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Q. If y=\frac{1}{1+x^{n-m}+x^{p-m}}+\frac{1}{1+x^{m-n}+x^{p-n}}+\frac{1}{1+x^{m-p}+x^{n-p}} , then \frac{dy}{dx} at e^{m^{n^{p}}} \text{(a) }e^{mnp} \text{(b) }e^{\frac{mn}{p}} \text{(c) }e^{\frac{np}{m}} \text{(d) } none ...
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1. In the isosceles triangle ABC, \left|\vec{AB} \right|=\left|\vec{BC} \right|=8 , a point E divides AB internally in the ratio 1:3, then the cosine of the angle between \vec{CE} & \vec{CA} is ( where \left| \vec{CA}\right|= ...
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Q. Use the fact that e^x>1+x to prove that e^\pi>\pi^e . ...
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*Image* P.S. : It is a doubt. ...
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Q. A current of 3.7A is passed for 6hrs. between Ni electrodes in 0.5L of 2M solution in Ni(NO3)2. What will be the molarity of the solution at the end of electrolysis? Ans: 2M. Shouldn't it be less than 2M since it started a ...
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1. A small 10W source of ultraviolet light of wavelength 99nm is held at a distance 0.1m from a metal surface. The radius of an atom of the metal is approximately 0.05nm. Find a) the average number of photons striking an atom ...
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This is a dig from the old threads of TargetIIT which was unsolved. Let f be a continuous function in [a,b]. Prove that there exists c \in \left(a,b \right) such that \int_{a}^{c}{f(x)dx}=\left(b-c \right)f(c) . ...
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1. Let f be defined on [0,1] be a twice differentiable function such that, \left|f''(x) \right|\leq 1 for all x \in \left[0,1 \right] . If f(0)=f(1) , then show that, \left|f'(x)\right|<1 for all x \in \left[0,1 \right] . ...
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Q. Let f(x) and g(x) be differentiable functions such that f'(x)g(x)\neq f(x)g'(x) for any real x. Show that between any two real solutions of f(x)=0 , there is atleast one real solution of g(x)=0 . ...
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1. Prove that (n!)! is divisible by (n!)^{(n-1)!} . 2. Show that 1!+2!+3!+ \dots + n! cannot be a perfect square for any n \in \mathbb{N}, \: n\geq 4 . ...
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Q. Find the hydration product: *Image* ...
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Q. An electron in the ground state of the He atom joins the Helium nucleus to form He+ ion. Assuming that electron had zero kinetic energy after the transition, find the wavelength of the photon emitted. ...
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* 1. Using the relation 2(1-\cos x)<x^2, x\neq 0 , or otherwise, prove that \sin(\tan x)\geq x, \text{ } \forall \text{ } x\in\left[0,\frac{\pi }{4} \right] . 2. Find the point on the curve 4x2+a2y2=4a2, 4<a2<8 that ...
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1. Show that x^2>(1+x)[ln(1+x)]^2 \text{ } \forall\text{ } x>0 2. Let a+b=4, where a<2 and let g(x) be a differentiable function. If dg/dx >0 for all x, prove that \int_{0}^{a}{g(x)dx}+\int_{0}^{b}{g(x)dx} increas ...
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1. \int \left( \sqrt{x+\sqrt{x^2+a^2}}\right)dx ...
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1. A coil with inductance L=0.70H and active resistance r=20Ω is connected in series with an inductance-free resistance R. An alternating voltage with effective value V=220V and frequency ω=314 rad/s is applied across the t ...
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1. Find the largest term of the following sequence: \text{a) } a_{n}=\frac{n^2}{n^3+200} 2. ABC is an isosceles triangle inscribed in a circle of radius r, AB=AC and h is the altitude from A to BC. If the triangle ABC has per ...
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1. A bulb is placed at a depth of 2 7 m in water and a floating opaque disc is placed over the bulb so that the bulb is not visible from the surface. What is the minimum diameter of the disc? 2. A fish rising vertically to th ...
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1. A current loop of any arbitrary shape lies in a uniform magnetic field B. Show that the net magnetic force acting on the loop is zero. 2. Prove that the force acting on a current carrying wire, joining two fixed points a a ...
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We say \lim_{x\rightarrow 2}x^{2}=4 . Let the limit be L . So is L=4 or L<4 or L> 4 ? Having addressed this question, can we conclude on the following statements: 1) \lim_{x\rightarrow 0}\frac{\sin x}{x}=1 \Rightarrow \ ...
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1.\lim_{x\rightarrow \infty} \frac{\left(\ln x \right)^{2}}{x} NOTE: Without L' Hospital Rule 2.\lim_{x\rightarrow \infty} \left(1+\frac{1}{x} +\frac{1}{x^{2}}\right)^{2x} 3.\lim_{x\rightarrow 1} \left(\frac{x}{x-1} -\frac{1} ...
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1. *Image* Unlike other aromatic amines, why is this amine strongly basic? 2. Compare the dipole moments of the following compounds: *Image* 3. What is the major elimination product? *Image* 4.Identify the bond which will und ...
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*Image* Q18 (HCV). Figure shows a typical circuit for low-pass filter. An AC input Vi=10mV is applied at the left end and the output Vo is received at the right end. Find the output voltages for ν= 10kHz, 100kHz, 1.0MHz and ...
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Q1. In a uniform magnetic field of induction B, a wire in the form of a semicircle of radius r rotates about the diameter of the circle with an angular frequency ω. The axis of rotation is perpendicular to the field. If the ...
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Consider these two explanations for this result, energy stored in a capacitor = 1/2 CV2 ! Explanation 1: Suppose the plates of capacitor of surface area A and charged with charges +Q and -Q are almost in contact. Now, the pla ...
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Q. A metal sphere of radius R is charged to a potential V. a) Find the electrostatic energy stored in the electric field within the concentric sphere of radius 2R. b) Show that the electrostatic field energy stored outside th ...
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Q51(HCV). A block of mass m and having charge q is placed on a smooth horizontal table and is connected to a wall through an unstressed spring of spring constant k as shown in the figure. A horizontal electric field E paralle ...
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Subhomoy da you know this is directed towards you. So you please explain what the method of image charges is and how it can be used to solve problems. Everyone and anyone else is also invited to take part! ...
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1. Is the following substitution reaction feasible? R-Ph + CH3- → R-CH3 + Ph- 2. Compare the rate of reaction by SN1 mechanism: a) *Image* b) *Image* 3. State whether the following statement is true or false : Hyperconjugat ...
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\cot^{-1} \frac{y}{\sqrt{1-x^{2}-y^{2}}}=2\tan^{-1} \sqrt{\frac{3}{4x^{2}}-1}-\tan^{-1} \sqrt{\frac{3}{x^{2}}-4} &\\\\ \text{Q. Express the above as a rational integral equation between x and y.} My attempt went dirty after a ...
