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Given below are two statements : Statement (I) : Dimensions of specific heat is $$[L^2 T^{-2} K^{-1}]$$. Statement (II) : Dimensions of gas constant is $$[ML^2 T^{-1} K^{-1}]$$. In the light of the above statements, choose the most appropriate answer from the options given below.
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A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in $$t_1$$. If it is projected vertically downwards from the same point with the same speed, it reaches the ground in $$t_2$$. Time required to reach the ground, if it is dropped from the top of the tower, is :
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A body of weight $$200 \text{ N}$$ is suspended from a tree branch through a chain of mass $$10 \text{ kg}$$. The branch pulls the chain by a force equal to (if $$g = 10 \text{ m/s}^2$$) :
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A car of $$800 \text{ kg}$$ is taking turn on a banked road of radius $$300 \text{ m}$$ and angle of banking $$30°$$. If coefficient of static friction is $$0.2$$ then the maximum speed with which car can negotiate the turn safely: $$(g = 10 \text{ m/s}^2, \sqrt{3} = 1.73)$$
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When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be:
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Assuming the earth to be a sphere of uniform mass density, a body weighed $$300 \text{ N}$$ on the surface of earth. How much it would weigh at $$R/4$$ depth under surface of earth?
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Pressure inside a soap bubble is greater than the pressure outside by an amount : (given : $$R$$ = Radius of bubble, $$S$$ = Surface tension of bubble)
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A total of $$48 \text{ J}$$ heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by $$2°C$$. The work done by the gas is: Given, $$R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1}$$.
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Energy of 10 non rigid diatomic molecules at temperature $$T$$ is :
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Two identical conducting spheres $$P$$ and $$S$$ with charge $$Q$$ on each, repel each other with a force $$16 \text{ N}$$. A third identical uncharged conducting sphere $$R$$ is successively brought in contact with the two spheres. The new force of repulsion between $$P$$ and $$S$$ is :
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The number of electrons flowing per second in the filament of a $$110 \text{ W}$$ bulb operating at $$220 \text{ V}$$ is : (Given $$e = 1.6 \times 10^{-19} \text{ C}$$)
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Match List-I with List-II :
Choose the correct answer from the options given below :
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In a coil, the current changes from $$-2 \text{ A}$$ to $$+2 \text{ A}$$ in $$0.2 \text{ s}$$ and induces an emf of $$0.1 \text{ V}$$. The self inductance of the coil is :
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In the given electromagnetic wave $$E_y = 600 \sin(\omega t - kx) \text{ Vm}^{-1}$$, intensity of the associated light beam is (in $$\text{W/m}^2$$) : (Given $$\epsilon_0 = 9 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}$$)
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In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division $$= 49$$ MSD; 20 divisions on main scale in each cm. For mark on paper MSR $$= 8.45 \text{ cm}$$, VC $$= 26$$. For mark on paper seen through slab MSR $$= 7.12 \text{ cm}$$, VC $$= 41$$. For powder particle on the top surface of the glass slab MSR $$= 4.05 \text{ cm}$$, VC $$= 1$$. (MSR = Main Scale Reading, VC = Vernier Coincidence) Refractive index of the glass slab is :
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For the thin convex lens, the radii of curvature are at $$15 \text{ cm}$$ and $$30 \text{ cm}$$ respectively. The focal length the lens is $$20 \text{ cm}$$. The refractive index of the material is :
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When UV light of wavelength $$300 \text{ nm}$$ is incident on the metal surface having work function $$2.13 \text{ eV}$$, electron emission takes place. The stopping potential is: (Given $$hc = 1240 \text{ eVnm}$$)
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The longest wavelength associated with Paschen series is : (Given $$R_H = 1.097 \times 10^7 \text{ SI unit}$$)
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The acceptor level of a p-type semiconductor is $$6 \text{ eV}$$. The maximum wavelength of light which can create a hole would be : Given $$hc = 1242 \text{ eVnm}$$.
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In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $$4^{th}$$ division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is $$0.04 \text{ mm}$$ then how many main scale divisions are there in $$1 \text{ cm}$$?
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A particle moves in a straight line so that its displacement $$x$$ at any time $$t$$ is given by $$x^2 = 1 + t^2$$. Its acceleration at any time $$t$$ is $$x^{-n}$$ where $$n =$$ ___________
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Three balls of masses $$2 \text{ kg}$$, $$4 \text{ kg}$$ and $$6 \text{ kg}$$ respectively are arranged at centre of the edges of an equilateral triangle of side $$2 \text{ m}$$. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ___________ $$\text{kgm}^2$$.
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A wire of cross sectional area A, modulus of elasticity $$2 \times 10^{11} \text{ Nm}^{-2}$$ and length $$2 \text{ m}$$ is stretched between two vertical rigid supports. When a mass of $$2 \text{ kg}$$ is suspended at the middle, it sags lower from its original position making an angle $$\theta = \frac{1}{100}$$ radian on the points of support. The value of A is ___________ $$\times 10^{-4} \text{ m}^2$$ (consider $$x << L$$). (given : $$g = 10 \text{ m/s}^2$$)
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Two open organ pipes of lengths $$60 \text{ cm}$$ and $$90 \text{ cm}$$ resonate at $$6^{th}$$ and $$5^{th}$$ harmonics respectively. The difference of frequencies for the given modes is ___________ Hz. (Velocity of sound in air $$= 333 \text{ m/s}$$)
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A capacitor of capacitance $$10 \mu\text{F}$$ whose plates are separated by $$10 \text{ mm}$$ through air and each plate has area $$4 \text{ cm}^2$$ is now filled equally with two dielectric media of $$K_1 = 2$$, $$K_2 = 3$$ respectively as shown in figure. If new force between the plates is $$8 \text{ N}$$. The supply voltage is ___________ $$\text{ V}$$.
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In the given figure an ammeter A consists of a $$240 \Omega$$ coil connected in parallel to a $$10 \Omega$$ shunt. The reading of the ammeter is ___________ mA.
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A coil having 100 turns, area of $$5 \times 10^{-3} \text{ m}^2$$, carrying current of $$1 \text{ mA}$$ is placed in uniform magnetic field of $$0.20 \text{ T}$$ such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through $$90°$$ is ___________ $$\mu\text{J}$$.
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For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is $$2.5 \text{ nF}$$. If resistance of $$200 \Omega$$ and $$100 \text{ mH}$$ inductor is being used in the given circuit. The frequency of ac source is ___________ $$\times 10^3 \text{ Hz}$$. (given $$\pi^2 = 10$$)
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Two coherent monochromatic light beams of intensities I and 4I are superimposed. The difference between maximum and minimum possible intensities in the resulting beam is $$xI$$. The value of $$x$$ is ___________.
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In Franck-Hertz experiment, the first dip in the current-voltage graph for hydrogen is observed at 10.2 V. The wavelength of light emitted by hydrogen atom when excited to the first excitation level is ___________ nm. (Given $$hc = 1245 \text{ eVnm}$$, $$e = 1.6 \times 10^{-19} \text{ C}$$).
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Molality ($$m$$) of $$3M$$ aqueous solution of NaCl is : (Given : Density of solution $$= 1.25 \text{ g mL}^{-1}$$, Molar mass in $$\text{gmol}^{-1}$$: Na $$- 23$$, Cl $$- 35.5$$)
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The ratio $$\frac{K_P}{K_C}$$ for the reaction : $$CO_{(g)} + \frac{1}{2} O_{2(g)} \rightleftharpoons CO_{2(g)}$$ is :
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Match List - I with List - II.
Choose the correct answer from the options given below :
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The number of ions from the following that are expected to behave as oxidising agent is : $$Sn^{4+}, Sn^{2+}, Pb^{2+}, Tl^{3+}, Pb^{4+}, Tl^{+}$$
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Evaluate the following statements related to group 14 elements for their correctness. (A) Covalent radius decreases down the group from C to Pb in a regular manner. (B) Electronegativity decreases from C to Pb down the group gradually. (C) Maximum covalance of C is 4 whereas other elements can expand their covalance due to presence of d orbitals. (D) Heavier elements do not form $$p\pi - p\pi$$ bonds. (E) Carbon can exhibit negative oxidation states. Choose the correct answer from the options given below :
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The correct statement among the following, for a "chromatography" purification method is :
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The incorrect statement regarding the geometrical isomers of 2-butene is :
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How can an electrochemical cell be converted into an electrolytic cell?
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Arrange the following elements in the increasing order of number of unpaired electrons in it. (A) Sc (B) Cr (C) V (D) Ti (E) Mn. Choose the correct answer from the options given below :
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The correct IUPAC name of $$[PtBr_2(PMe_3)_2]$$ is :
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Given below are two statements : Statement I : $$PF_5$$ and $$BrF_5$$ both exhibit $$sp^3d$$ hybridisation. Statement II : Both $$SF_6$$ and $$[Co(NH_3)_6]^{3+}$$ exhibit $$sp^3d^2$$ hybridisation. In the light of the above statements, choose the correct answer from the options given below :
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Match List - I with List - II.
Choose the correct answer from the options given below :
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The correct arrangement for decreasing order of electrophilic substitution for above compounds is :
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Consider the given reaction, identify the major product "P". $$CH_3 - COOH \xrightarrow{(i) LiAlH_4 \; (ii) PCC \; (iii) HCN/OH^{-} \; (iv) H_2O/OH^{-}, \Delta}$$ "P"
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Consider the above chemical reaction. Product "A " is :
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The major products formed A and B respectively are:
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Identify the product (A) in the following reaction.
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During the detection of acidic radical present in a salt, a student gets a pale yellow precipitate soluble with difficulty in $$NH_4OH$$ solution when sodium carbonate extract was first acidified with dil. $$HNO_3$$ and then $$AgNO_3$$ solution was added. This indicates presence of :
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Match List - I with List - II.
Choose the correct answer from the options given below :
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The incorrect statements regarding enzymes are : (A) Enzymes are biocatalysts. (B) Enzymes are non-specific and can catalyse different kinds of reactions. (C) Most Enzymes are globular proteins. (D) Enzyme - oxidase catalyses the hydrolysis of maltose into glucose. Choose the correct answer from the option given below :
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Consider the following reactions $$NiS + HNO_3 + HCl \rightarrow A + NO + S + H_2O$$. $$A + NH_4OH + H_3C - C(=N-OH) - C(=N-OH) - H_3C \rightarrow B + NH_4Cl + H_2O$$. The number of protons that do not involve in hydrogen bonding in the product $$B$$ is ___________.
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For hydrogen atom, energy of an electron in first excited state is $$-3.4 \text{ eV}$$, K.E. of the same electron of hydrogen atom is $$x \text{ eV}$$. Value of $$x$$ is ___________ $$\times 10^{-1} \text{ eV}$$. (Nearest integer)
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An amine (X) is prepared by ammonolysis of benzyl chloride. On adding p-toluenesulphonyl chloride to it the solution remains clear. Molar mass of the amine (X) formed is ___________ $$\text{gmol}^{-1}$$. (Given molar mass in $$\text{gmol}^{-1}$$: C : 12, H : 1, O : 16, N : 14)
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For the reaction at $$298 \text{ K}$$, $$2A + B \rightarrow C$$. $$\Delta H = 400 \text{ kJ mol}^{-1}$$ and $$\Delta S = 0.2 \text{ kJ mol}^{-1} \text{K}^{-1}$$. The reaction will become spontaneous above ___________ K.
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Consider the two different first order reactions given below $$A + B \rightarrow C$$ (Reaction 1) $$P \rightarrow Q$$ (Reaction 2). The ratio of the half life of Reaction 1 : Reaction 2 is $$5 : 2$$. If $$t_1$$ and $$t_2$$ represent the time taken to complete $$\frac{2}{3}^{rd}$$ and $$\frac{4}{5}^{th}$$ of Reaction 1 and Reaction 2, respectively, then the value of the ratio $$t_1 : t_2$$ is ___________ $$\times 10^{-1}$$ (nearest integer). [Given : $$\log_{10}(3) = 0.477$$ and $$\log_{10}(5) = 0.699$$]
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Among $$VO_2^{+}$$, $$MnO_4^{-}$$ and $$Cr_2O_7^{2-}$$, the spin-only magnetic moment value of the species with least oxidising ability is ___________ BM (Nearest integer). (Given atomic number $$V = 23, Mn = 25, Cr = 24$$)
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Number of carbocations from the following that are not stabilized by hyperconjugation is __________
_
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When '$$x$$' $$\times 10^{-2}$$ mL methanol (molar mass $$= 32 \text{ g}$$; density $$= 0.792 \text{ g/cm}^3$$) is added to $$100 \text{ mL}$$ water (density $$= 1 \text{ g/cm}^3$$), the following diagram is obtained. $$x =$$ ___________ (nearest integer).
[Given : Molal freezing point depression constant of water at $$273.15 \text{ K}$$ is $$1.86 \text{ K kg mol}^{-1}$$]
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Total number of species from the following with central atom utilising $$sp^2$$ hybrid orbitals for bonding is ___________. $$NH_3$$, $$SO_2$$, $$SiO_2$$, $$BeCl_2$$, $$C_2H_2$$, $$C_2H_4$$, $$BCl_3$$, $$HCHO$$, $$C_6H_6$$, $$BF_3$$, $$C_2H_4Cl_2$$
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The ratio of number of oxygen atoms to bromine atoms in the product Q is ___________ $$\times 10^{-1}$$.
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If $$z_1, z_2$$ are two distinct complex numbers such that $$\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z}_2}\right| = 2$$, then
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Let $$0 \leq r \leq n$$. If $$^{n+1}C_{r+1} : ^{n}C_{r} : ^{n-1}C_{r-1} = 55 : 35 : 21$$, then $$2n + 5r$$ is equal to:
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If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $$315^{th}$$ position in this arrangement is :
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Let $$ABC$$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $$ABC$$ and the same process is repeated infinitely many times. If $$P$$ is the sum of perimeters and $$Q$$ is be the sum of areas of all the triangles formed in this process, then :
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A software company sets up $$m$$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $$m$$ is equal to:
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If $$P(6, 1)$$ be the orthocentre of the triangle whose vertices are $$A(5, -2)$$, $$B(8, 3)$$ and $$C(h, k)$$, then the point $$C$$ lies on the circle:
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If the locus of the point, whose distances from the point $$(2, 1)$$ and $$(1, 3)$$ are in the ratio $$5 : 4$$, is $$ax^2 + by^2 + cxy + dx + ey + 170 = 0$$, then the value of $$a^2 + 2b + 3c + 4d + e$$ is equal to :
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$$\lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \cdots + ((n-1)^2 - (n-1)) \cdot 1}{(1^3 + 2^3 + \cdots + n^3) - (1^2 + 2^2 + \cdots + n^2)}$$ is equal to :
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Let $$A = \{1, 2, 3, 4, 5\}$$. Let $$R$$ be a relation on $$A$$ defined by $$xRy$$ if and only if $$4x \leq 5y$$. Let $$m$$ be the number of elements in $$R$$ and $$n$$ be the minimum number of elements from $$A \times A$$ that are required to be added to $$R$$ to make it a symmetric relation. Then $$m + n$$ is equal to :
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If $$A$$ is a square matrix of order 3 such that $$\det(A) = 3$$ and $$\det(\text{adj}(-4 \text{adj}(-3 \text{adj}(3 \text{adj}((2A)^{-1}))))) = 2^m 3^n$$, then $$m + 2n$$ is equal to :
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Let $$f(x) = \frac{1}{7 - \sin 5x}$$ be a function defined on $$\mathbb{R}$$. Then the range of the function $$f(x)$$ is equal to :
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Suppose for a differentiable function $$h$$, $$h(0) = 0$$, $$h(1) = 1$$ and $$h'(0) = h'(1) = 2$$. If $$g(x) = h(e^x)e^{h(x)}$$, then $$g'(0)$$ is equal to:
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If the function $$f(x) = \left(\frac{1}{x}\right)^{2x}; x \gt 0$$ attains the maximum value at $$x = \frac{1}{e}$$ then :
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If $$\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) +$$ constant, then the maximum value of $$a \sin x + b \cos x$$, is :
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If the area of the region $$\{(x, y) : \frac{a}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2, 0 < a < 1\}$$ is $$(\log_e 2) - \frac{1}{7}$$ then the value of $$7a - 3$$ is equal to:
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Suppose the solution of the differential equation $$\frac{dy}{dx} = \frac{(2+\alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta\gamma - 4\alpha)}$$ represents a circle passing through origin. Then the radius of this circle is :
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Let $$\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$$, $$\vec{b} = ((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projection of $$\vec{a}$$ on $$\vec{b}$$ is :
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Let $$\vec{a} = 6\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$|\vec{c}| \geq 6$$, $$\vec{a} \cdot \vec{c} = 6|\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$\vec{a} \times \vec{b}$$ and $$\vec{c}$$ is $$60°$$, then $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is equal to:
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Let $$P(\alpha, \beta, \gamma)$$ be the image of the point $$Q(3, -3, 1)$$ in the line $$\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}$$ and $$R$$ be the point $$(2, 5, -1)$$. If the area of the triangle $$PQR$$ is $$\lambda$$ and $$\lambda^2 = 14K$$, then $$K$$ is equal to :
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If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
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Let $$\alpha, \beta$$ be roots of $$x^2 + \sqrt{2}x - 8 = 0$$. If $$U_n = \alpha^n + \beta^n$$, then $$\frac{U_{10} + \sqrt{2}U_9}{2U_8}$$ is equal to ___________
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If $$S(x) = (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 60(1+x)^{60}$$, $$x \neq 0$$, and $$(60)^2 S(60) = a(b)^b + b$$, where $$a, b \in N$$, then $$(a + b)$$ equal to ___________
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The length of the latus rectum and directrices of a hyperbola with eccentricity $$e$$ are 9 and $$x = \pm \frac{4}{\sqrt{13}}$$, respectively. Let the line $$y - \sqrt{3}x + \sqrt{3} = 0$$ touch this hyperbola at $$(x_0, y_0)$$. If $$m$$ is the product of the focal distances of the point $$(x_0, y_0)$$, then $$4e^2 + m$$ is equal to ___________
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In a triangle $$ABC$$, $$BC = 7$$, $$AC = 8$$, $$AB = \alpha \in \mathbb{N}$$ and $$\cos A = \frac{2}{3}$$. If $$49\cos(3C) + 42 = \frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to ___________
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If the system of equations $$2x + 7y + \lambda z = 3$$, $$3x + 2y + 5z = 4$$, $$x + \mu y + 32z = -1$$ has infinitely many solutions, then $$(\lambda - \mu)$$ is equal to ___________
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f : [0, \infty) \rightarrow \mathbb{R}$$ be a function defined by $$f(x) = \left[\frac{x}{2} + 3\right] - [\sqrt{x}]$$. Let $$S$$ be the set of all points in the interval $$[0, 8]$$ at which $$f$$ is not continuous. Then $$\sum_{a \in S} a$$ is equal to ___________
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Let $$[t]$$ denote the largest integer less than or equal to $$t$$. If $$\int_0^3 \left([x^2] + \left[\frac{x^2}{2}\right]\right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}$$, where $$a, b, c \in \mathbb{Z}$$, then $$a + b + c$$ is equal to ___________
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If the solution $$y(x)$$ of the given differential equation $$(e^y + 1)\cos x \, dx + e^y \sin x \, dy = 0$$ passes through the point $$\left(\frac{\pi}{2}, 0\right)$$, then the value of $$e^{y\left(\frac{\pi}{6}\right)}$$ is equal to ___________
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If the shortest distance between the lines $$\frac{x - \lambda}{3} = \frac{y - 2}{-1} = \frac{z - 1}{1}$$ and $$\frac{x + 2}{-3} = \frac{y + 5}{2} = \frac{z - 4}{4}$$ is $$\frac{44}{\sqrt{30}}$$, then the largest possible value of $$|\lambda|$$ is equal to ___________
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From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $$X$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$n - m$$ is equal to ___________
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