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If mass is written as $$m = kc^{P}G^{-1/2}h^{1/2}$$, then the value of $$P$$ will be : (Constants have their usual meaning with $$k$$ a dimensionless constant)
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Projectiles $$A$$ and $$B$$ are thrown at angles of $$45°$$ and $$60°$$ with vertical respectively from top of a 400 m high tower. If their times of flight are same, the ratio of their speeds of projection $$v_A : v_B$$ is:
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Three blocks $$A$$, $$B$$ and $$C$$ are pulled on a horizontal smooth surface by a force of 80 N as shown in figure. The tensions $$T_1$$ and $$T_2$$ in the string are respectively:
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A block of mass $$m$$ is placed on a surface having vertical cross section given by $$y = \frac{x^2}{4}$$. If coefficient of friction is 0.5, the maximum height above the ground at which block can be placed without slipping is:
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A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of $$60°$$ by a force of 10 N parallel to the inclined surface as shown in figure. When the block is pushed up by 10 m along inclined surface, the work done against frictional force is : $$g = 10$$ m s$$^{-2}$$
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Escape velocity of a body from earth is 11.2 km s$$^{-1}$$. If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the planet is:
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A block of ice at $$-10°C$$ is slowly heated and converted to steam at $$100°C$$. Which of the following curves represent the phenomenon qualitatively:
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Choose the correct statement for processes $$A$$ & $$B$$ shown in figure.
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If three moles of monoatomic gas $$\left(\gamma = \frac{5}{3}\right)$$ is mixed with two moles of a diatomic gas $$\left(\gamma = \frac{7}{5}\right)$$, the value of adiabatic exponent $$\gamma$$ for the mixture is:
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A particle of charge $$-q$$ and mass $$m$$ moves in a circle of radius $$r$$ around an infinitely long line charge of linear density $$+\lambda$$. Then time period will be given as: (Consider $$k$$ as Coulomb's constant)
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When a potential difference $$V$$ is applied across a wire of resistance $$R$$, it dissipates energy at a rate $$W$$. If the wire is cut into two halves and these halves are connected mutually parallel across the same supply, the energy dissipation rate will become:
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An alternating voltage $$V(t) = 220\sin 100\pi t$$ volt is applied to a purely resistive load of $$50 \Omega$$. The time taken for the current to rise from half of the peak value to the peak value is:
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Match List I with List II
Choose the correct answer from the options given below:
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A beam of unpolarised light of intensity $$I_0$$ is passed through a polaroid $$A$$ and then through another polaroid $$B$$ which is oriented so that its principal plane makes an angle of $$45°$$ relative to that of $$A$$. The intensity of emergent light is:
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If the total energy transferred to a surface in time $$t$$ is $$6.48 \times 10^5$$ J, then the magnitude of the total momentum delivered to this surface for complete absorption will be:
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For the photoelectric effect, the maximum kinetic energy $$E_k$$ of the photoelectrons is plotted against the frequency $$(\nu)$$ of the incident photons as shown in figure. The slope of the graph give
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An electron revolving in $$n^{th}$$ Bohr orbit has magnetic moment $$\mu_n$$. If $$\mu_n \propto n^x$$, the value of $$x$$ is:
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In a nuclear fission reaction of an isotope of mass $$M$$, three similar daughter nuclei of same mass are formed. The speed of a daughter nuclei in terms of mass defect $$\Delta M$$ will be:
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In the given circuit, the voltage across load resistance $$(R_L)$$ is:
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If 50 Vernier divisions are equal to 49 main scale divisions of a travelling microscope and one smallest reading of main scale is 0.5 mm, the Vernier constant of travelling microscope is:
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A vector has magnitude same as that of $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and is parallel to $$\vec{B} = 4\hat{i} + 3\hat{j}$$. The $$x$$ and $$y$$ components of this vector in first quadrant are $$x$$ and 3 respectively where $$x$$ = ____.
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Two discs of moment of inertia $$I_1 = 4$$ kg m$$^2$$ and $$I_2 = 2$$ kg m$$^2$$ about their central axes & normal to their planes, rotating with angular speeds 10 rad s$$^{-1}$$ & 4 rad s$$^{-1}$$ respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is _________ J.
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A big drop is formed by coalescing 1000 small identical drops of water. If $$E_1$$ be the total surface energy of 1000 small drops of water and $$E_2$$ be the surface energy of single big drop of water, the $$E_1 : E_2$$ is $$x : 1$$, where $$x$$ = ________.
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A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4 m, then the time period of small oscillations will be _________ s. [take $$g = \pi^2$$ m s$$^{-2}$$]
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A point source is emitting sound waves of intensity $$16 \times 10^{-8}$$ W m$$^{-2}$$ at the origin. The difference in intensity (magnitude only) at two points located at distances of 2 m and 4 m from the origin respectively will be ________ $$\times 10^{-8}$$ W m$$^{-2}$$.
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Two identical charged spheres are suspended by strings of equal lengths. The string make an angle of $$37°$$ with each other. When suspended in a liquid of density $$0.7$$ g cm$$^{-3}$$, the angle remains same. If density of material of the sphere is $$1.4$$ g cm$$^{-3}$$, the dielectric constant of the liquid is _____ ($$\tan 37° = \frac{3}{4}$$)
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Two resistance of $$100\Omega$$ and $$200\Omega$$ are connected in series with a battery of 4 V and negligible internal resistance. A voltmeter is used to measure voltage across $$100\Omega$$ resistance, which gives reading as 1 V. The resistance of voltmeter must be _______ $$\Omega$$.
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The current of 5 A flows in a square loop of sides 1 m is placed in air. The magnetic field at the centre of the loop is $$X\sqrt{2} \times 10^{-7}$$ T. The value of X is _________.
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A power transmission line feeds input power at 2.3 kV to a step down transformer with its primary winding having 3000 turns. The output power is delivered at 230 V by the transformer. The current in the primary of the transformer is 5 A and its efficiency is 90%. The winding of transformer is made of copper. The output current of transformer is ____ A.
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In an experiment to measure the focal length $$(f)$$ of a convex lens, the magnitude of object distance $$(x)$$ and the image distance $$(y)$$ are measured with reference to the focal point of the lens. The $$y - x$$ plot is shown in figure. The focal length of the lens is ____ cm.
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Given below are two statements:
Statement - I: Along the period, the chemical reactivity of the element gradually increases from group 1 to group 18.
Statement - II: The nature of oxides formed by group 1 element is basic while that of group 17 elements is acidic.
In the light of the above statements, choose the most appropriate from the options given below:
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Given below are two statements:
Statement-I: Since fluorine is more electronegative than nitrogen, the net dipole moment of $$NF_3$$ is greater than $$NH_3$$.
Statement-II: In $$NH_3$$, the orbital dipole due to lone pair and the dipole moment of NH bonds are in opposite direction, but in $$NF_3$$ the orbital dipole due to lone pair and dipole moments of N-F bonds are in same direction.
In the light of the above statements, choose the most appropriate 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: $$H_2Te$$ is more acidic than $$H_2S$$.
Reason R: Bond dissociation enthalpy of $$H_2Te$$ is lower than $$H_2S$$.
In the light of the above statements, choose the most appropriate from the options given below.
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IUPAC name of the following compound is:
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Which among the following purification methods is based on the principle of "Solubility" in two different solvents?
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The correct stability order of carbocations is
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Product A and B formed in the following set of reactions are:
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If a substance '$$A$$' dissolves in solution of a mixture of '$$B$$' and '$$C$$' with their respective number of moles as $$n_A$$, $$n_B$$ and $$n_C$$, mole fraction of $$C$$ in the solution is:
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The solution from the following with highest depression in freezing point/lowest freezing point is
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Reduction potential of ions are given below:
$$ClO_4^-$$: $$E° = 1.19$$ V; $$IO_4^-$$: $$E° = 1.65$$ V; $$BrO_4^-$$: $$E° = 1.74$$ V
The correct order of their oxidising power is:
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Choose the correct statements about the hydrides of group 15 elements.
A. The stability of the hydrides decreases in the order $$NH_3 > PH_3 > AsH_3 > SbH_3 > BiH_3$$
B. The reducing ability of the hydrides increases in the order $$NH_3 < PH_3 < AsH_3 < SbH_3 < BiH_3$$
C. Among the hydrides, $$NH_3$$ is strong reducing agent while $$BiH_3$$ is mild reducing agent.
D. The basicity of the hydrides increases in the order $$NH_3 < PH_3 < AsH_3 < SbH_3 < BiH_3$$
Choose the most appropriate from the options given below:
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The orange colour of $$K_2Cr_2O_7$$ and purple colour of $$KMnO_4$$ is due to
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$$A$$ and $$B$$ formed in the following reactions are:
$$CrO_2Cl_2 + 4NaOH \rightarrow A + 2NaCl + 2H_2O$$
$$A + 2HCl + 2H_2O_2 \rightarrow B + 3H_2O$$
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Alkaline oxidative fusion of $$MnO_2$$ gives "A" which on electrolytic oxidation in alkaline solution produces $$B$$. $$A$$ and $$B$$ respectively are:
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The molecule/ion with square pyramidal shape is:
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The coordination geometry around the manganese in decacarbonyldimanganese(0) is:
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Given below are two statements:
Statement - I: High concentration of strong nucleophilic reagent with secondary alkyl halides which do not have bulky substituents will follow $$S_N2$$ mechanism.
Statement - II: A secondary alkyl halide when treated with a large excess of ethanol follows $$S_N1$$ mechanism.
In the light of the above statements, choose the most appropriate from the options given below:
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Salicylaldehyde is synthesized from phenol, when reacted with
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m-chlorobenzaldehyde on treatment with 50% KOH solution yields
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The products A and B formed in the following reaction scheme are respectively:
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Number of spectral lines obtained in $$He^+$$ spectra, when an electron makes transition from fifth excited state to first excited state will be
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Two reactions are given below:
$$2Fe_{(s)} + \frac{3}{2}O_{2(g)} \rightarrow Fe_2O_{3(s)}$$, $$\Delta H° = -822$$ kJ/mol
$$C_{(s)} + \frac{1}{2}O_{2(g)} \rightarrow CO_{(g)}$$, $$\Delta H° = -110$$ kJ/mol
Then enthalpy change for following reaction:
$$3C_{(s)} + Fe_2O_{3(s)} \rightarrow 2Fe_{(s)} + 3CO_{(g)}$$
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The pH of an aqueous solution containing 1M benzoic acid ($$pK_a = 4.20$$) and 1M sodium benzoate is 4.5. The volume of benzoic acid solution in 300 mL of this buffer solution is _________ mL.
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Total number of species from the following which can undergo disproportionation reaction: $$H_2O_2$$, $$ClO_3^-$$, $$P_4$$, $$Cl_2$$, $$Ag$$, $$Cu^{+1}$$, $$F_2$$, $$NO_2$$, $$K^+$$
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Number of geometrical isomers possible for the given structure is/are ________.
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$$NO_2$$ required for a reaction is produced by decomposition of $$N_2O_5$$ in $$CCl_4$$ as by equation:
$$2N_2O_{5(g)} \rightarrow 4NO_{2(g)} + O_{2(g)}$$
The initial concentration of $$N_2O_5$$ is 3 mol L$$^{-1}$$ and it is 2.75 mol L$$^{-1}$$ after 30 minutes. The rate of formation of $$NO_2$$ is $$x \times 10^{-3}$$ mol L$$^{-1}$$ min$$^{-1}$$, value of $$x$$ is ________.
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Number of complexes which show optical isomerism among the following is _________.
$$cis-[Cr(ox)_2Cl_2]^{3-}$$, $$[Co(en)_3]^{3+}$$, $$cis-[Pt(en)_2Cl_2]^{2+}$$, $$cis-[Co(en)_2Cl_2]^+$$, $$trans-[Pt(en)_2Cl_2]^{2+}$$, $$trans-[Cr(ox)_2Cl_2]^{3-}$$
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2-chlorobutane + $$Cl_2 \rightarrow C_4H_8Cl_2$$ (isomers)
Total number of optically active isomers shown by $$C_4H_8Cl_2$$, obtained in the above reaction is ________.
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Number of metal ions characterized by flame test among the following is _________.
$$Sr^{2+}$$, $$Ba^{2+}$$, $$Ca^{2+}$$, $$Cu^{2+}$$, $$Zn^{2+}$$, $$Co^{2+}$$, $$Fe^{2+}$$
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The total number of correct statements, regarding the nucleic acids is _________.
A. RNA is regarded as the reserve of genetic information.
B. DNA molecule self-duplicates during cell division.
C. DNA synthesizes proteins in the cell.
D. The message for the synthesis of particular proteins is present in DNA.
E. Identical DNA strands are transferred to daughter cells.
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If $$z$$ is a complex number, then the number of common roots of the equation $$z^{1985} + z^{100} + 1 = 0$$ and $$z^3 + 2z^2 + 2z + 1 = 0$$, is equal to:
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Let $$a$$ and $$b$$ be two distinct positive real numbers. Let 11th term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{th}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to
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Suppose $$28 - p$$, $$p$$, $$70 - \alpha$$, $$\alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1 + x)^n$$. Then the value of $$2\alpha - 3p$$ equals
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For $$\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$$, let $$3\sin(\alpha + \beta) = 2\sin(\alpha - \beta)$$ and a real number $$k$$ be such that $$\tan\alpha = k\tan\beta$$. Then the value of $$k$$ is equal to
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If $$x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0$$ is the locus of a point, which moves such that it is always equidistant from the lines $$x + 2y + 7 = 0$$ and $$2x - y + 8 = 0$$, then the value of $$g + c + h - f$$ equals
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Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ be the points on the line $$5x + 7y = 50$$. Let the point $$P$$ divide the line segment $$AB$$ internally in the ratio $$7:3$$. Let $$3x - 25 = 0$$ be a directrix of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and the corresponding focus be $$S$$. If from $$S$$, the perpendicular on the $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to
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Let $$P$$ be a point on the hyperbola $$H: \frac{x^2}{9} - \frac{y^2}{4} = 1$$, in the first quadrant such that the area of triangle formed by $$P$$ and the two foci of $$H$$ is $$2\sqrt{13}$$. Then, the square of the distance of $$P$$ from the origin is
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Let $$R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$$ be a non-zero $$3 \times 3$$ matrix, where $$x\sin\theta = y\sin\left(\theta + \frac{2\pi}{3}\right) = z\sin\left(\theta + \frac{4\pi}{3}\right) \neq 0$$, $$\theta \in (0, 2\pi)$$.
For a square matrix $$M$$, let Trace($$M$$) denote the sum of all the diagonal entries of $$M$$. Then, among the statements:
(I) Trace($$R$$) = 0
(II) If Trace(adj(adj($$R$$))) = 0, then $$R$$ has exactly one non-zero entry.
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Consider the system of linear equations $$x + y + z = 5$$, $$x + 2y + \lambda^2 z = 9$$ and $$x + 3y + \lambda z = \mu$$, where $$\lambda, \mu \in R$$. Then, which of the following statement is NOT correct?
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If the domain of the function $$f(x) = \log_e\frac{2x+3}{4x^2+x-3} + \cos^{-1}\frac{2x-1}{x+2}$$ is $$(\alpha, \beta]$$, then the value of $$5\beta - 4\alpha$$ is equal to
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Let $$f: R \rightarrow R$$ be a function defined $$f(x) = \frac{x}{(1+x^4)^{1/4}}$$ and $$g(x) = f(f(f(f(x))))$$ then $$18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x) \, dx$$
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Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$$ be differentiable on $$R$$. Then, the value of $$\int_{-2}^{2} f(x) \, dx$$ equals
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Let $$f: R - \{0\} \rightarrow R$$ be a function satisfying $$f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$$ for all $$x, y$$, $$f(y) \neq 0$$. If $$f'(1) = 2024$$, then
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Let $$f(x) = (x+3)^{2}(x-2)^3$$, $$x \in [-4, 4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4, 4]$$, then the value of $$M - m$$ is:
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Let $$y = f(x)$$ be a thrice differentiable function in $$(-5, 5)$$. Let the tangents to the curve $$y = f(x)$$ at $$(1, f(1))$$ and $$(3, f(3))$$ make angles $$\frac{\pi}{6}$$ and $$\frac{\pi}{4}$$, respectively with positive x-axis. If $$27\int_1^3 \{f'(t)\}^2 + 1\} f''(t) \, dt = \alpha + \beta\sqrt{3}$$ where $$\alpha, \beta$$ are integers, then the value of $$\alpha + \beta$$ equals
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Let $$f: R \rightarrow R$$ be defined $$f(x) = ae^{2x} + be^x + cx$$. If $$f(0) = -1$$, $$f'(\log_e 2) = 21$$ and $$\int_0^{\log 4}(f(x) - cx) \, dx = \frac{39}{2}$$, then the value of $$|a + b + c|$$ equals:
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Let $$\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$$, $$\alpha, \beta \in R$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2 = 6$$. If $$\vec{a} \cdot \vec{b} = 3\sqrt{2}$$, then the value of $$(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2$$ is equal to
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}| = 1$$ and $$|\vec{b} \times \vec{a}| = 2$$. Then $$|(\vec{b} \times \vec{a}) - \vec{b}|^2$$ is equal to
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Let $$L_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})$$, $$\lambda \in R$$, $$L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + p\hat{k})$$, $$\mu \in R$$ and $$L_3: \vec{r} = \delta(l\hat{i} + m\hat{j} + n\hat{k})$$, $$\delta \in R$$ be three lines such that $$L_1$$ is perpendicular to $$L_2$$ and $$L_3$$ is perpendicular to both $$L_1$$ and $$L_2$$. Then the point which lies on $$L_3$$ is
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Bag $$A$$ contains 3 white, 7 red balls and bag $$B$$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $$A$$, if the ball drawn is white, is:
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The number of real solutions of the equation $$x(x^2 + 3|x| + 5|x-1| + 6|x-2|) = 0$$ is ______.
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In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is _________.
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Let $$S_n$$ be the sum to n-terms of an arithmetic progression $$3, 7, 11, \ldots$$, if $$40 < \frac{6}{n(n+1)}\sum_{k=1}^{n} S_k < 42$$, then $$n$$ equals ____________.
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Let $$\alpha = \sum_{k=0}^{n} \frac{{}^nC_k^2}{k+1}$$ and $$\beta = \sum_{k=0}^{n-1} \frac{{}^nC_k \cdot {}^nC_{k+1}}{k+2}$$. If $$5\alpha = 6\beta$$, then $$n$$ equals
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Consider two circles $$C_1: x^2 + y^2 = 25$$ and $$C_2: (x-\alpha)^2 + y^2 = 16$$, where $$\alpha \in (5, 9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin^{-1}\frac{\sqrt{63}}{8}$$. If the length of common chord of $$C_1$$ and $$C_2$$ is $$\beta$$, then the value of $$(\alpha\beta)^2$$ equals _________.
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If the variance $$\sigma^2$$ of the data
is $$k$$, then the value of $$[k]$$ is ______ (where $$.$$ denotes the greatest integer function)
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The number of symmetric relations defined on the set $$\{1, 2, 3, 4\}$$ which are not reflexive is _______.
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The area of the region enclosed by the parabola $$(y-2)^2 = x - 1$$, the line $$x - 2y + 4 = 0$$ and the positive coordinate axes is __________.
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Let $$Y = Y(X)$$ be a curve lying in the first quadrant such that the area enclosed by the line $$Y - y = Y'(x)(X - x)$$ and the co-ordinate axes, where $$(x, y)$$ is any point on the curve, is always $$\frac{-y^2}{2Y'(x)} + 1$$, $$Y'(x) \neq 0$$. If $$Y(1) = 1$$, then $$12Y(2)$$ equals ________.
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Let a line passing through the point $$(-1, 2, 3)$$ intersect the lines $$L_1: \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{-2}$$ at $$M(\alpha, \beta, \gamma)$$ and $$L_2: \frac{x+2}{-3} = \frac{y-2}{-2} = \frac{z-1}{4}$$ at $$N(a, b, c)$$. Then the value of $$\frac{(\alpha + \beta + \gamma)^2}{(a + b + c)^2}$$ equals ________________.
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