hydrogen bonded water molecule(s) in CuSO$$_4$$ . 5H$$_2$$O, respectively, are :
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Main Products formed during a reaction of 1-methoxy naphthalene with hydroiodic acid are:
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Consider the above reaction, the product X and Y respectively are :
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In the reaction of hypobromite with amide, the carbonyl carbon is lost as:
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An organic compound A on treatment with benzene sulfonyl chloride gives compound B. B is soluble in dil. NaOH solution. Compound A is :
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Match List - I with List - II :
| List-I (Class of Chemicals) | List-II (Example) | |
|---|---|---|
| (a) Antifertility drug | (i) | Meprobamate |
| (b) Antibiotic | (ii) | Alitame |
| (c) Tranquilizer | (iii) | Norethindrone |
| (d) Artificial Sweetener | (iv) | Salvarsan |
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Deficiency of vitamin K causes :
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Consider the above reaction where 6.1 g of benzoic acid is used to get 7.8 g of m-bromo benzoic acid. The percentage yield of the product is ___.
(Round off to the Nearest integer)
[Given : Atomic masses : C = 12.0 u, H : 1.0 u, O : 16.0 u, Br = 80.0 u]
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The number of species below that have two lone pairs of electrons in their central atom is ___ (Round off to the Nearest integer)
SF$$_4$$, BF$$_4^-$$, ClF$$_3$$, AsF$$_3$$, PCl$$_5$$, BrF$$_5$$, XeF$$_4$$, SF$$_6$$
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The gas phase reaction
$$2 A(g) \rightleftharpoons A_2(g)$$
at 400 K has $$\Delta G^\circ = +25.2$$ kJ mol$$^{-1}$$.
The equilibrium constant $$K_C$$ for this reaction is ___ $$\times 10^{-2}$$. (Round off to the Nearest integer)
Use : $$R = 8.3$$ J mol$$^{-1}$$ K$$^{-1}$$, ln 10 = 2.3, $$\log_{10} 2 = 0.30$$, 1 atm = 1 bar
antilog(-0.3) = 0.501
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The solubility of CdSO$$_4$$ in water is $$8.0 \times 10^{-4}$$ mol L$$^{-1}$$. Its solubility in 0.01 M H$$_2$$SO$$_4$$ solution is ___ $$\times 10^{-6}$$ mol L$$^{-1}$$ (Round off to the Nearest integer) (Assume that solubility is much less than 0.01 M)
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10.0 ml of Na$$_2$$CO$$_3$$ solution is titrated against 0.2 M HCl solution. The following values were obtained in 5 readings. 4.8 ml, 4.9 ml, 5.0 ml, 5.0 ml and 5.0 ml
Based on these readings, and convention of titrimetric estimation of concentration of Na$$_2$$CO$$_3$$ solution is ___ mM.
(Round off to the Nearest integer)
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A solute A dimerizes in water. The boiling point of a 2 molal solution of A is 100.52$$^\circ$$C. The percentage association of A is ___.
(Round off to the Nearest integer)
Use : K$$_b$$ for water = 0.52 K kg mol$$^{-1}$$
Boiling point of water = 100$$^\circ$$C
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The molar conductivity at infinite dilution of barium chloride, sulphuric acid and hydrochloric acid are 280, 860, 426 Scm$$^2$$mol$$^{-1}$$ respectively. The molar conductivity at infinite dilution of barium sulphate is ___ Scm$$^2$$ mol$$^{-1}$$ (Round off to the Nearest Integer).
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A reaction has a half life of 1 min. The time required for 99.9% completion of the reaction is ___ min.
(Round off to the Nearest integer)
[Use : ln 2 = 0.69, ln 10 = 2.3]
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A xenon compound A upon partial hydrolysis gives XeO$$_2$$F$$_2$$. The number of lone pair of electrons present in compound A is ___ (Round off to the Nearest integer)
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In Tollen's test for aldehyde, the overall number of electron(s) transferred to the Tollen's reagent formula [Ag(NH$$_3$$)$$_2$$]$$^+$$ per aldehyde group to form silver mirror is ___. (Round off to the Nearest integer)
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Let a complex number be $$w = 1 - \sqrt{3}i$$. Let another complex number $$z$$ be such that $$|zw| = 1$$ and $$\arg(z) - \arg(w) = \frac{\pi}{2}$$. Then the area of the triangle (in sq. units) with vertices origin, $$z$$ and $$w$$ is equal to
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Let $$S_1$$ be the sum of first $$2n$$ terms of an arithmetic progression. Let $$S_2$$ be the sum of first $$4n$$ terms of the same arithmetic progression. If $$(S_2 - S_1)$$ is 1000, then the sum of the first $$6n$$ terms of the arithmetic progression is equal to:
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If $$15 \sin^4 \alpha + 10 \cos^4 \alpha = 6$$, for some $$\alpha \in R$$, then the value of $$27 \sec^6 \alpha + 8\operatorname{cosec}^6 \alpha$$ is equal to :
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Let the centroid of an equilateral triangle $$ABC$$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $$x + y = 3$$. If $$R$$ and $$r$$ be the radius of circumcircle and incircle respectively of $$\triangle ABC$$, then $$(R + r)$$ is equal to :
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Let $$S_1 : x^2 + y^2 = 9$$ and $$S_2 : (x-2)^2 + y^2 = 1$$. Then the locus of center of a variable circle $$S$$ which touches $$S_1$$ internally and $$S_2$$ externally always passes through the points :
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Let a tangent be drawn to the ellipse $$\frac{x^2}{27} + y^2 = 1$$ at $$(3\sqrt{3}\cos\theta, \sin\theta)$$ where $$\theta \in \left(0, \frac{\pi}{2}\right)$$. Then the value of $$\theta$$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
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Consider a hyperbola $$H : x^2 - 2y^2 = 4$$. Let the tangent at a point $$P(4, \sqrt{6})$$ meet the x-axis at $$Q$$ and latus rectum at $$R(x_1, y_1)$$, $$x_1 > 0$$. If $$F$$ is a focus of $$H$$ which is nearer to the point $$P$$, then the area of $$\triangle QFR$$ (in sq. units) is equal to
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If $$P$$ and $$Q$$ are two statements, then which of the following compound statement is a tautology?
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Let in a series of $$2n$$ observations, half of them are equal to $$a$$ and remaining half are equal to $$-a$$. Also by adding a constant $$b$$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $$a^2 + b^2$$ is equal to :
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A pole stands vertically inside a triangular park $$ABC$$. Let the angle of elevation of the top of the pole from each corner of the park be $$\frac{\pi}{3}$$. If the radius of the circumcircle of $$\triangle ABC$$ is 2, then the height of the pole is equal to :
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Define a relation $$R$$ over a class of $$n \times n$$ real matrices $$A$$ and $$B$$ as "$$ARB$$" iff there exists a non-singular matrix $$P$$ such that $$PAP^{-1} = B$$. Then which of the following is true?
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Let the system of linear equations
$$4x + \lambda y + 2z = 0$$
$$2x - y + z = 0$$
$$\mu x + 2y + 3z = 0$$, $$\lambda, \mu \in R$$
has a non-trivial solution. Then which of the following is true?
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Let $$f : R - \{3\} \to R - \{1\}$$ be defined by $$f(x) = \frac{x-2}{x-3}$$. Let $$g : R \to R$$ be given as $$g(x) = 2x - 3$$. Then, the sum of all the values of $$x$$ for which $$f^{-1}(x) + g^{-1}(x) = \frac{13}{2}$$ is equal to
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Let $$f : R \to R$$ be a function defined as
$$$f(x) = \begin{cases} \frac{\sin(a+1)x + \sin 2x}{2x}, & \text{if } x < 0 \\ b, & \text{if } x = 0 \\ \frac{\sqrt{x+bx^3} - \sqrt{x}}{bx^{5/2}}, & \text{if } x > 0 \end{cases}$$$
If $$f$$ is continuous at $$x = 0$$, then the value of $$a + b$$ is equal to :
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Let $$g(x) = \int_0^x f(t)dt$$, where $$f$$ is continuous function in $$[0, 3]$$ such that $$\frac{1}{3} \le f(t) \le 1$$ for all $$t \in [0, 1]$$ and $$0 \le f(t) \le \frac{1}{2}$$ for all $$t \in (1, 3]$$.
The largest possible interval in which $$g(3)$$ lies is :
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The area (in sq. unit) bounded by the curve $$4y^2 = x^2(4-x)(x-2)$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = (y+1)\left((y+1)e^{x^2/2} - x\right)$$, $$0 < x < 2.1$$, with $$y(2) = 0$$. Then the value of $$\frac{dy}{dx}$$ at $$x = 1$$ is equal to
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In a triangle $$ABC$$, if $$|\overrightarrow{BC}| = 8$$, $$|\overrightarrow{CA}| = 7$$, $$|\overrightarrow{AB}| = 10$$, then the projection of the vector $$\overrightarrow{AB}$$ on $$\overrightarrow{AC}$$ is equal to :
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Let $$\vec{a}$$ and $$\vec{b}$$ be two non-zero vectors perpendicular to each other and $$|\vec{a}| = |\vec{b}|$$, If $$|\vec{a} \times \vec{b}| = |\vec{a}|$$, then the angle between the vectors $$\left(\vec{a} + \vec{b} + (\vec{a} \times \vec{b})\right)$$ and $$\vec{a}$$ is equal to :
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Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
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If $$f(x)$$ and $$g(x)$$ are two polynomials such that the polynomial $$P(x) = f(x^3) + xg(x^3)$$ is divisible by $$x^2 + x + 1$$, then $$P(1)$$ is equal to ___.
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If $$\sum_{r=1}^{10} r!(r^3 + 6r^2 + 2r + 5) = \alpha(11!)$$, then the value of $$\alpha$$ is equal to ___.
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The term independent of $$x$$ in the expansion of $$\left[\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right]^{10}$$, $$x \neq 1$$, is equal to ___.
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Let $${}^nC_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1+x)^n$$. If $$\sum_{k=0}^{10} (2^2 + 3k) {}^{10}C_k = \alpha \cdot 3^{10} + \beta \cdot 2^{10}$$, $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to ___.
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Let $$I$$ be an identity matrix of order $$2 \times 2$$ and $$P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$$. Then the value of $$n \in N$$ for which $$P^n = 5I - 8P$$ is equal to ___.
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Let $$f : R \to R$$ satisfy the equation $$f(x+y) = f(x) \cdot f(y)$$ for all $$x, y \in R$$ and $$f(x) \neq 0$$ for any $$x \in R$$. If the function $$f$$ is differentiable at $$x = 0$$ and $$f'(0) = 3$$, then $$\lim_{h \to 0} \frac{1}{h}(f(h) - 1)$$ is equal to ___.
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Let $$P(x)$$ be a real polynomial of degree 3 which vanishes at $$x = -3$$. Let $$P(x)$$ have local minima at $$x = -1$$, local maxima at $$x = 1$$ and $$\int_{-1}^{1} P(x)dx = 18$$, then the sum of all the coefficients of the polynomial $$P(x)$$ is equal to ___.
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Let $$y = y(x)$$ be the solution of the differential equation $$xdy - ydx = \sqrt{(x^2 - y^2)}dx$$, $$x \ge 1$$, with $$y(1) = 0$$. If the area bounded by the line $$x = 1$$, $$x = e^\pi$$, $$y = 0$$ and $$y = y(x)$$ is $$\alpha e^{2\pi} + \beta$$, then the value of $$10(\alpha + \beta)$$ is equal to ___.
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Let the mirror image of the point $$(1, 3, a)$$ with respect to the plane $$\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) - b = 0$$ be $$(-3, 5, 2)$$. Then the value of $$|a + b|$$ is equal to ___.
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Let $$P$$ be a plane containing the line $$\frac{x-1}{3} = \frac{y+6}{4} = \frac{z+5}{2}$$ and parallel to the line $$\frac{x-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}$$. If the point $$(1, -1, \alpha)$$ lies on the plane $$P$$, then the value of $$|5\alpha|$$ is equal to ___.
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