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Two trains $$A$$ and $$B$$ of length $$l$$ and $$4l$$ are travelling into a tunnel of length $$L$$ in parallel tracks from opposite directions with velocities $$108$$ km h$$^{-1}$$ and $$72$$ km h$$^{-1}$$, respectively. If train $$A$$ take $$35$$ s less time than train $$B$$ to cross the tunnel then, length $$L$$ of tunnel is:
(Given $$L = 60 \ l$$)
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A disc is rolling without slipping on a surface. The radius of the disc is $$R$$. At $$t = 0$$, the top most point on the disc is $$A$$ as shown in figure. When the disc completes half of its rotation, the displacement of point $$A$$ from its initial position is
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The ratio of powers of two motors is $$\frac{3\sqrt{x}}{\sqrt{x}+1}$$, that are capable of raising $$300$$ kg water in $$5$$ minutes and $$50$$ kg water in $$2$$ minutes respectively from a well of $$100$$ m deep. The value of $$x$$ will be
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A body of mass $$(5 \pm 0.5)$$ kg is moving with a velocity of $$(20 \pm 0.4)$$ m s$$^{-1}$$. Its kinetic energy will be
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Two bodies are having kinetic energies in the ratio $$16 : 9$$. If they have same linear momentum, the ratio of their masses respectively is:
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A bullet of $$10$$ g leaves the barrel of gun with a velocity of $$600$$ m s$$^{-1}$$. If the barrel of gun is $$50$$ cm long and mass of gun is $$3$$ kg, then value of impulse supplied to the gun will be:
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A planet having mass $$9 M_e$$ and radius $$4R_e$$, where $$M_e$$ and $$R_e$$ are mass and radius of earth respectively, has escape velocity in km s$$^{-1}$$ given by: (Given escape velocity on earth $$V_e = 11.2 \times 10^3$$ m s$$^{-1}$$)
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Under isothermal condition, the pressure of a gas is given by $$P = aV^{-3}$$, where $$a$$ is a constant and $$V$$ is the volume of the gas. The bulk modulus at constant temperature is equal to
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The figure shows a liquid of a given density flowing steadily in a horizontal tube of a varying cross-section. Cross-sectional area at A is $$1.5$$ cm$$^2$$, and that at B is $$25$$ mm$$^2$$, if the speed of liquid at B is $$60$$ cm s$$^{-1}$$ then $$(P_A - P_B)$$ is
(Given $$P_A$$ and $$P_B$$ are liquid pressures at A and B points. Density $$\rho = 1000$$ kg m$$^{-3}$$. A and B are on the axis of tube)
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The rms speed of oxygen molecule in a vessel at particular temperature is $$\left(1 + \frac{5}{x}\right)^{\frac{1}{2}}v$$, when $$v$$ is the average speed of the molecule. The value of $$x$$ will be:
(take $$\pi = \frac{22}{7}$$)
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Which graph represents the difference between total energy and potential energy of a particle executing SHM vs its distance from mean position?
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Two charges of each magnitude $$0.01$$ C and separated by a distance of $$0.4$$ mm constitute an electric dipole. If the dipole is placed in an uniform electric field $$\vec{E}$$ of $$10$$ dyne $$\cdot$$ C$$^{-1}$$ making $$30°$$ angle with $$\vec{E}$$, the magnitude of torque acting on dipole is:
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Different combination of 3 resistors of equal resistance $$R$$ are shown in the figures. The increasing order for power dissipation is:
(A)
(B)
(C)
(D)
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The source of time varying magnetic field may be
(A) a permanent magnet
(B) an electric field changing linearly with time
(C) direct current
(D) a decelerating charge particle
(E) an antenna fed with a digital signal
Choose the correct answer from the options given below.
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Which of the following Maxwell's equation is valid for time varying conditions but not valid for static conditions:
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A vessel of depth $$d$$ is half filled with oil of refractive index $$n_1$$ and the other half is filled with water of refractive index $$n_2$$. The apparent depth of this vessel when viewed from above will be-
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The difference between threshold wavelengths for two metal surfaces $$A$$ and $$B$$ having work function $$\phi_A = 9$$ eV and $$\phi_B = 4.5$$ eV in nm is:
{Given, $$hc = 1242$$ eV nm}
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$$^{238}_{92}$$A $$\to ^{234}_{90}$$B $$+ ^{4}_{2}$$D $$+ Q$$
In the given nuclear reaction, the approximate amount of energy released will be:
[Given, mass of $$^{238}_{92}$$A $$= 238.05079 \times 931.5$$ MeV c$$^{-2}$$, mass of $$^{234}_{90}$$B $$= 234.04363 \times 931.5$$ MeV c$$^{-2}$$, mass of $$^{4}_{2}$$D $$= 4.00260 \times 931.5$$ MeV c$$^{-2}$$]
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For the following circuit and given inputs $$A$$ and $$B$$, choose the correct option for output 'Y'
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Match List-I with List-II
| List-I (Layer of atmosphere) | List-II (Approximate height over earth's surface) | ||
|---|---|---|---|
| A | F$$_1$$-Layer | I | 10 km |
| B | D-Layer | II | 170 - 190 km |
| C | Troposphere | III | 100 km |
| D | E-Layer | V | 65 - 75 km |
Choose the correct answer from the options given below.
A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $$\pi : 22$$ then, the value of its angular speed will be _____ rad s$$^{-1}$$.
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The elastic potential energy stored in a steel wire of length $$20$$ m stretched through $$2$$ cm is $$80$$ J. The cross sectional area of the wire is _____ mm$$^2$$. (Given, $$Y = 2.0 \times 10^{11}$$ N m$$^{-2}$$)
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At a given point of time the value of displacement of a simple harmonic oscillator is given as $$y = A \cos(30°)$$. If amplitude is $$40$$ cm and kinetic energy at that time is $$200$$ J, the value of force constant is $$1.0 \times 10^x$$ N m$$^{-1}$$. The value of $$x$$ is _____.
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A thin infinite sheet charge and an infinite line charge of respective charge densities $$+\sigma$$ and $$+\lambda$$ are placed parallel at $$5$$ m distance from each other. Points P and Q are at $$\frac{3}{\pi}$$ m and $$\frac{4}{\pi}$$ m perpendicular distances from line charge towards sheet charge, respectively. $$E_P$$ and $$E_Q$$ are the magnitudes of resultant electric field intensities at point P and Q respectively. If $$\frac{E_P}{E_Q} = \frac{4}{a}$$ for $$2|\sigma| = |\lambda|$$, then the value of $$a$$ is _____.
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A potential $$V_0$$ is applied across a uniform wire of resistance $$R$$. The power dissipation is $$P_1$$. The wire is then cut into two equal halves and a potential of $$V_0$$ is applied across the length of each half. The total power dissipation across two wires is $$P_2$$. The ratio of $$P_2 : P_1$$ is $$\sqrt{x} : 1$$. The value of $$x$$ is _____.
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When a resistance of $$5 \ \Omega$$ is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of $$250$$ mA, however when $$1050 \ \Omega$$ resistance is connected with it in series, it gives full scale deflection for $$25$$ volt. The resistance of galvanometer is _____ $$\Omega$$.
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In the given figure, an inductor and resistor are connected in series with a battery of emf $$E$$ volt. $$\frac{E^a}{2b}$$ J s$$^{-1}$$ represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of $$\frac{b}{a}$$ will be _____.
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A fish rising vertically upward with a uniform velocity of $$8$$ m s$$^{-1}$$, observes that a bird is diving vertically downward towards the fish with the velocity of $$12$$ m s$$^{-1}$$. If the refractive index of water is $$\frac{4}{3}$$, then the actual velocity of the diving bird to pick the fish, will be _____ m s$$^{-1}$$.
The radius of 2$$^{\text{nd}}$$ orbit of He$$^+$$ of Bohr's model is $$r_1$$ and that of fourth orbit of Be$$^{3+}$$ is represented as $$r_2$$. Now the ratio $$\frac{r_2}{r_1}$$ is $$x : 1$$. The value of $$x$$ is _____.
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From the given transfer characteristic of a transistor in CE configuration, the value of power gain of this configuration is $$10^x$$, for $$R_B = 10$$ k$$\Omega$$, and $$R_C = 1$$ k$$\Omega$$. The value of $$x$$ is _____.
The energy of an electron in the first Bohr orbit of hydrogen atom is $$-2.18 \times 10^{-18}$$ J. Its energy in the third Bohr orbit is _____.
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Which of the following statements are not correct?
A. The electron gain enthalpy of F is more negative than that of Cl.
B. Ionization enthalpy decreases in a group of periodic table.
C. The electronegativity of an atom depends upon the atoms bonded to it.
D. Al$$_2$$O$$_3$$ and NO are examples of amphoteric oxides.
Choose the most appropriate answer from the options given below:
In which of the following processes, the bond order increases and paramagnetic character changes to diamagnetic one?
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ClF$$_5$$ at room temperature is a
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Among the following compounds, the one which shows highest dipole moment is
Given below are two statements:
Statement I: Permutit process is more efficient compared to the synthetic resin method for the softening of water.
Statement II: Synthetic resin method results in the formation of soluble sodium salts.
In the light of the above statements, choose the most appropriate answer from the options given below:
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Be(OH)$$_2$$ reacts with Sr(OH)$$_2$$ to yield an ionic salt. Choose the incorrect option related to this reaction from the following
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In the following reaction 'X' is
CH$$_3$$(CH$$_2$$)$$_4$$CH$$_3$$ $$\xrightarrow{\text{Anhy. AlCl}_3 / \text{HCl}, \Delta}$$ X (major product)
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The radical which mainly causes ozone depletion in the presence of UV radiations is :
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What happens when a lyophilic sol is added to a lyophobic sol?
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Which one of the following is most likely a mismatch?
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The incorrect statement from the following for borazine is:
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The pair of lanthanides in which both elements have high third-ionization energy is:
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The mismatched combinations are
A. Chlorophyll $$-$$ Co
B. Water hardness $$-$$ EDTA
C. Photography $$-$$ [Ag(CN)$$_2$$]$$^-$$
D. Wilkinson catalyst $$-$$ [(Ph$$_3$$P)$$_3$$RhCl]
E. Chelating ligand $$-$$ D $$-$$ Penicillamine
Choose the correct answer from the options given below.
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2-Methyl propyl bromide reacts with C$$_2$$H$$_5$$O$$^-$$ and gives 'A' whereas on reaction with C$$_2$$H$$_5$$OH it gives 'B'. The mechanism followed in these reactions and the products 'A' and 'B' respectively are:
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In the reaction given below
An amino alcohol reacts with (i) HCl (ii) KOH to give 'B' Major product. 'B' is:
In the above reaction, left hand side and right hand side rings are named as 'A' and 'B' respectively. They undergo ring expansion. The correct statement for this process is:
D $$-$$ (+) $$-$$ Glyceraldehyde $$\xrightarrow{\text{i) HCN}}$$ $$\xrightarrow{\text{ii) H}_2\text{O/H}^+}$$ $$\xrightarrow{\text{iii) HNO}_3}$$
The products formed in the above reaction are
A cyclic amide (lactam) reacts with (i) NaOH, $$\Delta$$ (ii) H$$^+$$ to give 'A' (Major Product).
'A' is:
Match the following
| Column-A | Column-B | ||
|---|---|---|---|
| a | Nylon 6 | I | Natural Rubber |
| b | Vulcanized Rubber | II | Cross Linked |
| c | cis-1, 4-polyisoprene | III | Caprolactam |
| d | Polychloroprene | IV | Neoprene |
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An organic compound gives $$0.220$$ g of CO$$_2$$ and $$0.126$$ g of H$$_2$$O on complete combustion. If the % of carbon is $$24$$ then the % of hydrogen is _____ $$\times 10^{-1}$$. (Nearest integer)
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A certain quantity of real gas occupies a volume of $$0.15$$ dm$$^3$$ at $$100$$ atm and $$500$$ K when its compressibility factor is $$1.07$$. Its volume at $$300$$ atm and $$300$$ K (When its compressibility factor is $$1.4$$) is _____ $$\times 10^{-4}$$ dm$$^3$$ (Nearest integer)
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A$$_2$$ + B$$_2$$ $$\to$$ 2AB. $$\Delta H_f = -200$$ kJ mol$$^{-1}$$
AB, A$$_2$$ and B$$_2$$ are diatomic molecules. If the bond enthalpies of A$$_2$$, B$$_2$$ and AB are in the ratio 1 : 0.5 : 1, then the bond enthalpy of A$$_2$$ is _____ kJ mol$$^{-1}$$ (Nearest integer)
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$$25.0$$ mL of $$0.050$$ M Ba(NO$$_3$$)$$_2$$ is mixed with $$25.0$$ mL of $$0.020$$ M NaF. K$$_{sp}$$ of BaF$$_2$$ is $$0.5 \times 10^{-6}$$ at 298K. The ratio of $$[\text{Ba}^{2+}][\text{F}^-]^2$$ and K$$_{sp}$$ is _____.
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KMnO$$_4$$ is titrated with ferrous ammonium sulphate hexahydrate in presence of dilute H$$_2$$SO$$_4$$. Number of water molecules produced for 2 molecules of KMnO$$_4$$ is _____.
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$$20$$ mL of calcium hydroxide was consumed when it was reacted with $$10$$ mL of unknown solution of H$$_2$$SO$$_4$$. Also $$20$$ mL standard solution of $$0.5$$ M HCl containing 2 drops of phenolphthalein was titrated with calcium hydroxide, the mixture showed pink colour when burette displayed the value of $$35.5$$ mL whereas the burette showed $$25.5$$ mL initially. The concentration of H$$_2$$SO$$_4$$ is _____ M. (Nearest integer)
Solution of $$12$$ g of non-electrolyte (A) prepared by dissolving it in $$1000$$ mL of water exerts the same osmotic pressure as that of $$0.05$$ M glucose solution at the same temperature. The empirical formula of A is CH$$_2$$O. The molecular mass of A is _____ g. (Nearest integer)
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A metal surface of $$100$$ cm$$^2$$ area has to be coated with nickel layer of thickness $$0.001$$ mm. A current of 2A was passed through a solution of Ni(NO$$_3$$)$$_2$$ for 'x' seconds to coat the desired layer. The value of x is _____. (Nearest integer)
($$\rho$$Ni (density of Nickel) is $$10$$ g mL$$^{-1}$$, Molar mass of Nickel is $$60$$ g mol$$^{-1}$$, F $$= 96500$$ C mol$$^{-1}$$)
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t$$_{87.5}$$ is the time required for the reaction to undergo $$87.5\%$$ completion and t$$_{50}$$ is the time required for the reaction to undergo $$50\%$$ completion. The relation between t$$_{87.5}$$ and t$$_{50}$$ for a first order reaction is
t$$_{87.5}$$ = x $$\times$$ t$$_{50}$$
The value of x is _____. (Nearest integer)
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For the given reaction
The total number of possible products formed by tertiary carbocation of A is _____.
Let $$s_1, s_2, s_3, \ldots, s_{10}$$ respectively be the sum of 12 terms of 10 A.P.s whose first terms are $$1, 2, 3, \ldots, 10$$ and the common differences are $$1, 3, 5, \ldots, 19$$ respectively. Then $$\sum_{i=1}^{10} s_i$$ is equal to
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Fractional part of the number $$\frac{4^{2022}}{15}$$ is equal to
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Let $$PQ$$ be a focal chord of the parabola $$y^2 = 36x$$ of length 100, making an acute angle with the positive $$x-$$axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that $$PM : MQ = 3 : 1$$. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $$PQ$$?
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Let the tangent and normal at the point $$(3\sqrt{3}, 1)$$ on the ellipse $$\frac{x^2}{36} + \frac{y^2}{4} = 1$$ meet the $$y-$$axis at the points $$A$$ and $$B$$ respectively. Let the circle $$C$$ be drawn taking $$AB$$ as a diameter and the line $$x = 2\sqrt{5}$$ intersect $$C$$ at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point $$(\alpha, \beta)$$, then $$\alpha^2 - \beta^2$$ is equal to
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The negation of the statement $$((A \wedge (B \vee C)) \Rightarrow (A \vee B)) \Rightarrow A$$ is
The number of symmetric matrices of order 3, with all the entries from the set $$\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$ is
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Let $$B = \begin{bmatrix} 1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4 \end{bmatrix}$$, $$\alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A| = 2$$. Then $$\begin{bmatrix} \alpha & -2\alpha & \alpha \end{bmatrix} B \begin{bmatrix} \alpha \\ -2\alpha \end{bmatrix}$$ is equal to
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For the system of linear equations
$$2x + 4y + 2az = b$$
$$x + 2y + 3z = 4$$
$$2x + 5y + 2z = 8$$
which of the following is NOT correct?
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For $$x \in \mathbb{R}$$, two real valued functions $$f(x)$$ and $$g(x)$$ are such that, $$g(x) = \sqrt{x} + 1$$ and $$fog(x) = x + 3 - \sqrt{x}$$. Then $$f(0)$$ is equal to
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For the differentiable function $$f : \mathbb{R} - \{0\} - \mathbb{R}$$, let $$3f(x) + 2f\left(\frac{1}{x}\right) = \frac{1}{x} - 10$$, then $$\left|f(3) + f'\left(\frac{1}{4}\right)\right|$$ is equal to
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$$\max_{0 \leq x \leq \pi} \left\{x - 2\sin x \cos x + \frac{1}{3}\sin 3x\right\} =$$
The set of all $$a \in \mathbb{R}$$ for which the equation $$x|x-1| + |x+2| + a = 0$$ has exactly one real root, is
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$$\int_0^{\infty} \frac{6}{e^{3x} + 6e^{2x} + 11e^x + 6} dx =$$
Among
$$(S1) : \lim_{n \to \infty} \frac{1}{n^2}(2 + 4 + 6 + \ldots + 2n) = 1$$
$$(S2) : \lim_{n \to \infty} \frac{1}{n^{16}}(1^{15} + 2^{15} + 3^{15} + \ldots + n^{15}) = \frac{1}{16}$$
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The area of the region enclosed by the curve $$f(x) = \max\{\sin x, \cos x\}$$, $$-\pi \leq x \leq \pi$$ and the $$x-$$axis is
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Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be the solution curves the differential equation $$\frac{dy}{dx} = y + 7$$ with initial conditions $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then the curves $$y = y_1(x)$$ and $$y = y_2(x)$$ intersect at
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Let $$\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$$, $$\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$$ and $$\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$$. If a vector $$\vec{d}$$ satisfies $$\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{d} \cdot \vec{a} = 24$$, then $$|\vec{d}|^2$$ is equal to
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Let the equation of plane passing through the line of intersection of the planes $$x + 2y + az = 2$$ and $$x - y + z = 3$$ be $$5x - 11y + bz = 6a - 1$$. For $$c \in \mathbb{Z}$$, if the distance of this plane from the point $$(a, -c, c)$$ is $$\frac{2}{\sqrt{a}}$$, then $$\frac{a+b}{c}$$ is equal to
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The distance of the point $$(-1, 2, 3)$$ from the plane $$\vec{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 10$$ parallel to the line of the shortest distance between the lines $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{k})$$ and $$\vec{r} = (2\hat{i} - \hat{j}) + \mu(\hat{i} - \hat{j} + \hat{k})$$ is
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A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $$X$$ denotes the number of tosses of the coin, then the mean of $$X$$ is
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Let $$w = z\bar{z} + k_1z + k_2iz + \lambda(1+i)$$, $$k_1, k_2 \in \mathbb{R}$$. Let $$Re(w) = 0$$ be the circle $$C$$ of radius 1 in the first quadrant touching the line $$y = 1$$ and the $$y$$-axis. If the curve $$Im(w) = 0$$ intersects $$C$$ at $$A$$ and $$B$$, then $$30(AB)^2$$ is equal to _____.
The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.
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The sum to 20 terms of the series $$2 \cdot 2^2 - 3^2 + 2 \cdot 4^2 - 5^2 + 2 \cdot 6^2 - \ldots$$ is equal to _____.
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Let $$\alpha$$ be the constant term in the binomial expansion of $$\left(\sqrt{x} - \frac{6}{x^{3/2}}\right)^n$$, $$n \leq 15$$. If the sum of the coefficients of the remaining terms in the expansion is $$649$$ and the coefficient of $$x^{-n}$$ is $$\lambda\alpha$$, then $$\lambda$$ is equal to _____.
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Let $$m_1$$ and $$m_2$$ be the slopes of the tangents drawn from the point $$P(4, 1)$$ to the hyperbola $$H : \frac{y^2}{25} - \frac{x^2}{16} = 1$$. If $$Q$$ is the point from which the tangents drawn to $$H$$ have slopes $$|m_1|$$ and $$|m_2|$$ and they make positive intercepts $$\alpha$$ and $$\beta$$ on the $$x-$$axis, then $$\frac{(PQ)^2}{\alpha\beta}$$ is equal to _____.
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Let the mean of the data
| $$x$$ | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| Frequency ($$f$$) | 4 | 24 | 28 | $$\alpha$$ | 8 |
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If $$S = \left\{x \in \mathbb{R} : \sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right) - \sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right) = \frac{\pi}{4}\right\}$$ then $$\sum_{x \in S}\left(\sin\left((x^2+x+5)\frac{\pi}{2}\right) - \cos\left((x^2+x+5)\pi\right)\right)$$ is equal to _____.
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Let for $$x \in \mathbb{R}$$, $$S_0(x) = x$$, $$S_k(x) = C_k x + k\int_0^x S_{k-1}(t)dt$$ where $$C_0 = 1$$, $$C_k = 1 - \int_0^1 S_{k-1}(x)dx$$, $$k = 1, 2, 3, \ldots$$ Then $$S_2(3) + 6C_3$$ is equal to _____.
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Let $$\vec{a} = 3\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{c} = 2\hat{i} - 3\hat{j} + 3\hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a} = \vec{b} \times \vec{c}$$ and $$|\vec{b}|^2 = 50$$, then $$\left|72 - |\vec{b} + \vec{c}|^2\right|$$ is equal to _____.
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Let the image of the point $$\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$$ in the plane $$x - 2y + z - 2 = 0$$ be $$P$$. If the distance of the point $$Q(6, -2, \alpha)$$, $$\alpha > 0$$, from $$P$$ is $$13$$, then $$\alpha$$ is equal to _____.
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