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The equation of a circle is given by $$x^2 + y^2 = a^2$$, where $$a$$ is the radius. If the equation is modified to change the origin other than $$(0, 0)$$, then find out the correct dimensions of $$A$$ and $$B$$ in a new equation: $$(x - At)^2 + (y - \frac{t}{B})^2 = a^2$$
The dimensions of $$t$$ is given as $$[T^{-1}]$$
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An object moves at a constant speed along a circular path in a horizontal plane with centre at the origin. When the object is at $$x = +2$$ m, its velocity is $$-4\hat{j}$$ m s$$^{-1}$$. The object's velocity ($$v$$) and acceleration ($$a$$) at $$x = -2$$ m will be
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The time taken by an object to slide down $$45°$$ rough inclined plane is $$n$$ times as it takes to slide down a perfectly smooth $$45°$$ incline plane. The coefficient of kinetic friction between the object and the incline plane is:
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Force acts for $$20$$ s on a body of mass $$20$$ kg, starting from rest, after which the force ceases and then body describes $$50$$ m in the next $$10$$ s. The value of force will be:
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Identify the correct statements from the following:
(A) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative
(B) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative
(C) Work done by friction on a body sliding down an inclined plane is positive
(D) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero
(E) Work done by the air resistance on an oscillating pendulum is negative
Choose the correct answer from the options given below:
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The time period of a satellite of earth is $$24$$ hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.
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A fully loaded boeing aircraft has a mass of $$5.4 \times 10^5$$ kg. Its total wing area is $$500$$ m$$^2$$. It is in level flight with a speed of $$1080$$ km h$$^{-1}$$. If the density of air $$\rho$$ is $$1.2$$ kg m$$^{-3}$$, the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be ($$g = 10$$ m s$$^{-2}$$)
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Heat energy of $$184$$ kJ is given to ice of mass $$600$$ g at $$-12°$$C, Specific heat of ice is $$2222.3$$ J kg$$^{-1}$$ °C$$^{-1}$$ and latent heat of ice is $$336$$ kJ kg$$^{-1}$$.
(A) Final temperature of system will be $$0°$$C
(B) Final temperature of the system will be greater than $$0°$$C
(C) The final system will have a mixture of ice and water in the ratio of $$5 : 1$$
(D) The final system will have a mixture of ice and water in the ratio of $$1 : 5$$
(E) The final system will have water only
Choose the correct answer from the options given below:
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At $$300$$ K, the rms speed of oxygen molecules is $$\sqrt{\frac{a+5}{\alpha}}$$ times to that of its average speed in the gas. Then, the value of $$\alpha$$ will be (use $$\pi = \frac{22}{7}$$)
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A point charge $$2 \times 10^{-2}$$ C is moved from $$P$$ to $$S$$ in a uniform electric field of $$30$$ N C$$^{-1}$$ directed along positive $$x$$-axis. If coordinates of $$P$$ and $$S$$ are $$(1, 2, 0)$$ m and $$(0, 0, 0)$$ m respectively, the work done by electric field will be
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With the help of potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is
(A) directly proportional to the length of the potentiometer wire
(B) directly proportional to the potential gradient of the wire
(C) inversely proportional to the potential gradient of the wire
(D) inversely proportional to the length of the potentiometer wire
Choose the correct option for the above statements:
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The electric current in a circular coil of four turns produces a magnetic induction $$32$$ T at its centre. The coil is unwound and is rewound into a circular coil of single turn, the magnetic induction at the centre of the coil by the same current will be:
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A square loop of area $$25$$ cm$$^2$$ has a resistance of $$10$$ $$\Omega$$. The loop is placed in uniform magnetic field of magnitude $$40.0$$ T. The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in $$1.0$$ sec, will be
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For the given figures, choose the correct options:
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Given below are two statements:
Statement I: Electromagnetic waves are not deflected by electric and magnetic field.
Statement II: The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other as $$E_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} B_0$$.
In the light of the above statements, choose the correct answer from the options given below:
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A scientist is observing a bacteria through a compound microscope. For better analysis and to improve its resolving power he should. (Select the best option)
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Substance $$A$$ has atomic mass number $$16$$ and half life of $$1$$ day. Another substance $$B$$ has atomic mass number $$32$$ and half life of $$\frac{1}{2}$$ day. If both $$A$$ and $$B$$ simultaneously start undergo radio activity at the same time with initial mass $$320$$ g each, how many total atoms of $$A$$ and $$B$$ combined would be left after $$2$$ days
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For the given logic gates combination, the correct truth table will be
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The modulation index for an A.M. wave having maximum and minimum peak to peak voltages of $$14$$ mV and $$6$$ mV respectively is:
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In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as $$5.25$$ mm and apparent thickness of the glass slab at $$5.00$$ mm. Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on Vernier scale is equal to 49 divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is $$\frac{x}{10} \times 10^{-3}$$, where $$x$$ is ______.
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A car is moving on a circular path of radius $$600$$ m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an initial speed of $$54$$ km h$$^{-1}$$ is $$t(1-e^{-\frac{\pi}{2}})$$ s. The value of $$t$$ is ______.
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A particle of mass $$100$$ g is projected at time $$t = 0$$ with a speed $$20$$ m s$$^{-1}$$ at an angle $$45°$$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $$t = 2$$ s is found to be $$\sqrt{K}$$ kg m$$^2$$ s$$^{-1}$$. The value of $$K$$ is ______.
(Take $$g = 10$$ m s$$^{-2}$$)
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A metal block of base area $$0.20$$ m$$^2$$ is placed on a table, as shown in the figure. A liquid film of thickness $$0.25$$ mm is inserted between the block and the table. The block is pushed by a horizontal force of $$0.1$$ N and moves with a constant speed. If the viscosity of the liquid is $$5.0 \times 10^{-3}$$ Pl, the speed of the block is ______ $$\times 10^{-3}$$ m s$$^{-1}$$.
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A particle of mass $$250$$ g executes a simple harmonic motion under a periodic force $$F = (-25x)$$ N. The particle attains a maximum speed of $$4$$ m s$$^{-1}$$ during its oscillation. The amplitude of the motion is ______ cm.
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For a charged spherical ball, electrostatic potential inside the ball varies with $$r$$ as $$V = 2ar^2 + b$$. Here, $$a$$ and $$b$$ are constant and $$r$$ is the distance from the center. The volume charge density inside the ball is $$-\lambda a\varepsilon$$. The value of $$\lambda$$ is ______.
$$\varepsilon$$ = permittivity of medium.
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A null point is found at $$200$$ cm in potentiometer when cell in secondary circuit is shunted by $$5$$ $$\Omega$$. When a resistance of $$15$$ $$\Omega$$ is used for shunting null point moves to $$300$$ cm. The internal resistance of the cell is ______ $$\Omega$$.
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An inductor of inductance $$2$$ $$\mu$$H is connected in series with a resistance, a variable capacitor and an AC source of frequency $$7$$ kHz. The value of capacitance for which maximum current is drawn into the circuit is $$\frac{1}{x}$$ F, where the value of $$x$$ is ______. (Take $$\pi = \frac{22}{7}$$)
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Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are $$2.8$$ (medium $$-1$$) and $$6.8$$ (medium $$-2$$), respectively. To satisfy the condition, so that the reflected and refracted rays are perpendicular to each other, the angle of incidence should be $$\tan^{-1}\left(1 + \frac{10}{\theta}\right)^{\frac{1}{2}}$$, the value of $$\theta$$ is ______.
(Given for dielectric media, $$\mu_r = 1$$)
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The ratio of de-Broglie wavelength of an $$\alpha$$-particle and a proton accelerated from rest by the same potential is $$\frac{1}{\sqrt{m}}$$, the value of $$m$$ is:
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When two resistance $$R_1$$ and $$R_2$$ connected in series and introduced into the left gap of a meter bridge and a resistance of $$10$$ $$\Omega$$ is introduced into the right gap, a null point is found at $$60$$ cm from left side. When $$R_1$$ and $$R_2$$ are connected in parallel and introduced into the left gap, a resistance of $$3$$ $$\Omega$$ is introduced into the right gap to get null point at $$40$$ cm from left end. The product of $$R_1 R_2$$ is ______ $$\Omega^2$$
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Given below are two statements:
Statement I: The decrease in first ionization enthalpy from B to Al is much larger than that from Al to Ga.
Statement II: The d orbitals in Ga are completely filled.
In the light of the above statements, choose the most appropriate answer from the options given below
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According to MO theory the bond orders for O$$_2^{2-}$$, CO and NO$$^+$$ respectively, are
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Which of the following relations are correct?
(A) $$\Delta U = q + p\Delta V$$
(B) $$\Delta G = \Delta H - T\Delta S$$
(C) $$\Delta S = \frac{q_{rev}}{T}$$
(D) $$\Delta H = \Delta U - \Delta nRT$$
Choose the most appropriate answer from the options given below:
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An indicator 'X' is used for studying the effect of variation in concentration of iodide on the rate of reaction of iodide ion with H$$_2$$O$$_2$$ at room temp. The indicator 'X' forms blue colored complex with compound 'A' present in the solution. The indicator 'X' and compound 'A' respectively are
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Given below are two statements:
Statement I: Nickel is being used as the catalyst for producing syn gas and edible fats.
Statement II: Silicon forms both electron rich and electron deficient hydrides.
In the light of the above statements, choose the most appropriate answer from the options given below:
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When a hydrocarbon A undergoes combustion in the presence of air, it requires $$9.5$$ equivalents of oxygen and produces $$3$$ equivalents of water. What is the molecular formula of A?
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The concentration of dissolved Oxygen in water for growth of fish should be more than X ppm and Biochemical Oxygen Demand in clean water should be less than Y ppm. X and Y in ppm are, respectively.
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Match List I with List II.
| List I | List II |
|---|---|
| A. van't Hoff factor, i | I. Cryoscopic constant |
| B. k$$_f$$ | II. Isotonic solutions |
| C. Solutions with same osmotic pressure | III. $$\frac{\text{Normal molar mass}}{\text{Abnormal molar mass}}$$ |
| D. Azeotropes | IV. Solutions with same composition of vapour above it |
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Match List-I and List-II
| List-I | List-II |
|---|---|
| A. Osmosis | I. Solvent molecules pass through semi permeable membrane towards solvent side. |
| B. Reverse osmosis | II. Movement of charged colloidal particles under the influence of applied electric potential towards oppositely charged electrodes |
| C. Electro osmosis | III. Solvent molecules pass through semi permeable membrane towards solution side |
| D. Electrophoresis | IV. Dispersion medium moves in an electric field. |
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The major component of which of the following ore is sulphide based mineral?
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A solution of CrO$$_5$$ in amyl alcohol has a ____colour
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The set of correct statements is:
(i) Manganese exhibits $$+7$$ oxidation state in its oxide.
(ii) Ruthenium and Osmium exhibit $$+8$$ oxidation in their oxides.
(iii) Sc shows $$+4$$ oxidation state which is oxidizing in nature.
(iv) Cr shows oxidising nature in $$+6$$ oxidation state.
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Correct order of spin only magnetic moment of the following complex ions is:
(Given At. No. Fe : 26, Co : 27)
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The one giving maximum number of isomeric alkenes on dehydrohalogenation reaction is (excluding rearrangement)
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Find out the major product for the following reaction
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Find out the major products from the following reaction sequence.
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Reaction of propanamide with Br$$_2$$ / KOH (aq) produces:
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Match List-I and List-II.
| List-I | List-II |
|---|---|
| A. Elastomeric polymer | I. Urea formaldehyde resin |
| B. Fibre polymer | II. Polystyrene |
| C. Thermosetting polymer | III. Polyester |
| D. Thermoplastic polymer | IV. Neoprene |
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A doctor prescribed the drug Equanil to a patient. The patient was likely to have symptoms of which disease?
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Following tetrapeptide can be represented as
(F, L, D, Y, I, Q, P are one letter codes for amino acids)
Assume that the radius of the first Bohr orbit of hydrogen atom is $$0.6$$ $$\text{\AA}$$. The radius of the third Bohr orbit of He$$^+$$ is ______ picometer. (Nearest Integer)
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At 298 K
$$\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g), K_1 = 4 \times 10^5$$
$$\text{N}_2(g) + \text{O}_2(g) \rightleftharpoons 2\text{NO}(g), K_2 = 1.6 \times 10^{12}$$
$$\text{H}_2(g) + \frac{1}{2}\text{O}_2(g) \rightleftharpoons \text{H}_2\text{O}(g), K_3 = 1.0 \times 10^{-13}$$
Based on above equilibria, the equilibrium constant of the reaction,
$$2\text{NH}_3(g) + \frac{5}{2}\text{O}_2(g) \rightleftharpoons 2\text{NO}(g) + 3\text{H}_2\text{O}(g)$$
is ______ $$\times 10^{-33}$$ (Nearest integer)
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The volume of HCl, containing $$73$$ g L$$^{-1}$$, required to completely neutralise NaOH obtained by reacting $$0.69$$ g of metallic sodium with water, is ______ mL. (Nearest Integer)
(Given: molar Masses of Na, Cl, O, H are 23, 35.5, 16 and 1 g mol$$^{-1}$$ respectively)
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On heating, LiNO$$_3$$ gives how many compounds among the following?
Li$$_2$$O, N$$_2$$, O$$_2$$, LiNO$$_2$$, NO$$_2$$
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When $$0.01$$ mol of an organic compound containing $$60\%$$ carbon was burnt completely, $$4.4$$ g of CO$$_2$$ was produced. The molar mass of compound is ______ g mol$$^{-1}$$ (Nearest integer)
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A metal M forms hexagonal close-packed structure. The total number of voids in $$0.02$$ mol of it is ______ $$\times 10^{21}$$ (Nearest integer)
(Given N$$_A = 6.02 \times 10^{23}$$)
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The equilibrium constant for the reaction Zn(s) + Sn$$^{2+}$$(aq) $$\rightleftharpoons$$ Zn$$^{2+}$$(aq) + Sn(s) is $$1 \times 10^{20}$$ at 298 K. The magnitude of standard electrode potential of Sn/Sn$$^{2+}$$ if E$$^0_{\text{Zn}^{2+}/\text{Zn}} = -0.76$$ V is ______ $$\times 10^{-2}$$ V. (Nearest integer)
Given: $$\frac{2.303 RT}{F} = 0.059$$ V
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For conversion of compound A $$\to$$ B, the rate constant of the reaction was found to be $$4.6 \times 10^{-5}$$ L mol$$^{-1}$$ s$$^{-1}$$. The order of the reaction is ______.
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Total number of acidic oxides among N$$_2$$O$$_3$$, NO$$_2$$, N$$_2$$O, Cl$$_2$$O$$_7$$, SO$$_2$$, CO, CaO, Na$$_2$$O and NO is ______.
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The denticity of the ligand present in the Fehling's reagent is ______.
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The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:
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The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is
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Let $$K$$ be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x)^{99}$$. Let $$a$$ be the middle term in the expansion of $$\left(2 + \frac{1}{\sqrt{2}}\right)^{200}$$. If $$\frac{^{200}C_{99}K}{a} = \frac{2^l m}{n}$$, where $$m$$ and $$n$$ are odd numbers, then the ordered pair $$(l, n)$$ is equal to:
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The set of all values of $$\lambda$$ for which the equation $$\cos^2 2x - 2\sin^4 x - 2\cos^2 x = \lambda$$
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If the tangent at a point P on the parabola $$y^2 = 3x$$ is parallel to the line $$x + 2y = 1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4} + \frac{y^2}{1} = 1$$ are perpendicular to the line $$x - y = 2$$, then the area of the triangle $$PQR$$ is:
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The statement $$B \Rightarrow ((\neg A) \vee B)$$ is not equivalent to:
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Let $$R$$ be a relation defined on $$\mathbb{N}$$ as $$a R b$$ is $$2a + 3b$$ is a multiple of $$5, a, b \in \mathbb{N}$$. Then $$R$$ is
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The set of all values of $$t \in \mathbb{R}$$, for which the matrix $$\begin{bmatrix} e^t & e^{-t}(\sin t - 2\cos t) & e^{-t}(-2\sin t - \cos t) \\ e^t & e^{-t}(2\sin t + \cos t) & e^{-t}(\sin t - 2\cos t) \\ e^t & e^{-t}\cos t & e^{-t}\sin t \end{bmatrix}$$ is invertible, is
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Let $$f$$ and $$g$$ be twice differentiable functions on $$R$$ such that
$$f''(x) = g''(x) + 6x$$
$$f'(1) = 4g'(1) - 3 = 9$$
$$f(2) = 3, g(2) = 12$$
Then which of the following is NOT true?
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The value of the integral $$\int_1^2 \left(\frac{t^4+1}{t^6+1}\right) dt$$ is:
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The value of the integral $$\int_{1/2}^{2} \frac{\tan^{-1}x}{x} dx$$ is equal to
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The area of the region $$A = \{(x,y) : |\cos x - \sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\}$$
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Let $$y = y(x)$$ be the solution of the differential equation $$x \log_e x \frac{dy}{dx} + y = x^2 \log_e x$$, $$(x > 1)$$. If $$y(2) = 2$$, then $$y(e)$$ is equal to
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If $$\vec{a} = \hat{i} + 2\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{c} = 7\hat{i} - 3\hat{j} + 4\hat{k}$$, $$\vec{r} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$$ and $$\vec{r} \cdot \vec{a} = 0$$ then $$\vec{r} \cdot \vec{c}$$ is equal to:
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Let $$\vec{a} = 4\hat{i} + 3\hat{j}$$ and $$\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$$ and $$\vec{c}$$ is a vector such that $$\vec{c} \cdot (\vec{a} \times \vec{b}) + 25 = 0$$, $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$$ and projection of $$\vec{c}$$ on $$\vec{a}$$ is $$1$$, then the projection of $$\vec{c}$$ on $$\vec{b}$$ equals:
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Shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$ and $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$ is:
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The plane $$2x - y + z = 4$$ intersects the line segment joining the points $$A(a, -2, 4)$$ and $$B(2, b, -3)$$ at the point $$C$$ in the ratio $$2 : 1$$ and the distance of the point $$C$$ from the origin is $$\sqrt{5}$$. If $$ab < 0$$ and $$P$$ is the point $$(a-b, b, 2b-a)$$ then $$CP^2$$ is equal to:
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If the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{1}$$ and $$\frac{x-a}{2} = \frac{y+2}{3} = \frac{z-3}{1}$$ intersects at the point $$P$$, then the distance of the point $$P$$ from the plane $$z = a$$ is:
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Let $$S = \{w_1, w_2, \ldots\}$$ be the sample space associated to a random experiment. Let $$P(w_n) = \frac{P(w_{n-1})}{2}$$, $$n \geq 2$$. Let $$A = \{2k + 3l; k, l \in \mathbb{N}\}$$ and $$B = \{w_n; n \in A\}$$. Then $$P(B)$$ is equal to
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Let $$\alpha_1, \alpha_2, \ldots, \alpha_7$$ be the roots of the equation $$x^7 + 3x^5 - 13x^3 - 15x = 0$$ and $$|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$$. Then, $$\alpha_1\alpha_2 - \alpha_3\alpha_4 + \alpha_5\alpha_6$$ is equal to ______.
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Let $$\alpha = 8 - 14i$$, $$A = \left\{z \in \mathbb{C} : \frac{\alpha\bar{z} - \bar{\alpha}z}{z^2 - (\bar{z})^2 - 112i} = 1\right\}$$ and $$B = \{z \in \mathbb{C} : |z + 3i| = 4\}$$. Then, $$\sum_{z \in A \cap B} (Re\ z - Im\ z)$$ is equal to ______.
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The total number of 4-digit numbers whose greatest common divisor with $$54$$ is $$2$$, is
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Let $$a_1 = b_1 = 1$$ and $$a_n = a_{n-1} + (n-1)$$, $$b_n = b_{n-1} + a_{n-1}$$, $$\forall n \geq 2$$. If $$S = \sum_{n=1}^{10} \left(\frac{b_n}{2^n}\right)$$ and $$T = \sum_{n=1}^{8} \frac{n}{2^{n-1}}$$ then $$2^7(2S - T)$$ is equal to
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Let $$\{a_k\}$$ and $$\{b_k\}$$, $$k \in \mathbb{N}$$, be two G.P.s with common ratio $$r_1$$ and $$r_2$$ respectively such that $$a_1 = b_1 = 4$$ and $$r_1 < r_2$$. Let $$c_k = a_k + b_k$$, $$k \in \mathbb{N}$$. If $$c_2 = 5$$ and $$c_3 = \frac{13}{4}$$ then $$\sum_{k=1}^{\infty} c_k - (12a_6 + 8b_4)$$ is equal to
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A circle with centre $$(2, 3)$$ and radius $$4$$ intersects the line $$x + y = 3$$ at the points $$P$$ and $$Q$$. If the tangents at $$P$$ and $$Q$$ intersect at the point $$S(\alpha, \beta)$$, then $$4\alpha - 7\beta$$ is equal to
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A triangle is formed by the tangents at the point $$(2, 2)$$ on the curves $$y^2 = 2x$$ and $$x^2 + y^2 = 4x$$, and the line $$x + y + 2 = 0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to
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Let $$X = \{11, 12, 13, \ldots, 40, 41\}$$ and $$Y = \{61, 62, 63, \ldots, 90, 91\}$$ be the two sets of observations. If $$\bar{x}$$ and $$\bar{y}$$ are their respective means and $$\sigma^2$$ is the variance of all the observations in $$X \cup Y$$, then $$|\bar{x} + \bar{y} - \sigma^2|$$ is equal to
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Let A be a symmetric matrix such that $$|A| = 2$$ and $$\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$$. If the sum of the diagonal elements of A is $$s$$, then $$\frac{\beta s}{\alpha^2}$$ is equal to ______.
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Consider a function $$f : \mathbb{N} \to \mathbb{R}$$, satisfying $$f(1) + 2f(2) + 3f(3) + \ldots + xf(x) = x(x+1)f(x)$$; $$x \geq 2$$ with $$f(1) = 1$$. Then $$\frac{1}{f(2022)} + \frac{1}{f(2028)}$$ is equal to
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If the equation of the normal to the curve $$y = \frac{x-a}{(x+b)(x-2)}$$ at the point $$(1, -3)$$ is $$x - 4y = 13$$ then the value of $$a + b$$ is equal to ______.
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