The diameter and height of a cylinder are measured by a meter scale to be $$12.6 \pm 0.1$$ cm and $$34.2 \pm 0.1$$ cm, respectively. What will be the value of its volume in appropriate significant figures?
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The diameter and height of a cylinder are measured by a meter scale to be $$12.6 \pm 0.1$$ cm and $$34.2 \pm 0.1$$ cm, respectively. What will be the value of its volume in appropriate significant figures?
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Two vectors $$\vec{A}$$ and $$\vec{B}$$ have equal magnitudes. The magnitude of $$(\vec{A} + \vec{B})$$ is '$$n$$' times the magnitude of $$(\vec{A} - \vec{B})$$. The angle between $$\vec{A}$$ and $$\vec{B}$$ is:
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Two forces P and Q, of magnitude 2F and 3F, respectively, are at an angle $$\theta$$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle $$\theta$$ is:
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A particle starts from the origin at time $$t = 0$$ and moves along the positive $$x$$-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time $$t = 5s$$?
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A particle which is experiencing a force, given by $$\vec{F} = 3\hat{i} - 12\hat{j}$$, undergoes a displacement of $$\vec{d} = 4\hat{i}$$. If the particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement?
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Two identical spherical balls of mass $$M$$ and radius $$R$$ each are stuck on two ends of a rod of length $$2R$$ and mass $$M$$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is:
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A rigid massless rod of length $$3l$$ has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be:
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Two stars of masses $$3 \times 10^{31}$$ kg each, and at distance $$2 \times 10^{11}$$ m rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the star's rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at O is: (Take Gravitational constant $$G = 6.67 \times 10^{-11}$$ N m$$^2$$ kg$$^{-2}$$)
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Half mole of an ideal monoatomic gas is heated at a constant pressure of 1 atm from 20$$^{\circ}$$C to 90$$^{\circ}$$C. Work done by the gas is
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An unknown metal of mass 192 g heated to a temperature of 100$$^{\circ}$$C was immersed into a brass calorimeter of mass 128 g containing 240 g of water at a temperature of 8.4$$^{\circ}$$C. Calculate the specific heat of the unknown metal if water temperature stabilizes at 21.5$$^{\circ}$$C. (Specific heat of brass is 394 J kg$$^{-1}$$K$$^{-1}$$)
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2 kg of a monoatomic gas is at a pressure of $$4 \times 10^4$$ N m$$^{-2}$$. The density of the gas is 8 kg m$$^{-3}$$. What is the order of energy of the gas due to its thermal motion?
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A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic material with their respective axes. But the magnetic moment of hoop is twice of solid cylinder. They are placed in a uniform magnetic field in such a manner that their magnetic moments make a small angle with the field. If the oscillation periods of hoop and cylinder are $$T_h$$ and $$T_c$$ respectively, then:
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A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is:
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A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $$\omega$$. If the radius of the bottle is 2.5 cm then $$\omega$$ is close to: (density of water $$= 10^3$$ kg/m$$^3$$)
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A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz).
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Charges $$-q$$ and $$+q$$, located at A and B, respectively, constitute an electric dipole. Distance $$AB = 2a$$, $$O$$ is the mid point of the dipole and $$OP$$ is perpendicular to $$AB$$. A charge $$Q$$ is placed at P where $$OP = y$$ and $$y \gg 2a$$. The charge $$Q$$ experiences an electrostatic force $$F$$. If $$Q$$ is now moved along the equatorial line to P' such that $$OP' = \frac{y}{3}$$, the force on $$Q$$ will be close to ($$\frac{y}{3} \ll 2a$$):
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Four equal point charges $$Q$$ each are placed in the $$xy$$ plane at $$(0, 2)$$, $$(4, 2)$$, $$(4, -2)$$ and $$(0, -2)$$. The work required to put a fifth charge $$Q$$ at the origin of the coordinate system will be:
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A parallel plate capacitor having capacitance 12 pF is charged by a battery to a potential difference of 10 V between its plates. The charging battery is now disconnected and a porcelain slab of dielectric constant 6.5 is slipped between the plates. The work done by the capacitor on the slab is:
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The actual value of resistance $$R$$, shown in the figure is 30$$\Omega$$. This is measured in an experiment as shown using the standard formula $$R = \frac{V}{I}$$, where V and I are the readings of the voltmeter and ammeter, respectively. If the measured value of $$R$$ is 5% less, then the internal resistance of the voltmeter is:
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The Wheatstone bridge shown in the figure below, gets balanced when the carbon resistor used as $$R_1$$ has the colour code (orange, red, brown). The resistors $$R_2$$ and $$R_4$$ are 80 $$\Omega$$ and 40 $$\Omega$$, respectively. Assuming that the colour code for the carbon resistors gives their accurate values, the colour code for the carbon resistor, used as $$R_3$$, would be:
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A current of 2 mA was passed through an unknown resistor which dissipated a power of 4.4 W. Dissipated power when an ideal power supply of 11 V is connected across it is:
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At some location the horizontal component of earth's magnetic field is $$18 \times 10^{-6}$$ T. At this location, magnetic needle of length 0.12 m and pole strength 1.8 Am is suspended from its mid-point using a thread, it makes 45$$^{\circ}$$ angles with horizontal in equilibrium. To keep this needle horizontal, the vertical force that should be applied at one of its ends is:
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The self induced emf of a coil is 25 volts. When the current in it is changed at uniform rate from 10A to 25A in 1s, the change in the energy of the inductance is:
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The electric field of a plane polarized electromagnetic wave in free space at time $$t = 0$$ is given by the expression $$\vec{E}(x, y) = 10\hat{j}\cos(6x + 8z)$$. The magnetic field $$\vec{B}(x, z, t)$$ is given by ($$c$$ is the velocity of light.)
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The eye can be regarded as a single refracting surface. The radius of curvature of this surface is equal to that of the cornea (7.8 mm). This surface separates two media of refractive indices 1 and 1.34. Calculate the distance from the refracting surface at which a parallel beam of light will come to focus.
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Consider a Young's double slit experiment as shown in figure. What should be the slit separation $$d$$ in terms of wavelength $$\lambda$$ such that the first minima occurs directly in front of the slit ($$S_1$$)?
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A metal plate of area $$1 \times 10^{-4}$$ m$$^2$$ is illuminated by a radiation of intensity 16 m W/m$$^2$$. The work function of the metal is 5 eV. The energy of the incident photons is 10 eV and only 10% of it produces photo electrons. The number of emitted photo electrons per second and their maximum energy, respectively, will be: [$$1eV = 1.6 \times 10^{-19}$$ J]
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Consider the nuclear fission, $$Ne^{20} \rightarrow 2He^4 + C^{12}$$. Given that the binding energy/nucleon of $$Ne^{20}$$, $$He^4$$ and $$C^{12}$$ are 8.03 MeV, 7.86 MeV, respectively. Identify the correct statement:
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For the circuit shown below, the current through the Zener diode is:
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The modulation frequency of an AM radio station is 250 kHz, which is 10% of the carrier wave. If another AM station approaches you for license what broadcast frequency will you allot?
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The 71$$^{st}$$ electron of an element X with an atomic number of 71 enters the orbital:
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The ground state energy of a hydrogen atom is $$-13.6$$ eV. The energy of second excited state of He$$^+$$ ion in eV is:
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The process with negative entropy change is:
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An ideal gas undergoes isothermal compression from 5 m$$^3$$ to 1 m$$^3$$ against a constant external pressure of 4 N m$$^{-2}$$. The heat released in this process is 24 J mol$$^{-1}$$ K$$^{-1}$$ and is used to increase the pressure of 1 mole of Al. The temperature of Al increases by:
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5.1 g $$NH_4SH$$ is introduced in 3.0 L evacuated flask at 327$$^{\circ}$$C. 30% of the solid $$NH_4SH$$ is decomposed to $$NH_3$$ and $$H_2S$$ as gases. The $$K_P$$ of the reaction at 327$$^{\circ}$$C is: ($$R = 0.082$$ L atm mol$$^{-1}$$ K$$^{-1}$$, Molar mass of S = 32 g mol$$^{-1}$$, Molar mass of N = 14 g mol$$^{-1}$$)
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In the reaction of oxalate with permanganate in acidic medium, the number of electrons involved in producing one molecule of CO$$_2$$ is:
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The number of 2-centre-2-electron and 3-centre-2-electron bonds in $$B_2H_6$$, respectively, are:
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What is the IUPAC name of the following compound?
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What will be the major product in the following mononitration reaction?
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The reaction that is not involved in the ozone layer depletion mechanism in the stratosphere is:
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A compound of formula $$A_2B_3$$ has the HCP lattice. Which atom forms the HCP lattice and what fraction of the tetrahedral voids are occupied by the other atoms?
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The amount of sugar ($$C_{12}H_{22}O_{11}$$) required to prepare 2L of its 0.1 M aqueous solution is:
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The elevation in boiling point for 1 molal solution of glucose is 2 K. The depression in freezing point for 2 molal solution of glucose in the same solvent is 2 K. The relation between $$K_b$$ and $$K_f$$ is:
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In the cell, Pt(s)|H$$_2$$(g, 1 bar)|HCl(aq)|AgCl(s)|Ag(s)|Pt(s), the cell potential is 0.92 V when a $$10^{-6}$$ molar HCl solution is used. The standard electrode potential of Ag|AgCl|Cl$$^-$$ electrode is: (Given, $$\frac{2.303RT}{F} = 0.06$$ V at 298 K)
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For an elementary chemical reaction, $$A_2 \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} 2A$$, the expression for $$\frac{d[A]}{dt}$$ is:
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The haemoglobin and the gold sol are examples of:
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The electrolytes usually used in the electroplating of gold and silver, respectively, are:
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Among the following reactions of hydrogen with halogens, the one that requires a catalyst is:
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The pair that contains two $$P - H$$ bonds in each of the oxoacids is:
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Sodium metal on dissolution in liquid ammonia gives a deep blue solution due to the formation of:
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A reaction of cobalt (III) chloride and ethylenediamine in a 1 : 2 mole ratio generates two isomeric products A (violet-coloured) and B (green-coloured). A can show optical activity, but, B is optically inactive. What type of isomers do A and B represent?
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The difference in the number of unpaired electrons of a metal ion in its high-spin and low-spin octahedral complexes is two. The metal ion is:
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The major product of the following reaction is:
The major product obtained in the following reaction is:
The major product of the following reaction is:
Which is the most suitable reagent for the following transformation?
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An aromatic compound 'A' having molecular formula $$C_7H_6O_2$$, on treating with aqueous ammonia and heating forms compound 'B'. The compound 'B' on reaction with molecular bromine and potassium hydroxide provides compound 'C' having molecular formula $$C_6H_7N$$. The structure of 'A' is:
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The major product of the following reaction is:
Which of the following tests cannot be used for identifying amino acids?
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The correct match between item I and item II is:
Item I (Compound) Item II (Reagent)
a. Lysine p. 1-naphthol
b. Furfural q. Ninhydrin
c. Benzyl alcohol r. KMnO$$_4$$
d. Styrene s. Ceric ammonium nitrate
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The value of $$\lambda$$ such that sum of the squares of the roots of the quadratic equation, $$x^2 + (3-\lambda)x + 2 = \lambda$$ has the least value is:
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Let $$z = \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^5 + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^5$$. If $$R(z)$$ and $$I(z)$$ respectively denote the real and imaginary parts of $$z$$, then:
If $$\sum_{r=0}^{25} \{^{50}C_r \cdot ^{50-r}C_{25-r}\} = K \cdot ^{50}C_{25}$$, then $$K$$ is equal to:
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The positive value of $$\lambda$$ for which the co-efficient of $$x^2$$ in the expansion $$x^2\left(\sqrt{x} + \frac{\lambda}{x^2}\right)^{10}$$ is 720, is:
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The value of $$\cos\frac{\pi}{2^2} \cdot \cos\frac{\pi}{2^3} \cdots \cos\frac{\pi}{2^{10}} \cdot \sin\frac{\pi}{2^{10}}$$ is:
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Two vertices of a triangle are $$(0, 2)$$ and $$(4, 3)$$. If its orthocenter is at the origin, then its third vertex lies in which quadrant?
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Two sides of a parallelogram are along the lines, $$x + y = 3$$ and $$x - y + 3 = 0$$. If its diagonals intersect at $$(2, 4)$$, then one of its vertex is:
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If the area of an equilateral triangle inscribed in the circle $$x^2 + y^2 + 10x + 12y + c = 0$$ is $$27\sqrt{3}$$ sq. units, then $$c$$ is equal to:
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The length of the chord of the parabola $$x^2 = 4y$$ having equation $$x - \sqrt{2}y + 4\sqrt{2} = 0$$ is:
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Let $$S = \left\{(x, y) \in R^2 : \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\right\}$$, where $$r \neq \pm 1$$. Then $$S$$ represents:
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Consider the following three statements:
P: 5 is a prime number
Q: 7 is a factor of 192
R: LCM of 5 and 7 is 35
Then the truth value of which one of the following statements is true?
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If the mean and standard deviation of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ are 10 and 3, respectively, then the variance of 6 observations $$x_1, x_2, \ldots, x_5$$ and $$-50$$ is equal to:
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With the usual notation in $$\triangle ABC$$, if $$\angle A + \angle B = 120^{\circ}$$, $$a = \sqrt{3} + 1$$ units and $$b = \sqrt{3} - 1$$ units, then the ratio $$\angle A : \angle B$$ is:
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Let $$A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$$, where $$b \gt 0$$. Then the minimum value of $$\frac{\det(A)}{b}$$ is:
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The number of values of $$\theta \in (0, \pi)$$ for which the system of linear equations
$$x + 3y + 7z = 0$$
$$-x + 4y + 7z = 0$$
$$(\sin 3\theta)x + (\cos 2\theta)y + 2z = 0$$
has a non-trivial solution, is:
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Let $$a_1, a_2, a_3, \ldots, a_{10}$$ be in G.P. with $$a_i > 0$$ for $$i = 1, 2, \ldots, 10$$ and $$S$$ be the set of pairs $$(r, k)$$, $$r, k \in N$$ (the set of natural numbers) for which $$\begin{vmatrix} \log_e a_1^r a_2^k & \log_e a_2^r a_3^k & \log_e a_3^r a_4^k \\ \log_e a_4^r a_5^k & \log_e a_5^r a_6^k & \log_e a_6^r a_7^k \\ \log_e a_7^r a_8^k & \log_e a_8^r a_9^k & \log_e a_9^r a_{10}^k \end{vmatrix} = 0$$. Then the number of elements in $$S$$, is:
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The value of $$\cot\left(\sum_{n=1}^{19} \cot^{-1}\left(1 + \sum_{p=1}^{n} 2p\right)\right)$$ is:
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Let $$N$$ be the set of natural numbers and two functions $$f$$ and $$g$$ be defined as $$f, g: N \to N$$ such that $$f(n) = \begin{cases} \frac{n+1}{2}, & \text{if n is odd} \\ \frac{n}{2}, & \text{if n is even} \end{cases}$$ and $$g(n) = n - (-1)^n$$. Then $$fog$$ is:
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Let $$f: (-1, 1) \to R$$ be a function defined by $$f(x) = \max\left\{-|x|, -\sqrt{1-x^2}\right\}$$. If $$K$$ be the set of all points at which $$f$$ is not differentiable, then $$K$$ has exactly:
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A helicopter is flying along the curve given by $$y - x^{3/2} = 7$$, $$(x \geq 0)$$. A soldier positioned at the point $$\left(\frac{1}{2}, 7\right)$$, who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
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The tangent to the curve, $$y = xe^{x^2}$$ passing through the point $$(1, e)$$ also passes through the point:
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If $$\int x^5 e^{-4x^3}dx = \frac{1}{48}e^{-4x^3}f(x) + C$$, where $$C$$ is a constant of integration, then $$f(x)$$ is equal to:
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The value of $$\int_{-\pi/2}^{\pi/2} \frac{dx}{[x] + [\sin x] + 4}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is:
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If $$\int_0^x f(t)dt = x^2 + \int_x^1 t^2 f(t)dt$$, then $$f'\left(\frac{1}{2}\right)$$ is:
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A curve amongst the family of curves represented by the differential equation, $$(x^2 - y^2)dx + 2xy \; dy = 0$$ which passes through $$(1, 1)$$, is:
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Let $$f(x)$$ be a differentiable function such that $$f'(x) = 7 - \frac{3}{4}\frac{f(x)}{x}$$, $$(x > 0)$$ and $$f(1) \neq 4$$. Then $$\lim_{x \to 0^+} xf\left(\frac{1}{x}\right)$$:
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Let $$\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$$ and $$\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$$, be two given vectors where vectors $$\vec{a}$$ and $$\vec{b}$$ are non-collinear. The value of $$\lambda$$ for which vectors $$\vec{\alpha}$$ and $$\vec{\beta}$$ are collinear, is:
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The plane which bisects the line segment joining the points $$(-3, -3, 4)$$ and $$(3, 7, 6)$$ at right angles, passes through which one of the following points?
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On which of the following lines lies the point of intersection of the line, $$\frac{x-4}{2} = \frac{y-5}{2} = \frac{z-3}{1}$$ and the plane, $$x + y + z = 2$$?
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If the probability of hitting a target by a shooter, in any shot is $$\frac{1}{3}$$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $$\frac{5}{6}$$, is:
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