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$$a$$, $$b$$, $$c$$, $$d$$ are digits in 2014 such that $$\{a, b, c, d\} = \{2, 0, 1, 4\}$$. Then the number of different values $$\left(\left(a^b\right)^c\right)^d$$ takes is
$$\left(\left(a^b\right)^c\right)^d = a^{bcd}$$. If $$a = 0$$, the exponent is $$2 \times 1 \times 4 = 8$$ and the value is 0. If $$a$$ is 1, 2 or 4, then 0 is one of $$b$$, $$c$$, $$d$$, so the exponent is 0 and the value is 1. Only the values 0 and 1 occur, so there are 2 different values.
The maximum value of $$n$$ such that $$2^n$$ divides $$21!$$ is
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A circle C has radius 6. Two circles P and Q having equal radii are touching each other externally and touching C internally. Another circles S is touching the circle C internally and the circles P, Q externally. The radius of the circle S is
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If $$f \colon \mathbb{R} \rightarrow \mathbb{R}$$ is a function defined by $$f(x) = \frac{x^2}{x^2 + 1}$$ then the number of integral values taken by $$f$$ is
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$$\sqrt[3]{20 + 14\sqrt{2}} + \sqrt[3]{20 - 14\sqrt{2}} =$$ _______
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A circle of radius 1 is concentric with circles of radii $$a$$ and $$b$$ with $$b > a > 1$$. The area of the smallest circle equals the area of the ring with outer radius $$a$$ and inner radius 1 and also equals the area of the ring with outer radius $$b$$ and inner radius $$a$$. Then $$a + b$$ is ___________
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The sides of a cyclic quadrilateral lie along the lines $$a_1 x + b_1 y = 2013$$, $$a_2 x + b_2 y = 2014$$, $$a_1 x + b_1 y = 2015$$, $$a_2 x + b_2 y = 2018$$ then $$a_1 a_2 + b_1 b_2 =$$ _____________
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Given $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$, the value of $$\cos 15^\circ =$$ _______________
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State True or False. There exists a function $$f$$ from the set of integers $$\mathbb{Z}$$ to the set of natural numbers $$\mathbb{N}$$ that is an onto function.
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State True or False. The quadratic equation the sum of whose roots is the sum of the roots of $$5x^2 + 4x - 3 = 0$$ and the product of the roots is the product of the roots of $$7x^2 + 24x + 2 = 0$$ has real roots.
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State True or False. In a circle if a chord of length $$\sqrt{2}$$ subtends an angle of $$45^\circ$$ at a point on the circumference of the circle then the radius of the circle is 1.
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State True or False. It is not possible to write the polynomial $$x^4 + 1$$ as a product of two quadratic polynomials.
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In a circle having $$A(2, 2)$$ and $$B(10, 8)$$ as extremities of a diameter if a triangle $$ABC$$ is inscribed then what is the maximum possible area of the triangle?
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If $$f \colon \mathbb{R} \rightarrow [0, \infty)$$ is defined by $$f(x) = x^2$$ and $$g \colon [0, \infty) \rightarrow \mathbb{R}$$ is defined by $$g(x) = \sqrt{x}$$ then does the function $$gof(x)$$ exist? If so what is the value of $$gof(-4)$$?
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A square is inscribed in a sector having angle $$60^\circ$$ at its vertex. Find the ratio of the area of the square to that of the sector.
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If $$[x]$$ denotes the greatest integer less than or equal to $$x$$ then what is the value of $$\sum_{k=1}^{2014} \left[\frac{k}{1007}\right]$$?
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A car travels between places A and B covering first $$\frac{1}{3}$$ of the distance at $$25\text{ kmph}$$ and the remaining distance at $$40\text{ kmph}$$ for some time and $$60\text{ kmph}$$ for an equal amount of time. Find the average speed (in kmph) of the car.
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A mirror produces inverted twice magnified image of an object at a distance of $$30\text{ cm}$$ from the object. Find the focal length (in cm) of the mirror.
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A block is made with equal volumes of two materials whose densities are in the ratio $$1 \colon 2$$. This block floats with $$\frac{3}{4}$$ of its volume submerged in a certain liquid. Find the ratio of the volumes of the materials (lighter to denser) for $$\frac{2}{3}$$ of the volume of the block to be just submerged.
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A ray of light is incident at a height $$h$$ above the principal axis of a lens (position indicated by dotted line) and deviates through $$30^\circ$$ as it passes through the lens. Find the focal length of the lens.
[image]
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In the figure shown resistance between any two corners connected by short wires is half of the resistance of the wires connected by long wires. Terminals A, B and C are connected to the corners by resistanceless wires. If the equivalent resistance between A and C is $$R$$, find the equivalent resistance between A and B.
[image]
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If 4.2 milligram of mass were to be completely converted into heat energy and this heat is used to raise the temperature of a million kg of water, find the temperature raise (in degree Celsius) of water.
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A bullet moving horizontally at $$v$$ strikes a stationary block lying on a smooth horizontal surface and gets embedded in it. The block moves at $$V$$ after the bullet strikes it. Find the ratio of masses of the bullet and the block.
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Wavelength of light in a certain medium is $$\lambda$$ where its velocity is $$2.5 \times 10^8\text{ m/s}$$. If the wavelength of this light in some other medium is $$0.6\lambda$$, find the velocity of light in this medium. What is the ratio of frequencies of the light in the two media.
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A positively charged particle to the east of a long vertical current carrying wire having a velocity directed eastward experiences a vertical upward force. Find the direction of the force on a positively charged particle located south of the conductor and having a velocity directed vertically upward.
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Two small identical charged conducting spheres have charges in the ratio $$1 \colon 3$$ and exert a force $$F$$ on each other when they are some distance apart. The spheres are brought in to contact and then they are arranged at a distance twice the earlier. Knowing that the total charge on the two spheres divides equally between them when they make contact, find the force between them.
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First four ionization energy values of an element are 191, 578, 872 and 5972 k.cals. The number of valency electrons in the element is
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Correct set of four quantum numbers for the valence electron of rubidium (Z = 37) is
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List I - A) $$\text{NH}_4^+$$, B) $$\text{H}_3\text{O}^+$$, C) $$\text{XeO}_3$$, D) $$\text{SO}_3$$<br>List II - 1) $$sp^3$$ - two lone pairs, 2) $$sp^2$$ - one lone pair, 3) $$sp^3$$ - no lone pairs, 4) $$sp^3$$ - one lone pair, 5) $$sp^2$$ - no lone pairs<br>Correct match is
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Alkaline earth metal compounds are less soluble in water than the corresponding alkali metal compounds, because alkali metals have
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$$\text{NH}_4^+$$ ion in aqueous solution will behave as.
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Here the six transition metal atoms are given Sc, Ti, V, Co, Ni, Cu. Among these how many pairs are having same no of unpaired electrons ________________
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$${}^{209}_{83}\text{Bi}$$ is produced from a parent element by the emission of $$2\alpha$$, $$4\beta$$ and one positron what will be the parent element________________
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Alkyl halide used in preparation of n-hexane in Wurtz reaction is. _______________
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The weight percentage of NaCl solution is 10. If the weight of solution is 150 gr. Calculate weight of NaCl (in grams) _________________
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Number of grms NaOH present in 5 liters of millimolar solution is ____________
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Name of the compound. Alkaline hydrolysis is of white phosphorous gives sodium salt of phosphorous and a gas with rotten fishy smell, the gas is.
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Name of the compound. Beryllium carbide on hydrolysis produces a gas. The name of the gas is.
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Name of the compound. Chemical name of Bleaching powder is.
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Name of the compound. Lithium nitrate on decomposition produce two paramagnetic gases one is colourless gas and the other is brown gas. The brown gas is.
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Name of the compound. Name of the lead compound used for making protective paint for iron and steel from rusting.
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The hydrogen ion concentration of a solution is 0.001 M at $$25^\circ\text{C}$$. What will be the hydroxyl ion concentration (in M) of solution at same temperature.
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Arrange in the increasing order of hydration energies of $$\text{Be}^{2+}$$, $$\text{Ca}^{2+}$$, $$\text{Mg}^{2+}$$, $$\text{Ba}^{2+}$$, $$\text{Sr}^{2+}$$
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A radioactive element from VII A group emits a $$\alpha$$ particle a $$\beta$$ particle and a positron. What will be the position of new element in the periodic table.
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Give the balanced equation. Lead nitrate on decomposition produce two paramagnetic gases.
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Give the balanced equation. Sodium bicarbonate on reaction with zinc sulphate gives two metal salts and a colourless gas.
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