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Match List I with List II
| List I | List II |
|---|---|
| A. Torque | I. kg m$$^{-1}$$ s$$^{-2}$$ |
| B. Energy density | II. kg m s$$^{-1}$$ |
| C. Pressure gradient | III. kg m$$^{-2}$$ s$$^{-2}$$ |
| D. Impulse | IV. kg m$$^2$$ s$$^{-2}$$ |
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A vehicle travels $$4$$ km with speed of $$3$$ km h$$^{-1}$$ and another $$4$$ km with speed of $$5$$ km h$$^{-1}$$, then its average speed is:
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An object is allowed to fall from a height $$R$$ above the earth, where $$R$$ is the radius of earth. Its velocity when it strikes the earth's surface, ignoring air resistance, will be:
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A block of $$\sqrt{3}$$ kg is attached to a string whose other end is attached to the wall. An unknown force $$F$$ is applied so that the string makes an angle of $$30°$$ with the wall. The tension $$T$$ is: (Given $$g = 10$$ m s$$^{-2}$$)
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A machine gun of mass $$10$$ kg fires $$20$$ g bullets at the rate of $$180$$ bullets per minute with a speed of $$100$$ m s$$^{-1}$$ each. The recoil velocity of the gun is:
A force is applied to a steel wire $$A$$, rigidly clamped at one end. As a result elongation in the wire is $$0.2$$ mm. If same force is applied to another steel wire $$B$$ of double the length and a diameter $$2.4$$ times that of the wire $$A$$, the elongation in the wire $$B$$ will be (wires having uniform circular cross sections)
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Efficiency of a reversible heat engine will be highest at $$-273°$$C temperature of cold reservoir.
Reason R: The efficiency of Carnot's engine depends not only on temperature of cold reservoir but it depends on the temperature of hot reservoir too and is given as $$\eta = \left(1 - \frac{T_2}{T_1}\right)$$
In the light of the above statements, choose the correct answer from the options given below:
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A flask contains hydrogen and oxygen in the ratio of $$2 : 1$$ by mass at temperature $$27°$$C. The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is:
For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is $$1$$ kg, the angular frequency is $$\omega_1$$. When the mass block is $$2$$ kg the angular frequency is $$\omega_2$$. The ratio $$\frac{\omega_2}{\omega_1}$$ is:
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As shown in the figure, a point charge $$Q$$ is placed at the centre of conducting spherical shell of inner radius $$a$$ and outer radius $$b$$. The electric field due to charge $$Q$$ in three different regions I, II and III is given by:
(I: $$r < a$$, II: $$a < r < b$$, III: $$r > b$$)
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As shown in the figure, a current of $$2$$ A flowing in an equilateral triangle of side $$4\sqrt{3}$$ cm. The magnetic field at the centroid $$O$$ of the triangle is:
(Neglect the effect of earth's magnetic field)
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A current carrying rectangular loop $$PQRS$$ is made of a uniform wire. The length $$PR = QS = 5$$ cm and $$PQ = RS = 100$$ cm. If the ammeter current reading changes from $$I$$ to $$2I$$, then the ratio of the magnetic forces per unit length on the wire $$PQ$$ due to the wire $$RS$$ in the two cases respectively $$(f^I_{PQ} : f^{2I}_{PQ})$$ is:
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In the given circuit, the rms value of current $$(I_{rms})$$ through the resistor $$R$$ is:
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A thin prism $$P_1$$ with an angle $$6°$$ and made of glass of refractive index $$1.54$$ is combined with another prism $$P_2$$ made from glass of refractive index $$1.72$$ to produce dispersion without average deviation. The angle of prism $$P_2$$ is:
A point source of $$100$$ W emits light with $$5\%$$ efficiency. At a distance of $$5$$ m from the source, the intensity produced by the electric field component is:
An electron accelerated through a potential difference $$V_1$$ has a de-Broglie wavelength of $$\lambda$$. When the potential is changed to $$V_2$$, its de-Broglie wavelength increases by $$50\%$$. The value of $$\left(\frac{V_1}{V_2}\right)$$ is equal to:
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The nuclear density of nuclides $$^{10}_5$$B, $$^{6}_3$$Li, $$^{56}_{26}$$Fe, $$^{20}_{10}$$Ne and $$^{209}_{83}$$Bi can be arranged as
$$\rho^N_{Bi} > \rho^N_{Fe} > \rho^N_{Ne} > \rho^N_{B} > \rho^N_{Li}$$
Reason R: The radius $$R$$ of nucleus is related to its mass number $$A$$ as $$R = R_0 A^{\frac{1}{3}}$$, where $$R_0$$ is a constant.
In the light of the above statement, choose the correct answer from the options given below:
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The output $$Y$$ for the inputs $$A$$ and $$B$$ of circuit is given by
Truth table of the shown circuit is:
Match List I with List II
| List I | List II |
|---|---|
| A. Attenuation | I. Combination of a receiver and transmitter |
| B. Transducer | II. Process of retrieval of information from the carrier wave at receiver |
| C. Demodulation | III. Converts one form of energy into another |
| D. Repeater | IV. Loss of strength of a signal while propagating through a medium |
A stone tied to $$180$$ cm long string at its end is making $$28$$ revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is $$\frac{1936}{x}$$ m s$$^{-2}$$. The value of $$x$$ ______. [Take $$\pi = \frac{22}{7}$$]
A body of mass $$2$$ kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power $$P$$. Its displacement in $$4$$ s is $$\frac{1}{3}\alpha^2\sqrt{P}$$ m. The value of $$\alpha$$ will be ______.
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A uniform disc of mass $$0.5$$ kg and radius $$r$$ is projected with a velocity $$18$$ m s$$^{-1}$$ at $$t = 0$$ s on a rough horizontal surface. It starts off with a purely sliding motion at $$t = 0$$ s. After $$2$$ s it acquires a purely rolling motion. The total kinetic energy of the disc after $$2$$ s will be ______ J.
(given, coefficient of friction is $$0.3$$ and $$g = 10$$ m s$$^{-2}$$).
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A faulty thermometer reads $$5°$$C in melting ice and $$95°$$C in steam. The correct temperature on absolute scale will be ______ K when the faulty thermometer reads $$41°$$C.
The velocity of a particle executing SHM varies with displacement $$(x)$$ as $$4v^2 = 50 - x^2$$. The time period of oscillations is $$\frac{x}{7}$$ s. The value of $$x$$ is ______. [Take $$\pi = \frac{22}{7}$$]
As shown in figure, a cuboid lies in a region with electric field $$E = 2x^2\hat{i} - 4y\hat{j} + 6\hat{k}$$ N C$$^{-1}$$. The magnitude of charge within the cuboid is $$n\varepsilon_0$$ C. The value of $$n$$ is ______ (if dimension of cuboid is $$1 \times 2 \times 3$$ m$$^3$$)
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The equivalent resistance between $$A$$ and $$B$$ is ______ $$\Omega$$.
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If the potential difference between $$B$$ and $$D$$ is zero, the value of $$x$$ is $$\frac{1}{n}$$ $$\Omega$$. The value of $$n$$ is ______.
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In an ac generator, a rectangular coil of $$100$$ turns each having area $$14 \times 10^{-2}$$ m$$^2$$ is rotated at $$360$$ rev min$$^{-1}$$ about an axis perpendicular to a uniform magnetic field of magnitude $$3.0$$ T. The maximum value of the emf produced will be ______ V. [Take $$\pi = \frac{22}{7}$$]
In a Young's double slit experiment, the intensities at two points, for the path difference $$\frac{\lambda}{4}$$ and $$\frac{\lambda}{3}$$ ($$\lambda$$ being the wavelength of light used) are $$I_1$$ and $$I_2$$ respectively. If $$I_0$$ denotes the intensity produced by each one of the individual slits, then $$\frac{I_1+I_2}{I_0} =$$ ______.
A radioactive nucleus decays by two different process. The half life of the first process is $$5$$ minutes and that of the second process is $$30$$ s. The effective half-life of the nucleus is calculated to be $$\frac{\alpha}{11}$$ s. The value of $$\alpha$$ is ______.
The wave function $$(\Psi)$$ of $$2s$$ is given by
$$\Psi_{2s} = \frac{1}{2\sqrt{2\pi}}\left(\frac{1}{a_0}\right)^{1/2}\left(2 - \frac{r}{a_0}\right)e^{-r/2a_0}$$
At $$r = r_0$$, radial node is formed. Thus, $$r_0$$ in terms of $$a_0$$
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Maximum number of electrons that can be accommodated in shell with $$n = 4$$ are:
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Bond dissociation energy of E $$-$$ H bond of the "H$$_2$$E" hydrides of group 16 elements (given below), follows order.
(A) O
(B) S
(C) Se
(D) Te
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Chlorides of which metal are soluble in organic solvents:
Which of the following reaction is correct?
Boric acid in solid, whereas BF$$_3$$ is gas at room temperature because of
The most stable carbocation for the following is:
The correct order of pK$$_a$$ values for the following compounds is:
Match List I with List II:
| List I (Mixture) | List II (Separation Technique) |
|---|---|
| (A) CHCl$$_3$$ + C$$_6$$H$$_5$$NH$$_2$$ | I. Steam distillation |
| (B) C$$_6$$H$$_{14}$$ + C$$_5$$H$$_{12}$$ | II. Differential extraction |
| (C) C$$_6$$H$$_5$$NH$$_2$$ + H$$_2$$O | III. Distillation |
| (D) Organic compound in H$$_2$$O | IV. Fractional distillation |
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A:
can be easily reduced using Zn $$-$$ Hg/HCl to
.
Reason R: Zn $$-$$ Hg/HCl is used to reduce carbonyl group to $$-$$CH$$_2-$$ group.
In the light of the above statements, choose the correct answer from the options given below:
The water quality of a pond was analysed and its BOD was found to be 4. The pond has
Given below are two statements:
Statement I: During Electrolytic refining, the pure metal is made to act as anode and its impure metallic form is used as cathode.
Statement II: During the Hall-Heroult electrolysis process, purified Al$$_2$$O$$_3$$ is mixed with Na$$_3$$AlF$$_6$$ to lower the melting point of the mixture.
In the light of the above statements, choose the most appropriate answer from the options given below:
Formulae for Nessler's reagent is:
KMnO$$_4$$ oxidises I$$^-$$ in acidic and neutral/faintly alkaline solution, respectively to
Match List I with List II:
| List I (Complexes) | List II (Hybridisation) |
|---|---|
| (A) [Ni(CO)$$_4$$] | I. sp$$^3$$ |
| (B) [Cu(NH$$_3$$)$$_4$$]$$^{2+}$$ | II. dsp$$^2$$ |
| (C) [Fe(NH$$_3$$)$$_6$$]$$^{2+}$$ | III. sp$$^3$$d$$^2$$ |
| (D) [Fe(H$$_2$$O)$$_6$$]$$^{2+}$$ | IV. d$$^2$$sp$$^3$$ |
1 L, 0.02M solution of [Co(NH$$_3$$)$$_5$$ SO$$_4$$]Br is mixed with 1 L, 0.02M solution of [Co(NH$$_3$$)$$_5$$Br]SO$$_4$$. The resulting solution is divided into two equal parts (X) and treated with excess AgNO$$_3$$ solution and BaCl$$_2$$ solution respectively as shown below:
1 L Solution (X) + AgNO$$_3$$ solution (excess) $$\to$$ Y
1 L Solution (X) + BaCl$$_2$$ solution (excess) $$\to$$ Z
The number of moles of Y and Z respectively are
The Cl $$-$$ Co $$-$$ Cl bond angle values in a fac-[Co(NH$$_3$$)$$_3$$Cl$$_3$$] complex is/are:
Decreasing order towards S$$_N$$1 reaction for the following compounds is:
Given below are two statements: One is labelled as Assertion A and the other labelled as Reason R.
Assertion A: Antihistamines do not affect the secretion of acid in stomach.
Reason R: Antiallergic and antacid drugs work on different receptors.
In the light of the above statements, choose the correct answer from the options given below:
1 mole of ideal gas is allowed to expand reversibly and adiabatically from a temperature of $$27°$$C. The work done is $$3$$ kJ mol$$^{-1}$$. The final temperature of the gas is ______ K (Nearest integer). Given $$C_v = 20$$ J mol$$^{-1}$$ K$$^{-1}$$
Consider the following equation:
$$2\text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2\text{SO}_3(g), \Delta H = -190$$ kJ.
The number of factors which will increase the yield of SO$$_3$$ at equilibrium from the following is ______.
A. Increasing temperature
B. Increasing pressure
C. Adding more SO$$_2$$
D. Adding more O$$_2$$
E. Addition of catalyst
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The strength of $$50$$ volume solution of hydrogen peroxide is ______ g/L (Nearest integer).
Given: Molar mass of H$$_2$$O$$_2$$ is 34 g mol$$^{-1}$$. Molar volume of gas at STP = 22.7 L.
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Iron oxide FeO, crystallises in a cubic lattice with a unit cell edge length of $$5.0$$ $$\text{\AA}$$. If density of the FeO in the crystal is $$4.0$$ g cm$$^{-3}$$, then the number of FeO units present per unit cell is ______ (Nearest integer)
Given: Molar mass of Fe and O is 56 and 16 g mol$$^{-1}$$ respectively.
N$$_A = 6.0 \times 10^{23}$$ mol$$^{-1}$$
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Lead storage battery contains $$38\%$$ by weight solution of H$$_2$$SO$$_4$$. The van't Hoff factor is $$2.67$$ at this concentration. The temperature in Kelvin at which the solution in the battery will freeze is ______ (Nearest integer).
Given K$$_f = 1.8$$ K kg mol$$^{-1}$$
The electrode potential of the following half cell at 298 K
X|X$$^{2+}$$(0.001M)||Y$$^{2+}$$(0.01M)|Y is ______ $$\times 10^{-2}$$ V (Nearest integer)
Given: E$$^0_{X^{2+}/X} = -2.36$$ V
E$$^0_{Y^{2+}/Y} = +0.36$$ V
$$\frac{2.303 RT}{F} = 0.06$$ V
An organic compound undergoes first order decomposition. If the time taken for the $$60\%$$ decomposition is $$540$$ s, then the time required for $$90\%$$ decomposition will be ______ s. (Nearest integer).
Given: $$\ln 10 = 2.3$$; $$\log 2 = 0.3$$
The graph of $$\log \frac{x}{m}$$ vs $$\log p$$ for an adsorption process is a straight line inclined at an angle of $$45°$$ with intercept equal to $$0.6020$$. The mass of gas adsorbed per unit mass of adsorbent at the pressure of $$0.4$$ atm is ______ $$\times 10^{-1}$$ (Nearest integer)
Given: $$\log 2 = 0.3010$$
Number of compounds from the following which will not dissolve in cold NaHCO$$_3$$ and NaOH solutions but will dissolve in hot NaOH solution is ______.
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A short peptide on complete hydrolysis produces 3 moles of glycine (G), two moles of leucine (L) and two moles of valine (V) per mole of peptide. The number of peptide linkages in it are ______.
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The number of ways of selecting two numbers $$a$$ and $$b$$, $$a \in \{2, 4, 6, \ldots, 100\}$$ and $$b \in \{1, 3, 5, \ldots, 99\}$$ such that $$2$$ is the remainder when $$a + b$$ is divided by $$23$$ is
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Let $$a, b, c > 1$$, $$a^3, b^3$$ and $$c^3$$ be in A.P. and $$\log_a b$$, $$\log_c a$$ and $$\log_b c$$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $$\frac{a+4b+c}{3}$$ and the common difference is $$\frac{a-8b+c}{10}$$ is $$-444$$, then $$abc$$ is equal to
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Let $$x = \left(8\sqrt{3} + 13\right)^{13}$$ and $$y = \left(7\sqrt{2} + 9\right)^{9}$$. If $$[t]$$ denotes the greatest integer $$\leq t$$, then
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The parabolas: $$ax^2 + 2bx + cy = 0$$ and $$d^2 + 2ex + fy = 0$$ intersect on the line $$y = 1$$. If $$a, b, c, d, e, f$$ are positive real numbers and $$a, b, c$$ are in G.P., then
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Let $$A$$ be a point on the $$x$$-axis. Common tangents are drawn from $$A$$ to the curves $$x^2 + y^2 = 8$$ and $$y^2 = 16x$$. If one of these tangents touches the two curves at $$Q$$ and $$R$$, then $$(QR)^2$$ is equal to
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Let $$f, g$$ and $$h$$ be the real valued functions defined on $$\mathbb{R}$$ as
$$f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$$, $$g(x) = \begin{cases} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{cases}$$ and $$h(x) = 2[x] - f(x)$$, where $$[x]$$ is the greatest integer $$\leq x$$. Then the value of $$\lim_{x \to 1} g(h(x-1))$$ is
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Consider the following statements:
$$P$$: I have fever
$$Q$$: I will not take medicine
$$R$$: I will take rest
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:
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Let $$S$$ be the set of all values of $$a_1$$ for which the mean deviation about the mean of $$100$$ consecutive positive integers $$a_1, a_2, a_3, \ldots, a_{100}$$ is $$25$$. Then $$S$$ is
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If $$P$$ is a $$3 \times 3$$ real matrix such that $$P^T = aP + (a-1)I$$, where $$a > 1$$, then
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For $$\alpha, \beta \in \mathbb{R}$$, suppose the system of linear equations
$$x - y + z = 5$$
$$2x + 2y + \alpha z = 8$$
$$3x - y + 4z = \beta$$
has infinitely many solutions. Then $$\alpha$$ and $$\beta$$ are the roots of
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Let $$a_1 = 1, a_2, a_3, a_4, \ldots$$ be consecutive natural numbers. Then $$\tan^{-1}\left(\frac{1}{1+a_1a_2}\right) + \tan^{-1}\left(\frac{1}{1+a_2a_3}\right) + \ldots + \tan^{-1}\left(\frac{1}{1+a_{2021}a_{2022}}\right)$$ is equal to
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The range of the function $$f(x) = \sqrt{3-x} + \sqrt{2+x}$$ is
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If the functions $$f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$$ and $$g(x) = \frac{x^3}{3} + ax + bx^2$$, $$a \neq 2b$$ have a common extreme point, then $$a + 2b + 7$$ is equal to
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$$\lim_{n \to \infty} \frac{3}{n}\left\{4 + \left(2 + \frac{1}{n}\right)^2 + \left(2 + \frac{2}{n}\right)^2 + \ldots + \left(3 - \frac{1}{n}\right)^2\right\}$$ is equal to
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Let $$q$$ be the maximum integral value of $$p$$ in $$[0, 10]$$ for which the roots of the equation $$x^2 - px + \frac{5}{4}p = 0$$ are rational. Then the area of the region $$\{(x,y) : 0 \leq y \leq (x-q)^2, 0 \leq x \leq q\}$$ is
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The solution of the differential equation $$\frac{dy}{dx} = -\left(\frac{x^2+3y^2}{3x^2+y^2}\right)$$, $$y(1) = 0$$ is
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Let $$\lambda \in \mathbb{R}$$, $$\vec{a} = \lambda\hat{i} + 2\hat{j} - 3\hat{k}$$, $$\vec{b} = \hat{i} - \lambda\hat{j} + 2\hat{k}$$. If $$\left((\vec{a}+\vec{b}) \times (\vec{a} \times \vec{b})\right) \times (\vec{a}-\vec{b}) = 8\hat{i} - 40\hat{j} - 24\hat{k}$$ then $$\left|\lambda(\vec{a}+\vec{b}) \times (\vec{a}-\vec{b})\right|^2$$ is equal to
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors. Let $$|\vec{a}| = 1$$, $$|\vec{b}| = 4$$ and $$\vec{a} \cdot \vec{b} = 2$$. If $$\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$$, then the value of $$\vec{b} \cdot \vec{c}$$ is
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A vector $$\vec{v}$$ in the first octant is inclined to the $$x$$ axis at $$60°$$, to the $$y$$-axis at $$45°$$ and to the $$z$$-axis at an acute angle. If a plane passing through the points $$(\sqrt{2}, -1, 1)$$ and $$(a, b, c)$$, is normal to $$\vec{v}$$, then
If a plane passes through the points $$(-1, k, 0)$$, $$(2, k, -1)$$, $$(1, 1, 2)$$ and is parallel to the line $$\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$$, then the value of $$\frac{k^2+1}{(k-1)(k-2)}$$ is
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If the value of real number $$\alpha \gt 0$$ for which $$x^2 - 5\alpha x + 1 = 0$$ and $$x^2 - \alpha x - 5 = 0$$ have a common real roots is $$\frac{3}{\sqrt{2\beta}}$$ then $$\beta$$ is equal to ______.
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The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is ______.
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The $$8^{th}$$ common term of the series
$$S_1 = 3 + 7 + 11 + 15 + 19 + \ldots$$
$$S_2 = 1 + 6 + 11 + 16 + 21 + \ldots$$ is
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$$50^{th}$$ root of a number $$x$$ is $$12$$ and $$50^{th}$$ root of another number $$y$$ is $$18$$. Then the remainder obtained on dividing $$(x + y)$$ by $$25$$ is ______.
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Let $$P(a_1, b_1)$$ and $$Q(a_2, b_2)$$ be two distinct points on a circle with center $$C(\sqrt{2}, \sqrt{3})$$. Let $$O$$ be the origin and $$OC$$ be perpendicular to both $$CP$$ and $$CQ$$. If the area of the triangle $$OCP$$ is $$\frac{\sqrt{35}}{2}$$, then $$a_1^2 + a_2^2 + b_1^2 + b_2^2$$ is equal to ______.
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Let $$A = \{1, 2, 3, 5, 8, 9\}$$. Then the number of possible functions $$f : A \to A$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in A$$ with $$m \cdot n \in A$$ is equal to
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If $$\int \sqrt{\sec 2x - 1} dx = \alpha \log_e \left|\cos 2x + \beta + \sqrt{\cos 2x\left(1 + \cos\frac{1}{\beta}x\right)}\right|$$ + constant, then $$\beta - \alpha$$ is equal to
Let $$A$$ be the area of the region $$\{(x,y) : y \geq x^2, y \geq (1-x)^2, y \leq 2x(1-x)\}$$. Then $$540A$$ is equal to
Let a line $$L$$ pass through the point $$P(2, 3, 1)$$ and be parallel to the line $$x + 3y - 2z - 2 = 0 = x - y + 2z$$. If the distance of $$L$$ from the point $$(5, 3, 8)$$ is $$\alpha$$, then $$3\alpha^2$$ is equal to ______.
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A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $$p$$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $$q$$. If $$p : q = m : n$$, where $$m$$ and $$n$$ are co-prime, then $$m + n$$ is equal to
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