Iit Jee Mains Advance Materials

► Standard formulations exist for the sums of powers of the first n natural numbers. The sum of the first n natural numbers is n * (n + 1) / 2, the sum of their squares is n * (n + 1) * (2n + 1) / 6, and the sum of their cubes is equal to the square of the sum of the first n natural numbers. ► The method of differences, also known as the V-N method, is an advanced telescoping technique used to find the sum of complex series. It works by expressing the general term of the series as a difference of two consecutive terms of another sequence, such as V(n) - V(n - 1) or V(n - 1) - V(n). ► When the V-N method is applied to a summation, all intermediate terms in the expanded series cancel out perfectly in pairs. This leaves behind only the boundary values from the beginning and the end of the expansion, allowing for a rapid calculation of the total sum.

Important 🔖Sequence & Series 🔢 ► An arithmetic progression (AP) is a sequence in which the difference between consecutive terms remains constant. The general term or nth term of an AP is given by the formula a + (n - 1)d, where a is the first term and d is the common difference. ► The sum of the first n terms of an arithmetic progression can be calculated using two formulas. It is given by (n / 2) * [2a + (n - 1)d] or alternatively by (n / 2) * [a + l], where l represents the last term of the progression. ► When solving problems that involve an arithmetic progression, choosing terms symmetrically simplifies algebraic calculations. For three terms in an AP, it is ideal to select them as (a - d), a, and (a + d), while for four terms, the selection should be (a - 3d), (a - d), (a + d), and (a + 3d). ► If every term of an arithmetic progression is increased, decreased, multiplied, or divided by a non-zero constant, the resulting sequence remains an arithmetic progression. This property demonstrates that the sequence properties adapt uniformly without breaking the arithmetic nature. ► A geometric progression (GP) is a sequence where the ratio of any term to its preceding term remains constant. The general term or nth term of a GP is defined as a * r^(n - 1), where a is the first term and r is the common ratio. ► The sum of the first n terms of a geometric progression is given by the formula a * (r^n - 1) / (r - 1) provided that the common ratio r is not equal to one. For an infinite geometric progression, the sum converges to a / (1 - r), which is valid only under the condition that the absolute value of r is strictly less than one. ► Symmetrical selection of terms is also useful for simplifying product-based geometric progression problems. For three terms in a GP, it is best to choose them as a/r, a, and ar, whereas for four terms, the ideal selection is a/r^3, a/r, ar, and ar^3. ► The arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are single mean values inserted between two positive numbers an and b. Their respective formulas are (a + b) / 2 for the AM, the square root of a * b for the GM, and 2ab / (a + b) for the HM. ► When multiple means are inserted between two positive numbers a and b, specific algebraic properties emerge. The total sum of n arithmetic means inserted between them equals n times their single arithmetic mean, while the total product of n geometric means equals the nth power of their single geometric mean. ► The AM-GM-HM inequality states that for any set of positive real numbers, the arithmetic mean is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean. Strict equality holds if and only if all the numbers in the set are equal, and this principle is a core tool in JEE for finding the maximum or minimum values of expressions. ► An arithmetico-geometric progression (AGP) is a specialized series formed by multiplying the corresponding terms of an arithmetic progression and a geometric progression. The standard form is expressed as a, (a + d)r, (a + 2d)r^2, (a + 3d)r^3, and so on. ► To find the sum of an arithmetico-geometric progression, a specific algorithmic method is required. The entire series is multiplied by the common ratio r, the resulting series is shifted forward by one term, and it is then subtracted from the original series to reduce it into a standard geometric progression. ► The sum of an infinite arithmetico-geometric progression can be evaluated directly through a derived formula. It is given by [a / (1 - r)] + [d * r / (1 - r)^2], which holds true only when the absolute value of the common ratio r is strictly less than one. ► Sigma notation possesses linear algebraic properties that help break down complex series summation. The summation of a constant from 1 to n equals the constant multiplied by n, the summation can be distributed across separate terms, and any constant multiplier can be pulled outside the summation operator.

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Important Notes 📖- Electrochemical Cells🔋 ► An electrochemical cell can convert electrical energy to chemical energy and can also convert electrical energy to chemical energy. There are two types of electrochemical cells- Galvanic cell and Electrolytic cell. ► Cathodes are usually metal electrodes. It is the electrode where reduction takes place. The cathode is the positive electrode in a galvanic cell and a negative electrode in an electrolytic cell. Electrons move into the cathode. ► A half-cell is half of an electrochemical cell (electrolytic or galvanic), where either oxidation or reduction occurs. At equilibrium, there is no transfer of electrons across the half cells. Therefore, the potential difference between them is nil. ► A salt bridge is a device used to connect the oxidation and reduction half-cells of a galvanic cell (a type of electrochemical cell). Strong electrolytes are generally used to make the salt bridges in electrochemical cells. Since ZnSO4 is not a strong electrolyte, it is not used to make salt bridges. ► Emf of a cell is equal to the maximum potential difference across its electrodes, which occurs when no current is drawn through the cell. It can also be defined as the net voltage between the oxidation and reduction half-reactions. ► Cell potential is an intensive property as it is independent of the amount of material present. Gibbs free energy is defined for an electrochemical cell and is an extensive property as it depends on the quantity of the material. ► Electrode potential is the tendency of an electrode to accept or to lose electrons. Electrode potential depends on the nature of the electrode, temperature of the solution and the concentration of metal ions in the solution. It doesn’t depend on the size of the electrode. ► The salt bridge connects the two half-cell solutions to complete the circuit of the electrochemical cell. The electrolytes of the salt bridge are generally prepared in agar-agar or gelatin so that the electrolytes are kept in a semi-solid phase and do not mix with the half-cell solutions and interfere with the electrochemical reaction. ► A salt bridge is a junction that connects the anodic and cathodic compartments in a cell or electrolytic solution. It maintains electrical neutrality within the internal circuit, preventing the cell from rapidly running its reaction to equilibrium. ► A Voltaic or Galvanic cell is a type of electrochemical cell that converts chemical energy into electrical energy. Photovoltaic cells are used to convert light energy into electrical energy. An Electrolytic cell is a type of electrochemical cell that converts electrical energy into chemical energy. A fuel cell is an electrochemical cell that converts the chemical energy of a fuel and an oxidizing agent into electricity. ► For all spontaneous chemical reactions, the change in Gibbs free energy (ΔG°) is always negative. For a spontaneous reaction in an electrolytic cell, the cell potential (E°cell) should be positive. ► In an electrochemical cell, when an opposing externally potential is applied and increased slowly, the reaction continues to take place. When the external potential is equal to the potential of the cell, the reaction stops. Once the externally applied potential is greater than the potential of the cell, the reaction goes in the opposite direction and the cell behaves like an electrolytic cell. ► Primary cells cannot be used again and again. Since there is no fluid inside, these cells are also known as dry cells. The internal resistance is high and the chemical reaction is irreversible. Their initial cost is cheap. ► A secondary battery (a series of cells) is one which can be charged, discharged into a load, and recharged many times. Nickel-cadmium cell, Lead storage cell and Mercury cell are examples of secondary cells. Leclanche cell is an example of a primary cell.

━━━━━━━━━━━━━━━━━━━━ ✅ QUICK REVISION ✅ ✅ FORMULAS ✅ 🎯1. Formulas related to force: 🔰F = ma 🔰F = kx 🔰F = m(vf² - vi²/2S) 🔰F = mv/t 🔰F = md/t² 🔰F = m(vf - vi)/t 🔰F = Area × density × velocity² 🔰F = 1/2 mv²/d 🔰F = 1/2 Pv/d 🔰F = Power/velocity 🔰Fc = mv²/r 🔰Fc = mrw² 🔰Fc/2 = mv²/2r 🔰Fc = 2K.E/r 🔰F = Area × Stress 🔰F = pir² × stress 🔰F = YA × Strain 🔰F = YAl/L 🔰F = pressure × area 🔰F = change in momentum × time interval 🔰F = - 2mVx × Vx/2l 🔰F2 = F1/A1 × A2 🔰F = qE 🔰F = kQ/r² 🔰F = ILB sintheta 🔰F = q (v × B) 🔰F = qE + q(v × B) 🎯2. Formulas related to energy and work 👉Fd = k.e 👉mgh = 1/2 mv² 👉E = 1/2 kx² 👉E = Ve 👉E = nhf 👉E = nhc/lambda 👉E = Pc 👉K.e = hf - work function = hf - hf° = hf - hc/w° (here w° is cutt off wavelength) 👉E = 1/2 Pv 👉mv²/2r= Fc/2 👉K.E/r = Fc/2 👉K.E = Fc×/r/2 👉K.e = 1.5 KT 👉E = VQ 👉E = Power × time 👉E = Fvt 👉% loss in K.e = v1² - v2²/v1² × 100 👉% loss in P.e = h1² - h²/h1² × 100 👉Energy lost due to air friction(Fh) = 1/2mv² - mgh (when body is thrown upward) 👉Energy lost due to air friction(FS) = mgh - 1/2mv² (when body is thrown downward) 👉E = 1/2 CV² (capacitor) 👉E = R × hc (R is Rydberg' constant) J = m-¹ × Js ms-¹ 👉hf kalpha x rays = EL - Ek hf kbeta x rays = EM - Ek 👉Binding energy = mass defect × c² 👉W = Fd Costheta 👉W = nmgh (when person is climbing stairs) 👉W = n(m+m) gh (when person is climbing stairs with some load) 👉W = 0mgh + 1mgh + 2mgh + 3mgh ....... (in case of stacking bricks. For ist brick h=0. For 2nd brick h=1. For 3rd brick h=2 and so on) 👉W = Fd = PA × change in V 👉W = Q - change in U 👉Q = mc × change in T T/273.16 = Q/Q3 (Thermodynamic scale) 👉W = I²Rt 👉W = emf×charge 👉W = VQ 👉W = 1/2 lF 👉W = YAl²/2L 👉W = StressAl²/2Strain 👉W = PressureAl²/2Strain 👉W = Fl²/2Strain 🎯3. Formulas related to Power 💥P = Fv 💥P = E/t 💥P = n(mgh/t) 💥P = Fd/t 💥P = mv²/2t 🎯4. Formulas related to distance, displacement, velocity and accelration 📝d = vt 📝d = at² 📝d = (vf + vi/2) ×t 📝d = 5t² (for distance in 'n' seconds) 📝d = 5(2tn - 1) (for distance in 'nth' second) 📝d = 1/2 mv²/F 📝d = vit + 5t² 📝d = v × underroot 2H/g 📝d = vt = x°wt = x°2pi/T × t = x°2pift 📝x = x° Sin wt 📝x = x° Sin (underroot k/m) t vf = vi + at 📝2as = vf² - vi² 📝2as = (vi + at)² - vi² 📝2as = vf² - (vf - at) ² 📝v = underroot Vfx² + Vfy² 📝v = Power/Force 📝v = 2×K.E/momentum (k.e = 1/2 Pv) 📝v² = 2×Power×time/mass (P = mv²/2t) v = underroot 2as v = underroot gr (speed at highest point in a verticle circle) v = underroot 5gr (speed at lowest point in a verticle circle) 📝v² = 2FS/m 📝v² = 2E/m 📝v² = 2Ve/m 📝v = eBr/m (velocity of particle under action of magnetic force along circular path) 📝v² = Force/Area.Density 📝v = w underroot x°² - x² 📝v = underroot k/m × underroot x°² - x² 📝v = x°w (at mean position where x=0) 📝v = x° underoot k/m 📝v = v° underroot 1 - x²/x°² (for determining ratio b/w inst. Velocity and maxi. Velocity) 📝v= x°2pif = x°2pi/T 📝a = x°w² = x°w.w = vw = v.2pif Common velocity = m1v1/m1+m2 📝vi² = Rg/Sin2theta 📝v = underoot Tension×length/mass 📝V = 2pi ke²/nh (speed of e- in nth orbit) 📝Vn = V/n 📝v = nh/2pimr (lambda = 2pir and lambda=h/p) 📝ma = kx 📝a = kx/m (SHM) 📝a = - gx/l (Simple pendulum) 📝ac = v²/r 🎯5. Formulas related to wavelength 'w' ❌w = v/f ❌w = 1/wave number ❌w1 = 2l (when pipe is opened at both ends) ❌w1 = 4l (when pipe is opened at one end) ❌Delta w = Us/f (doppler shift) Wavelength for obs. = w - delta w = v/f - Us/f ❌w = hc/Ve ❌w = hc/E ❌w = h/mv ❌w = h/P as P = underroot 2mE so ❌w = h/underroot 2mE (de Broglie wavelength) ❌w = underroot 150/V A° (short method for de Broglie wavelength. This formula is applicable only for e-) ❌1/w = RH (1/p²-1/n²) ❌Wmaxi/Wmini = n²/n²-p² (for determining ratio b/w maxi. Wavelength to mini. Wavelength for series of atomic spectrum) ❌w = 2pir/n (n is no. of loops in a circle

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🔰Carbon and its Compounds🔰 ✍️ Introduction ➖Carbon plays very important roles for all living beings. ➖The amount of carbon in the earth’s crust is merely 0.02%, which is available in the form of minerals such as carbonates, hydrogen-carbonates, coal, and petroleum. ➖The presence of carbon in the atmosphere of the earth is 0.03%, in the form of carbon dioxide. ✍️ Compounds of Carbon ➖Almost all carbon compounds (except a few) are poor conductors of the electricity. ➖The diamond and graphite both are formed by carbon atoms; however, the difference lies between them in the manner in which the carbon atoms are bonded to one another. ➖In diamond, each atom of the carbon, is bonded to four other carbon atoms and form a rigid three-dimensional structure. ➖In graphite, each atom of the carbon, is bonded to three other carbon atoms in the same plane, which gives a hexagonal array. ➖There is also difference in some physical structure of diamond and graphite. ➖Diamond is the hardest substance known whereas graphite is smooth and slippery substance. ➖Graphite is good conductor of electricity whereas diamond is not. ➖The compounds, which has identical molecular formula, but different structures, are known as structural isomers. ➖The saturated hydrocarbons are known as alkanes. ➖The unsaturated hydrocarbons, which comprise of one or more double bonds, are known as alkenes. ➖The unsaturated hydrocarbons, which comprise of one or more triple bonds, are known as alkynes. ✍️ Use of Alcohol as Fuel ➖Sugarcane plants very efficient convert sunlight into chemical energy and its juice can be used to prepare molasses. ➖When molasses is fermented, it produces alcohol (ethanol). ➖Some of the countries now using alcohol as an additive in petrol, as it is a cleaner fuel. ➖These alcohol, on burning in sufficient air (oxygen), gives rise to only carbon dioxide and water. ✍️ Esters ➖Esters are sweet-smelling substances, which are most commonly formed by reaction of an acid and an alcohol. ➖When esters react in the presence of an acid or a base, it gives back the alcohol and carboxylic acid. ➖The reaction of esters with an acid or a base, is known as saponification because it is used in the preparation of soap. ➖The molecules of soap normally are sodium or potassium salts of long-chain carboxylic acids. ➖Interestingly, the ionic-end of soap dissolves in water whereas the carbon chain dissolves in oil. This typical features of the soap molecules forms structures known as micelles. ➖In micelles, one end of the molecules is towards the oil droplet whereas the ionic-end remains outside. ➖The soap micelle helps in dissolving the dirt in water; likewise, the clothes get cleaned. ➖On the other hand, detergents are usually ammonium or sulphonate salts of long chain carboxylic acids, which remain effective even in hard water. ➖Detergents are customarily used to make shampoos and some other products for cleaning clothes.

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The difference of number of sigma bonds in benzene and number of pi electrons in cyclopentadienyl anion is 'n'. An isomer 'A' of alkane CnH2n+2 contains 2 tertiary H atoms. On monochlorination, 'A' gives a major product 'B', and on monobromination, it gives a major product 'C'. A. In DNA structure, the number of hydrogen bonds stabilizing a Guanine-Cytosine (G-C) base pair is 'n'/2, while the number stabilizing an Adenine-Thymine (A-T) pair is 'n'/3. B. Due to significant steric hindrance, product 'C' is highly resistant to SN1 reactions but readily undergoes solvolysis via an SN2 mechanism in a protic solvent like ethanol. C. Consider the reaction of B with NaI in acetone solvent the time required for the concentration of B to decrease from 0.2 M to 0.1 M is observed to be 60 minutes. If the reaction of B started with an initial concentration of 0.8 M, then the first half-life is 30 minutes D. a salt containing a cation Fe^x+ (where x = n/2 )is subjected to qualitative analysis. the original salt solution is treated with ammonium thiocyanate (NH4SCN) solution, a blood-red coloration appears.

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