EXERCISE -1
1) EXPAND:
a) (3x+2y)⁴. 81x⁴+216x³y+ 216x²y²+ 96xy³+ 16y⁴
b) (2x-3y)⁴. 16x⁴-96x³y+ 216x²y³- 216xy²+ 81y⁴
c) (x+ 1/y)¹¹. x¹¹+ 11x¹⁰/y + 55x⁹/y² + 165x⁸/y³ + 330x⁷/y⁴ + 462x⁶/y⁵ + 462x⁵/y⁶ + 330x⁴/y⁷ + 165x³/y⁸ + 55x²/y⁹ + 11x/y¹⁰ + 1/y¹¹
d) (√x + √y)¹⁰. x⁵+ 10x⁹⁾²y¹⁾²+ 45x⁴y+ 120x⁷⁾²y³⁾² + 210x³y²+252⁵⁾² y⁵⁾² + 210x²y³ + 120x³⁾²y⁷⁾²+ 45xy⁴ + 10x¹⁾²y⁹⁾² + y⁵
e) (x + 1/x)⁵. x⁵+5x³+ 10x+ 10/x+ 5/x³+ 1/x⁵
f) (x - 1/x)⁶. x⁶-6x⁴+15x²- 20+ 15/x²- 6/x⁴+ 1/x⁶
g) (2x/3 - 3/2x)⁶. 64x⁶/729 - 32x⁴/27 + 20x²/3 - 20+ 135/4x² - 243/8x⁴+ 729/64x⁶
h) (x² - 2/x)⁷. x¹⁴ -14x¹¹ + 84x⁸ - 280x⁵ + 560x² - 672/x + 448/x⁴ - 128/x⁷
I) (1+x)⁵. 1+ 5x + 10x²+ 10x³+ 5x⁴ + x⁵
j) (1- 3x)⁷. 1- 21x+ 189x² - 945x³ + 2835x⁴ - 5103x⁵ + 5103x⁶- 2187x⁷
k) (a² - 2bc)⁵. a¹⁰-10a⁸bc+ 40a⁶b²c²- 80a⁴b³c³+80a²b⁴c⁴-32b⁵c⁵
L) (x³ + 2/x²)⁵. x¹⁵+ 10x¹⁰+ 40x⁵ + 80+ 80/x⁵ + 32/x¹⁰
m) (1+ 2x- 3x²)⁵. 1+ 10x+ 25x²- 40x³ - 190x⁴+ 92x⁵+ 570x⁶- 360x⁷ - 675x⁸ + 810x⁹ - 243x¹⁰
2) EVALUATE::
a) (√2+1)⁶+(√2-1)⁶. 198
b) (2+√3)⁷+(2 - √3)⁷. 9884
c) (√3+1)⁵ +(√3 - 1)⁵ 88√3
d) (x+ √(x²-1)⁶+ (x- √(x²-1)⁶. 64x⁶ - 96x⁴ + 36x² - 2.
e) (2a+b)⁶ - 6b(2a+b)⁵+ 15b²(2a+b)⁴ - 20b³(2a+b)³+15b⁴(2a+b)²- 6b⁵(2a+b) + b⁶. 64a⁶
f) (1+2√x)⁵+(1-2√x)⁵. 2(1+40x+80x²)
g) (3+√2)⁵-(3-√2)⁵. 1178√2
h) (√3+1)⁵-(√3-1)⁵. 152
I) (96)³. 884736
j) (102)⁵. 11040808032
k) (101)⁴. 104060401
l) (98)⁵. 9039207968
m) (994)⁴ 976215137296
n) (1.01)⁵. 1.0510100501
o) (1001)⁵. 1005010010005001
3) FIND:
a) 7th term of (4x/5 + 5/2x)⁸. 4375/x⁴
b) 10th term of (a/b - 2b/a²)¹². -366080b⁵/a¹⁴
c) 16th term of (√x - √y)¹⁷. -136xy¹⁵⁾²
d) 4th term of (x/y - y/x)¹⁰. -120x⁴/y⁴
e) 5th term of (x²+ 3/x³)¹⁰. 17010
f) 19th term of (2√x - √y)²⁰. 760xy⁹
g) 11th term of (2x - 1/x²)²⁵. ²⁵C₁₀(2¹⁵/x⁵)
h) 7th term in (3x² - 1/x³)¹⁰. 17010/x¹⁰
I) 7th term in (4x/5 + 5/2x)⁸. 4375/x⁴
j) 4th term in (x+ 2/x)⁹. 672x³
4) Find the coefficient of:
a) x⁷ in (x²+ 1/x)¹¹. 462
b) x² in (3x- 1/x)⁶. 1215
c) x¹⁸ in (x² - 3a/x)¹⁵. 110565a⁴
d) x¹⁰ in (x² - 2)¹¹. 29568
e) x⁶ in (3x² - 1/3x)⁹. 378
f) x¹⁰ in (2x² -1/x)²⁰. ²⁰C₁₀ 2¹⁰
g) x⁷ in (x - 1/x²). -⁴⁰C₁₁
h) 1/x¹⁵ in (3x² - a/3x³)¹⁰. -40a⁷/27
I) x⁹ in (x² - 1/3x)⁹. -28/9
j) 1/x² in (2x³ - 1/x²)⁶. 60
k) x¹² in (ax⁴ - bx)⁹. 9ab⁸
l) x³² in (x⁴ - 1/x³)¹⁵. 1365
m) 1/x in (2x² - 1/x)¹⁰. -960
n) 1/x¹¹ in (x² - 1/x³)¹². -792
5) Find the term and coefficient independent of x in the expansion of::
a) (x+1/x)¹⁰. 252
b) (x² +1/x)¹². 495
c) (2x+1/3x²)⁹. 1792/9
d) (x - 1/x)¹². 924
e) (x² - 2/x³)¹⁵. 320320
f) (3x²/2 -1/3x)⁹. 7/18
g) (9x²-1/3x)¹². 495
h) (√x - √c/√x)¹⁰. -252√c⁵
I) (2x²- 3/x³)²⁵. ²⁵C₁₀(2¹⁵× 3¹⁰)
j) {√(x/3)+3/2x²)¹⁰. -3003×3¹⁰×2⁵
6) Find the middle term in the expansion of:
a) (3 + x)⁶. 540x³
b) (2x/3 - 3/2x)²⁰. ²⁰C₁₀
c) (1 - x²/2)¹⁴. -429x¹⁴/16
d) (a/x + bx)¹². 924a⁶b⁶
e) (x² - 2/x)¹⁰. -8064x⁵
f) (a/3 + 9b)⁸. 5670a⁴b⁴
g) (x/a - a/x)¹⁰. -252
h) (3x/ - x³/6)⁹. 189x¹⁷/8, -21x¹⁹/16
i) (2x² - 1/x)⁷. -560x⁵, 280x²
j) (3x - 2/x²)¹⁵. -6435×3⁸×2⁷/x⁶, 6437×3⁷×2⁸/x⁹
k) (x⁴ - 1/x³)¹¹. -462x⁹, 462x²
I) (x - 1/x)¹⁰. -252
m) (2x - x²/4)⁹. 63x¹³/4, -63x¹⁴/32
7) a) Find the 5th term from the end in the expansion of (x - 1/x)¹². 495x⁴
b) Find the 4th term from the end in the expansion of (4x/5 - 5/2x)⁹. 10500/x³
c) (x⁴ + 1/x³)¹⁵, 4th term from the end. 455x³⁹ or 455/x²⁴
d) (3/x² - x³/6)³, 4th term from the end. 35x⁶/48
e) (2x - 1/x²)²⁵, 11th term from the end. - ²⁵C₁₅ 2¹⁰/x ²⁰
f) 11th term from end (2x - 1/x²)²⁵. ²⁵C₁₅ (2¹⁰/x²⁰)
g) 5th term from the end of (3x -1/x²)¹⁰. 17010/x⁸
h) 4th term from the end in (x+2/x)⁹. 5376/x³
I) 4th term from the end of (4x/5 - 5/2x)⁹. 10500/x³
j) 7th term from the end in (2x² - 3/2x)⁸. 4032x¹⁰
EXERCISE -2
1) Using Binomial theorem, prove
A) 6ⁿ - 5n leaves remainder 1 when divided by 25.
B) 2³ⁿ - 7n -1 is divisible by 49, where n belongs to N.
C) 3ⁿ⁺² - 8n - 9 is divisible by 64, where n belongs to N.
D) 3³ⁿ - 26n -1 is divisible by 676.
E) 9ⁿ⁺¹ - 8n - 9 is divisible by 64, where n belongs to N.
2) Show that the middle term in the expansion of
A) (1+ x)²ⁿ is :
{1.3.5....(2n -1) 2ⁿ. xⁿ}/n!
B) (x+ 1/x)²ⁿ is :
{1.3.5....(2n -1) 2ⁿ}/n!
3) Find the general term in the expansion
a) (x² - y)⁶.
4) Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion (⁴√2 + 1/⁴√3)ⁿ is √6: 1. 10
5) If 17th and 18th term in the expansion of (2+ a)⁵⁰ are equal, then find a.
6) If the fourth term in the expansion of (ax + 1/x))ⁿ is 5/2, then find the value of a and n. 1/2, 6
7) Find the value of a so that the term independent of x in (√x + a/x²)¹⁰ is 405. ±3
8) Find a positive value of n for which the Coefficient of x² in the expansion of (1+ x)ⁿ is 6. 4
9) Find the coefficient of x⁷ in (ax² + 1/bx)¹¹ and 1/x⁷ in (ax - 1/bx²)¹¹. 11C5 a⁶/b⁵
10)A) Prove that the coefficient of (1+ x)²ⁿ is equal to the sum of the coefficient of middle terms in the expansion of (1+ x)²ⁿ⁻¹.
B) Find the value of k for which the coefficient of the middle term in (1+ Kx)⁴ and (1- Kx)⁶ are equal. -3/10
11) The sum of the coefficient of first three terms in the expansion of (x - 3/x²)ⁿ, x≠ 0, n being a natural number, is 559. Find the term of the expansion containing x³. 12
12) If the coefficient of (2r+4)th and (r -2)th terms in the expansion of (1+ x)¹⁸ are equal, find r. 6
13) If the coefficient of (2r+1)th and (r +2)th terms in the expansion of (1+ x)⁴³ are equal, find r. 14
14) If the coefficient of (r - 5)th and (2r -1)th terms in the expansion of (1+ x)³⁴ are equal, find r. 14
15) The co-efficients of 5th, 6th, 7th terms in the expansion (1+ x)ⁿ are in AP, find n. 7 or 14
16) The co-efficients of 2nd, 3rd, 4th terms in the expansion (1+ x)²ⁿ are in AP, show that 2n² - 9n +7= 0.
17) If the coefficient of (r - 1)th, r th and (r +1)th terms in the expansion of (1+ x)ⁿ are in the ratio 1:3:5 then find n, r. 7, 3
18) The 3rd, 4th, and 5th terms in the expansion of (x + a)ⁿ are respectively 84, 280 and 560, find the value of x, a, n. 1, 7, 2
19) If the coefficient of three consecutive terms in the expansion of (1+ x)ⁿ be 76, 95 and 76, find n. 8
MISCELLANEOUS - A
1) Find the coefficient of x¹⁶ in the expansion of x¹⁰(x -2)¹⁰. 3360
2) Find the coefficient of x¹⁰ in the expansion (1- x²)(1- x)¹⁵. -3432
3) Find the coefficient of x in the expansion of (1- x²+ 2x⁴)(1- 1/x)⁶. -34
4) Find the term independent of x in the expansion of
a) (x + 1/x)²ⁿ. (2n)!/(n!)²
b) (1+ x)³(x - 1/x)⁶. 25
c) x¹⁰(x - 1/x)¹⁶. -550
d) (1+ 4x)ᵖ(1+ 1/4x)ᑫ. (p and q are positive integers). (p+ q)!/p! q!
e) (1+ x)ᵖ(1+ 1/x)ᑫ. (p and q are positive integers). (p+ q)!/p! q!
f) (1- x²)(x + 1/x)⁷. -70
5) If the coefficient of the (4r +5)-th and (2r+ 1)-th terms in the expansion (1+ x)¹⁰ are equal, find the value of r. 1
6) If the term independent of x in the expansion of (√x - √c/x²)¹⁰ is 405, find the value of c. 9
7) If the coefficient of four successive terms in the expansion of (1+ x)ⁿ be a₁, a₂, a₃ and a₄ respectively, show that,
a₁/(a₁+ a₂) + a₃/(a₃ + a₄) = 2a₂/(a₂ + a₃).
8) Find numerically the greatest term in each of the following expansions:
a) (2+ 3x)⁸ when x= 1/2. 6048
b) (2x - 3y)⁹ when x= 5/9, y= 3/5. 672000
c) (1+ 2x)⁹ when x= 1/3. 4th, 224/9
d) (2a/b + 3b/a)¹⁰ when a= 1/2, b= 1/3. 5th, 3360 x 3⁶
e) (a - 4b)⁸ when a= 1/2, b= 1/3. 7th, 28672/729
f) (1- x/5)¹³ when x= 1/3. 1st, 1
9) Find the power of x in that term of the expansion of (2+ 5x/2)¹² which has the greatest numerical coefficient. Also find the value of the coefficient. 7, 198x 5⁷
10) If m and n be positive integers, show that the coefficient of xᵐ and xⁿ in the expansion of (1+ x)ᵐ⁺ⁿ are equal.
11) The coefficient of x¹⁴ and x⁴³ in the expansion of (1+ x)⁵⁷ are equal. T/ F
12) Show that the coefficient of (p+1)-th term in the expansion of (1+ x)ⁿ⁺¹ is equal to the sum of the coefficient of the pth and (p+1)th terms in the expansion of (1+ x)ⁿ.
13) Show that the coefficient of xⁿ in the expansion of (1+ x)²ⁿ is twice the coefficient of xⁿ in the expansion of (1+ x)²ⁿ⁻¹.
14) Show that the coefficient of the middle term in the expansion of (1+ x)²ⁿ is equal to the sum of the coefficient of the two middle terms in the expansion of (1+ x)²ⁿ⁻¹.
15) Show that the coefficient of the middle term in the expansion of (1+ x)¹² is equal to the sum of the coefficient of the two middle terms in the expansion of (1+ x)¹¹.
16) If the coefficient of x and x² in the expansion of (1+ ax))ⁿ are 12 and 60 respectively, find the values of a and b. 2,6
17) The second and third terms in the expansion of (x +3))ⁿ are 240 and 720 respectively. Find x when n= 5. 2
18) If n be positive integer, show that
a) (1+ a)ⁿ + ⁿC₁(1+ a)ⁿ⁻¹(1- a)+ ⁿC₂(1+ a)ⁿ⁻² (1- a)²+ ....+ (1- a)ⁿ= 2ⁿ.
b) 2ⁿ - n/1! . 2ⁿ⁻¹ + n(n -1)/2! . 2ⁿ⁻² - .....+ (-1)ⁿ= 1.
19) If in the expansion of (x + a)ⁿ the sum of the odd terms is P and the sum of the even terms is Q, then show that
a) P²- Q²= (x²- a²)ⁿ.
b) 4PQ= (x + a)²ⁿ - (x - a)²ⁿ.
20) Find the coefficient of x⁴ in the expansion of (1+ x + x²+ x³)¹¹. 990
21) If the coefficient of the pth and (p+2)th terms in the expansion of (1+ x))ⁿ are in AP, then show that, n² -(4p +1)n + 4p½= 2.
22) If the sum of the coefficient of the first, second and third terms in the expansion of (x²+ 1/x))ⁿ is 46, find the term independent of x. 84
23) If the coefficient of three successive terms in the expansion of (1+ x))ⁿ are 66, 220 and 495 respectively, find the value of n. 12
24) If the coefficient of 2nd, 3rd, 4th and 5th terms in the expansion of (1+ x))ⁿ are a,b,c,d respectively, then show that, a/(a+ b) + c/(c + d) = 2b/(b + c).
25) If the 6th, 7th, 8th and 9th terms in the expansion of (x + p))ⁿ, when expanded in ascending powers of x, be a,b,c,d respectively, then show that (b²- ac)/(c²- bd) = 4a/3c.
26) Find numerically the greatest coefficient in the expansion of:
a) (3+ 2x)⁸. 56 x 3⁵. 2³
b) (3a - 5b)¹⁰. 210 x 3⁴5⁶
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