Questions: Binomial Theorem

Binomial Theorem26 min readLesson 4 of 5

in hinglish

Chapter ka recap. Binomial theorem se (a+b)^n ka expansion milta hai, jisme n+1 terms aur binomial coefficients nCr hote hain. Pascal triangle coefficients deta hai, aur general term T_{r+1} = nCr a^{n−r}b^r hai. Particular power ya term independent of x nikalne ke liye power ko barabar rakh kar r nikalo. Middle term even n me ek, odd n me do. Coefficient sums ke liye x = 1 ya x = −1 substitute karo. Approximation me 100 ± chhota number wala split use karo. Divisibility me base ko 1 + a likh kar expand karo. Ab das sawal solve karo, phir answer milao. Yaad rakho: general term wale question me r nikaalne ke baad middle term ke liye (n+1)/2 check karo, aur jab do middle terms hon to unka product bhi pooch sakte hain. Har answer ko binomial expansion ke pehle teen terms se verify karo.

Q1. Expand (x + 2) to the fourth power. Answer: The coefficients of the fourth row are 1, 4, 6, 4 and 1. Multiply by the powers of x and 2: x to the fourth plus 4 times x cubed times 2 plus 6 times x squared times 4 plus 4 times x times 8 plus 16. This gives x to the fourth plus 8x cubed plus 24x squared plus 32x plus 16. A quick check at x equal to 1 gives 1 plus 8 plus 24 plus 32 plus 16 equal to 81, which is 3 to the fourth, the same as substituting x equal to 1 in (x plus 2) to the fourth. Hinglish: Fourth row ke coefficients 1, 4, 6, 4, 1 hain. Powers of x aur 2 multiply karo: x^4 + 8x³ + 24x² + 32x + 16.

Q2. Find the coefficient of x to the fifth in (1 + x) to the eighth power. Answer: The general term is 8Cr times x to the r, so the coefficient of x to the fifth is 8C5. By symmetry 8C5 equals 8C3, which is 56. Hinglish: General term 8Cr x^r hai, isliye x^5 ka coefficient 8C5 = 8C3 = 56.

Q3. Find the term independent of x in (2x squared + 1 by x) to the tenth power. Answer: The general term is 10Cr times (2x squared) to the (10 minus r) times (1 by x) to the r. The power of x is 20 minus 3r. For the term independent of x, set 20 minus 3r equal to 0, which gives r equal to 20 by 3. Since r must be a whole number between 0 and 10, no such term exists and the constant coefficient is 0. Do not copy the r equal to 6 value from the x squared question: that value belongs to a different required power. Hinglish: General term me x ki power 20 - 3r hai. Constant term ke liye 20 - 3r = 0 chahiye, par r = 20/3 integer nahi hai, isliye constant term exist nahi karta. r = 6 wala case x² ke liye tha, confuse mat karo.

Q4. Find the coefficient of x squared in (2x squared + 1 by x) to the tenth power. Answer: Using the same general term, set 20 minus 3r equal to 2 to get r equal to 6. The coefficient is 10C6 times 2 to the fourth, that is 210 times 16 equal to 3360. Hinglish: 20 - 3r = 2 se r = 6 milta hai. Coefficient 10C6 \times 2^4 = 210 \times 16 = 3360.

Q5. Find the middle term in the expansion of (x by a + a by x) to the tenth power. Answer: The index is even, so there is one middle term, at position 10 by 2 plus 1 equal to 6. Its coefficient is 10C5, and the powers of x and a combine to give a constant term equal to 10C5 with a minus sign depending on the sign inside the bracket. Here the expansion is a sum, so the middle term is 10C5 equal to 252. Hinglish: n = 10 even hai, isliye middle term 6th hai. Coefficient 10C5 = 252, aur x/a aur a/x ki powers cancel ho jaati hain.

Q6. Evaluate 96 cubed using the binomial theorem. Answer: Write 96 as 100 minus 4 and expand (100 minus 4) cubed as 100 cubed minus 3 times 100 squared times 4 plus 3 times 100 times 16 minus 64. That is 1000000 minus 120000 plus 4800 minus 64, which equals 884736. Hinglish: 96 = 100 - 4 likho. (100-4)³ = 100³ - 3\cdot100²\cdot4 + 3\cdot100\cdot16 - 64 = 884736.

Q7. Find the sum of all coefficients in the expansion of (1 + 2x) to the fifth power. Answer: Put x equal to 1 in the expression. The expansion then sums to (1 plus 2) to the fifth, which is 3 to the fifth equal to 243. The answer is the value of the whole expression at x equal to 1. Hinglish: Coefficients ka sum nikalne ke liye x = 1 rakho: (1 + 2)^5 = 3^5 = 243.

Q8. Show that the coefficient of x to the power r in (1 + x) to the n equals the coefficient of x to the power n minus r. Answer: The coefficient of x to the power r is nCr and the coefficient of x to the power n minus r is nC(n minus r). The identity nCr equals nC(n minus r) completes the proof. The combinatorial reason is that choosing r objects from n is the same as rejecting the other n minus r. Hinglish: Coefficient nCr aur nC(n-r) hote hain, aur identity nCr = nC(n-r) se dono barabar. Reason: r choose karna aur n-r reject karna same baat hai.

Q9. Find the coefficient of x in the expansion of (1 minus 2x) to the fifth power. Answer: The general term is 5Cr times 1 to the (5 minus r) times (minus 2x) to the r, so the x term has r equal to 1 and coefficient 5 times minus 2 equal to minus 10. Hinglish: x wale term ke liye r = 1, coefficient 5C1 \times (-2) = -10.

Q10. Using the binomial theorem, show that (1.01) to the million is greater than 10000. Answer: Write (1.01) to the million as (1 plus 0.01) to the million. Its first two terms are 1 and 1000000 times 0.01 equal to 10000, and every other term is positive. Hence the value is greater than 1 plus 10000, and therefore certainly greater than 10000. The same argument shows why the first two terms alone decide the comparison even though the power is enormous: the remaining terms cannot be negative, so they only increase the total. Hinglish: (1 + 0.01)^1000000 ke pehle do terms 1 aur 10000 hain, aur baaki sab positive hain. Isliye value 10000 se badi hai. Power bada hone se baaki terms sirf value badhate hain, ghatate nahi.

Common mistakes on this chapter Writing all signs negative when the bracket has a minus is the first error. Forgetting that the general term is the (r plus 1)th term and reporting the term number wrongly is the second. Forcing a non-integer r to fit a required power is the third, when the honest answer is that the term does not exist. Using the general binomial series for a non-integral index is outside the syllabus and is the fourth. Read the final answer against these four checks. Quiz Quiz below: five multiple choice questions on coefficients, the general term, middle terms and coefficient sums. Attempt them after the notes above, then move to the previous year page.