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A thin 1 m long rod has a radius of 5 mm. A force of $$50\pi \times 10^3$$ N is applied at one end to determine its Young's modulus. Assume that the force is exactly known. If the least count in the measurement of all lengths is 0.01 mm, which of the following statements is false?
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$$A$$, $$B$$, $$C$$, and $$D$$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $$AD = C\ln(BD)$$ holds true. Then which of the combination is not a meaningful quantity?
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A particle of mass $$m$$ is acted upon by a force $$F$$ given by the empirical law $$F = \frac{R}{t^2}v(t)$$. If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot
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Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute (rpm) to ensure proper mixing is close to:
(Take the radius of the drum to be 1.25 m and its axle to be horizontal):
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The velocity-time graph of a particle of mass 10 kg is shown in the figure. The net work done on the particle in the first two seconds of the motion is
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A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by $$n^2Rt^2$$, where $$n$$ is a constant. The power delivered to the particle by the force acting on it, is:
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In the figure shown $$ABC$$ is a uniform wire. If the center of mass of the wire lies vertically below point $$A$$, then $$\frac{BC}{AB}$$ is close to
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An astronaut of mass $$m$$ is working on a satellite orbiting the earth at a distance $$h$$ from the earth's surface. The radius of the earth is $$R$$, while its mass is $$M$$. The gravitational pull $$F_G$$ on the astronaut is
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A bottle has an opening of radius a and length b. A cork of length b and radius $$(a + \Delta a)$$ where $$(\Delta a \ll a)$$, is compressed to fit into the opening completely (see figure). If the bulk modulus of the cork is B and the coefficient of friction between the bottle and the cork is $$\mu$$, then the force needed to push the cork into the bottle is
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A Carnot freezer takes heat from water at $$0°C$$ inside it and rejects it to the room at a temperature of $$27°C$$. The latent heat of ice is $$336 \times 10^3$$ J kg$$^{-1}$$. If 5 kg of water at $$0°C$$ is converted into ice at $$0°C$$ by the freezer, then the energy consumed by the freezer is close to:
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Which of the following shows the correct relationship between the pressure $$P$$ and density $$\rho$$ of an ideal gas at constant temperature?
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In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to:
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A toy-car, blowing its horn, is moving with a steady speed of 5 m s$$^{-1}$$, away from a wall. An observer, towards whom the toy car is moving, is able to hear 5 beats per second. If the velocity of sound in air is 340 m s$$^{-1}$$, the frequency of the horn of the toy car is close to
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Within a spherical charge distribution of charge density $$\rho(r)$$, N equipotential surfaces of potential $$V_0$$, $$V_0 + \Delta V$$, $$V_0 + 2\Delta V$$, ... $$V_0 + N\Delta V$$ $$(\Delta V > 0)$$, are drawn and have increasing radii $$r_0$$, $$r_1$$, $$r_2$$, ... $$r_N$$, respectively. If the difference in the radii of the surfaces is constant for all values of $$V_0$$ and $$\Delta V$$ then:
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The figure shows a network of capacitors where the number indicates capacitances in micro Farad. The value of capacitance C if the equivalent capacitance between point A and B is to be 1 $$\mu F$$ is:
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A galvanometer has a 50 division scale. Battery has no internal resistance. It is found that there is deflection of 40 divisions when R.B. = 2400 $$\Omega$$. Deflection becomes 20 divisions when resistance taken from resistance box is 4900 $$\Omega$$. Then we can conclude:
Note: This question is awarded as the bonus. Now the question is corrected.
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The resistance of an electrical toaster has a temperature dependence given by $$R(T) = R_0[1 + \alpha(T - T_0)]$$ in its range of operation. At $$T_0 = 300$$ K, $$R = 100$$ $$\Omega$$ and at $$T = 500$$ K, $$R = 120$$ $$\Omega$$. The toaster is connected to a voltage source at 200 V and its temperature is raised at a constant rate from 300 to 500 K in 30 s. The total work done in raising the temperature is:
Note: This question was awarded as the bonus since all options were incorrect in the exam.
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A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15 m and height 5 m is flying towards east over Delhi. Its speed is 240 m s$$^{-1}$$. The earth's magnetic field over Delhi is $$5 \times 10^{-5}$$ T with the declination angle ~0° and dip of $$\theta$$ such that $$\sin \theta = \frac{2}{3}$$. If the voltage developed is $$V_B$$ between the lower and upper side of the plane and $$V_W$$ between the tips of the wings then $$V_B$$ and $$V_W$$ are close to:
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A conducting metal circular-wire-loop of radius $$r$$ is placed perpendicular to a magnetic field which varies with time as $$B = B_0 e^{-t/\tau}$$, where $$B_0$$ and $$\tau$$ are constants at time $$t = 0$$. If the resistance of the loop is $$R$$, then the heat generated in the loop after a long time $$(t \to \infty)$$ is
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Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed v in a uniform magnetic field B going into the plane of the paper (see the figure below). If the charge densities $$\sigma_1$$ and $$\sigma_2$$ are induced on the left and right surfaces respectively of the sheet, then (ignore fringe effects)
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Consider an electromagnetic wave propagating in vacuum. Choose the correct statement:
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To determine refractive index of glass slab using a travelling microscope, minimum number of readings required are:
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A hemispherical glass body of radius 10 cm and refractive index 1.5 is silvered on its curved surface. A small air bubble is 6 cm below the flat surface inside it along the axis. The position of the image of the air bubble made by the mirror is seen:
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Two stars are 10 light years away from the earth. They are seen through a telescope of objective diameter 30 cm. The wavelength of light is 600 nm. To see the stars just resolved by the telescope, the minimum distance between them should be (1 light year = $$9.46 \times 10^{15}$$ m) of the order of:
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A photoelectric surface is illuminated successively by monochromatic light of wavelengths $$\lambda$$ and $$\frac{\lambda}{2}$$. If the maximum kinetic energy of the emitted photoelectrons in the second case is 3 times that in the first case, the work function of the surface is:
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A neutron moving with a speed 'v' makes a head on collision with a stationary hydrogen atom in ground state. The minimum kinetic energy of the neutron for which perfectly inelastic collision will take place is:
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The ratio (R) of output resistance $$r_0$$, and the input resistance $$r_i$$ in measurements of input and output characteristics of a transistor is typically in the range:
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To get an output of 1 from the circuit shown in the figure the input must be:
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A realistic graph depicting the variation of the reciprocal of input resistance in an input characteristics measurement in a common emitter transistor configuration is:
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A modulated signal $$C_m(t)$$ has the form $$C_m(t) = 30\sin 300\pi t + 10(\cos 200\pi t - \cos 400\pi t)$$. The carrier frequency $$f'_c$$, the modulating frequency (message frequency) $$f_\omega$$ and the modulation index $$\mu$$ are respectively given by:
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Aqueous solution of which of the following salts will not contain ions with the electronic configuration $$1s^2 2s^2 2p^6 3s^2 3p^6$$?
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The following statements concern elements in the periodic table. Which of the following is true?
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The bond angle H-X-H is the greatest in the compound:
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Initially, the root-mean-square (RMS) velocity of $$N_2$$ molecules at certain temperature is u. If this temperature is doubled and all the nitrogen molecules dissociate into nitrogen atoms, then the new RMS velocity will be:
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If 100 mole of $$H_2O_2$$ decompose at 1 bar and 300 K, the work is done (kJ) by one mole of $$O_2(g)$$ as it expands against 1 bar pressure is:
$$2H_2O_2(l) \rightleftharpoons 2H_2O(l) + O_2(g)$$
$$(R = 8.3$$ J K$$^{-1}$$ mol$$^{-1})$$
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A solid XY kept in an evacuated sealed container undergoes decomposition to form a mixture of gases X and Y at temperature T. The equilibrium pressure is 10 bar in this vessel. $$K_p$$ for this reaction is?
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The volume of 0.1 M strong dibasic acid sufficient to neutralize 1 g of a base that furnishes 0.04 mole of $$OH^-$$ in aqueous solution is:
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What is the commercial name for calcium oxide?
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Identify the reaction which does not liberate hydrogen
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Assertion: Among the carbon allotropes, diamond is an insulator, whereas, graphite is a good conductor of electricity.
Reason: Hybridization of carbon in diamond and graphite are sp$$^3$$ and sp$$^2$$ respectively.
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Sodium extract is heated with concentrated $$HNO_3$$ before testing for halogens because:
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Consider the reaction sequence below:
X is:
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Which one of the following substances used in dry cleaning is a better strategy to control environmental pollution?
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An aqueous solution of a salt $$MX_2$$ at certain temperature has a Van't Hoff factor of 2. What is the degree of dissociation for this solution of the salt?
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Oxidation of succinate ion produces ethylene and carbon dioxide gases. On passing 0.2 Faraday electricity through an aqueous solution of potassium succinate, what is the total volume of gases (at both cathode and anode) at STP (1 atm and 273 K)?
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Identify the correct statement:
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The rate law for the reaction below is given by the expression k[A][B]
$$A + B \to$$ Product
If the concentration of B is increased from 0.1 to 0.3 mol, keeping the value of A at 0.1 mol, the rate constant will be:
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Gold numbers of some colloids are Gelatin: 0.005 - 0.01, Gum Arabic: 0.15 - 0.25, Oleate: 0.04 - 1.0, Starch: 15 - 25. Which among the following is a better protective colloid?
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Extraction of copper by smelting uses silica as an additive to remove.
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Identify the incorrect statement:
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Which of the following are the transition metal ions responsible for colour in ruby and emerald respectively?
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Which of the following is an example of homoleptic complex?
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Fluorination of an aromatic ring is easily accomplished by treating a diazonium salt with $$HBF_4$$. Which of the following conditions is correct about this reaction?
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Which one of the following reagents is not suitable for the elimination reaction?
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Bromination of cyclohexene under conditions given below yields:
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The correct statement about the synthesis of erythritol $$(C(CH_2OH)_4)$$ used in the preparation of PETN is
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The N which contributes least to the basicity of the compound is:
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Which of the following polymers is synthesized using a free radical polymerization technique?
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Which of the following is a bactericidal antibiotic?
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Observation of "Rhumann's purple" is a confirmatory test for the presence of:
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If $$x$$ is a solution of the equation $$\sqrt{2x+1} - \sqrt{2x-1} = 1$$, $$\left(x \geq \frac{1}{2}\right)$$, then $$\sqrt{4x^2 - 1}$$ is equal to:
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Let $$z = 1 + ai$$, be a complex number, $$a > 0$$, such that $$z^3$$ is a real number. Then, the sum $$1 + z + z^2 + \ldots + z^{11}$$ is equal to:
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If $$\frac{^{n+2}C_6}{^{n-2}P_2} = 11$$, then $$n$$ satisfies the equation:
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Let $$a_1, a_2, a_3, \ldots a_n, \ldots$$, be in A.P. If $$a_3 + a_7 + a_{11} + a_{15} = 72$$, then the sum of its first 17 terms is equal to:
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The sum $$\sum_{r=1}^{10} (r^2 + 1) \times (r!)$$, is equal to:
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If the coefficients of $$x^{-2}$$ and $$x^{-4}$$, in the expansion of $$\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{18}$$, $$(x \gt 0)$$, are $$m$$ and $$n$$ respectively, then $$\frac{m}{n}$$ is equal to
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If $$A > 0$$, $$B > 0$$ and $$A + B = \frac{\pi}{6}$$, then the minimum positive value of $$(\tan A + \tan B)$$ is:
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Let $$P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$$ and $$Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\}$$, be two sets. Then
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A straight line through origin $$O$$ meets the lines $$3y = 10 - 4x$$ and $$8x + 6y + 5 = 0$$ at points $$A$$ and $$B$$ respectively. Then, $$O$$ divides the segment $$AB$$ in the ratio
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A ray of light is incident along a line which meets another line $$7x - y + 1 = 0$$ at the point $$(0, 1)$$. The ray is then reflected from this point along the line $$y + 2x = 1$$. Then the equation of the line of incidence of the ray of light is:
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Equation of the tangent to the circle, at the point $$(1, -1)$$, whose center is the point of intersection of the straight lines $$x - y = 1$$ and $$2x + y = 3$$ is:
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$$P$$ and $$Q$$ are two distinct points on the parabola, $$y^2 = 4x$$, with parameters $$t$$ and $$t_1$$, respectively. If the normal at $$P$$ passes through $$Q$$, then the minimum value of $$t_1^2$$, is
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A hyperbola whose transverse axis is along the major axis of the conic $$\frac{x^2}{3} + \frac{y^2}{4} = 4$$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $$\frac{3}{2}$$, then which of the following points does not lie on the hyperbola?
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$$\lim_{x \to 0} \frac{(1 - \cos 2x)^2}{2x\tan x - x\tan 2x}$$ is
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The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is
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The mean of 5 observations is 5 and their variance is 12.4. If three of the observations are 1, 2 & 6; then the value of the remaining two is:
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The angle of elevation of the top of a vertical tower from a point A, due east of it is 45°. The angle of elevation of the top of the same tower from a point B, due south of A is 30°. If the distance between A and B is $$54\sqrt{2}$$ m, then the height of the tower (in meters), is:
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Let $$A$$, be a $$3 \times 3$$ matrix, such that $$A^2 - 5A + 7I = O$$.
Statement - I: $$A^{-1} = \frac{1}{7}(5I - A)$$.
Statement - II: The polynomial $$A^3 - 2A^2 - 3A + I$$, can be reduced to $$5(A - 4I)$$. Then:
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If $$A = \begin{bmatrix} -4 & -1 \\ 3 & 1 \end{bmatrix}$$, then the determinant of the matrix $$(A^{2016} - 2A^{2015} - A^{2014})$$ is:
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Let $$a, b \in R$$, $$(a \neq 0)$$. If the function $$f$$, defined as
$$f(x) = \begin{cases} \frac{2x^2}{a}, & 0 \leq x \lt 1 \\ a, & 1 \leq x \lt \sqrt{2} \\ \frac{2b^2 - 4b}{x^3}, & \sqrt{2} \leq x \lt 8 \end{cases}$$
is continuous in the interval $$[0, \infty)$$, then an ordered pair $$(a, b)$$ can be
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Let C be a curve given by $$y(x) = 1 + \sqrt{4x - 3}$$, $$x > \frac{3}{4}$$. If $$P$$ is a point on C, such that the tangent at $$P$$ has slope $$\frac{2}{3}$$, then a point through which the normal at $$P$$ passes, is:
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Let $$f(x) = \sin^4 x + \cos^4 x$$. Then, $$f$$ is an increasing function in the interval:
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The integral $$\int \frac{dx}{(1+\sqrt{x})\sqrt{x - x^2}}$$ is equal to
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For $$x \in R$$, $$x \neq 0$$, if $$y(x)$$ is a differentiable function such that $$x\int_1^x y(t)dt = (x+1)\int_1^x ty(t)dt$$, then $$y(x)$$ equals (where C is a constant)
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The value of the integral $$\int_4^{10} \frac{[x^2]}{[x^2 - 28x + 196] + [x^2]}dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is
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The solution of the differential equation $$\frac{dy}{dx} + \frac{y}{2}\sec x = \frac{\tan x}{2y}$$, where $$0 \leq x < \frac{\pi}{2}$$ and $$y(0) = 1$$, is given by
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The number of distinct real values of $$\lambda$$, for which the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{\lambda^2}$$ and $$\frac{x-3}{1} = \frac{y-2}{\lambda^2} = \frac{z-1}{2}$$, are coplanar is
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$$ABC$$ is a triangle in a plane with vertices $$A(2, 3, 5)$$, $$B(-1, 3, 2)$$ and $$C(\lambda, 5, \mu)$$. If the median through $$A$$ is equally inclined to the coordinate axes, then the value of $$(\lambda^3 + \mu^3 + 5)$$ is
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Let $$ABC$$ be a triangle whose circumcentre is at $$P$$. If the position vectors of $$A$$, $$B$$, $$C$$ and $$P$$ are $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ and $$\frac{\vec{a}+\vec{b}+\vec{c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:
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An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is
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