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Given below are two statements:
Statement (I) : Planck's constant and angular momentum have the same dimensions.
Statement (II) : Linear momentum and moment of force have the same dimensions. In light of the above statements, choose the correct answer from the options given below :
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Position of an ant (S in metres) moving in $$Y - Z$$ plane is given by $$S = 2t^2 \hat{j} + 5\hat{k}$$ (where $$t$$ is in second). The magnitude and direction of velocity of the ant at $$t = 1$$ s will be :
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A train is moving with a speed of $$12 \text{ m s}^{-1}$$ on rails which are $$1.5$$ m apart. To negotiate a curve radius $$400$$ m, the height by which the outer rail should be raised with respect to the inner rail is (Given, $$g = 10 \text{ m s}^{-2}$$):
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Two bodies of mass $$4$$ g and $$25$$ g are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is :
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A body of mass $$1000$$ kg is moving horizontally with a velocity $$6 \text{ m s}^{-1}$$. If $$200$$ kg extra mass is added, the final velocity (in $$\text{m s}^{-1}$$) is:
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The acceleration due to gravity on the surface of earth is $$g$$. If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :
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Given below are two statements :
Statement (I) : Viscosity of gases is greater than that of liquids.
Statement (II) : Surface tension of a liquid decreases due to the presence of insoluble impurities. In the light of the above statements, choose the most appropriate answer from the options given below :
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$$0.08$$ kg air is heated at constant volume through $$5°C$$. The specific heat of air at constant volume is $$0.17 \text{ kcal kg}^{-1} \text{ °C}^{-1}$$ and $$1 \text{ J} = 4.18 \text{ joule cal}^{-1}$$. The change in its internal energy is approximately.
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The average kinetic energy of a monatomic molecule is $$0.414$$ eV at temperature: (Use $$K_B = 1.38 \times 10^{-23} \text{ J mol}^{-1} \text{ K}^{-1}$$)
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An electric charge $$10^{-6}$$ $$\mu$$C is placed at origin $$(0, 0)$$ m of $$X - Y$$ co-ordinate system. Two points $$P$$ and $$Q$$ are situated at $$(\sqrt{3}, \sqrt{3})$$ m and $$(\sqrt{6}, 0)$$ m respectively. The potential difference between the points $$P$$ and $$Q$$ will be :
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A wire of resistance $$R$$ and length $$L$$ is cut into $$5$$ equal parts. If these parts are joined parallely, then resultant resistance will be :
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A wire of length $$10$$ cm and radius $$\sqrt{7} \times 10^{-4}$$ m connected across the right gap of a meter bridge. When a resistance of $$4.5 \; \Omega$$ is connected on the left gap by using a resistance box, the balance length is found to be at $$60$$ cm from the left end. If the resistivity of the wire is $$R \times 10^{-7} \; \Omega$$ m, then value of $$R$$ is :
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A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $$\vec{E}$$ and $$\vec{B}$$ represent the electric and magnetic fields respectively, then the region of space may have :
(A) $$E = 0, B = 0$$; (B) $$E = 0, B \neq 0$$; (C) $$E \neq 0, B = 0$$; (D) $$E \neq 0, B \neq 0$$. Choose the most appropriate answer from the options given below :
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A rectangular loop of length $$2.5$$ m and width $$2$$ m is placed at $$30°$$ to a magnetic field of $$4$$ T. The loop is removed from the field in $$10$$ sec. The average emf induced in the loop during this time is
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A plane electromagnetic wave propagating in $$x$$-direction is described by $$E_y = (200 \text{ V m}^{-1}) \sin[1.5 \times 10^7 t - 0.05x]$$. The intensity of the wave is : (Use $$\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}$$)
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If the refractive index of the material of a prism is $$\cot\left(\frac{A}{2}\right)$$, where $$A$$ is the angle of prism then the angle of minimum deviation will be
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A convex lens of focal length $$40$$ cm forms an image of an extended source of light on a photoelectric cell. A current $$I$$ is produced. The lens is replaced by another convex lens having the same diameter but focal length $$20$$ cm. The photoelectric current now is
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The radius of third stationary orbit of electron for Bohr's atom is $$R$$. The radius of fourth stationary orbit will be:
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Which of the following circuits is reverse-biased?
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Identify the physical quantity that cannot be measured using spherometer :
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A particle starts from origin at $$t = 0$$ with a velocity $$5\hat{i} \text{ m s}^{-1}$$ and moves in $$x - y$$ plane under action of a force which produces a constant acceleration of $$(3\hat{i} + 2\hat{j}) \text{ m s}^{-2}$$. If the $$x$$-coordinate of the particle at that instant is $$84$$ m, then the speed of the particle at this time is $$\sqrt{\alpha} \text{ m s}^{-1}$$. The value of $$\alpha$$ is _______.
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Four particles, each of mass $$1$$ kg are placed at four corners of a square of side $$2$$ m. The moment of inertia of the system about an axis perpendicular to its plane and passing through one of its vertex is ______ kg m$$^2$$.
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If average depth of an ocean is $$4000$$ m and the bulk modulus of water is $$2 \times 10^9 \text{ N m}^{-2}$$, then fractional compression $$\frac{\Delta V}{V}$$ of water at the bottom of ocean is $$\alpha \times 10^{-2}$$. The value of $$\alpha$$ is _______, (Given, $$g = 10 \text{ m s}^{-2}, \rho = 1000 \text{ kg m}^{-3}$$)
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A particle executes simple harmonic motion with an amplitude of $$4$$ cm. At the mean position, velocity of the particle is $$10 \text{ cm s}^{-1}$$. The distance of the particle from the mean position when its speed becomes $$5 \text{ cm s}^{-1}$$ is $$\sqrt{\alpha}$$ cm, where $$\alpha =$$ ______.
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A thin metallic wire having cross sectional area of $$10^{-4} \text{ m}^2$$ is used to make a ring of radius $$30$$ cm. A positive charge of $$2\pi$$ C is uniformly distributed over the ring, while another positive charge of $$30$$ pC is kept at the centre of the ring. The tension in the ring is _______ N; provided that the ring does not get deformed (neglect the influence of gravity). (Given, $$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9$$ SI units)
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The charge accumulated on the capacitor connected in the following circuit is ______ $$\mu$$C. (Given $$C = 150 \; \mu$$F)
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Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is $$5.0$$ cm. The magnitude of the magnetic field at a point P midway between the wires is ______ $$\mu$$T. (Given: $$\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$$)
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Two coils have mutual inductance $$0.002$$ H. The current changes in the first coil according to the relation $$i = i_0 \sin \omega t$$, where $$i_0 = 5$$ A and $$\omega = 50\pi \text{ rad s}^{-1}$$. The maximum value of emf in the second coil is $$\frac{\pi}{\alpha}$$ V. The value of $$\alpha$$ is
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Two immiscible liquids of refractive indices $$\frac{8}{5}$$ and $$\frac{3}{2}$$ respectively are put in a beaker as shown in the figure. The height of each column is $$6$$ cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $$\frac{\alpha}{4}$$ cm. The value of $$\alpha$$ is _______.
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In a nuclear fission process, a high mass nuclide $$(A \approx 236)$$ with binding energy $$7.6$$ MeV/Nucleon dissociated into two middle mass nuclides $$(A \approx 118)$$, having binding energy of $$8.6$$ MeV/Nucleon. The energy released in the process would be _______ MeV.
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The electronic configuration for Neodymium is: [Atomic Number for Neodymium 60]
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Which of the following electronic configuration would be associated with the highest magnetic moment?
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Choose the polar molecule from the following :
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Which of the following is strongest Bronsted base?
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Given below are two statements : Statement (I) : Aqueous solution of ammonium carbonate is basic.
Statement (II) : Acidic/basic nature of salt solution of a salt of weak acid and weak base depends on $$K_a$$ and $$K_b$$ value of acid and the base forming it.
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) : Melting point of Boron (2453 K) is unusually high in group 13 elements.
Reason (R) : Solid Boron has very strong crystalline lattice.
In the light of the above statements, choose the most appropriate answer from the options given below:
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IUPAC name of the following compound (P) is :
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Cyclohexene is _________ type of an organic compound.
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Which of the following has highly acidic hydrogen?
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The ascending order of acidity of $$-OH$$ group in the following compounds is :
(A) Bu-OH
Choose the correct answer from the options given below :
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Highest enol content will be shown by :
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A solution of two miscible liquids showing negative deviation from Raoult's law will have :
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Element not showing variable oxidation state is :
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NaCl reacts with conc. $$H_2SO_4$$ and $$K_2Cr_2O_7$$ to give reddish fumes (B), which react with NaOH to give yellow solution (C). (B) and (C) respectively are :
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Given below are two statements :
Statement (I) : The 4f and 5f-series of elements are placed separately in the Periodic table to preserve the principle of classification. Statement (II) : s-block elements can be found in pure form in nature.
In light of the above statements, choose the most appropriate answer from the options given below:
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Yellow compound of lead chromate gets dissolved on treatment with hot NaOH solution. The product of lead formed is a :
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Consider the following complex ions $$P = [FeF_6]^{3-}$$, $$Q = [V(H_2O)_6]^{2+}$$, $$R = [Fe(H_2O)_6]^{2+}$$. The correct order of the complex ions, according to their spin only magnetic moment values (in B.M.) is :
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The correct statement regarding nucleophilic substitution reaction in a chiral alkyl halide is :
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Given below are two statements :
Statement (I) : p-nitrophenol is more acidic than m-nitrophenol and o-nitrophenol.
Statement (II) : Ethanol will give immediate turbidity with Lucas reagent.
In the light of the above statements, choose the correct answer from the options given below :
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Two nucleotides are joined together by a linkage known as :
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Mass of methane required to produce $$22$$ g of CO after complete combustion is _______ g. (Given Molar mass in g mol$$^{-1}$$, $$C = 12.0, H = 1.0, O = 16.0$$)
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The number of electrons present in all the completely filled subshells having $$n = 4$$ and $$s = +\frac{1}{2}$$ is ______ (Where $$n$$ = principal quantum number and $$s$$ = spin quantum number)
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Sum of bond order of CO and $$NO^+$$ is _______.
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If three moles of an ideal gas at $$300$$ K expand isothermally from $$30 \text{ dm}^3$$ to $$45 \text{ dm}^3$$ against a constant opposing pressure of $$80$$ kPa, then the amount of heat transferred is _________ J.
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Among the following, total number of meta directing functional groups is (Integer based) $$-OCH_3$$, $$-NO_2$$, $$-CN$$, $$-CH_3$$, $$-NHCOCH_3$$, $$-COR$$, $$-OH$$, $$-COOH$$, $$-Cl$$
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Among the given organic compounds, the total number of aromatic compounds is _______.
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3-Methylhex-2-ene on reaction with HBr in presence of peroxide forms an addition product (A). The number of possible stereoisomers for 'A' is _______.
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The mass of silver (Molar mass of Ag : $$108 \text{ g mol}^{-1}$$) displaced by a quantity of electricity which displaces $$5600$$ mL of $$O_2$$ at S.T.P. will be ______ g.
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Consider the following data for the given reaction $$2HI_{(g)} \rightarrow H_{2(g)} + I_{2(g)}$$. $$[HI] \text{ (mol L}^{-1}\text{)}$$: $$0.005, \; 0.01, \; 0.02$$. Rate $$\text{(mol L}^{-1} \text{ s}^{-1}\text{)}$$: $$7.5 \times 10^{-4}, \; 3.0 \times 10^{-3}, \; 1.2 \times 10^{-2}$$. The order of the reaction is _______.
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From the given list, the number of compounds with $$+4$$ oxidation state of Sulphur: $$SO_3, H_2SO_3, SOCl_2, SF_4, BaSO_4, H_2S_2O_7$$
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If $$S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}$$, then $$n(S)$$ is:
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The number of common terms in the progressions $$4, 9, 14, 19, \ldots$$ up to $$25^{th}$$ term and $$3, 6, 9, 12, \ldots$$ up to $$37^{th}$$ term is :
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If $$A$$ denotes the sum of all the coefficients in the expansion of $$(1 - 3x + 10x^2)^n$$ and $$B$$ denotes the sum of all the coefficients in the expansion of $$(1 + x^2)^n$$, then :
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$${}^{n-1}C_r = (k^2 - 8) \; {}^{n}C_{r+1}$$ if and only if :
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The portion of the line $$4x + 5y = 20$$ in the first quadrant is trisected by the lines $$L_1$$ and $$L_2$$ passing through the origin. The tangent of an angle between the lines $$L_1$$ and $$L_2$$ is :
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Four distinct points $$(2k, 3k), (1, 0), (0, 1)$$ and $$(0, 0)$$ lie on a circle for $$k$$ equal to :
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If the shortest distance of the parabola $$y^2 = 4x$$ from the centre of the circle $$x^2 + y^2 - 4x - 16y + 64 = 0$$ is $$d$$, then $$d^2$$ is equal to :
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The length of the chord of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$, whose mid point is $$(1, \frac{2}{5})$$, is equal to:
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If $$a = \lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$$ and $$b = \lim_{x \to 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$$, then the value of $$ab^3$$ is :
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Let $$a_1, a_2, \ldots, a_{10}$$ be 10 observations such that $$\sum_{k=1}^{10} a_k = 50$$ and $$\sum_{\forall k < j} a_k \cdot a_j = 1100$$. Then the standard deviation of $$a_1, a_2, \ldots, a_{10}$$ is equal to :
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Let $$S = \{1, 2, 3, \ldots, 10\}$$. Suppose $$M$$ is the set of all the subsets of $$S$$, then the relation $$R = \{(A, B) : A \cap B \neq \phi; \; A, B \in M\}$$ is :
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Consider the matrix $$f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$$. Given below are two statements : Statement I: $$f(-x)$$ is the inverse of the matrix $$f(x)$$. Statement II: $$f(x) f(y) = f(x + y)$$. In the light of the above statements, choose the correct answer from the options given below
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The function $$f : \mathbb{N} - \{1\} \rightarrow \mathbb{N}$$; defined by $$f(n) =$$ the highest prime factor of $$n$$, is :
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Consider the function $$f(x) = \begin{cases} \frac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|}, & x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}}, & x > 3 \\ b, & x = 3 \end{cases}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. If $$S$$ denotes the set of all ordered pairs $$(a, b)$$ such that $$f(x)$$ is continuous at $$x = 3$$, then the number of elements in $$S$$ is :
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If $$\int_0^1 \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx = a + b\sqrt{2} + c\sqrt{3}$$, where $$a, b, c$$ are rational numbers, then $$2a + 3b - 4c$$ is equal to :
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If $$(a, b)$$ be the orthocentre of the triangle whose vertices are $$(1, 2), (2, 3)$$ and $$(3, 1)$$, and $$I_1 = \int_a^b x \sin(4x - x^2) \, dx$$, $$I_2 = \int_a^b \sin(4x - x^2) \, dx$$, then $$36 \frac{I_1}{I_2}$$ is equal to :
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Let $$x = x(t)$$ and $$y = y(t)$$ be solutions of the differential equations $$\frac{dx}{dt} + ax = 0$$ and $$\frac{dy}{dt} + by = 0$$ respectively, $$a, b \in \mathbb{R}$$. Given that $$x(0) = 2$$; $$y(0) = 1$$ and $$3y(1) = 2x(1)$$, the value of $$t$$, for which $$x(t) = y(t)$$, is :
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If $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})$$ and $$\vec{c}$$ be the vector such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{a} \cdot ((\vec{c} \times \vec{b}) - \vec{b} - \vec{c})$$ is equal to
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The distance, of the point $$(7, -2, 11)$$ from the line $$\frac{x-6}{1} = \frac{y-4}{0} = \frac{z-8}{3}$$ along the line $$\frac{x-5}{2} = \frac{y-1}{-3} = \frac{z-5}{6}$$, is :
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If the shortest distance between the lines $$\frac{x-4}{1} = \frac{y+1}{2} = \frac{z}{-3}$$ and $$\frac{x-\lambda}{2} = \frac{y+1}{4} = \frac{z-2}{-5}$$ is $$\frac{6}{\sqrt{5}}$$, then the sum of all possible values of $$\lambda$$ is :
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If $$\alpha$$ satisfies the equation $$x^2 + x + 1 = 0$$ and $$(1 + \alpha)^7 = A + B\alpha + C\alpha^2$$, $$A, B, C \geq 0$$, then $$5(3A - 2B - C)$$ is equal to _______.
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If $$8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^2}(3 + 2p) + \frac{1}{4^3}(3 + 3p) + \ldots \infty$$, then the value of $$p$$ is _______.
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Let the set of all $$a \in \mathbb{R}$$ such that the equation $$\cos 2x + a \sin x = 2a - 7$$ has a solution be $$[p, q]$$ and $$r = \tan 9° - \tan 27° - \frac{1}{\cot 63°} + \tan 81°$$, then $$pqr$$ is equal to _______.
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Let$$A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$$,$$B = [B_1 \; B_2 \; B_3]$$, where$$B_1, B_2, B_3$$ are column matrices, and $$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$, $$AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$$, $$AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$. If $$\alpha = |B|$$ and $$\beta$$ is the sum of all the diagonal elements of $$B$$, then $$\alpha^3 + \beta^3$$ is equal to _______.
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Let $$f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3), \; x \in \mathbb{R}$$. Then $$f'(10)$$ is equal to _______.
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Let for a differentiable function $$f : (0, \infty) \rightarrow \mathbb{R}$$, $$f(x) - f(y) \geq \log_e\left(\frac{x}{y}\right) + x - y, \; \forall x, y \in (0, \infty)$$. Then $$\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)$$ is equal to _______.
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Let the area of the region $$\{(x, y) : x - 2y + 4 \geq 0, \; x + 2y^2 \geq 0, \; x + 4y^2 \leq 8, \; y \geq 0\}$$ be $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime numbers. Then $$m + n$$ is equal to _______.
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If the solution of the differential equation $$(2x + 3y - 2)dx + (4x + 6y - 7)dy = 0$$, $$y(0) = 3$$, is $$\alpha x + \beta y + 3\log_e|2x + 3y - \gamma| = 6$$, then $$\alpha + 2\beta + 3\gamma$$ is equal to _______.
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The least positive integral value of $$\alpha$$, for which the angle between the vectors $$\alpha\hat{i} - 2\hat{j} + 2\hat{k}$$ and $$\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$$ is acute, is _______.
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A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required and let $$a = P(X = 3)$$, $$b = P(X \geq 3)$$ and $$c = P(X \geq 6 \mid X > 3)$$. Then $$\frac{b + c}{a}$$ is equal to _______.
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