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The radius $$r$$, length $$l$$ and resistance $$R$$ of a metal wire was measured in the laboratory as $$r = 0.35 \pm 0.05$$ cm, $$R = 100 \pm 10$$ ohm, $$l = 15 \pm 0.2$$ cm. The percentage error in resistivity of the material of the wire is :
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The dimensional formula of angular impulse is :
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A particle moving in a circle of radius $$R$$ with uniform speed takes time $$T$$ to complete one revolution. If this particle is projected with the same speed at an angle $$\theta$$ to the horizontal, the maximum height attained by it is equal to $$4R$$. The angle of projection $$\theta$$ is then given by :
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Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is $$0.04$$, the acceleration of the system in m s$$^{-2}$$ is: (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless):
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A simple pendulum of length $$1$$ m has a wooden bob of mass $$1$$ kg. It is struck by a bullet of mass $$10^{-2}$$ kg moving with a speed of $$2 \times 10^{2}$$ m s$$^{-1}$$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $$g = 10$$ m s$$^{-2}$$)
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A ball of mass $$0.5$$ kg is attached to a string of length $$50$$ cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $$400$$ N. The maximum possible value of angular velocity of the ball in rad s$$^{-1}$$ is :
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If $$R$$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $$g = \pi^2$$ m s$$^{-2}$$, then the length of the second's pendulum at a height $$h = 2R$$ from the surface of earth will be:
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With rise in temperature, the Young's modulus of elasticity
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The pressure and volume of an ideal gas are related as $$PV^{3/2} = K$$ (Constant). The work done when the gas is taken from state $$A(P_1, V_1, T_1)$$ to state $$B(P_2, V_2, T_2)$$ is :
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Two moles of a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :
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Two identical capacitors have same capacitance $$C$$. One of them is charged to the potential $$V$$ and other to the potential $$2V$$. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is :
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The reading in the ideal voltmeter $$V$$ shown in the given circuit diagram is:
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A galvanometer has a resistance of $$50\ \Omega$$ and it allows maximum current of $$5$$ mA. It can be converted into voltmeter to measure upto $$100$$ V by connecting in series a resistor of resistance.
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A parallel plate capacitor has a capacitance $$C = 200$$ pF. It is connected to $$230$$ V ac supply with an angular frequency $$300$$ rad s$$^{-1}$$. The rms value of conduction current in the circuit and displacement current in the capacitor respectively are :
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In series LCR circuit, the capacitance is changed from $$C$$ to $$4C$$. To keep the resonance frequency unchanged, the new inductance should be :
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A monochromatic light of wavelength $$6000\ \mathring{A}$$ is incident on the single slit of width $$0.01$$ mm. If the diffraction pattern is formed at the focus of the convex lens of focal length $$20$$ cm, the linear width of the central maximum is :
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The de Broglie wavelengths of a proton and an $$\alpha$$ particle are $$\lambda$$ and $$2\lambda$$ respectively. The ratio of the velocities of proton and $$\alpha$$ particle will be :
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The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly :
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In the given circuit if the power rating of Zener diode is $$10$$ mW, the value of series resistance $$R_s$$ to regulate the input unregulated supply is:
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$$10$$ divisions on the main scale of a Vernier calliper coincide with $$11$$ divisions on the Vernier scale. If each division on the main scale is of $$5$$ units, the least count of the instrument is :
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A particle is moving in one dimension (along $$x$$ axis) under the action of a variable force. Its initial position was $$16$$ m right of origin. The variation of its position $$x$$ with time $$t$$ is given as $$x = -3t^3 + 18t^2 + 16t$$, where $$x$$ is in m and $$t$$ is in s. The velocity of the particle when its acceleration becomes zero is _________ m s$$^{-1}$$.
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The identical spheres each of mass $$2M$$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $$4$$ m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $$\frac{4\sqrt{2}}{x}$$, where the value of $$x$$ is ________.
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A plane is in level flight at constant speed and each of its two wings has an area of $$40$$ m$$^2$$. If the speed of the air is $$180$$ km h$$^{-1}$$ over the lower wing surface and $$252$$ km h$$^{-1}$$ over the upper wing surface, the mass of the plane is ________kg. (Take air density to be $$1$$ kg m$$^{-3}$$ and $$g = 10$$ m s$$^{-2}$$)
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A tuning fork resonates with a sonometer wire of length $$1$$ m stretched with a tension of $$6$$ N. When the tension in the wire is changed to $$54$$ N, the same tuning fork produces $$12$$ beats per second with it. The frequency of the tuning fork is _______ Hz.
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Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $$\theta$$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $$1.5$$ g/cc, the dielectric constant of water will be ______. (Take density of water $$= 1$$ g/cc)
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The current in a conductor is expressed as $$I = 3t^2 + 4t^3$$, where $$I$$ is in Ampere and $$t$$ is in second. The amount of electric charge that flows through a section of the conductor during $$t = 1$$ s to $$t = 2$$ s is ____________ C.
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A regular polygon of $$6$$ sides is formed by bending a wire of length $$4\pi$$ meter. If an electric current of $$4\pi\sqrt{3}$$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $$x \times 10^{-7}$$ T. The value of $$x$$ is ______.
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A rectangular loop of sides $$12$$ cm and $$5$$ cm, with its sides parallel to the $$x$$-axis and $$y$$-axis respectively moves with a velocity of $$5$$ cm s$$^{-1}$$ in the positive $$x$$ axis direction, in a space containing a variable magnetic field in the positive $$z$$ direction. The field has a gradient of $$10^{-3}$$ T cm$$^{-1}$$ along the negative $$x$$ direction and it is decreasing with time at the rate of $$10^{-3}$$ T s$$^{-1}$$. If the resistance of the loop is $$6$$ m$$\Omega$$, the power dissipated by the loop as heat is ______ $$\times 10^{-9}$$ W.
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The distance between object and its $$3$$ times magnified virtual image as produced by a convex lens is $$20$$ cm. The focal length of the lens used is ________ cm.
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The radius of a nucleus of mass number $$64$$ is $$4.8$$ fermi. Then the mass number of another nucleus having radius of $$4$$ fermi is $$\frac{1000}{x}$$, where $$x$$ is _________.
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According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron $$\lambda$$ and momentum of electron $$p$$? (Four graphs are shown: (1) $$\lambda$$ vs $$p$$ showing a rectangular hyperbola, (2) $$p$$ vs $$\lambda$$ showing a rectangular hyperbola, (3) $$\lambda$$ vs $$1/p$$ showing a straight line through origin, (4) $$\lambda$$ vs $$p$$ showing a straight line with negative slope)
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In case of isoelectronic species the size of $$F^-$$, $$Ne$$ and $$Na^+$$ is affected by:
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Arrange the bonds in order of increasing ionic character in the molecules: $$LiF$$, $$K_2O$$, $$N_2$$, $$SO_2$$ and $$ClF_3$$.
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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: $$PH_3$$ has lower boiling point than $$NH_3$$. Reason R: In liquid state $$NH_3$$ molecules are associated through Vander Waal's forces, but $$PH_3$$ molecules are associated through hydrogen bonding. In the light of the above statements, choose the most appropriate answer from the options given below:
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Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following :
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Which of the following reactions are disproportionation reactions? (1) $$Cu^+ \rightarrow Cu^{2+} + Cu$$ (2) $$3MnO_4^{2-} + 4H^+ \rightarrow 2MnO_4^- + MnO_2 + 2H_2O$$ (3) $$2KMnO_4 \rightarrow K_2MnO_4 + MnO_2 + O_2$$ (4) $$2MnO_4^- + 3Mn^{2+} + 2H_2O \rightarrow 5MnO_2 + 4H^+$$. Choose the correct answer from the options given below:
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In acidic medium, $$K_2Cr_2O_7$$ shows oxidising action as represented in the half reaction $$Cr_2O_7^{2-} + XH^+ + Ye^- \rightarrow 2A + ZH_2O$$. X, Y, Z and A are respectively:
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Given below are two statements: Statement (I): Potassium hydrogen phthalate is a primary standard for standardisation of sodium hydroxide solution. Statement (II): In this titration phenolphthalein can be used as indicator. In the light of the above statements, choose the most appropriate answer from the options given below:
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Given below are two statements: Statement (I): Aminobenzene and aniline are same organic compounds. Statement (II): Aminobenzene and aniline are different organic compounds. In the light of the above statements, choose the most appropriate answer from the options given below:
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Ionic reactions with organic compounds proceed through: (A) Homolytic bond cleavage (B) Heterolytic bond cleavage (C) Free radical formation (D) Primary free radical (E) Secondary free radical. Choose the correct answer from the options given below:
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In Kjeldahl's method for estimation of nitrogen, $$CuSO_4$$ acts as :
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Which of the following compound will most easily be attacked by an electrophile?
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We have three aqueous solutions of NaCl labelled as 'A', 'B' and 'C' with concentration $$0.1$$ M, $$0.01$$ M and $$0.001$$ M, respectively. The value of van't Hoff factor $$i$$ for these solutions will be in the order:
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Which of the following complex is homoleptic?
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Given below are two statements: Statement (I): A solution of $$[Ni(H_2O)_6]^{2+}$$ is green in colour. Statement (II): A solution of $$[Ni(CN)_4]^{2-}$$ is colourless. In the light of the above statements, choose the most appropriate answer from the options given below:
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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: Haloalkanes react with KCN to form alkyl cyanides as a main product while with AgCN form isocyanide as the main product. Reason R: KCN and AgCN both are highly ionic compounds. In the light of the above statement, choose the most appropriate answer from the options given below:
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Identify A and B in the following sequence of reaction:
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Choose the correct answer from options given below:
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Given below are two statements: Statement (I): The $$-NH_2$$ group in Aniline is ortho and para directing and a powerful activating group. Statement (II): Aniline does not undergo Friedel-Craft's reaction (alkylation and acylation). In the light of the above statements, choose the most appropriate answer from the options given below:
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If one strand of a DNA has the sequence ATGCTTCA, sequence of the bases in complementary strand is:
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Consider the following reaction: $$3PbCl_2 + 2(NH_4)_3PO_4 \rightarrow Pb_3(PO_4)_2 + 6NH_4Cl$$. If $$72$$ mmol $$PbCl_2$$ is mixed with $$50$$ mmol of $$(NH_4)_3PO_4$$, then amount of $$Pb_3(PO_4)_2$$ formed in mmol is (nearest integer):
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Lowest oxidation number of an atom in a compound $$A_2B$$ is $$-2$$. The number of electrons in its valence shell is:
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The number of molecules/ion/s having trigonal bipyramidal shape is: $$PF_5$$, $$BrF_5$$, $$PCl_5$$, $$[PtCl_4]^{2-}$$, $$BF_3$$, $$Fe(CO)_5$$
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$$K_a$$ for $$CH_3COOH$$ is $$1.8 \times 10^{-5}$$ and $$K_b$$ for $$NH_4OH$$ is $$1.8 \times 10^{-5}$$. The pH of ammonium acetate solution will be:
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Number of optical isomers possible for 2-chlorobutane is:
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Total number of deactivating groups in aromatic electrophilic substitution reaction among the following is:
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The potential for the given half cell at 298K is $$- x \times 10^{-2}$$ V.
$$2H^+_{(aq)} + 2e^- \rightarrow H_2(g)$$, $$[H^+] = 1$$ M, $$P_{H_2} = 2$$ atm.
(Given $$2.303\ RT/F = 0.06$$ V, $$\log 2 = 0.3$$). The value of $$x$$ is:
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The ratio of $$\frac{^{14}C}{^{12}C}$$ in a piece of wood is $$\frac{1}{8}$$ part that of atmosphere. If half life of $$^{14}C$$ is $$5730$$ years, the age of wood sample is _____ years.
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Among the following oxides of p-block elements: $$Cl_2O_7$$, $$CO$$, $$PbO_2$$, $$N_2O$$, $$NO$$, $$Al_2O_3$$, $$SiO_2$$, $$N_2O_5$$, $$SnO_2$$, the number of amphoteric oxides is:
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The number of white coloured salts among the following is: (A) $$SrSO_4$$ (B) $$MgNH_4PO_4$$ (C) $$BaCrO_4$$ (D) $$Mn(OH)_2$$ (E) $$PbSO_4$$ (F) $$PbCrO_4$$ (G) $$AgBr$$ (H) $$PbI_2$$ (I) $$CaC_2O_4$$ (J) $$Fe(OH)_2(CH_3COO)$$
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Let $$S = \{x \in R : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10$$}. Then the number of elements in $$S$$ is:
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Let $$S = z \in \mathbb{C} :| z - 1 |= 1 \text{ and } (\sqrt{2} - 1)(z + \bar{z}) - i(z - \bar{z}) = 2\sqrt{2}$$. Let $$z_1, z_2 \in S$$ be such that $$z_1 = \max_{z \in S}z$$ and $$z_2 = \min_{z \in S}z$$. Then $$|\sqrt{2}z_1 - z_2^2|$$ equals:
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If $$n$$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $$n$$ is equal to:
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Let $$3, a, b, c$$ be in A.P. and $$3, a-1, b+1, c+9$$ be in G.P. Then, the arithmetic mean of $$a$$, $$b$$ and $$c$$ is:
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If $$\tan A = \frac{1}{\sqrt{x}\sqrt{x^2+x+1}}$$, $$\tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$$ and $$\tan C=(x^{-3}+x^{-2}+x^{-1})^{\frac{1}{2}}$$, $$0 < A, B, C < \frac{\pi}{2}$$, then $$A + B$$ is equal to:
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Let $$C: x^2 + y^2 = 4$$ and $$C': x^2 + y^2 - 4\lambda x + 9 = 0$$ be two circles. If the set of all values of $$\lambda$$ so that the circles $$C$$ and $$C'$$ intersect at two distinct points, is $$\mathbb{R} - \{a, b\} $$, then the point $$8a + 12, 16b - 20$$ lies on the curve:
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Let $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$ be an ellipse, whose eccentricity is $$\frac{1}{\sqrt{2}}$$ and the length of the latus rectum is $$\sqrt{14}$$. Then the square of the eccentricity of $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ is:
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For $$0 < \theta < \pi/2$$, if the eccentricity of the hyperbola $$x^2 - y^2\csc^2\theta = 5$$ is $$\sqrt{7}$$ times eccentricity of the ellipse $$x^2\csc^2\theta + y^2 = 5$$, then the value of $$\theta$$ is:
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Let the median and the mean deviation about the median of 7 observations $$170, 125, 230, 190, 210, a, b$$ be $$170$$ and $$\frac{205}{7}$$ respectively. Then the mean deviation about the mean of these 7 observations is:
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If $$A = \begin{pmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{pmatrix}$$, $$B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$, $$C = ABA^T$$ and $$X = A^TC^2A$$, then $$\det X$$ is equal to:
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If the system of equations $$\\2x + 3y - z = 5\\ x + \alpha y + 3z = -4\\ 3x - y + \beta z = 7\\$$ has infinitely many solutions, then $$13\alpha\beta$$ is equal to:
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Let $$f: \mathbb{R} \to \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \log_e x, & x \gt 0 \\ e^{-x}, & x \leq 0 \end{cases}$$ and $$g(x) = \begin{cases} x, & x \geq 0 \\ e^x, & x \lt 0 \end{cases}$$. Then, $$g \circ f: \mathbb{R} \to \mathbb{R}$$ is:
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \frac{a - b\cos 2x}{x^2}, & x < 0 \\ x^2 + cx + 2, & 0 \leq x \leq 1 \\ 2x + 1, & x > 1 \end{cases}\\$$ If $$f$$ is continuous everywhere in $$\mathbb{R}$$ and $$m$$ is the number of points where $$f$$ is NOT differentiable then $$m + a + b + c$$ equals:
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If $$5f(x) + 4f\left(\frac{1}{x}\right) = x^2 - 2$$, $$\forall x \neq 0$$ and $$y = 9x^2 f(x)$$, then $$y$$ is strictly increasing in:
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The value of the integral $$\int_0^{\pi/4} \frac{x\,dx}{\sin^4 2x + \cos^4 2x}$$ equals:
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The area enclosed by the curves $$xy + 4y = 16$$ and $$x + y = 6$$ is equal to:
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1$$, $$y(0) = 1$$. Then, $$\left(\frac{1}{\sqrt{2}} + y\left(\frac{1}{\sqrt{2}}\right)\right)^2$$ equals:
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Let $$\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$$, $$\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$$ and $$\vec{c} = ((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$$. Then $$\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})$$ is equal to:
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If the shortest distance between the lines $$\frac{x-\lambda}{-2} = \frac{y-2}{1} = \frac{z-1}{1}$$ and $$\frac{x-\sqrt{3}}{1} = \frac{y-1}{-2} = \frac{z-2}{1}$$ is $$1$$, then the sum of all possible values of $$\lambda$$ is:
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A bag contains $$8$$ balls, whose colours are either white or black. $$4$$ balls are drawn at random without replacement and it was found that $$2$$ balls are white and other $$2$$ balls are black. The probability that the bag contains equal number of white and black balls is:
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Let $$P = \{z \in \mathbb{C} : |z + 2 - 3i| \leq 1\}$$ and $$Q = \{z \in \mathbb{C} : z(1+i) + \bar{z}(1-i) \leq -8\}$$. Let in $$P \cap Q$$, $$|z - 3 + 2i|$$ be maximum and minimum at $$z_1$$ and $$z_2$$ respectively. If $$|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ equals:
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Let $$3, 7, 11, 15, \ldots, 403$$ and $$2, 5, 8, 11, \ldots, 404$$ be two arithmetic progressions. Then the sum of the common terms in them is equal to:
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If the coefficient of $$x^{30}$$ in the expansion of $$\left(1 + \frac{1}{x}\right)^6 (1+x^2)^7 (1-x^3)^8$$; $$x \neq 0$$ is $$\alpha$$, then $$|\alpha|$$ equals:
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Let the line $$L: \sqrt{2}x + y = \alpha$$ pass through the point of the intersection $$P$$ (in the first quadrant) of the circle $$x^2 + y^2 = 3$$ and the parabola $$x^2 = 2y$$. Let the line $$L$$ touch two circles $$C_1$$ and $$C_2$$ of equal radius $$2\sqrt{3}$$. If the centres $$Q_1$$ and $$Q_2$$ of the circles $$C_1$$ and $$C_2$$ lie on the $$y$$-axis, then the square of the area of the triangle $$PQ_1Q_2$$ is equal to:
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Let $$\{x\}$$ denote the fractional part of $$x$$ and $$f(x) = \frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\} - \{x\}^3}$$, $$x \neq 0$$. If $$L$$ and $$R$$ respectively denotes the left hand limit and the right hand limit of $$f(x)$$ at $$x = 0$$, then $$\frac{32}{\pi^2}(L^2 + R^2)$$ is equal to:
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The number of elements in the set $$S = \{(x,y,z) : x,y,z \in \mathbb{Z}, x + 2y + 3z = 42, x,y,z \geq 0\}$$ equals:
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Let $$A = \{1, 2, 3, \ldots, 20\}$$. Let $$R_1$$ and $$R_2$$ be two relations on $$A$$ such that $$R_1 = \{(a,b) : b \text{ is divisible by } a\}$$ and $$R_2 = \{(a,b) : a \text{ is an integral multiple of } b\}$$. Then, number of elements in $$R_1 - R_2$$ is equal to:
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If $$\int_{-\pi/2}^{\pi/2} \frac{8\sqrt{2}\cos x\,dx}{(1+e^{\sin x})(1+\sin^4 x)} = \alpha\pi + \beta\log_e(3+2\sqrt{2})$$, where $$\alpha, \beta$$ are integers, then $$\alpha^2 + \beta^2$$ equals:
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If $$x = x(t)$$ is the solution of the differential equation $$(t+1)dx = (2x + (t+1)^4)dt$$, $$x(0) = 2$$, then $$x(1)$$ equals:
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Let the line of the shortest distance between the lines $$L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$$ and $$L_2: \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k})$$ intersect $$L_1$$ and $$L_2$$ at $$P$$ and $$Q$$ respectively. If $$(\alpha, \beta, \gamma)$$ is the midpoint of the line segment $$PQ$$, then $$2(\alpha + \beta + \gamma)$$ is equal to:
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