BOOSTER- A
2) How many terms of the series 1/2 + 1/3+ 1/6+..... must be taken so that the sum may be -3/2? 9
3) Find the sum of 1- 3+ 5- 7 + 9 -11....to n terms. n when n is odd; - n when n is even
4) How many even numbers are there between 15 and 115? Find the sum of all these numbers. 50,3250
5) Find the sum of all numbers between 200 and 300 which are multiples of 7. 3479
6) If (p +1)th term of an AP be a. Find the sum of first (2p +1) terms of the AP. (2p+1)a
7) If the 11th term of an AP be 25, find the sum of first 21 terms of the AP. 525
8) There are (2n +1) terms in an AP. Show that the ratio of the sum of odd terms and the sum of even terms is (n +1): n.
9) Find the 99th term of the series 2+7+14+23+34+..... 9998
10) How many terms are there in the series 1+3+6+10+15+21+...+5050? 100
11) The sum of four numbers in AP is 20 and the sum of their squares is 120; find the numbers. 2,4,6,8 or 8,6,4,2
12) The sum of six numbers in AP is 345 and the difference between the first and the sixth is 55; find the numbers. 30,41,52,63,74,85 or 85,74,63,52,41,30
13) The fourth term of an AP is thrice the first term and the seventh term exceeds twice the third term by 2. Find the sum of first ten terms of the AP. 240
14) If the sum of first n terms of an AP is 40, the common difference is 2 and the last term is 13, find the value of n. 4 or 10
15) The sum of first n terms of two AP 's are in the ratio (3n +1): (3n -1). Find the ratio of their tenth terms. 29:28
16) Show that the sum of n arithmetic means between two numbers is n times the arithmetic mean of those two numbers.
17) n arithmetic means are inserted between 1 and 41. If the sum of the means and the two numbers between 189, find the value of n. 7
18) If (b+ c)/a, (c + a)/b, (a+ b)/c, are in AP and a+ b + c ≠ 0, show that 1/a, 1/b, 1/c are in AP.
19) If b²+ bc+ c², c²+ ca+ a², a²+ ab+ b² are in AP, show that a,b,c are in AP.
20) If a,b,c are in AP, show that (1/a) (1/b + 1/c), (1/b)(1/c + 1/a), (1/c)(1/a + 1/b) are in AP.
21) If a,b,c,d are in AP, show that, ab+ cd+ ad= 3bc.
BOOSTER- B
1) A farmer undertakes to pay off a debt of Rs 2700 by monthly instas. He pays Rs 200 as the first installment and increases every subsequent instalment by Rs 25 over the immediate previous installment. In how many instalments his debt will be cleared up? 9
2) The sum of first n terms of an AP is denoted by Sₙ. If Sₙ = n²p and Sₘ = m²p, m≠ n, then show that, Sₚ= p³.
3) There are 100 terms in an AP. If the sum of terms occupying the even places is 1600 and the sum of the terms occupying the odd places is 1300. Find the two middle terms of the AP. 26,32
4) If the sum of first p terms of an AP be P and the sum of first q terms be Q, show that the first term of the progression will be {Pq(q -1) - Qp(p -1)}/{pq(q - p)}.
5) If the first and the last terms of an AP are a and l respectively and the sum of the terms is S, show that the common difference is (l²- a²)/(2S - (l + a)).
6) The sum of three numbers in GP is 7 and the sum of their squares is 21; find the sum of their cubes. 73
7) The sum of three positive numbers in GP is 65 and the product of the first and the third number is 225. Find the numbers. 5,15,45 or 45,15,5
8) Find the sum of the GP 2+4+8+.....to 20 terms. 2(2²⁰-1)
9) If (a+ bx)/(a - bx) = (b + cx)/(b - cx)= (c + dx)/(c - dx) (x≠ 0), show that a,b,c,d are in GP.
10) If a,b,c,d are in GP, show that, 1/(a²+ b²), 1/(b²+ c²), 1/(c²+ d²) are in GP.
11) The 5th term of a GP is 8 and the 13th term is 128. Find the sum upto 12th term. 126(√2+1)
12) The sum of three consecutive terms of an AP is 15; if 1 is substracted from the second term, the resulting numbers form a GP. Find the terms. 2,5,8 or 8,5,2
13) The second, third and sixth terms of an AP are three consecutive terms of a GP. Find the common ratio of the GP. 3
14) Find the sum of:
a) 1+ 11+111+1111+....to n terms. (10/81)(10ⁿ -1) - π/9
b) 0.8+0.88+0.888+....to n terms. (8n/8) - (7/81) (1- 1/10ⁿ)
c) 1+4+10+22+.....to n terms. 3.2ⁿ - 3 - 2n
d) 1+ (1+2)+ (1+2+2²)+ ....to n terms. 2ⁿ⁺¹ - 2 - n
e) 1+ (1+ x)+ (1+ x + x²)+ .....to n terms. n/+1- x) x(1- xⁿ)/(1- x)²
f) 1/2 + 3/2²+ 5/2³+ ......+ (2n -1)/2ⁿ. 3- (2n +3)/2ⁿ
15) If the first, second and last terms of a GP are a,b, and l respectively, show that the sum of the terms is (bl - a²)/+l - a).
16) If u₁, u₂, u₃,.......are in GP whose common ratio is k, then express the value of u₁u₂ + u₂u₃ + .....+ uₙuₙ₊₁ in terms of k and u₁. u₁²k
17) If the pth, qth and rth terms of an AP are in GP, then show that the common ratio of the GP is (q - r)/(p - q).
18) If the pth, qth, rth and sth terms of an AP are in GP, then show that p - q, q- r, r- s, are in GP.
19) Solve: 1+ 3+ 3²+........3ˣ= 1093. 6
20) If a,b,c,d are in AP and a, c,d are in GP, show that a²- d²= 3(b²- ad).
21) The product of three numbers in GP is 512. If 8 be added to the first and 6 to the second, the resulting numbers with the third number form an AP. Find the numbers. 4, 8,16 or 16,8,3
BOOSTER - C
1) If a,b,c are in AP and a, b - a, c - a are in GP, show that, a= b/3 = c/5.
2) If (a - x)/px= (a - y)/qy= (a - z)/rz and p,q,r are three consecutive terms of an AP, show that 2/y = 1/x + 1/z.
3) There are even number of terms in a GP. If the sum of the progression is thrice the sum of the odd terms, then find the common ratio of the GP. 2
4) Show that the product of n geometric means between a and b is (ab)ⁿ⁾².
5) The arithmetic mean of two numbers is A and their geometric mean is G; find the numbers. A+ √(A²- G²) and A - √(A²- G²)
6) Find the least value of n for which 1+3+3²+....3ⁿ⁻¹ > 900. 7
7) If tₙ be the nth term of a GP and t₁ = 2, find the minimum value of 2t₂ + 3t₃. Find also the value of the common ratio for which the expression will be minimum. -2/3,-1.3
8) A man saves Rs4 in the first month. Now from the second month he saves in every month twice the savings of the preceding month. Thus after 8 months, from the 9th month he saves in every month Rs4 less than the savings of the immediate previous month. What will be his total savings in 16 months? Rs4972
9) If {(1+ i)/(1- i)}³ - {(1- i)/(1+ i)}³= p + iq, find (p,q). (0,-2)
10) If x= 2 + 3i and y= 2- 3i, find the value of (x³- y³)/(x³+ y³). -9i/46
11) If x= 1+ 3i, find the value of x⁴- 5x³+ 18x²- 34x +2. -18
12) If x= -5+ 2√-4, find the value of x⁴+ 9x³+ 35x²- x +4. -160
13) If x= 2+ i, find the value of x²- 5x²+ 9x -5. 0
14) Find the modulus of:
a) 2/(4+ 3i) - 4/(3- 4i). √5/5
b) {(2- 3i)(√5+ 2i)}/3i(3- 2i). 1
15) If z₁ = (√3 -1) + (√3+ 1) and z₂ = - √3+ i, find the values of amp z₁, amp z₂ and amp (z₁z₂). 5π/12, 5π/6, -3π/4
16) If z= sin(6π/5) + i (1+ cos(6π/5)), find |z| and amp z. 2 cos(2π/5), 9π/10
17) If z= x + iy and 2|z -1|= |z -2|, show that, 3(x²+ y²)= 4x.
18) If arg{(z -2)/(z - 4i)}= π/4, show that the locus of z in the complex plane is a circle.
19) Find the square root of:
a) 9i. ±(3/√2) (1+ i)
b) 1+ 2√-6. ±(√3+ i √2)
c) y + √(y²- x²) (x¹> y²). ±(1/√2) {√(x +y) + i √(x - y)}
d) (7- 24i)/(3+ 4i). ±(1- 2i)
e) a²+ 1/a²+ 2(a + 1/a)i + 1. ±(a + 1/a + i)
BOOSTER - D
1) Show that:
a) √(1+ i) + √(1- i)= √{2(√2+1)}
b) √{2+ i.3√5} + √{2- i.3√5} = 3√2
c) (19 + 5√-24)¹⁾² - (19 - 5√-24)¹⁾²= 2√-6.
d) (4 + 3√-29)¹⁾² + (4 - 3√-24)¹⁾²= 6.
2) If ω be an imaginary cube root of 1, show that,
a) (1- ω²)+1- ω⁴)(1- ω⁸)(1- ω¹⁰)= 8.
b) 1/(1+ 2ω) - 1/(1+ ω) + 1/(2+ ω)= 0.
c) (x + yω + zω²)⁴ + (xω + yω² + z)⁴ + (xω² + y + zω)⁴= 0.
3) If a= cosθ + i sinθ, show that,
1+ a + a²= (1+ 2 cosθ)(cosθ + i sinθ).
4) If √(a + bi + ci²+ di³)= p + iq, show that
(p²+ q²)²= (a - c)²+ (b - d)².
5) Let z₁ = a + ib and z₂ = p + iq be two complex numbers such that, Iₘ(z₁. Conjugate z₂)= 1. If ω₁ = a + ip and ω₂ = b + iq, show that, Iₘ (ω₁. Conjugate of ω₂)= 1.
6) If x¹⁾³ = ωa¹⁾³ + ω²b¹⁾³ (ω is an imaginary cube root of unity),
show that, (x - a - b)³ = 27abx.
7) If the roots of the equation x² - ax+ b=0 are real and the difference of the roots is less than 1, show that, (a²-1)/4 < b < a²/4.
8) If the roots of the equation px² - 2qx+ p =0 are real and unequal, show that the roots of the equation qx² - 2px+ q =0 are imaginary. (p,q are real).
9) If the roots of the equation ax² + 2bx+ c =0 are imaginary, show that the roots of the equation ax² +2(a + b)x+ a+ 2b + c =0 are also imaginary. (a,b,c are real)
10) Show that the roots of the equation (1- ac)x² - (a²+ c²)x- (1+ ac) =0 are real and distinct.
11) If the roots of x² - 8x+ a² - 6a=0 are real, show that, -2< a < î.
12) If α, β are the roots of 5x²+ 7x +3=0, find the value of (α³+ β³)/(α⁻¹ + β⁻¹). 12/125
13) If α and β are the roots of the equation 6x²- 6x +1=0, show that,
(1/2) (a + bα + cα²+ dα³) + (1/2) (a+ bβ + cβ²+ dβ⅗)= a/1 + b/2 + c/3 + d/4.
14) If one root of the equation ax²+ bx + c=0 be n times the other, show that, nb²= ac (1+ n)².
15) If one root of the equation x²+ (5a +2)x + (5a +2) =0 be five times the other, find the numerical value of a. 28/25
16) If one root of the equation px²+ qx + p =0 be the square of the others, show that, q³+ 2p³= 3p²q.
17) If sum of the roots of the equation x² - px + q =0 be m times their difference, show that, (m²-1)p²= 4m²q.
18) If the ratio of the roots of the equation ax²+ bx + c=0 be r, show that, acr²+ 2(ac - b²)r + ac= 0.
19) If the ratio of the roots of the equation a₁x²+ b₁x + c₁=0 be equal to the ratio of the roots of the equation a₂x²+ b₂x + c₂ =0, show that, b₁²/b₂²= a₁c₁/a₂c₂.
20) If the difference between the corresponding roots of the equation x²+ ax + b =0 and x²+ bx + a =0 (a≠ b) is the same, find a+ b. -4
21) If the sum of the roots of the equation ax²+ bx + c=0 be equal to the sum of their squares, show that b/c, a/b, a/c are in AP.
BOOSTER - E
1) If x₁, x₂ are the roots of x² - 3x + A =0 and x₃, x₄ are the roots of x² - 12x + B =0 and x₁, x₂, x₃, x₄ are in increasing GP, then find the values of A and B. 2,32
2) The constant term in the equation x² + px + q =0 is misprinted 40 for 24 and the roots are therefore obtained as 4 and 10. Find the roots of the original equation. 2,12
3) If the roots of the equation x² + 6x + 13 =0 are p and q, find the equation whose roots are pq and p²+ q². x² - 23x + 130 =0
4) If the roots of the equation x² - px + q =0 are α and β, find the equation whose roots are mα + nβ and nα + mβ. x² - (m + n)px + mnp½+ (m - n)²q =0
5) If α and β are the roots of 2x² - 3x - 5 =0, find the equation whose roots are 2α + 1/β and 2β + 1/α. 5x² - 12x - 32 =0
6) Form the equation whose roots α and β satisfy the relations α²+ β²= 240 and αβ = 80. x² ± 20x +80 =0,
7) If α and β are the roots of the equation x² + px + q =0, show that the roots of the equation x² + (α+ β - αβ)x - (α+ β)αβ =0 are p and q.
8) If the equation x² -11x + a =0 and x² - 14x + 2a =0 have a common root, find the values of a. 0,24
9) For what values of m the equation 3x² + 4mx + 2 =0 and 2x² + 3x -2 =0 will have a common root? 7/4,-11/8
10) Show that the equations (b - c)a⅖+ (c - a)x + (a - b)= 0 and (c - a)x² + (a - b)x + (b - c) =0 have a common root.
11) Find the condition so that the equation mx² + x + 1 =0 and x² + x + m =0 may have a common root. (mn -1)²= (m -1)(n -1)
12) If one root of the equation ax² + bx + c =0 is the reciprocal of one root of the equation ox² + qx + r =0, show that, (bp - cq)(aq - br)= (cr - ap)².
13) Form a quadratic equation with rational co-efficients whose one root is 2pq/{p+ q - √(p²+ q²)}, (p,q are rational and p²+ q² is not a perfect square).
14) If the difference of the roots of a quadratic equation is a and the ratio of the roots be b (>1), form the equation. (b -1)²x² - a(b²-1)x + ab =0
15) If one root of the equation 4x² + 2x + 1 =0 be cosα, show that its other root is cos3α.
16) If α, β are the roots of x² - ax + b =0 and γ , δ are the roots of x² - px + q =0, find the equation whose roots are αγ + βδ and αδ+ βγ. x¹ - apx + q(a²- 2b) + b(p²- pq )=0
17) If p and q are the roots of ax² + 2bx + c =0, find the equation whose roots are pω + qω½ and pω²+ qω (ω is imaginary cube roots of 1)
18) If the roots of the equation x² + px + q =0 be α ± √β, show that the roots of the equation (p¹- 4q)(p²x² + 4px) - 16q= 0 are 1/α ± 1/√β.
19) If a and x are real, show that the greatest value of 2(a - x) (x + √(x²+ a²)) is 2a².
20) How many odd numbers of six digits can be formed with the digits 0,1,4,5,6,7, none of the digits being repeated in any number? 288
21) From 10 different things 6 things are taken at a time so that a particular thing is always included. Find the number of such permutations. 90720
BOOSTER - F
1) How many different arrangements of the letters of the word FAILURE can be made so that the vowels are always together? 576
2) In how many ways can the letters of the word STATION be arranged so that the vowels are always together? 360
3) How many words can be formed by the letters of the word PEOPLE taken all together so that the two P's are not together? 120
4) In how many ways 6 books be arranged on a shelf of an almirah so that 2 particular book will not be together? 480
5) Find the values 10 electric lamps of which 4 are defective. Find the number of samples of six lamps taken at random from the box which will contain two defective lamps. 90
6) A committee of 5 is to be formed from 6 boys and 4 girls. How many different committees can be formed so that each committee contains atleast two girls ? 186
7) An examiner has to answer 6 questions out of 12 questions. The questions are divided into two groups, each group containing 6 questions. The examiner is not permitted to answer more than 4 questions from any group. In how many ways can he answer in all 6 questions? 850
8) The Indian cricket eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that team contains atleast three bowlers ? 1260
9) In a plane 5 points out of 12 points are collinear, no three of the remaining points are collinear. Find the number of straight lines formed by joining these points. 57
10) If n be natural number then show by the mathematical induction that
a) n³+ 11n is divisible by 6.
b) 3⁴ⁿ⁺¹ + 16n -3 is divisible by 256.
c) 4ⁿ + 15n -1 is divisible by 9.
d) 3²ⁿ +1 is even but not divisible by 4.
11) Find the co-efficient of x⁴ in the expansion of (x/2 - 3/x²)¹⁰. 405/256
12) Find the co-efficient of 1/x³ in the expansion of (2x²+ 1/x)¹². 1760
13) Find the co-efficient of x⁷ in the expansion of (1- x)²(x - 1/x)¹¹. -110
14) If n be positive integer, find the co-efficient of 1/x in the expansion of (1+ x)ⁿ(1+ 1/x)ⁿ. (2n)!/{(n -1)!(n +1)!}
15) Find the term independent of x in the expansion of (2x - 1/3x²)⁹. -1792/9
16) Find the term independent of x in the expansion of {√(x/3) - √3/2x²}¹⁰. 5/12
17) Show that the term independent of x in (x - 1/x²)³ⁿ is (-1)ⁿ. (3n)!/{n! (2n)!}
18) If the co-efficient of x⁷ and x⁸ in the expansion of (3+ x/2)ⁿ are equal, find n. 55
19) If the term independent of x in the expansion of (√x - √c/x²)¹⁰ is 405, find the value of c. 9
20) If the term free from X in the expansion of (px²/3 - 3/2x)⁹ be 2268, find the value of p. 4
21) The fifth term in the expansion of (q/2x - px)⁹ is free from X and the term is 1120. Show that pq= 4.
BOOSTER - G
1) Find the fifth term from the end in the expansion of (3x - 1/x²)¹⁰. 17010/x⁸
2) If the co-efficient of pth term in the expansion of (1+ x)ⁿ is p and that of (p +1)th term is q, find n. p+ q -1
3) Find the middle term (or middle terms) in the expansion of (x³- 3x½ + 3x -1)³. 126x⁵, - 126x⁴
4) If n be positive integer, find the middle term in the expansion of (2x - 1/3x)²ⁿ. (-1)ⁿ (2/3)ⁿ (2n)!/(n!)²
5) Show that if n≥ 2 be an integer, than 2³ⁿ - 3n -1 is always divisible by 9.
6) Using binomial theorem find the value of (0.99)¹⁵ correct to four decimal places. 9.8601
7) Show that xⁿ/n! + xⁿ⁻¹/(n -1)!1! + (xⁿ⁻².a²)/(n -2)!2! + ....aⁿ/n! = (x + a)ⁿ/n!.
8) Show that, ⁿC₁ x(1- x)ⁿ⁻¹ + 2. ⁿC₂ x²(1- x)ⁿ⁻² + 3. ⁿC₃ x³(1- x)ⁿ⁻³ +.....+ r. ⁿCᵣ xʳ(1- x)ⁿ⁻ʳ + .....+ n. ⁿCₙ xⁿ= nx, where n is any positive integer.
9) Two consecutive terms in the expansion of (2+ 1/2)⁹ are equal. Find the values of these two terms. 1152
10) Using binomial theorem show that 9¹¹+ 11⁹ is divisible by 10.
11) If the ratio of the three consecutive terms in the expansion of (1+ x)ⁿ be 1:2:3, find the value of n. 14
12) The co-efficients of three consecutive terms in the expansion of (1+ x)ⁿ are 165,330 and 462, find n. 11
13) If the co-efficient of the second, third and fourth terms in the expansion of (1+ x)²ⁿ are in AP, show that, 2n²- 9n +7=0.
14) If the 3rd, 4th, 5th and 6th terms in the expansion of (x + p)ⁿ, when expanded in ascending powers of x, be a,b,c and d respectively, show that, (b²- ac)/(c²- bd) = 5c/3c.
15) if n be positive integer, show that
x - ⁿC₁(x + y) + ⁿC₂(x + 2y) - ⁿC₃(x + 3y) + .....(-1)ⁿ (x + ny)= 0.
16) Show that
1- ⁿC₁ (1+ x)/(1+ nx) + ⁿC₂ (1+ 2x)/(1+ nx)² - ⁿC₃ (1+ 3x)/(1+ nx)³ +....= 0.
17) If (1+ x)ⁿ= C₀ + C₁x + C₂x² + ....+ Cₙ xⁿ, show that,
a) C₀/1 + C₂/3 + C₄/5 + .....= 2ⁿ/(n+1).
b) C₀C₂ + C₁C₃ + C₂C₄ + .....+ Cₙ₋₂Cₙ = (2n)!/{(n -2)!(n+2)!}.
18) If the sum of all the co-efficients in the expansion of (x² + 1/x)ⁿ be 1024, find the co-efficient of x⁵ in the expansion. 252
19) If -1/3 < x < 1/3, find the (r+1)th term in the expansion of (1- 3x)⁻¹⁾³. {1.4.7...(3r-2)}xʳ/r!
20) If |x|< 1, find the co-efficient of xʳ in the expansion of (1+ 2x + 3x² + 4x³ +..... To ∞)². {(r+1)(r+2)(r+3)}/6
21) Using binomial theorem find the value of √2 correct to three decimal places. 1.414
BOOSTER- H
1) Using binomial theorem find the values of
a) √24. 4.89898
b) ³√998. 9.99333
correct to five decimal places.
2) If n be positive integer and |x| < 1, show that the co-efficient of nth term in the expansion of (1- x)⁻ⁿ is twice the co-efficient of (n -1)th term.
3) Find the co-efficient of x⁵ in the expansion of (1+ x)/(1- x). 2
4) Find the co-efficient of x⁴ in the expansion of {(1- x)/(1+ x)}². 16
5) Find the co-efficient of xⁿ in the expansion of {(1+ x)/(1- x)}². 4n
6) Find the co-efficient of x⁴ in the expansion of (1+ x + x² + x³ + .....∞)¹⁾². 35/128
7) Find the sum of the following series:
a) 1+ 1/4 + 1.4/4.8 + 1.4.7/4.8.12 + .....∞. ³√4
b) 1- 3/4 + 3.5/4.8 - 3.5.7/4.8.12. + to ∞. 2/3 √(2/3)
c) 1- 1/8 + 1.5/8.16 - 1.5.9/8.16.24 + .....∞. ⁴√(2/3)
8) Show that: 8 = 1+ 9/8 + 9.15/8.16 + 9.15.21/8.16.24. + .....to ∞.
9) Show that: √2= (7/5) [1+ 1/10² + 1.3/1.2 . 1/10⁴. + 1.3.5/1.2.3 . 1/10⁶ + .....∞].
10) Express in terms of e:
(1+ 1/2! + 1/4! + 1/6! +....) ÷ (1+ 1/3! + 1/5! + 1/7!+......). (e²+1)/(e²-1)
11) Find the co-efficient of xⁿ in the expansion of (a + bx)/eˣ. (-1)ⁿ(a - bn)/n!
12) Find the co-efficient of xⁿ in the expansion of (1+ x + x²)/e²ˣ. (-1)ⁿ2ⁿ⁻²/n! (4 - 3n + n²)
13) Show that 1+ 3/2! + 5/4! + 7/6! + .....∞= e.
14) Show that: 1+ 2/3! + 3/5! + 4/7! +.....∞ = e/2.
15) Find the co-efficient of xⁿ in the expansion of [n -2)x - x²]e⁻ˣ. (-1)ⁿ⁻¹
16) Find the sum of:
a) 1/2! + 2²/4! + 2⁴/6! + 2⁶/8! +......∞. (1/8) (e - 1/e)²
b) 5/1! + 7/3! + 9/5! + 11/7!+.....∞. (1/2) (5e - 3/e)
b) 5/1! + 7/3! + 9/5! + 11/7!+.....∞. (1/2) (5e - 3/e)
c) 2.3/3! + 3.5/5! + 4.7/7! + 5.9/9! +......∞. (1/4) (3e + 1/e)-1
d) 1²/2! + 2²/3! + 3²/4! +.....to ∞. e -1
e) 1²/3! + 2²/4! + 3²/5! +.......∞. 2e -5
f) 1⁴/1! + 2⁴/2! + 3⁴/3! + 4⁴/4! +......∞. 15e
g) 2/1! + (2+4)/2! + (2+4+6)/3! + .....∞. 3e
h) 1+ (1+3)x/2! + (1+3+5)x⅖/3! +.....∞. (x +1)eˣ.
i) 2/1! + 7/2! + 15/3! + 26/4! + 40/5! +.....∞. 7e/2
BOOSTER - I
1) Show that: 1+ (1+3)/2! + (1+3+3²)/3! +.....to ∞ = (1/2) (e³ - e)
2) Expand: (x + x³/3! + x⁵/5! + .....)² in ascending powers of x. (1/2) (2²x²/2! + 2⁴x⁴/4! + 2⁶x⁶/6! +.....∞
3) If ax² < 1 and a/x² < 1, show that
a(x² + 1/x²) - (a²/2)(x⁴ + 1/x⁴) + (a³/3)(x⁶ + 1/x⁶) - ....∞ = logₑ(1+ a² + ax² + a/x²).
4) Show that: 2xy/(x²+ y²) + (1/3) 2xy/(x² + y²)³ + (1/5) 2xy/(x² + y²)⁵ +.....∞ = logₑ{(x + y)/(x - y)}.
5) Show that, logₑ{(1+ x)¹⁺ˣ (1- x)¹⁻ˣ}= 2{x²/1.2 + x⁴/3.4 + x⁶/5.6 + ....∞.
6) Write down the expansion of logₑ(1- x) as an infinite series and state the condition of validity. Using the series find the expansion of logₑ2.
7) If y= 2x² -1, show that,
1/x² + 1/2x⁴ + 1/3x⁶ +.....∞ = 2/y + 2/3y³ + 2/5y⁵+......∞
8) Write down the expansion of logₑ(1+x)/(1- x). From it, choosing suitable value of x, find the expansion of logₑ3 and show that, logₑ3= 1.0958(approx)
9) Show that: logₑ(9/5)= 2{2/7 + (1/3)(2/7)³+ (1/5)(2/7)⁵+.....∞}
10) Show that if n> 1, logₑn - logₑ(n -1) = 1/n + 1/2n² + 1/3n³ +....∞
11) Show that:
a) 1/2.1 + 1/2².2! + 1.2/2³.3! + 1.2.3/2⁴.4! +.....∞= logₑ2.
b) 4/1.3 - 6/2.4 + 12/5.7 - 14/6.8 +.....∞= logₑ2.
c) 1/3.4 + 1/5.6 + 1/7.8 + .....∞= logₑ2 - 1/2
11) Show that: 1/3 - 1/2 . 1/3² + 1/3. 1/3³ - 1/4. 1/3⁴ +.....∞= logₑ(4/3).
12) Show that: 1/5 + 1/3 (1/5)³ + 1/5 (1/5)⁵+.....(1/2)logₑ(3/2).
13) Show that: 1+ (1/2+ 1/3). 1/3² + (1/4+ 1/5). 1/3⁴ + (1/6+ 1/7). 1/3⁶+....∞= logₑ3.
14) Find the sum of 1/2(1/2)² + 2/3(1/2)³ + 3/4(1/2)⁴+.....∞.
15) Expand:logₑ(1+ x + x²+ x³) in ascending powers of x and hence find the co-efficient of x²ⁿ and x²ⁿ⁺¹.
BOOSTER - J
1) If a,b,c are in AP, show that, (a- c)²= 4(b²- ac).
2) There are (p -1) arithmetic means between a and b. The common difference of the AP is
a) (a+ b)/(p -1) b) (b - a)/p c) (b - a)/(p -1) d) (b - a)/(p -2). b
3) The fourth term of a GP is the square of the first term. If the seventh term of the progression is 216, find the first term. 6
4) The 7th term of an AP is 40. Find the sum of the first 13 terms? 520
5)
αβ ω γ φ δ ψ ₁₂₃₄₅ ₓ ₐ ᵢ ₙ ₋ ₊
∞ ³
ⁿⁿ⁺¹ⁿ²ⁿ
