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Match List I with List II:
| List-I (Physical Quantity) | List-II (Dimensional Formula) |
|---|---|
| A. Pressure gradient | I. $$[M^0L^2T^{-2}]$$ |
| B. Energy density | II. $$[M^1L^{-1}T^{-2}]$$ |
| C. Electric Field | III. $$[M^1L^{-2}T^{-2}]$$ |
| D. Latent heat | IV. $$[M^1L^1T^{-3}A^{-1}]$$ |
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A stone is projected at angle $$30°$$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be:
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A car is moving on a horizontal curved road with radius $$50$$ m. The approximate maximum speed of car will be, if friction between tyres and road is $$0.34$$. [Take $$g = 10$$ m s$$^{-2}$$]
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A block of mass $$m$$ slides down the plane inclined at angle $$30°$$ with an acceleration $$\frac{g}{4}$$. The value of coefficient of kinetic friction will be:
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Two particles of equal mass $$m$$ move in a circle of radius $$r$$ under the action of their mutual gravitational attraction. The speed of each particle will be:
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Surface tension of a soap bubble is $$2.0 \times 10^{-2}$$ N m$$^{-1}$$. Work done to increase the radius of soap bubble from $$3.5$$ cm to $$7$$ cm will be: [Take $$\pi = \frac{22}{7}$$]
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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: If $$dQ$$ and $$dW$$ represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics $$dQ = dU - dW$$.
Reason R: First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the option given below:
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A bicycle tyre is filled with air having pressure of $$270$$ kPa at $$27°$$C. The approximate pressure of the air in the tyre when the temperature increases to $$36°$$C is
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A person observes two moving trains, $$A$$ reaching the station and $$B$$ leaving the station with equal speed of $$30$$ m s$$^{-1}$$. If both trains emit sounds with frequency $$300$$ Hz, (Speed of sound: $$330$$ m s$$^{-1}$$) approximate difference of frequencies heard by the person will be:
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In a cuboid of dimension $$2L \times 2L \times L$$, a charge $$q$$ is placed at the centre of the surface $$S$$ having area of $$4L^2$$. The flux through the opposite surface to $$S$$ is given by
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Ratio of thermal energy released in two resistor $$R$$ and $$3R$$ connected in parallel in an electric circuit is:
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A single current carrying loop of wire carrying current $$I$$ flowing in anticlockwise direction seen from $$+ve$$ z direction and lying in $$xy$$ plane is shown in figure. The plot of $$\hat{j}$$ component of magnetic field $$(B_y)$$ at a distance $$a$$ (less than radius of the coil) and on $$yz$$ plane vs $$z$$ coordinate looks like
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The magnitude of magnetic induction at mid-point $$O$$ due to current arrangement as shown in figure will be
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Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $$R$$ is placed inside a large square loop of wire of side $$L(L \gg R)$$. The loops are coplanar and their centres coincide:
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Which of the following are true?
A. Speed of light in vacuum is dependent on the direction of propagation.
B. Speed of light in a medium is independent of the wavelength of light.
C. The speed of light is independent of the motion of the source.
D. The speed of light in a medium is independent of intensity.
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The threshold wavelength for photoelectric emission from a material is $$5500$$ $$\text{\AA}$$. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a
A. 75 W infra-red lamp
B. 10 W infra-red lamp
C. 75 W ultra-violet lamp
D. 10 W ultra-violet lamp
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If a radioactive element having half-life of $$30$$ min is undergoing beta decay, the fraction of radioactive element remains undecayed after $$90$$ min will be:
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Which of the following statement is not correct in the case of light emitting diodes?
A. It is a heavily doped p-n junction.
B. It emits light only when it is forward biased.
C. It emits light only when it is reverse biased.
D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.
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If the height of transmitting and receiving antennas are $$80$$ m each, the maximum line of sight distance will be:
Given: Earth's radius $$= 6.4 \times 10^6$$ m.
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A tennis ball is dropped on to the floor from a height of $$9.8$$ m. It rebounds to a height $$5.0$$ m. Ball comes in contact with the floor for $$0.2$$ s. The average acceleration during contact is ______ m s$$^{-2}$$. [Given $$g = 10$$ m s$$^{-2}$$]
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A $$0.4$$ kg mass takes $$8$$ s to reach ground when dropped from a certain height $$P$$ above surface of earth. The loss of potential energy in the last second of fall is ______ J. [Take $$g = 10$$ m s$$^{-2}$$]
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A solid sphere of mass $$2$$ kg is making pure rolling on a horizontal surface with kinetic energy $$2240$$ J. The velocity of centre of mass of the sphere will be ______ m s$$^{-1}$$.
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A body cools from $$60°$$C to $$40°$$C in $$6$$ minutes. If, temperature of surroundings is $$10°$$C. Then, after the next 6 minutes, its temperature will be ______ °C.
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Two simple harmonic waves having equal amplitudes of $$8$$ cm and equal frequency of $$10$$ Hz are moving along the same direction. The resultant amplitude is also $$8$$ cm. The phase difference between the individual waves is ______ degree.
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A point charge $$q_1 = 4q_0$$ is placed at origin. Another point charge $$q_2 = -q_0$$ is placed at $$x = 12$$ cm. Charge of proton is $$q_0$$. The proton is placed on $$x$$-axis so that the electrostatic force on the proton in zero. In this situation, the position of the proton from the origin is ______ cm.
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A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field $$B = 0.8$$ T. When released the radius of the loop starts shrinking at a constant rate of $$2$$ cm s$$^{-1}$$. The induced emf in the loop at an instant when the radius of the loop is $$10$$ cm will be ______ mV.
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In a Young's double slit experiment, two slits are illuminated with a light of wavelength $$800$$ nm. The line joining $$A_1P$$ is perpendicular to $$A_1A_2$$ as shown in the figure. If the first minimum is detected at $$P$$, the value of slits separation $$a$$ will be:
The distance of screen from slits $$D = 5$$ cm. Answer in mm.
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As shown in figures, three identical polaroids $$P_1$$, $$P_2$$ and $$P_3$$ are placed one after another. The pass axis of $$P_2$$ and $$P_3$$ are inclined at angle of $$60°$$ and $$90°$$ with respect to axis of $$P_1$$. The source $$S$$ has an intensity of $$256$$ W m$$^{-2}$$. The intensity of light at point $$O$$ is ______ W m$$^{-2}$$.
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A radioactive element $$^{242}_{92}$$X emits two $$\alpha$$-particles, one electron and two positrons. The product nucleus is represented by $$^{234}_{P}$$Y. The value of $$P$$ is ______.
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In a metre bridge experiment the balance point is obtained if the gaps are closed by $$2$$ $$\Omega$$ and $$3$$ $$\Omega$$. A shunt of $$X$$ $$\Omega$$ is added to $$3$$ $$\Omega$$ resistor to shift the balancing point by $$22.5$$ cm. The value of $$X$$ is ______.
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The shortest wavelength of hydrogen atom in Lyman series is $$\lambda$$. The longest wavelength in Balmer series of He$$^+$$ is
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The bond dissociation energy is highest for
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For 1 mol of gas, the plot of pV vs p is shown below. p is the pressure and V is the volume of the gas.
What is the value of compressibility factor at point A?
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Which of the given compounds can enhance the efficiency of hydrogen storage tank?
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The magnetic behaviour of Li$$_2$$O, Na$$_2$$O$$_2$$ and KO$$_2$$, respectively, are
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The correct order of hydration enthalpies is
(A) K$$^+$$
(B) Rb$$^+$$
(C) Mg$$^{2+}$$
(D) Cs$$^+$$
(E) Ca$$^{2+}$$
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During the borax bead test with CuSO$$_4$$, a blue green colour of the bead was observed in oxidising flame due to the formation of
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Compound that will give positive Lassaigne's test for both nitrogen and halogen is
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Correct statement about smog is
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The standard electrode potential (M$$^{3+}$$/M$$^{2+}$$) for V, Cr, Mn & Co are $$-0.26$$ V, $$-0.41$$ V, $$+1.57$$ V and $$+1.97$$ V, respectively. The metal ions which can liberate H$$_2$$ from a dilute acid are
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Following figure shows dependence of molar conductance of two electrolytes on concentration. $$\Lambda^0_m$$ is the limiting molar conductivity.
The number of Incorrect statement(s) from the following is ______
(A) $$\Lambda^0_m$$ for electrolyte A is obtained by extrapolation
(B) For electrolyte B, $$\Lambda_m$$ Vs $$\sqrt{c}$$ graph is a straight line with intercept equal to $$\Lambda^0_m$$
(C) At infinite dilution, the value of degree of dissociation approach zero for electrolyte B.
(D) $$\Lambda_m$$ for any electrolyte A or B can be calculated using $$\lambda°$$ for individual ions.
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Which of the following salt solutions would coagulate the colloid solution formed when FeCl$$_3$$ is added to NaOH solution, at the fastest rate?
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The reaction representing the Mond process for metal refining is ______
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"A" obtained by Ostwald's method involving air oxidation of NH$$_3$$, upon further air oxidation produces "B". "B" on hydration forms an oxoacid of Nitrogen along with evolution of "A". The oxoacid also produces "A" and gives positive brown ring test
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Chiral complex from the following is:
Here en = ethylene diamine
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Identify the correct order for the given property for following compounds
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The increasing order of pK$$_a$$ for the following phenols is
(A) 2,4-Dinitrophenol
(B) 4-Nitrophenol
(C) 2,4,5-Trimethylphenol
(D) Phenol
(E) 3-Chlorophenol
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Match List I with List II.
| List I (Reaction) | List II (Reagents) |
|---|---|
| (A) Hoffmann Degradation | (I) Conc. KOH, $$\Delta$$ |
| (B) Clemmensen reduction | (II) CHCl$$_3$$, NaOH/H$$_3$$O$$^+$$ |
| (C) Cannizzaro reaction | (III) Br$$_2$$, NaOH |
| (D) Reimer-Tiemann reaction | (IV) Zn $$-$$ Hg/HCl |
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The major product 'P' for the following sequence of reactions is:
Match List I with List II
| List I (Antimicrobials) | List II (Names) |
|---|---|
| (A) Narrow Spectrum Antibiotic | (I) Furacin |
| (B) Antiseptic | (II) Sulphur dioxide |
| (C) Disinfectants | (III) Penicillin-G |
| (D) Broad spectrum antibiotic | (IV) Chloramphenicol |
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Number of cyclic tripeptides formed with 2 amino acids A and B is:
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The number of molecules or ions from the following, which do not have odd number of electrons are ______.
(A) NO$$_2$$
(B) ICl$$_4^-$$
(C) BrF$$_3$$
(D) ClO$$_2$$
(E) NO$$_2^+$$
(F) NO
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Consider the following reaction approaching equilibrium at $$27°$$C and 1 atm pressure
$$\text{A} + \text{B} \underset{K_r=10^2}{\overset{K_f=10^3}{\rightleftharpoons}} \text{C} + \text{D}$$
The standard Gibb's energy change $$(\Delta_r G°)$$ at $$27°$$C is $$(-)$$ ______ kJ mol$$^{-1}$$. (Nearest integer).
(Given: R $$= 8.3$$ J K$$^{-1}$$ mol$$^{-1}$$ and $$\ln 10 = 2.3$$)
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Water decomposes at 2300 K
$$\text{H}_2\text{O}(g) \to \text{H}_2(g) + \frac{1}{2}\text{O}_2(g)$$
The percent of water decomposing at 2300 K and 1 bar is ______ (Nearest integer). Equilibrium constant for the reaction is $$2 \times 10^{-3}$$ at 2300 K
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Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is $$x \times 10^{-1}$$. The value of $$x$$ is ______ (Nearest integer). Assume complete dissociation.
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Following chromatogram was developed by adsorption of compound 'A' on a 6 cm TLC glass plate. Retardation factor of the compound 'A' is ______ $$\times 10^{-1}$$.
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17 mg of a hydrocarbon (M.F. C$$_{10}$$H$$_{16}$$) takes up 8.40 mL of the H$$_2$$ gas measured at 0°C and 760 mm of Hg. Ozonolysis of the same hydrocarbon yields
The number of double bond/s present in the hydrocarbon is
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Solid Lead nitrate is dissolved in 1 litre of water. The solution was found to boil at $$100.15°$$C. When $$0.2$$ mol of NaCl is added to the resulting solution, it was observed that the solution froze at $$-0.8°$$C. The solutibility product of PbCl$$_2$$ formed is ______ $$\times 10^{-6}$$ at 298 K. (Nearest integer)
Given: K$$_b = 0.5$$ K kg mol$$^{-1}$$ and K$$_f = 1.8$$ kg mol$$^{-1}$$. Assume molality to be equal to molarity in all cases.
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For certain chemical reaction X $$\to$$ Y, the rate of formation of product is plotted against the time as shown in the figure. The number of Correct statement/s from the following is ______
(A) Over all order of this reaction is one
(B) Order of this reaction can't be determined
(C) In region-I and III, the reaction is of first and zero order respectively
(D) In region-II, the reaction is of first order
(E) In region-II, the order of reaction is in the range of 0.1 to 0.9.
The sum of bridging carbonyls in W(CO)$$_6$$ and Mn$$_2$$(CO)$$_{10}$$ is ______.
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Let $$\lambda \neq 0$$ be a real number. Let $$\alpha, \beta$$ be the roots of the equation $$14x^2 - 31x + 3\lambda = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$35x^2 - 53x + 4\lambda = 0$$. Then $$\frac{3\alpha}{\beta}$$ and $$\frac{4\alpha}{\gamma}$$ are the roots of the equation:
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For two non-zero complex numbers $$z_1$$ and $$z_2$$, if $$\text{Re}(z_1 z_2) = 0$$ and $$\text{Re}(z_1 + z_2) = 0$$, then which of the following are possible?
(A) Im $$(z_1) > 0$$ and Im $$(z_2) > 0$$
(B) Im $$(z_1) < 0$$ and Im $$(z_2) > 0$$
(C) Im $$(z_1) > 0$$ and Im $$(z_2) < 0$$
(D) Im $$(z_1) < 0$$ and Im $$(z_2) < 0$$
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Let $$f(\theta) = 3\left(\sin^4\left(\frac{3\pi}{2} - \theta\right) + \sin^4(3\pi + \theta)\right) - 2\left(1 - \sin^2 2\theta\right)$$ and $$S = \left\{\theta \in [0, \pi] : f'(\theta) = -\frac{\sqrt{3}}{2}\right\}$$. If $$4\beta = \sum_{\theta \in S} \theta$$ then $$f(\beta)$$ is equal to
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A light ray emits from the origin making angle $$30°$$ with the positive $$x$$-axis. After getting reflected by the line $$x + y = 1$$, if this ray intersects x-axis at Q, then the abscissa of Q is
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Let $$B$$ and $$C$$ be the two points on the line $$y + x = 0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y - 2x = 2$$ such that $$\triangle ABC$$ is an equilateral triangle. Then, the area of the $$\triangle ABC$$ is
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Let the tangents at the points $$A(4, -11)$$ and $$B(8, -5)$$ on the circle $$x^2 + y^2 - 3x + 10y - 15 = 0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to
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Let $$x = 2$$ be a root of the equation $$x^2 + px + q = 0$$ and $$f(x) = \begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}$$. Then $$\lim_{x \to 2p^+} [f(x)]$$
where $$[.]$$ denotes greatest integer function, is
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If $$p, q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p$$, $$q$$ and $$r$$ makes the logical expression $$\{(p \vee q) \wedge ((\neg p) \vee r)\} \to ((\neg q) \vee r)$$ false?
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Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a $$3 \times 3$$ matrix $$A$$ such that $$A^2 = 3A + \alpha I$$. If $$A^4 = 21A + \beta I$$, then
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Consider the following system of equations
$$\alpha x + 2y + z = 1$$
$$2\alpha x + 3y + z = 1$$
$$3x + \alpha y + 2z = \beta$$
For some $$\alpha, \beta \in \mathbb{R}$$. Then which of the following is NOT correct.
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The domain of $$f(x) = \frac{\log_{(x+1)}(x-2)}{e^{2\log_e x^2 - (2x+3)}}$$, $$x \in R$$ is
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Let $$f : R \to R$$ be a function such that $$f(x) = \frac{x^2+2x+1}{x^2+1}$$. Then
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Let $$f(x) = x + \frac{a}{\pi^2-4}\sin x + \frac{b}{\pi^2-4}\cos x$$, $$x \in \mathbb{R}$$ be a function which satisfies $$f(x) = x + \int_0^{\pi/2} \sin(x+y)f(y)dy$$. Then $$(a+b)$$ is equal to
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Let $$[x]$$ denote the greatest integer $$\leq x$$. Consider the function $$f(x) = \max\{x^2, 1 + [x]\}$$. Then the value of the integral $$\int_0^2 f(x) dx$$ is:
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Let $$A = \left\{(x,y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4-(x-1)^2}\right\}$$ and
$$B = \left\{(x,y) \in \mathbb{R} \times \mathbb{R} : 0 \leq y \leq \min\left\{2x, \sqrt{4-(x-1)^2}\right\}\right\}$$. Then the ratio of the area of $$A$$ to the area of $$B$$ is
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Let $$\Delta$$ be the area of the region $$\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}$$. Then $$\frac{1}{2}\left(\Delta - 21\sin^{-1}\frac{2}{\sqrt{7}}\right)$$ is equal to
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Let $$y = f(x)$$ be the solution of the differential equation $$y(x+1)dx - x^2 dy = 0$$, $$y(1) = e$$. Then $$\lim_{x \to 0^+} f(x)$$ is equal to
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If the vectors $$\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$$, $$\vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$$ are coplanar and the projection of $$\vec{a}$$ on the vector $$\vec{b}$$ is $$\sqrt{54}$$ units, then the sum of all possible values of $$\lambda + \mu$$ is equal to
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Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $$\sigma^2$$ represent mean and variance of X, respectively, then $$10(\mu^2 + \sigma^2)$$ is equal to
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If all the six digit numbers $$x_1x_2x_3x_4x_5x_6$$ with $$0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$72^{th}$$ number is ______.
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Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is
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Let $$a_1, a_2, a_3, \ldots$$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $$9$$ and the sum of fifth and seventh terms is $$24$$, then $$a_1a_9 + a_2a_4a_9 + a_5 + a_7$$ is equal to
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Let the coefficients of three consecutive terms in the binomial expansion of $$(1 + 2x)^n$$ be in the ratio $$2 : 5 : 8$$. Then the coefficient of the term, which is in the middle of these three terms, is
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If the co-efficient of $$x^9$$ in $$\left(\alpha x^3 + \frac{1}{\beta x}\right)^{11}$$ and the co-efficient of $$x^{-9}$$ in $$\left(\alpha x - \frac{1}{\beta x^3}\right)^{11}$$ are equal, then $$(\alpha\beta)^2$$ is equal to
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y) = f(x) + f(y) - 1, \forall x, y \in \mathbb{R}$$. If $$f'(0) = 2$$, then $$|f(-2)|$$ is equal to
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Suppose f is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x, y \in \mathbb{N}$$ and $$f(1) = \frac{1}{5}$$. If $$\sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$$ then m is equal to ______.
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A$$, $$B$$, $$C$$ and $$D$$ be $$\vec{a} - \vec{b} + \vec{c}$$, $$\lambda\vec{a} - 3\vec{b} + 4\vec{c}$$, $$-\vec{a} + 2\vec{b} - 3\vec{c}$$ and $$2\vec{a} - 4\vec{b} + 6\vec{c}$$ respectively. If $$\vec{AB}$$, $$\vec{AC}$$ and $$\vec{AD}$$ are coplanar, then $$\lambda$$ is:
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Let the equation of the plane P containing the line $$x + 10 = \frac{8-y}{2} = z$$ be $$ax + by + 3z = 2(a+b)$$ and the distance of the plane P from the point $$(1, 27, 7)$$ be $$c$$. Then $$a^2 + b^2 + c^2$$ is equal to
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Let the co-ordinates of one vertex of $$\triangle ABC$$ be $$A(0, 2, \alpha)$$ and the other two vertices lie on the line $$\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\triangle ABC$$ is $$21$$ sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ______.
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