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A person travels $$x$$ distance with velocity $$v_1$$ and then $$x$$ distance with velocity $$v_2$$ in the same direction. The average velocity of the person is $$v$$, then the relation between $$v$$, $$v_1$$ and $$v_2$$ will be
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Two projectiles are projected at 30$$^\circ$$ and 60$$^\circ$$ with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:
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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: An electric fan continues to rotate for some time after the current is switched off.
Reason R: Fan continues to rotate due to inertia of motion.
In the light of above statements, choose the most appropriate answer from the options given below.
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Given below are two statements:
Statement I: Rotation of the earth shows effect on the value of acceleration due to gravity $$(g)$$.
Statement II: The effect of rotation of the earth on the value of $$g$$ at the equator is minimum and that at the pole is maximum.
In the light of the above statements, choose the correct answer from the options given below
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The time period of a satellite, revolving above earth's surface at a height equal to R will be (Given $$g = \pi^2$$ m s$$^{-2}$$, $$R$$ = radius of earth)
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Young's moduli of the material of wires $$A$$ and $$B$$ are in the ratio of 1 : 4, while its area of cross sections are in the ratio of 1 : 3. If the same amount of load is applied to both the wires, the amount of elongation produced in the wires A and B will be in the ratio of [Assume length of wires A and B are same]
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A gas is compressed adiabatically, which one of the following statement is NOT true?
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A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature $$T$$. Neglecting all vibrational modes, the total internal energy of the system will be:
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For a periodic motion represented by the equation $$y = \sin\omega t + \cos\omega t$$ the amplitude of the motion is
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The distance between two plates of a capacitor is $$d$$ and its capacitance is $$C_1$$, when air is the medium between the plates. If a metal sheet of thickness $$\frac{2d}{3}$$ and of the same area as plate is introduced between the plates, the capacitance of the capacitor becomes $$C_2$$. The ratio $$\frac{C_2}{C_1}$$ is
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In a metallic conductor, under the effect of applied electric field, the free electrons of the conductor
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A bar magnet is released from rest along the axis of a very long vertical copper tube. After some time the magnet will
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Given below are two statements:
Statement I: For diamagnetic substance $$-1 \leq x < 0$$, where x is the magnetic susceptibility.
Statement II: Diamagnetic substance when placed in an external magnetic field, tend to move from stronger to weaker part of the field. In the light of the above statements, choose the correct answer from the options give below.
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The amplitude of magnetic field in an electromagnetic wave propagating along $$y$$-axis is $$6.0 \times 10^{-7}$$ T. The maximum value of electric field in the electromagnetic wave is
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The ratio of intensities at two points P and Q on the screen in a Young's double slit experiment where phase difference between two waves of same amplitude are $$\frac{\pi}{3}$$ and $$\frac{\pi}{2}$$, respectively are
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The variation of stopping potential $$V_0$$ as a function of the frequency $$(\nu)$$ of the incident light for a metal is shown in figure. The work function of the surface is
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The half life of a radioactive substance is $$T$$. The time taken, for disintegrating $$\frac{7^{th}}{8}$$ part of its original mass will be:
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If each diode has a forward bias resistance of 25$$\Omega$$ in the below circuit, which of the following options is correct?
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A message signal of frequency 3 kHz is used to modulate a carrier signal of frequency 1.5 MHz. The bandwidth of the amplitude modulated wave is
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In an experiment with vernier callipers of least count 0.1 mm, when two jaws are joined together the zero of vernier scale lies right to the zero of the main scale and 6$$^{th}$$ division of vernier scale coincides with the main scale division. While measuring the diameter of a spherical bob, the zero of vernier scale lies in between 3.2 cm and 3.3 cm marks and 4$$^{th}$$ division of vernier scale coincides with the main scale division. The diameter of bob is measured as
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If the maximum load carried by an elevator is 1400 kg (600 kg-Passengers + 800 kg-elevator), which is moving up with a uniform speed of 3 m s$$^{-1}$$ and the frictional force acting on it is 2000 N, then the maximum power used by the motor is _______ kW. $$g = 10$$ m s$$^{-2}$$
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A force of $$-P\hat{k}$$ acts on the origin of the coordinate system. The torque about the point (2, -3) is $$P(a\hat{i} + b\hat{j})$$. The ratio of $$\frac{a}{b}$$ is $$\frac{x}{2}$$. The value of x is _______
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Figure below shows a liquid being pushed out of the tube by a piston having area of cross section 2.0 cm$$^2$$. The area of cross section at the outlet is 10 mm$$^2$$. If the piston is pushed at a speed of 4 cm s$$^{-1}$$, the speed of outgoing fluid is _______ cm s$$^{-1}$$
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A rectangular block of mass 5 kg attached to a horizontal spiral spring executes simple harmonic motion of amplitude 1 m and time period 3.14 s. The maximum force exerted by spring on block is _______ N.
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An electron revolves around an infinite cylindrical wire having uniform linear charge density $$2 \times 10^{-8}$$ C m$$^{-1}$$ in circular path under the influence of attractive electrostatic field as shown in the figure. The velocity of electron with which it is revolving is _______ $$\times 10^6$$ m s$$^{-1}$$. Given mass of electron $$= 9 \times 10^{-31}$$ kg
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A rectangular parallelopiped is measured as 1 cm $$\times$$ 1 cm $$\times$$ 100 cm. If its specific resistance is $$3 \times 10^{-7}$$ $$\Omega$$ m, then the resistance between its two opposite rectangular faces will be _______ $$\times 10^{-7}$$ $$\Omega$$.
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A straight wire carrying a current of 14 A is bent into a semicircular arc of radius 2.2 cm as shown in the figure. The magnetic field produced by the current at the centre O of the arc is _______ $$\times 10^{-4}$$ T
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A square loop of side 2.0 cm is placed inside a long solenoid that has 50 turns per centimetre and carries a sinusoidally varying current of amplitude 2.5 A and angular frequency 700 rad s$$^{-1}$$. The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is $$x \times 10^{-4}$$ V. The value of x is _______ (Take, $$\pi = \frac{22}{7}$$)
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A point object O is placed in front of two thin symmetrical coaxial convex lenses $$L_1$$ and $$L_2$$ with focal length 24 cm and 9 cm respectively. The distance between two lenses is 10 cm and the object is placed 6 cm away from lens $$L_1$$ as shown in the figure. The distance between the object and the image formed by the system of two lenses is _______ cm.
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If 917 $$\mathring{\text{A}}$$ be the lowest wavelength of Lyman series then the lowest wavelength of Balmer series will be _______ $$\mathring{\text{A}}$$.
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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: 3.1500 g of hydrated oxalic acid dissolved in water to make 250.0 mL solution will result in 0.1 M oxalic acid solution.
Reason R: Molar mass of hydrated oxalic acid is 126 g mol$$^{-1}$$.
In the light of the above statements, choose the correct answer from the options given below:
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Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| A | 16 g of CH$$_4$$(g) | I | Weighs 28g |
| B | 1 g of H$$_2$$(g) | II | $$60.2 \times 10^{23}$$ electrons |
| C | 1 mole of N$$_2$$(g) | III | Weighs 32g |
| D | 0.5 mol of SO$$_2$$(g) | IV | Occupies 11.4 L volume at STP |
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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: The energy required to form Mg$$^{2+}$$ from Mg is much higher than that required to produce Mg$$^+$$
Reason R: Mg$$^{2+}$$ is small ion and carry more charge than Mg$$^+$$
In the light of the above statements, choose the correct answer from the options given below.
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The correct order of metallic character is
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Gibbs energy vs T plot for the formation of oxides is given below. For the given diagram, the correct statement is-
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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: Physical properties of isotopes of hydrogen are different.
Reason R: Mass difference between isotopes of hydrogen is very large.
In the light of the above statements, choose the correct answer from the options given below:
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Number of water molecules in washing soda and soda ash respectively are:
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The decreasing order of hydride affinity for following carbocations is:
(a)
(b)
(c)
(d)
Choose the correct answer from the options given below:
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The correct order for acidity of the following hydroxyl compound is
(A) CH$$_3$$OH
(B) (CH$$_3$$)$$_3$$COH
(C)
(D)
(E)
Choose the correct answer from the options given below:
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In Carius tube, an organic compound 'X' is treated with sodium peroxide to form a mineral acid 'Y'. The solution of BaCl$$_2$$ is added to 'Y' to form a precipitate 'Z'. 'Z' is used for the quantitative estimation of an extra element. 'X' could be
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The delicate balance of CO$$_2$$ and O$$_2$$ is NOT disturbed by
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The correct relationships between unit cell edge length 'a' and radius of sphere 'r' for face-centred and body-centred cubic structures respectively are:
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Ferric chloride is applied to stop bleeding because
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Match List-I with List-II.
| List-I Complex | List-II Crystal Field splitting energy ($$\Delta_o$$) | ||
|---|---|---|---|
| A. | $$[Ti(H_2O)_6]^{2+}$$ | I. | -1.2 |
| B. | $$[V(H_2O)_6]^{2+}$$ | II. | -0.6 |
| C. | $$[Mn(H_2O)_6]^{3+}$$ | III. | 0 |
| D. | $$[Fe(H_2O)_6]^{3+}$$ | IV. | -0.8 |
Choose the correct answer from the options given below:
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The correct order of the number of unpaired electrons in the given complexes is
(A) $$[Fe(CN)_6]^{3-}$$
(B) $$FeF_6^{3-}$$
(C) $$CoF_6^{3-}$$
(D) $$[Cr(oxalate)_3]^{3-}$$
(E) $$Ni(CO)_4$$
Choose the correct answer from the options given below:
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The major product 'P' formed in the given reaction is
Incorrect method of preparation for alcohols from the following is:
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In the reaction given below, the product 'X' is:
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Buna-S can be represented as:
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The reaction used for preparation of soap from fat is:
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The electron in the n$$^{th}$$ orbit of Li$$^{2+}$$ is excited to (n+1) orbit using the radiation of energy $$1.47 \times 10^{-17}$$ J (as shown in the diagram). The value of n is _______ Given: $$R_H = 2.18 \times 10^{-18}$$ J
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For a metal ion, the calculated magnetic moment is 4.90 BM. This metal ion has _______ number of unpaired electrons
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The number of molecules from the following which contain only two lone pair of electrons is _______
H$$_2$$O, N$$_2$$, CO, XeF$$_4$$, NH$$_3$$, NO, CO$$_2$$, F$$_2$$
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$$A(g) \rightleftharpoons 2B(g) + C(g)$$
For the given reaction, if the initial pressure is 450 mmHg and the pressure at time t is 720 mmHg at a constant temperature T and constant volume V. The fraction of A(g) decomposed under these conditions is $$x \times 10^{-1}$$. The value of x is (nearest integer) _______
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The number of endothermic process/es from the following is _______.
A. $$I_2(g) \to 2I(g)$$
B. $$HCl(g) \to H(g) + Cl(g)$$
C. $$H_2O(l) \to H_2O(g)$$
D. $$C(s) + O_2(g) \to CO_2(g)$$
E. Dissolution of ammonium chloride in water
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In alkaline medium, the reduction of permanganate anion involves a gain of _______ electrons.
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An aqueous solution of volume 300 cm$$^3$$ contains 0.63 g of protein. The osmotic pressure of the solution at 300 K is 1.29 mbar. The molar mass of the protein is _______ g mol$$^{-1}$$.
Given: R = 0.083 L bar K$$^{-1}$$ mol$$^{-1}$$
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The specific conductance of 0.0025M acetic acid is $$5 \times 10^{-5}$$ S cm$$^{-1}$$ at a certain temperature. The dissociation constant of acetic acid is _______ $$\times 10^{-7}$$. (Nearest integer)
Consider limiting molar conductivity of CH$$_3$$COOH as 400 S cm$$^2$$ mol$$^{-1}$$
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The number of incorrect statement/s from the following is _______
A. The successive half lives of zero order reactions decreases with time.
B. A substance appearing as reactant in the chemical equation may not affect the rate of reaction
C. Order and molecularity of a chemical reaction can be a fractional number
D. The rate constant units of zero and second order reaction are mol L$$^{-1}$$ s$$^{-1}$$ and mol$$^{-1}$$ L s$$^{-1}$$ respectively
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The difference in the oxidation state of Xe between the oxidised product of Xe formed on complete hydrolysis of XeF$$_4$$ and XeF$$_4$$ is _______
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Let $$S = \{z = x + iy: \frac{2z - 3i}{4z + 2i} \text{ is a real number}\}$$. Then which of the following is NOT correct?
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Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
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If $$S_n = 4 + 11 + 21 + 34 + 50 + \ldots$$ to $$n$$ terms, then $$\frac{1}{60}\left(S_{29}-S_9\right)$$ is equal to
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Let the number $$(22)^{2022} + (2022)^{22}$$ leave the remainder $$\alpha$$ when divided by 3 and $$\beta$$ when divided by 7. Then $$(\alpha^2 + \beta^2)$$ is equal to
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If the coefficients of $$x$$ and $$x^2$$ in $$(1 + x)^p(1 - x)^q$$ are 4 and -5 respectively, then $$2p + 3q$$ is equal to
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Let $$S = \{x \in [-\frac{\pi}{2}, \frac{\pi}{2}]: 9^{1-\tan^2 x} + 9^{\tan^2 x} = 10\}$$ and $$\beta = \sum_{x \in S} \tan^2 \frac{x}{3}$$, then $$\frac{1}{6}(\beta - 14)^2$$ is equal to
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Let $$A$$ be the point (1, 2) and $$B$$ be any point on the curve $$x^2 + y^2 = 16$$. If the centre of the locus of the point $$P$$, which divides the line segment AB in the ratio 3:2 is the point $$C(\alpha, \beta)$$, then the length of the line segment $$AC$$ is
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Let a circle of radius 4 be concentric to the ellipse $$15x^2 + 19y^2 = 285$$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
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The statement $$\sim p \vee \sim p \wedge q$$ is equivalent to
Let $$\mu$$ be the mean and $$\sigma$$ be the standard deviation of the distribution
| $$X_i$$ | 0 | 1 | 2 | 3 | 4 | 5 |
| $$f_i$$ | $$k+2$$ | $$2k$$ | $$k^2-1$$ | $$k^2-1$$ | $$k^2+1$$ | $$k-3$$ |
where $$\Sigma f_i = 62$$. If $$[x]$$ denotes the greatest integer $$\leq x$$, then $$[\mu^2 + \sigma^2]$$ is equal to
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Let $$A = \{2, 3, 4\}$$ and $$B = \{8, 9, 12\}$$. Then the number of elements in the relation $$R = \{(a_1, b_1, a_2, b_2) \in A \times B, A \times B: a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\}$$ is
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If $$A = \frac{1}{5!6!7!} \begin{vmatrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{vmatrix}$$, then adj $$2A$$ is equal to
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Let $$g(x) = f(x) + f(1-x)$$ and $$f''(x) > 0$$, $$x \in (0, 1)$$. If $$g$$ is decreasing in the interval $$(0, \alpha)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to
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For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int \frac{x^{2x}}{e} + \frac{e^{2x}}{x} \log_e x \, dx = \frac{1}{\alpha e} x^{\beta x} - \frac{1}{\gamma x} e^{\delta x} + C$$, where $$e = \sum_{n=0}^\infty \frac{1}{n!}$$ and C is constant of integration, then $$\alpha + 2\beta + 3\gamma - 4\delta$$ is equal to
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Let f be a continuous function satisfying $$\int_0^{t^2} f(x) + x^2 dx = \frac{4}{3}t^3$$, $$\forall t > 0$$. Then $$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to
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Let $$\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} + 5\hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d} = 12$$. Then $$(-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$$ is equal to
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If the points $$P$$ and $$Q$$ are respectively the circumcenter and the orthocentre of a $$\triangle ABC$$, then $$\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}$$ is equal to
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Let the image of the point P(1, 2, 6) in the plane passing through the points A(1, 2, 0) and B(1, 4, 1) C(0, 5, 1) be $$Q(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ equal to
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Let the line $$\frac{x}{1} = \frac{6-y}{2} = \frac{z+8}{5}$$ intersect the lines $$\frac{x-5}{4} = \frac{y-7}{3} = \frac{z+2}{1}$$ and $$\frac{x+3}{6} = \frac{3-y}{3} = \frac{z-6}{1}$$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane $$2x - 2y + z = 14$$ is
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Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $$\frac{k}{2^{15}}$$, then k is equal to
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The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _______.
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Suppose $$a_1, a_2, 2, a_3, a_4$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$$, then $$a_4$$ is equal to _______.
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Let the equations of two adjacent sides of a parallelogram $$ABCD$$ be $$2x - 3y = -23$$ and $$5x + 4y = 23$$. If the equation of its one diagonal $$AC$$ is $$3x + 7y = 23$$ and the distance of $$A$$ from the other diagonal is $$d$$, then $$50d^2$$ is equal to _______.
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Let $$S$$ be the set of values of $$\lambda$$, for which the system of equations $$6\lambda x - 3y + 3z = 4\lambda^2$$, $$2x + 6\lambda y + 4z = 1$$ and $$3x + 2y + 3\lambda z = \lambda$$ has no solution. Then $$12\sum_{\lambda \in S} \lambda$$ is equal to _______.
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If the domain of the function $$f(x) = \sec^{-1}\left(\frac{2x}{5x+3}\right)$$ is $$[\alpha, \beta) \cup (\gamma, \delta]$$, then $$3\alpha + 10\beta + \gamma + 21\delta$$ is equal to _______.
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In the figure, $$\theta_1 + \theta_2 = \frac{\pi}{2}$$ and $$\sqrt{3}BE = 4AB$$. If the area of $$\triangle CAB$$ is $$2\sqrt{3} - 3$$ unit$$^2$$, when $$\frac{\theta_2}{\theta_1}$$ is the largest, then the perimeter (in unit) of $$\triangle CED$$ is equal to _______.
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Let the quadratic curve passing through the point (-1, 0) and touching the line $$y = x$$ at (1, 1) be $$y = f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha + 1)$$ in the first quadrant is _______.
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If the area of the region $$\{(x, y): |x^2 - 2| \leq y \leq x\}$$ is A, then $$6A + 16\sqrt{2}$$ is equal to _______.
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Let the tangent at any point P on a curve passing through the points (1, 1) and ($$\frac{1}{10}$$, 100), intersect positive x-axis and y-axis at the points A and B respectively. If PA : PB = 1 : k and $$y = y(x)$$ is the solution of the differential equation $$e^{\frac{dy}{dx}} = kx + \frac{k}{2}$$, $$y(0) = k$$, then $$4y(1) - 5\log_e 3$$ is equal to _______.
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Let the foot of perpendicular from the point A(4, 3, 1) on the plane P: $$x - y + 2z + 3 = 0$$ be N. If $$B(5, \alpha, \beta)$$, $$\alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3\sqrt{2}$$, then $$\alpha^2 + \beta^2 + \alpha\beta$$ is equal to _______.
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