In the density measurement of a cube, the mass and edge length are measured as $$(10.00 \pm 0.10)$$ kg and $$(0.10 \pm 0.01)$$ m, respectively. The error in the measurement of density is:
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In the density measurement of a cube, the mass and edge length are measured as $$(10.00 \pm 0.10)$$ kg and $$(0.10 \pm 0.01)$$ m, respectively. The error in the measurement of density is:
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A ball is thrown vertically up (taken as +z-axis) from the ground. The correct momentum-height (p-h) diagram is:
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The stream of a river is flowing with a speed of 2 km h$$^{-1}$$. A swimmer can swim at a speed of 4 km h$$^{-1}$$. The direction of the swimmer with respect to the flow of the river, to cross the river straight, is:
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A uniform cable of mass $$M$$ and length $$L$$ is placed on a horizontal surface such that its $$\left(\frac{1}{n}\right)^{th}$$ part is hanging below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be:
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A body of mass 2 kg makes an elastic collision with a second body at rest and continues to move in the original direction but with one fourth of its original speed. What is the mass of the second body?
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A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $$\theta$$, where $$\theta$$ is the angle by which it has rotated, is given as $$k\theta^2$$. If its moment of inertia is I then the angular acceleration of the disc is:
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The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R, (ii) a solid cylinder of radius $$\frac{R}{2}$$ and (iii) a solid sphere of radius $$\frac{R}{4}$$. If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is:
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A solid sphere of mass M and radius a is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance 3a from the centre will be:
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If 'M' is the mass of water that rises in a capillary tube of radius 'r', then mass of water which will rise in a capillary tube of radius '2r' is:
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Following figure shows two processes A and B for a gas. If $$\Delta Q_A$$ and $$\Delta Q_B$$ are the amount of heat absorbed by the system in two cases, and $$\Delta U_A$$ and $$\Delta U_B$$ are changes in internal energies, respectively, then:
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For given gas at 1 atm pressure, rms speed of the molecules is 200 m/s at 127°C. At 2 atm pressure and at 227°C, the rms speed of the molecules will be:
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An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is $$\bar{v}$$, m is its mass and k$$_B$$ is Boltzmann's constant, then its temperature will be:
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A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $$\frac{1}{16}$$th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:
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A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is $$y = 0.3 \sin(0.157x) \cos(200\pi t)$$. The length of the string is: (All quantities are in SI units.)
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The pressure wave, $$P = 0.01 \sin[1000t - 3x]$$ N m$$^{-2}$$, corresponds to the sound produced by a vibrating blade on a day when atmospheric temperature is 0°C. On some other day when temperature is T, the speed of sound produced by the same blade and at the same frequency is found to be 336 m s$$^{-1}$$. Approximate value of T is:
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A system of three charges are placed as shown in the figure:
If $$D >> d$$, the potential energy of the system is best given by:
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A capacitor with capacitance 5 $$\mu$$F is charged to 5 $$\mu$$C. If the plates are pulled apart to reduce the capacitance to 2 $$\mu$$F, how much work is done?
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A wire of resistance R is bent to form a square ABCD as shown in the figure. The effective resistance between E and C is: (E is mid-point of arm CD)
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Determine the charge on the capacitor in the following circuit:
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A moving coil galvanometer has resistance 50 $$\Omega$$ and it indicates full deflection at 4 mA current. A voltmeter is made using this galvanometer and a 5 k$$\Omega$$ resistance. The maximum voltage, that can be measured using this voltmeter, will be close to:
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A rigid square loop of side 'a' and carrying current $$I_2$$ is lying on a horizontal surface near a long current $$I_1$$ carrying wire in the same plane as shown in figure. The net force on the loop due to the wire will be:
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A rectangular coil (Dimension 5 cm $$\times$$ 2.5 cm) with 100 turns, carrying a current of 3 A in the clock-wise direction, is kept centered at the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through 45° about Z-axis, then the torque on the coil is:
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The total number of turns and cross-section area in a solenoid is fixed. However, its length L is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to:
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The magnetic field of a plane electromagnetic wave is given by $$\vec{B} = B_0[\hat{i}\cos(kz - \omega t)] + B_1[\hat{j}\cos(kz + \omega t)]$$, where $$B_0 = 3 \times 10^{-5}$$ T and $$B_1 = 2 \times 10^{-6}$$ T. The RMS value of the force experienced by a stationary charge $$Q = 10^{-4}$$ C at z = 0 is closest to:
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A concave mirror for face viewing has a focal length of 0.4 m. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of 5 is:
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The figure shows a Young's double slit experimental setup. It is observed that when a thin transparent sheet of thickness t and refractive index $$\mu$$ is put in front of one of the slits, the central maximum gets shifted by a distance equal to n fringe width. If the wavelength of light used is $$\lambda$$ then t will be:
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The electric field of light wave is given as $$\vec{E} = 10^{-3} \cos\left(\frac{2\pi x}{5 \times 10^{-7}} - 2\pi \times 6 \times 10^{14}t\right) \hat{x} \frac{N}{C}$$. This light falls on a metal plate of work function 2 eV. The stopping potential of the photo-electrons is: Given, E (in eV) = $$\frac{12375}{\lambda(\text{in } \mathring{A})}$$
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Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be:
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An NPN transistor is used in common emitter configuration as an amplifier with 1 k$$\Omega$$ load resistance. Signal voltage of 10 mV is applied across the base-emitter. This produces a 3 mA change in the collector current and 15 $$\mu$$A change in the base current of the amplifier. The input resistance and voltage gain are:
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A signal A cos$$\omega$$t is transmitted using $$v_0 \sin\omega_0 t$$ as carrier wave. The correct amplitude modulated (AM) signal is:
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For a reaction, N$$_2$$(g) + 3H$$_2$$(g) $$\rightarrow$$ 2NH$$_3$$(g), identify di-hydrogen (H$$_2$$) as a limiting reagent in the following reaction mixtures.
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For any given series of spectral lines of atomic hydrogen, let $$\Delta\bar{v} = \bar{v}_{max} - \bar{v}_{min}$$ be the difference in maximum and minimum wave number in cm$$^{-1}$$. The ratio $$\Delta\bar{v}_{Lyman}/\Delta\bar{v}_{Balmer}$$ is:
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The element having the greatest difference between its first and second ionization energies, is:
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Among the following, the molecule expected to be stabilized by anion formation is: C$$_2$$, O$$_2$$, NO, F$$_2$$
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Consider the van der Waal's constants, a and b, for the following gases.
Which gas is expected to have the highest critical temperature?
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Among the following, the set of parameters that represents path functions, is:
i) q + w
ii) q
iii) w
iv) H - TS
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Magnesium powder burns in air to give:
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C$$_{60}$$, an allotrope of carbon contains:
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The correct IUPAC name of the following compound is:
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The increasing order of reactivity of the following compounds towards aromatic electrophilic substitution reaction is:
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The major product of the following reaction is:
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The major product of the following reaction is:
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Excessive release of CO$$_2$$ into the atmosphere results in:
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The osmotic pressure of a dilute solution of an ionic compound XY in water is four times that of a solution of 0.01 M BaCl$$_2$$ in water. Assuming complete dissociation of the given ionic compounds in water, the concentration of XY (in mol L$$^{-1}$$) in solution is:
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Liquid M and liquid N form an ideal solution. The vapour pressures of pure liquids M and N are 450 and 700 mmHg, respectively, at the same temperature. Then correct statements is:
(x$$_M$$ = Mole fraction of 'M' in solution; x$$_N$$ = Mole fraction of 'N' in solution; y$$_M$$ = Mole fraction of 'M' in vapour phase; y$$_N$$ = Mole fraction of 'N' in vapour phase)
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The standard Gibbs energy for the given cell reaction in kJ mol$$^{-1}$$ at 298 K is:
Zn(s) + Cu$$^{2+}$$(aq) $$\rightarrow$$ Zn$$^{2+}$$(aq) + Cu(s),
E$$^0$$ = 2 V at 298 K
(Faraday's constant, F = 96000 C mol$$^{-1}$$)
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The given plots represent the variation of the concentration of a reactant R with time for two different reactions (i) and (ii). The respective orders of the reaction are:
The aerosol is a kind of colloid in which:
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The ore that contains the metal in the form of fluoride is known as which of the following?
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Match the catalysts (Column I) with products (Column II)
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The correct order of the oxidation states of nitrogen in NO, N$$_2$$O, NO$$_2$$ and N$$_2$$O$$_3$$ is:
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The number of water molecules not coordinated to copper ion directly in CuSO$$_4$$ . 5H$$_2$$O, is:
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The degenerate orbitals of [Cr(H$$_2$$O)$$_6$$]$$^{3+}$$ are:
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The one that will show optical activity is (en = ethane-1, 2-diamine):
The major product of the following reaction is:
1. PBr$$_3$$
2. KOH (alc.)
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The organic compound that gives following qualitative analysis is:
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The major product of the following reaction is:
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The major product of the following reaction is:
CH$$_3$$CH = CHCO$$_2$$CH$$_3$$ $$\xrightarrow{\text{LiAlH}_4}$$
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Aniline dissolved in dilute HCl is reacted with sodium nitrite at 0°C. This solution was added dropwise to a solution containing an equimolar mixture of aniline and phenol in dilute HCl. The structure of the major product is:
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Which of the following statements is not true about sucrose?
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Let $$p, q \in Q$$. If $$2 - \sqrt{3}$$ is a root of the quadratic equation $$x^2 + px + q = 0$$, then:
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All the points in the set $$S = \left\{\frac{\alpha + i}{\alpha - i}, \alpha \in R\right\}$$, $$i = \sqrt{-1}$$ lie on a:
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A committee of 11 members is to be formed from 8 males and 5 females. If $$m$$ is the number of ways the committee is formed with at least 6 males and $$n$$ is the number of ways the committee is formed with at least 3 females, then:
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Let the sum of the first $$n$$ terms of a non-constant A.P., $$a_1, a_2, a_3, \ldots, a_n$$ be $$50n + \frac{n(n-7)}{2}A$$, where A is a constant. If $$d$$ is the common difference of this A.P., then the ordered pair $$(d, a_{50})$$ is equal to:
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If the fourth term in the Binomial expansion of $$\left(\frac{2}{x} + x^{\log_8 x}\right)^6$$, $$(x > 0)$$ is $$20 \times 8^7$$, then a value of $$x$$ is:
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The value of $$\cos^2 10° - \cos 10° \cos 50° + \cos^2 50°$$ is:
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Let $$S = \{\theta \in [-2\pi, 2\pi] : 2\cos^2\theta + 3\sin\theta = 0\}$$. Then the sum of the elements of S is:
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Slope of a line passing through $$P(2, 3)$$ and intersecting the line $$x + y = 7$$ at a distance of 4 units from $$P$$, is:
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If a tangent to the circle $$x^2 + y^2 = 1$$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:
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If one end of a focal chord of the parabola, $$y^2 = 16x$$ is at $$(1, 4)$$, then the length of this focal chord is:
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If the line $$y = mx + 7\sqrt{3}$$ is normal to the hyperbola $$\frac{x^2}{24} - \frac{y^2}{18} = 1$$, then a value of $$m$$ is:
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For any two statement $$p$$ and $$q$$, the negative of the expression $$p \lor (\sim p \land q)$$ is:
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If the standard deviation of the numbers $$-1, 0, 1, k$$ is $$\sqrt{5}$$ where $$k \gt 0$$, then $$k$$ is equal to:
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If $$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \cdots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix}$$, then the inverse of $$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$$ is:
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Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^2 + x + 1 = 0$$. Then for $$y \neq 0$$ in R, $$\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha \end{vmatrix}$$ is equal to:
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If the function $$f: R - \{1, -1\} \rightarrow A$$ defined by $$f(x) = \frac{x^2}{1 - x^2}$$, is surjective, then $$A$$ is equal to:
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Let $$f(x) = 15 - |x - 10|$$; $$x \in R$$. Then the set of all values of $$x$$, at which the function $$g(x) = f(f(x))$$ is not differentiable, is:
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If the function $$f$$ defined on $$\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$$ by $$f(x) = \begin{cases} \frac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$$ is continuous, then $$k$$ is equal to:
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Let $$\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)$$, where the function $$f$$ satisfies $$f(x + y) = f(x)f(y)$$ for all natural numbers $$x$$, $$y$$ and $$f(1) = 2$$. Then the natural number 'a' is:
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If $$f(x)$$ is a non-zero polynomial of degree four, having local extreme points at $$x = -1, 0, 1$$; then the set $$S = \{x \in R : f(x) = f(0)\}$$ contains exactly:
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If the tangent to the curve, $$y = x^3 + ax - b$$ at the point $$(1, -5)$$ is perpendicular to the line, $$-x + y + 4 = 0$$, then which one of the following points lies on the curve?
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Let $$S$$ be the set of all values of $$x$$ for which the tangent to the curve $$y = f(x) = x^3 - x^2 - 2x$$ at $$(x, y)$$ is parallel to the line segment joining the points $$(1, f(1))$$ and $$(-1, f(-1))$$, then $$S$$ is equal to:
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$$\int \sec^2 x \cdot \cot^{4/3} x \, dx$$ is equal to:
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The value of $$\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x} dx$$ is:
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The area (in sq. units) of the region $$A = \{(x, y) : x^2 \le y \le x + 2\}$$ is:
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The solution of the differential equation $$x\frac{dy}{dx} + 2y = x^2$$, $$(x \neq 0)$$ with $$y(1) = 1$$, is:
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Let $$\vec{\alpha} = 3\hat{i} + \hat{j}$$ and $$\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$$. If $$\vec{\beta} = \vec{\beta_1} - \vec{\beta_2}$$, where $$\vec{\beta_1}$$ is parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ is perpendicular to $$\vec{\alpha}$$, then $$\vec{\beta_1} \times \vec{\beta_2}$$ is equal to:
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A plane passing through the points $$(0, -1, 0)$$ and $$(0, 0, 1)$$ and making an angle $$\frac{\pi}{4}$$ with the plane $$y - z + 5 = 0$$, also passes through the point:
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If the line, $$\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}$$ meets the plane, $$x + 2y + 3z = 15$$ at a point P, then the distance of P from the origin is:
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Four persons can hit a target correctly with probabilities $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$ and $$\frac{1}{8}$$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:
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