DETERMINANTS
1) a+ x a x
If a- x a x= 0
a- x a -x
Then x is
a) 0 b) a c) 3 d) 2a
2) 0 p- q p- r
q - p 0 q - r
r -p r- q 0 is equal to
a) p+ q+ r b) 0 c) p - q - r d) -p + q + r
3) If a≠ b ≠ c such that
a²-1 b³-1 c³ -1
a b c = 0 then
a² b² c²
a) ab+ bc+ ca= 0 b) a+ b+ c=0 c) abc=1 d) a+ b+ c=1
4) 1+ x 1 1
1 1+ x 1 is equal to
1 1 1+x
a) x²(x+3) b) 3x³ c) 0 d) x³
5) 6i - 3i 1
If 4 3i -1 = x + iy then find x,upy
20 3 i
a) 3,1 b) 1,3 c) 0,3 d) 0,0
6) xp+ y x y
yp+ z y z= 0
0 xp+ y yp+z
foe all p ∈ E if
a) x,y,z are in AP
b) x,y,z are in GP
c) x,y,z are in HP
d) xy, yz, zx are in AP
7) a a+ d a+ d
∆= a² (a+d)² (a+2d)²=0. Then
2a+3d 2(a+d) 2a+d
a) d=0 b) a+ d=0 c) d=0 or a+ d= 0 d) none
8) The value of the determinant
bc ca ab
p q r
1 1 1
Where a,b,c are pth, qth and rth terms of a HP is
a) ap+ bq+ cr
b) (a+ b+ c)(p+ q+ r)
c) 0 d) none
9) The sum of two non integral roots of
x 2 5
3 x 3 =0
5 4 x is
a) 5 b) -5 c) -18 d) none
10) If x,y,z are integers in AP lying between 1 and 9 and x51, y41 and z31 are three digit numbers then the value of
5 4 3
x51 y41 z31
x y z is
a) x+ y+ z b) x - y + z c) 0 d) none
11) 1 1 1 1 bc a
If ∆₁=a b c & ∆₂= 1 ca b
a² b² c² 1 ab c then
a) ∆₁+ ∆₂=0 b) ∆₁+ 2∆₂=0 c) ∆₁= ∆₂ d) none
12) Two non-zero distinct numbers a,b are used as elements to make determinants of the third order. The number of determiners whose values is zero for all a,b is
a) 24 b) 32 c) a+ b d) none
13) The value of
a₁x+ b₁y a₂x+ b₂y a₃x + b₃y
b₁x+ a₁y b₂x+ a₂y b₃x+ a₃y
b₁x+ a₁ b₂x+ a₂ b₃x+ a₃
is equal to
a) x² + y² b) 0 c) a₁a₂a₃x²+ b₁b₂b₃y² d) none
14) x₁ y₁ 1 1 1 1
If x₂ y₂ 1 = b₁ b₂ b₃
x₃ y₃ 1 a₁ a₂ a₃
Then the two triangles whose vertices are (x₁, y₁), (x₂, y₂),(x₃, y₃ ) and (a₁,b₁),(a₂, b₂), (,a₃, ,b₃) are
a) congruent b) similar c) equal in area d) none
15) If α,β are nonreal numbers satisfying x³-1=0 then the value of
λ+1 α β
α λ+β 1
β 1 λ+ α
is equal to
a) 0 b) λ³ c) λ³+1 d) none
16) The value of
¹⁰C₄ ¹⁰C₅ ¹¹Cₘ
¹¹C₆ ¹¹C₇ ¹²Cₘ₊₂
¹²C₈ ¹²C₉ ¹³Cₘ₊₄
is equal to zero when m is
a) 6 b) 4 c) 5 d) none
17) If x> 0 and ≠ 1, y> 0 and ≠ 1, z> 0 and ≠ 1 and the value of
1 logₓy logₓk
logᵧx 1 logᵧk
logₖx logₖy 1
is
a) 0 b) 1 c) -1 d) none
18) The value of
1 1 1
(2ˣ+2⁻ˣ) (3ˣ+3⁻ˣ)² (5ˣ+ 5⁻ˣ)²
(2ˣ -2⁻ˣ)² (3ˣ - 3⁻ˣ)² (5ˣ - 5⁻ˣ)² is
a) 0 b) 30ˣ c) 30⁻ˣ d) none
19) The value of the determinant
⁵C₀ ⁵C₃ 14
⁵C₁ ⁵C₄ 1
⁵C₂ ⁵C₅ 1 is
a) 0 b) -(6!) c) 80 d) none
20) cosC tanA 0
sinB 0 -tanA
0 sinB cosC
has the value
a) 0 b) 1 c) sinA sinB sinC d) none
21) The value of
x x² - yz 1
y y² - zx 1
z z²- xy 1
a) 1 b) -1 c) 0 d) -xyz
22) If √-1= i and ω is a nonreal cube root of unity then the value of
1 ω² 1+ i+ ω²
-i -1 -1- i + ω
1- i ω²-1 -1
is equal to
a) 1 b) I c) ω d) 0
23) 1 x x+1
If f(x)= 2x x(x -1) x(x+1)
3x(x-1) x(x-1)(x-2) x(x²-1)
then f(100) is equal to
a) 0 b) 1 c) 100 d) -100
24) The value of
iᵐ iᵐ⁺¹ iᵐ⁺²
iᵐ⁺⁵ iᵐ⁺⁴ iᵐ⁺³
iᵐ⁺⁶ iᵐ⁺⁷ iᵐ⁺⁸ where i=√-1, is
a) 1 if m is a multiple of 4
b) 0 for all real m
c) - I if m is a multiple of 3 d) none
25) 7 x 2 x 2 7
If ∆₁= -5 x+1 3 , ∆₂= x+1 3 -5
4 x 7 x 7 4
Then ∆₁ - ∆₂ = 0 for
a) x=2 b) all real x c) x=0 d) none
26) 10 4 3 4 x+5 3
If ∆₁=17 7 4, ∆₂=7 x+12 4
4 -5 7 -5 x-1 7
such that ∆₁+ ∆₂= 0 then
a) x=5 b) x has no real value c) x=0 d) none
27) λ²+3λ λ-1 λ+3
Let λ+1 -2λ λ-4
λ-3 λ+4 3λ
= pλ⁴+ qλ³+ rλ²+ sλ + t be an identity in λ, where p,q,r,s,t are independent of λ. Then the value of t is
a) 4 b) 0 c) 1 d) none
28) 1+ x x x²
Let x 1+x x²
x² x 1+x
= ax⁵+ bx⁴ + cx³ + dx² + λx + μ be an identity in x, where a,b,c,d, λ, μ are independent of x. Then the value of λ is
a) 3 b) 2 c) 4 d) none
29) Using the factor theorem it is found that b+ c, c+ a and a+ b are three factors of the determinant
-2a a+ b a+ c
b+ a -2b b+ c
c+ a c+ b -2
The factor in the value of the determinant is
a) 4 b) 2 c) a+ b+ c d) none
30) If the determinant
Cos2x sin²x cos4x
Sin²x cos2x cos²x
Cos4x cos²x cos2x is independent in powes of sinx then the constant term in the expansion is
a) 1 b) 2 c) -1 d) none
31) 1 cosx 1- cosx
If ∆(x)= 1+ sinx cosx 1+sinx - cosx
Then ∫ ∆(x) dx at (π/2,0) is equal to
a) 1/4 b) 1/2 c) 0 d) -1/2
32) If I= √-1 and ⁴√1= α, β, γ, δ then
α β γ δ
β γ δ α
γ δ α β
δ α β γ
is equal to
a) I b) - I c) 1 d) 0
33) The roots of
x a b 1
λ x b 1 = 0
λ μ x 1
λ μ v 1
are independent of
a) λ, μ, v b) a,b c) λ, μ, v, a, b d) none
34) The value of
1 0 0 0 0
2 2 0 0 0
4 4 3 0 0
5 5 5 4 0
6 6 6 6 5 is
a) 6! b) 5! c) 1.2².3.4³.5⁴.6⁴ d) none
35) b²+ c² ab ac
If. bc c²+a² bc
ca cb a²+ b²
= square of a determinant ∆ of the third order then∆ is equal to
a) 0 c b b) a b c
c 0 a b c a
b a 0 c a b
c) 0 -c b
c 0 -a
-b -a 0 d) none
36) The system of equations ax+ 4y + z=0, bx+ 3y+ z=0, CX + 2y+ z=0 has non-trivial solutions if a, b,c c are in
a) AP b) GP c) HP d) none
37) If the equation a(y+ z)= x, b(z + x)= y and c(x + y)= z, where a≠ 1, b≠ -1, c≠ -1, admits of non-trivial solutions then
(1+ a)⁻¹ + (1+ b)⁻¹ + (1- c)⁻¹ is
a) 2 b) 1 c) 1/2 d) none
38) The system of equations 2x- y + z=0; x - 2y + z =0; λx - y + 2z =0 has infinite number of non-trivial solutions for
a) λ= 1 b) λ= 5 c) λ= -5 d) non real value of λ
39) The equation x+ y+ z = 6; x+ 2y + 3z = 10, x+ 2y + mz = n give infinite number of values of the triplet (x,y,z) if
a) m= 3, n ∈R b) m= 3, n ≠ 10 c) m= 3, n = 10 d) none
40) The system of equations 2x+ 3y= 8; 7x - 5y +3=0; 4x - 6y + λ =0 is solvable if λ is
a) 6 b) 8 c) -8 d) -6
41) If the system of equations ax+ by+ c=0; bx+ cy+ a=0; cx+ ay + b=0 has a solution then the system of equations
(b + c)x + (c + a)y + (a+ b)z =0
(c + a)x + (a + b)y + (b+ c)z =0
(a+ b)x + (b+ c)y + (c+ a)z =0 has
a) only one solution
b) no solution
c) infinite number of solutions d) none
42) Let {∆₁, ∆₂, ∆₃, .....∆ₖ} be the set of third order determinants that can be made with the distinct non-zero real numbers a₁, a₂, a₃, ....a₉. then
a) k=9! b) ᵏᵢ₌₀∑ ∆ᵢ = 0
c) atleast one ∆ᵢ= 0 d) none
43) x² (y+ z)² yz
y² (z +x)² zx
z² (x +y)² xy is divisible by
a) x² + y²+ z² b) x - y c) x - y - z d) x+ y+ z
44) The equation
1 x x²
x² 1 1 = 0 has
x x² 1
a) exactly two distinct roots
b) one pair of equal real roots
c) modulus of each root 1
d) three pairs of equal roots
45) n n+1 n+2
If f(n)= ⁿPₙ ⁿ⁺¹Pₙ₊₁ ⁿ⁺²Pₙ₊₂
ⁿCₙ ⁿ⁺¹Cₙ₊₁ ⁿ⁺²Cₙ₊₂
Where the symbols have their usual meanings. The f(n) is divisible by
a) n² + n+ 1 b) (n +1)! c) n! d) none
46) Let x≠ 1 and let a,b,c be non-zero real numbers. Then the determinants
a(1+ x) b c
a b(1+x) c
a b c(1+ x) is divisible by
a) abc b) (1+ x)² c) (1+ x)³ d) x(1+ x)²
47) The arbitrary constant on which the value of the determinant
1 α α²
cos(p - d)a cos pa cos(p - d)a
sin(p - d)a sin pa sin(p -d)a
does not depend is
a) α b) p c) d d) a
48) x+ a x+ b x+ a+ c
Let ∆(x)= x+b x+ c x- 1
x+ c x+ d x- b+d and
²₀∫ ∆(x) dx = -16, where a,b,c,d are in AP, then the common difference of the AP is
a) 1 b) 2 c) -2 d) none
49) If A+ B + C=π, eⁱᶿ = cosθ + i sinθ and
e²ⁱᴬ e⁻ⁱᶜ e⁻ⁱᴮ
z= e⁻ⁱᶜ e²ⁱᴮ e⁻ⁱᴬ
e⁻ⁱᴮ e⁻ⁱᴬ e⁻²ⁱᶜ then
a) Re(z)= 4 b) I'm(z)=0 c) Re(z)= -4 d) Im(z)= -1
50) a+ x a- x a- x
If a-x a+ x a- x =0
a- x a- x a+ x
Then x is
a) 0 b) a c) 3a d) 2a
51) A value of c for which the system of equations
x+ y=1;
(c + 2)x + (c +4)y = 6
(c+2)²x + (c +4)²y = 36 is solvable (consistent) is
a) 1 b) 2 c) 4 d) none
52) Eliminating a,b,c from x= a/(b - c), y= b/(c - a), z= c/(a - b) we get
a) 1 -x x b) 1 - x x
1 -y y= 0 1 1 -y =0
1 -z z 1 z 1
c) 1 -x x
y 1 -y= 0
-z z 1 d) none
53) The system of equations : 6x + 5y + λz= 0; 3x - y + 4z= 0; x + 2y - 3z= 0 has
a) only a trivial solution for λ ∈ R
b) exactly one non-trivial solution for some real λ
c) infinite number of non-trivial solutions for one value of λ
d) only one solution for λ ≠ -5
1a 2b 3c 4a 5d 6b 7c 8c 9b 10c 11a 12b 13b 14c 15b 16c 17a 18a 19b 20a 21c 22d 23a 24b 25b 26a 27b 28a 29a 30c 31d 32d 33b 34b 35a 36a 37a 38c 39c 40b 41c 42ab 43abd 44bcd 45ac 46abd 47a 48bc 49bc 50ac 51bc 52bc 53cd
COMPLEX NUMBERS
EXERCISE - A
1) Under what condition is the sum of two complex numbers x₁ + iy₁ and x₂ + iy₂
a) real number . If y₁ + y₂ = 0
b) a purely imaginary number? If x₁ + x₂ = 0
2) A student writes the formula √(ab)= √a √b. Then substitutes a= -1 and b = -1 and finds 1= -1. Where he is wrong.
3) Is the following computation correct? If not, give the correct computation:
" √-2. √-3 = √{(-2)(-3)}= √6 "
4) If x + iy = √{(a + ib)/(c + id)}, show that (x² + y²)²= (a²+ b²)/(c²+ d²).
5) If x = -5 + 2 √-4, find the value of x⁴ + 9x³ + 35x² - x +4. -160
6) If x= (3+ 5 √-1)/2, find the value of 2x³ + 2x² - 7x + 72 and show that it will be unaltered if x= (3- 5 √-1)/2. 4
7) If x= (-5+ i√3)/2, show that (x² + 5x)² + x(x +5)= 42.
8) If z= x + iy and z¹⁾³ = a - ib Then show that x/a - y/b = 4(a²- b²).
9) If the points on the Argand plane given by aeᶥᵅ , beᶥᵝ , ceᶥᵞ are collinear, then show that ∑ bc sin(β - γ) = 0.
10) Show that the equation
A²/(x - a) + B²/(x - b) + C²/(x - c) + .....+ H²/(x - h)= k has no imaginary roots.
11) Show that x⁴ + 4= (x + 1 + i)(x + 1 - i)(x -1+ i)(x -1 - i).
12) Simplify the following:
a) (i)⁴⁵⁷. i
b) (- √-1)⁴ⁿ⁺³ (n, a+ive integer). i
c) (1+ i)/(1- i) - (1- i)/(1+ i). 2i
d) (1+2i)/{1- (1- i)²)}. 1
12) Simplify the following:
a) {(1+ i)/(1- i)}⁴ⁿ⁺¹ (n, a +ive integer). i
b) (1+ i)ⁿ/(1- i)ⁿ⁻². 2iⁿ⁻¹
c) (1- i)³/(1- i³). -2
13) Simplify the following:
a) (3- i)/(2+ i) + (3+ i)/(2- i). 2
b) 3/(1+ i) - 2/(2- i) + 2/(1- i). (1/10) (17- 9i)
14) Compute:
a) (1+ i)⁻¹. (1- i)/2
b) [(√3 + i)(√3 - i)⁻³⁾². 1/8
15) Show the following in the form A+ iB:
a) (1+ i)²/(3- i). -1/5 + 3i/5
b) (5+ 4i)/(4+ 5i). 40/41 - 9i/41
c) {(3- 2i)(2+ 3i)}/{(1+ 2i)(2- i)}. 63/25 - 16i/25
d) {1/(1- 2i) + 3/(1+ i)}{(3+ 4i)/(2- 4i)}. 1/4 + 9i/4
EXERCISE - B
1) Show the following in the form of:
a) (a + ib)²/(a - ib) - (a - ib)²/(a + ib). 2b(3a²- b²)/(a²+ b²)
b) 1/1- cosθ + 2i sinθ). (1- 2i cot(θ/2))/(5+ 3 cosθ)
2) Find the lengths of the segments connecting the points represented by the following pairs of numbers:
a) 5, -3. 8
b) 3, -4i. 5
c) -6i, 3i. 9
d) -1 - i, 2+ 3i. 5
e) 3 - 2 i, 3+ 5i. 7
3) Find the square of root of the numbers:
a) 5+ 2i.
b) -5+ 12i. 3+2i
c) {√5+12i) + √(5- 12i)}/{√(5+ 12i) - √(5- 12i)}.
d) - 8 - 6i.
e) - 7 - 24i.
f) a²- 1 + 2a √-1
g) 4√-5 -1
h) 1+ 4√-3
i) i
j) - i
k) 1- i
EXERCISE - C
1) Find the value of
a) [4 + 3 √-20]¹⁾² + [4+ - 3 √-20]¹⁾². 6
b) √(8+ 6i) + √(8- 6i). 6
c) √(-35+ 12i) - √(-35 - 12i). 12i
2) Find the square root of the following:
a) x + i √(x⁴+ x²+1). ±(1/√2) [√(x²+x+1) + √(x²- x +1)}
b) 4ab - 2(a²- b²)√-1. ±[(a + b) - (a - b)i]
3) Find real values of x and y for which of the following are satisfied:
a) (1- i)x + (1+ i)y= 1- 3i. 2,-1
b) (x -1)/(3+ i) + (y -1)/(3- i) = i. -4,6
c) {(1+ i)x - 2i}/(3+ i) + {(2- 3i)+ i}/(3- i)= i. 3,-1
d) (x + iy)(2- 3i)= 4 + i. 5/13,14/13
e) (1+ i)²+ 6 + i = (2+ i)x. 5,±2
f) √(x²- 2x + 8) + (x + 4) i = y(2+ i). 3,2
g) (x⁴+ 2xi) - (3x²+ yi)= (3- 5i)+ (1+ 2yi). -2,1/3
EXERCISE - D
4) If a,b,c,.....k are the roots of the equation xⁿ+ p₁xⁿ⁻¹ + p₂xⁿ⁻²....+ pₙ₋₁x + pₙ = 0
(p₁, p₂,....pₙ are real), then show that
(1+ a²)(1+ b²)......(1+ k²)= (1- p₂ + p₄ - .....)² + (p₁ - p₃ + p₅ - ....)².
5) Express (1+ x²)(1+ y²)(1+ z²) as sum of two squares.
6) Solve the equation x⁴ - 4x² + 8x + 35=0 having given that one root is 2+ √-3. -2±i
7) Determine the condition that the equation z² + (p + iq)z + r + is=0 has real roots only. s²- pqs + rq²
8) Find real θ such that (3+ 2i sinθ)/(1- 2i sinθ) is
a) real. nπ where n is an integer
b) purely imaginary. nπ±π/3, where n is an integer
9) if [tanθ - i {sin(θ/2) + cos(θ/2)}]/(1+ 2i sin(θ/2)) is purely imaginary, then determine the general value of θ. nπ+ π/4
10) Show that a real value of x satisfy the equation (1- ix)/(1+ ix) = a - ib if a²+ b²= 1 (a,b real)
11) Show the following identities:
a) (x² + a²)⁴= (x⁴ - 6x²a² + a⁴)²+ (4x³a - 4xa³)².
b) (x² + a²)⁷ = (x⁷ - 21x⁵a²+ 35x³a⁴ - 7xa⁶)² + (7x⁶a - 35x⁴a³ +21x²a⁵ - a⁷)².
EXERCISE - E
1) If ω is an imaginary cube root of unity, show that
a) (1- ω + ω²)(1+ ω - ω²)= 4.
b) (1+ ω)³ - (1+ ω²)³ = 0.
c) (1- ω + ω²)³= (1+ ω - ω²)³= - 8
2) If ω is an imaginary cube root of unity, then (1+ ω - ω²)⁷ equals to
a) 128ω b) -128ω c) 128ω² d) - 128ω². d
3) If ω is an imaginary cube root of unity
a) (1- ω + ω²)⁵ + (1+ ω - ω²)⁵ = 32.
b) (2+ 5ω + 2ω²)⁶ = (2+ 2ω + 5ω²)⁶ = 729
4) Prove:
a) (1- ω)(1- ω²)(1- ω⁴)(1- ω⁸)= 9.
b) (1+ ω)(1+ ω²)(1+ ω⁴)(1+ ω⁸)= 1.
c) (1- ω + ω²)(1- ω² + ω⁴)(1- ω⁴ + ω⁸)......to 2n factors= 2²ⁿ.
5) a) x³ + y³ = (x + y)(ωx + ω²y)(ω²x + ωy).
b) x³ - y³ = (x - y)(ωx - ω²y)(ω²x - ωy).
c) (a + b+c)(a + bω + cω²)(a + bω² + cω)= a³ + b³ + c³.
d) (x + y)²+ (xω + yω²)²+ (xω² + yω)²= 6xy
e) (a + bω + cω²)³ + (a + bω²+ cω³) = (2a - b - c)(2b - c - a)(2c - a - b)= 27abc if a+ b + c =0.
EXERCISE - F
1) If α, β, γ are the cube roots of p, p< 0, then for any x, y, z.
a) (xα + yβ + zγ)/(xβ + yγ + zα)= ____. ω²
b) (a + bω+ cω²)/(c + aω + bω²) + (a + bω + cω²)/(b + cω + aω²)= ___. -1
2) If α and β are the complex cube roots of unity, show that
α⁴β⁴+ α⁻¹β⁻¹ = 0.
3) Show that 1/(1+ 2ω) + 1/(2+ ω) - 1/(1+ ω) = 0. Where ω is an imaginary cube root of unity.
4) If ω(≠1) be a cube root of unity and (1+ ω²)ⁿ = (1+ ω⁴)ⁿ, then the least +ve value of n is
a) 2 b) 3 c) 4 d) 5. b
5) If x = a+ npb, y= aα + bβ and z= aβ + bα where α and β are complex cube root of unity, show that
a) xyz = a³ + b³.
b) x³ + y³ + z³ = 3(a³ + b³).
6) Show that:
a) {(-1+ √-3)/2}¹⁰⁰ + {(-1- √-3)/2}¹⁰⁰ = -1
b) {(-1+ √-3)/2}³ⁿ + {(-1- √-3)/2}³ⁿ = 2.
c) If I = √-1, then
4+ 5(-1/2 + i√3/2)³³⁴ + 3(-1/2 + i√3/2)³⁶⁵ is equal to
a) 1- i √3 b) -1+ i √3 c) i √3 d) - i √3. c
7) If α is a complex number such that α¹+ α +1 =0 , then show that α³¹ is equal to α.
8) If α and β are the roots of x²- x +1=0, then α¹⁰³⁰ + β¹⁰³⁰ is equal to
a) 1030 b) 0 c) 1 d) -1
9) If α and β are the roots of x²- 2x +4=0, then α/β is equal to (1.2) (-1 ± √3i).
10) Show that the cube root of unity 1, ω, ω¹ are the vertices of an equilateral triangle.
EXERCISE - G
1) If z= (√3+ i)/2 then show that (z¹⁰¹ + i¹⁰³)¹⁰⁵ equals to z³.
2) Prove that for all odd integral values of n {(√3+ i)/(√3- i)}³ⁿ +1=0.
3) Show that {(-1+ √3i)/2)ⁿ + {(-1+ √3i)/2)ⁿ = -1
4) It is given that n is an odd integer greater than 3 but not a multiple of 3. Show that x³ + x² + x is a factor of (x +1)ⁿ - xⁿ -1.
5) If f(z) be divided by z - i and z + i, the remainders are respectively i and 1+ i. Determine the remainder when f(z) by z³ +1.
6) If A(x) and B(x) be two polynomials and f(x)= A(x³)+ x(B). If f(x) is divisible by x²+ x +1 then show that it is divisible by x -1 also.
7) If (1+ x + x)ⁿ = a₀ + a₁x + a₂x²+ a₃x²+....+ a₂ₙx²ⁿ, then show that (a₀ + a₃+ a₆+...)= (a₁ + a₄ + a₇+.....)= (a₂ + a₅+ a₈+.....)= 3ⁿ⁻¹.
8) Show that:
(x + y)ⁿ - xⁿ - yⁿ is divisible by xy(x + y)(x² + xy + y²), if n is odd but a multiple of 3.
9) Show that that the polynomial x⁴ˡ + x⁴ᵐ⁺¹ + x⁴ⁿ⁺² + x⁴ᵖ⁺³ is divisible by x³ + x² + x +1 where l, m, n, p are positive integers.
10) Find the common roots of the equations z³+ 2z²+ 2z +1=0 ; z¹⁹⁸⁵ + z¹⁰⁰ +1=0. ω,ω²
EXERCISE - H
1) Show that the values of ⁵ₙ₌₁∑ (xⁿ + 1/xⁿ)² when x² - x +1=0 is 7.
2) If t² + t + 1= 0, then show that the value of (t + 1/t)² + (t² + 1/t²) + .....+ (t²⁷ + 1/t²⁷)² = 54.
3) If t² + t + 1=0, then the value of the expression
(t + 1/t) + (t² + 1/t²)+.....+(t²⁷ + 1/t²⁷) equals to
a) 0 b) 1 c) -1 d) none. a
4) If α, β, γ are the roots of x³- 3x²+ 3x +7=0 and ω is cube root of unity, then the value of
(α -1)/(β-1) + (β -1)/(γ -1) + (γ -1)/(α -1) is
a) ω² b) 2ω½ c) 3ω² d) -3ω² c
5) If the argument of (z - a)(conjugate of z - b) is equal to that of {(√3+ i)(1+ √3i)}/(1+ i), where a,b are two real numbers, find the values of a and b so that the locus becames a circle having its centre at (1/2) (3+ i). 2,1
6) If 1/(a+ ω) + 1/(b + ω) + 1/(c + ω) + 1/(d + ω) = 2ω² and 1/(a+ ω⅖) + 1/(b + ω²) + 1/(c + ω²) + 1/(d + ω²) = 2ω then show that 1/(a+ 1) + 1/(b + 1) + 1/(c + 1) + 1/(d + 1) = 2.
7) If ω be complex cube root of unity satisfying the equation 1/(a+ ω) + 1/(b + ω) + 1/(c + ω) = 2ω² and 1/(a+ ω¹) + 1/(b + ω²) + 1/(c + ω²) = 2ω, then show that 1/(a+ 1) + 1/(b + 1) + 1/(c + 1) is equal to
a) 2 -2 c) -1+ ω¹ d) -1+ ω
8) If ax + cy+ bz= X, cx+ by+ az= Y, bx+ ay+ cz= Z, show that
a) (a²+ b²+ c² - bc - ca - ab)(x²+ y²+ z² - yz - zx- xy) = X²+ Y²+ Z² - YZ- ZX- XY
b) (a³+ b³+ c³ - 3abc)(x³+ y³+ z³ - 3xyz)= X³+ Y³+ Z³ - 3XYZ.
9) Given z₁ + z₂ + z₃ = A, z₁ + z₂ω + z₃ω² = B, z₁ + z₂ω²+ z₃ω= C
a) Express z₁, z₂, z₃ in terms of A,B,C.
b) |A|² + |B|² + |C|² = 3(|z₁|² + |z₂|² + |z₃|²).
10)a) For positive integer n₁, n₂ the value of the expression
(1+ i)ⁿ₁ + (1+ i³)ⁿ₂ + (1+ i⁵)ⁿ₃ + (1+ i⁷)ⁿ₇, where i = √-1 is a real number if and only if
a) n₁ = n₂ + 1 b) n₁ = n₂ -1 c) n₁ =n₂ d) n₁ > 0, n₂ > 0
b) z¹⁴ + 1/z¹⁴ = -1, where z is a root of the equation z + 1/z = 1.
11) For what real valus of x and y are the numbers
-3 + ix²y and x²+ y + 4i conjugate complex?
12) For what real values of x and y are the complex numbers x²- 7x + 9yi and y²i + 20i - 12 equal ?
13) The complex numbers sinx + i cos2x and cosx - i sin2x are conjugate to each other for x = .....
First of all remember the formula for modulus and amplitude of complex numbers :
Modulus
1) |z₁ + z₂| ≤ |z₁| + |z₂|
2) |z₁ - z₂| ≥ |z₁| - |z₂|
3) |z₁| - |z₂| ≤ |z₁ + z₂| ≤ |z₁|+ |z₂|
4) ||z₁| - |z₂||≤ |z₁ - z₂|
||z₁| - |z₂|| ≤ |z₁ + z₂| (by z₂ ---> - z₂)
5) a) |z₁z₂ ....|= |z₁||z₂| .....
b) |z₁/z₂| = |z₁|/|z₂|
6) |z| = |i z|= |- z|
7) a) |z²|= |z|²
b) |zⁿ|= |z|ⁿ
8) a) any unimodular number can be written as eᶥᶿ ∀ θ
b) |eᵃ⁺ⁱᵇ|= |eᵃ. eⁱᵇ|= eᵃ. 1
9) a) | 1- eᶥᶿ | = 2 sin(θ/2)
b) | 1+ eᶥᶿ | = 2 cos(θ/2)
So, (1± cosθ)²+ sin²θ = 2(1 ± cosθ)
= 4 sin²(θ/2) or 4 cos²(θ/2)
Amplitude
10) Amplitude θ is chosen such that -π≤ θ ≤ π
11) amp(z₁/z₂)= amp z₁ - amp z₂
12) amp(z₁z₂)= amp z₁ + amp z₂
13)a) amp(z)=0 or π==> z is purely real
b) amp(z)=π/2 => z is purely imaginary
c) amp(z)=π/4 => tan⁻¹(y/x)=π/4
So, x = y => R.P = I. P
14) Representation of certain complex numbers of the form a + ib in the form
r(cos θ + i sin θ)
= r = + √(a²+ b²) always and tan θ= b/a
= tanα say . So, θ = α.
a) a, + ive, b + ive, tanθ = b/a = tan α
So, θ = α
b) a, + ive, b - ive, then θ = -α. (Conjugate)
c) a, - ive, b + ive, then θ = π - α
d) a, - ive, b - ive then, θ = -(π - α)
EXERCISE - I
1) Find the modulus and the principal value of the argument of the following numbers:
a) 1- i. √2, -(π/4)
b) -1- √3 i. 2, -2π/3
c) 1+ √2 + i. √(4+ 2√2)), π/8
d) 3. 3, 0
e) -5. 5,π
f) 6i. 6, π/2
g) - 2i. 2, -π/2
h) (1+ i)²ⁿ⁺¹/(1- i)²ⁿ⁻¹ , n ∈ N. 2,π
i) (1+ 7i)/(2- i)². √2, 3π/4
j) 2.5(cos 300° + i sin 30°). 5/2√2, π/4
k) sin(6π/5) + i (1+ cos(6π/3)). 9π/10
l) (i -1)/{i(1- cos(2π/5))+ sin(2π/5)}. 11π/20
m) - √3 + i. 2,5π/6
n) (1+ i √3)/{2i( cos(π/3)+ i sin(π/3))}. 1,-π/2
o) i(√3 + i)⁶/4(1- i √3)². 4, π/6
p) -5(cos40° - i sin 40°). 5,140°
q) 4(cos 330° - i sin330°). 4,30°
r) 1+ i tanα (-π < α < π, α ≠ ± π/2).
EXERCISE - J
1) Find the modulus and argument of the complex number.
z₁ = z⅖- z if z= cosθ + i sinθ.
2) Show that -3 - 4i = 5 ₑi(π + tan⁻¹(4/3)).
3) Show that ₑ2mi cot⁻¹p[ (pi +1)/(pi - 1)]ᵐ = 1.
4) For any two non-zero complex numbers z₁ and z₂ if |z₁ + z₂|= |z₁| + |z₂|, then show that arg z₁ - arg z₂ is zero.
5) Show the above result if we have |z₁ - z₂|= |z₁| - |z₂ |
6) If arg(z)< 0, then arg (-z) - arg(z)=
a) π b) -π c) -π/2 d) π/2 a
7) If |aᵢ |< 2, i ∈ {1,2,3,....n}. Show that for no z, |z|< 1/3 and ⁿᵢ₌₁ᵢ∑ aᵢ zⁱ = 1 can occur simultaneously.
8) Show the following inequalities:
a) |z/|z| - 1|≤ arg z.
b) |z -1|≤ ||z| - 1| + |z|| arg z|
9) If |z|= 1, show that (z -1)/(z +1) (z ≠ 1), is a pure imaginary number. What will you conclude if z= 1 ?
10) The complex number z is such that |z|= 1, z≠ 1 and ω = (z -1)/(z +1). Then real part of ω is
a) 1/|z +1|² b) -1/|z +1|² c) √2/|z +1|² d) 0
11) If the number (z -1)/(z +1) is a pure imaginary then show that |z|= 1.
12) If P is the affix of z in the Argand diagram and P moves so that (z - i)/(z -1) is always purely imaginary, then show that locus of z is a circle of centre (1/2, 1/2) radius 1/√2.
EXERCISE - K
1) If arg (z¹⁾³)= (1/2) arg (z + conjugate of z. z¹⁾³), then show that |z|= 1.
2) If |(z₁+ z₂)/(z₁ - z₂)|= 1, then show that z₁/z₂ is purely imaginary number. What is condition that it may be zero ?
3) For two complex numbers z₁ and z₂, it is given that |(z₁ - z₂)/(z₁ + z₂)|= 1. Show that iz₁/z₂ = λ where λ is real. Also determine the angle between the lines drawn from origin to points z₁ + z₂ and z₁ - z₂. - 2tan⁻¹λ
4) If z be any point on the circle |z -1|= 1 then show that (z -2)/z = i tan(arg z).
5) Demonstrate that the complex number x + iy whose modulus is unity, y≠ 0, can be represented as
x + iy= (a + i)/(a - i). Where a is real number.
6) Show that for any complex number z,
|Re(z)| + | Um (z)| ≤ |z| √2
OR
|x| + |y| ≤ √2 |x + iy|
7) If z₁, z₂, z₃ are three complex numbers, show that
z₁ I'm (conjugate of z₂. z₃) + z₂ I'm (conjugate of z₃ . z₁) + z₃ Im(conjuctiva of z₁. z₂)= 0
Where Im (w)= imaginary part of w, w being a complex number.
8) If iz³ + z² - z + i =0, then show that |z|= 1.
9) The two complex numbers satisfying the equation
z. Conjugate of z - (1+ i)z - (3+ 2i) conjugate of z + (1+ 5i)= 0 are
a) 1+ i, 3 - 2i. b) 1+ i, 3 + 2i
c) 1- i, 3 + 2i d) 1- i, 3 - 2i. c
10) Prove that the sum and product of two complex numbers are real if and only if they are conjugate of each other.
11) If z₁, z₂ are conjugate complex numbers, and z₃, z₄ are also conjugate, then show that
Arg (z₃/z₂)= arg(z₁/z₄)
12) If (z₁, z₂) and (z₃, z₄) are two pairs of conjugate complex numbers, then show that
Arg(z₁/z₃)+ Arg(z₂/z₄)= 0.
13) z₁, z₂ , z₃ are three complex numbers whose modulii are a,b,c respectively and are such that the determinant
a b c
b c a = 0
c a b
If z₁ ≠ z₂, then show that
Arg{(z₃ - z₁)/(z₂ - z₁)}²= Arg(z₃/z₂)
14) Let z₁ = 10+ 6i and z₂ = 4+ 6i. If z is a complex number such that the argument of (z - z₁)/(z - z₂) is π/4, then show that
|z - 7 - 9i|= 3√2.
15) If Arg{(z -2)/(z +2)}= π/4, then show that |z - 2i|= 2√2
16) Find the complex numbers z which simultaneously satisfy the equations
|(z -12)/(z - 8i)|= 5/3 and
|(z -4)/(z -8)|= 1
17) Find all complex numbers z for which
Arg{(3z - 6 - 3i)/(2z - 8 - 6i)}=π/4 and.
|z - 3 + i |= 3. x²+ y²- 8x - 2y +13=0; (4+ 4/√5, 1- 2/√5)
EXERCISE - L
1) The complex numbers z = x + iy which satisfy the equation
|(z - 5i)/(z + 5i)|= 1 lie on the x-axis of x.
2) The locus of the point z satisfying the condition Arg{(z -1)/(z +1)}=π/2 is the circle x²+ y² - 2y/√3 - 1= 0. x²+ y²- (2/√3)y -1=0, circle
3) The region of the z-plane for which |(z - a)/z + conjugate of a)|= 1 (Re a ≠ 0) is y-axis.
4) If z= x + iy and ω = (1- iz)/(z - i), then |ω|= 1 implies that, in the complex plane z lies on the real axis..
5) If ω = {(z - i)/(1+ iz)}ⁿ, n integral , then show ω lies on the unit circle for all n.
6) Locate the complex numbers z = x + iy such that
a) |z -1| + |z +1| ≤ 4
Show that the point z satisfying the above equation represents the interior and boundary of ellipse x²/4 + y²/3 = 1
b) |z - i |, arg {z/(z + i)}= π/2
c) If the imaginary part of|(2z +1)/(iz +1} is -2, then the locus of the point representing z in the complex plane is a straight line x + 2y - 2= 0.
7) If |z|=√2, then the points given by 3+ 4z will lie on a circle. What is the centre and radius of this circle ? (3,0), 4√2
8) Determine the locus of the point z such that z²/(z -1) is always real. x²+ y²- 2x = 0
9) Find the centre and radius of the circle formed by all the points represented by z= x + iy satisfying the relation |z + α|/|z - β|= k (k ≠ 1) where α and β are constant complex numbers given by α = α₁ + iα₂ , β = β₁ + iβ₂.
10) For complex numbers z and w, prove that |z|²w - |w|²z = z - w if and only if z = w or z. Conjugate of w = 1.
11) If λ be real , show that the equation |z - a|² + |z - b |²=λ represents a circle.
Determine its centre and radius.
12) If P, A, B represent the complex numbers x + iy, 6i and 3 respectively and P moves in such a manner that PA= 2PB then show that z . Conjugate z = (4+ 2i)z + (4 - 2i). Conjugate of z
Also show that the locus of the point P is a circle whose centre is the point (4- 2i) and radius √20.
13) If z= 2+ k + i √(3- k²) where k is real such that k²< 3. Show that |(z +1)/(z -1)| is independent of k. Also show that the locus of the point z for different values of k is a part of the circle. What is its centre and radius ? (2,0),√3
14) Find all complex numbers satisfying the equation
2 |z|²+ z² - 5 + i √3= 0. (√(3/2), -1/2),(- √(3/2), 1/√2)
15) The number of solutions of the system of equations given by
|z|= 3 and |z +1 - i|=√2 is equal to
a) 4 b) 2 c) 1 d) no d
16) If |z|≤ 1, |w|≤ 1, show that
|z - w|² ≤ (|z| - |w|)²+ (Arg z - Arg w)²
17) If (5z₂/7z₁) is a purely imaginary number, then prove that|(2z₁ + 3z₂)/(2z₁ - 3z₂)| is equal to 1.
18) Show that the roots of the cubic equation (z + ab)³= a³, a ≠ 0 represent the vertices of a triangle of sides of length √3 |a|.
19) Show that the triangle with vertices at the points z₁, z₂ and (1- i)z₁ + iz₂ is right angled and isosceles.
20) Determine the condition so that the equation z²+ (a + ib)z + (c + id)= 0 has
i) one root real. d²- adb + b²c
ii) both roots equal. a²- b²= 4c and ab= 2d
21) If z₁ = 1+ 2i, z₂ = 2 + 3i, z₃ = 3+ 4i, then z₁, z₂ and z₃ are collinear.
EXERCISE - M
1) Given that eⁱᴬ, eⁱᴮ, eⁱᶜ are in AP., where A, B, C are the angles of a triangle then the triangle is
a) isosceles b) equilateral c) right angled d) none
2) Show that the locus of r for which the two curves arg(z)= π/6 and |z - 2√3 i| = r Intersect is 3.
3) Find the equation in complex variables of all the circles which are orthogonal to |z|= 1 and |z -1|= 4.
4) For all complex numbers z₁, z₂ satisfying |z₁|= 12 and |z₂ - 3 - 4i|= 5, The minimum value of |z₁ - z₂| is
a) 0 b) 2 c) 7 d) 17. b
5) Show that
|z₁ + z₂|²+ |z₁ - z₂|²= 2|z₁|²+ 2|z₂|².
Interpret the result geometrically and deduce that
| α + √(α½- β²)| + | α - √(α² - β²)|= |α + β| + |α - β|,
All numbers involved being complex.
6) Show that
|z₁| + |z₂|= |(1/2) (z₁ + z₂)+ √(z₁z₂)| + |(1/2) (z₁ + z₂) - √(z₁z₂)|
7) For any two complex numbers z₁, z₂ and any real numbers a and b
|az₁ - bz₂|² + |bz₁ + az₂|²= (a²+ b²) [|z₁|²+ |z₂|²]. T/F
8) If z₁ and z₂ are complex numbers, show that |z₁ + z₂|²= |z₁|²+ |z₂|² if and only if z₁ . Conjugate z₂ is purely imaginary.
9) Show that the inequality:
|z₁ + z₂|² ≤ (1+ λ)|z₁|² + (1+ 1/λ) |z₂|² where λ > 0.
10) Show that
|(z₁ - z₂)/(1- conjugate z₁. z₂)|< 1 if |z₁|< 1 and |z₂|< 1.
11) Show that
|(1- z₁. Conjugate z₂)/(z₁ - z₂)|< 1 if |z₁|< 1 < |z₂|.
12) If z₁ and z₂ are two complex numbers such that |(z₁ - z₂)/(1- z₁. Conjugate z₂)|= 1, then show that both z₁ and z₂ are unimodular.
13) Show that
|1- conjugate z₁. z₂| - |z₁ - z₂|² = (1- |z₁|²)(1- |z₂|²)
14) If |(az₁ - bz₂)/(ab - z₁. Conjugate z₂)|= 1 and |z₁|≠ b then show that |z₂| is a, where a, b are real numbers.
15) Let z₁, z₂ be two complex numbers such tha
TRIGONOMETRY
EXERCISE - B
1) 3[sin³(3π/2 - x) + sin³(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π- x)] is equal to
a) 0 b) b) 1 c) 3 d) sin4x + sin6x e) none
2) Show that
a) sin⁶x + cos⁶x + 3 sin²x cos²x =1
b) 3(sinx - cosx)⁴+ 6(sinx + cosx)²+ 4(sin⁶+ cos⁶x) is independent of x.
c) (sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2 sin²x cos²x).
d) (3+ cos4x) cos2x = 4(cos⁸x - sin⁸x).
e) If sinx + cosx = a, then find the values of |sinx - cosx| and cos⁴x + sin⁴x).
f) If sinx + cosecx = 2, then sin²x + cosec²x is equal to 2. T/F
g) f(x)= cos²x + sec²x≥ 2. T/ F.
OR minimum value of f(x) is 2
h) Given A= sin²x + cos⁴x, then for all real x.
a) 1≤ A≤ 2 b) 3/4 ≤ A≤ 1 c) 13/16≤ A≤ 1 d) 3/4≤ A≤ 13/16
i) Let A= sin⁸x + cos¹⁴x, then for all real x
a) A≥ 1 b) 0< A≤ 1 c) 1/2 < A≤ 3/2 d) none
j) If x, y are acute, sinx = 1/2, cosy= 1/3, then (x + y) ∈
a) (π/3,π/2) b) (π/2, 2π/3) c) (2π/3, 5π/6) d) (5π/6,π)
k) sec⁴x(1- sin⁴x) - 2 tan²x = 1
l) tan²x - sin²x = sin⁴x sec²x= tan²x sin²x.
m) (cotA+ tanB)/(cotB + tanA)= cotA tanB.
n) (sinx + cosx)(tanx + cotx)= secx + cosecx.
o) (cosx cosecx - sinx secx)/(cosx + sinx)= cosecx - secx.
p) (1+ cotA- cosecA)(1+ tanA + secA)= 2.
q) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
r) (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.
s) {cot²x(secx -1)/(1+ sinx)= sec²x {(1- sinx)/(1+ secx).
t) (secx +1- tanx)/(tanx - secx +1)= (1+ cosx)/sinx.
u) cosx/(1- tanx) + sinx/(1- cotx)= sinx + cosx.
EXERCISE - B
1) If tₙ = sinⁿx + cosⁿx, then (t₃ - t₅)/t₁ = (t₅ - t₇)t₃
2) Show that
a) tanx/(1- cotx) + cotx/(1- tanx)= secx cosecx +1.
b) (sinx + cosecx)²+ (cosx + secx)²= tan²x + cot²x +7.
c) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - cosecx/sec²x.
d) (secx - cosecx)(1+ tanx + cotx)= tanx secx - cotx cosecx.
e) {2sinx tanx(1- tanx)+ 2 sinx sec²x}/(1+ tanx)² = 3sinx/(1+ tanx).
f) (tanx + cosecy)² - (coty - secx)²= 2 tanx coty (cosec x + secy)
g) {(1+ sinx - cosx)/(1+ sinx + cosx)}² = (1- cosx)/(1+ cosx).
h) If 2sinx/(1+ sinx - cosx) = y, then (1- cosx + sinx)/(1+ sinx) is also y.
i) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)} sin²x cos²x = (1- sin²x cos²x)/(2+ sin²xcos²x).
j) (cosecx - secx)(cotx - tanx)= (cosecx + secx)(secx cosecx -2)
3) If tanx + sinx = m and tanx - sin x = n, then show that m²- n²= 4 √(mn)
4) Eliminate x from the relations a secx = 1- b tanx ; a² sec²x = 5 + b² tan²x.
EXERCISE - 3
1) If cosecx - sinx = m, and secx - cosx = n, eliminate x
2) If cosecx - sinx = a³, secx - cosx = b³, then a²b²(a²+ b²)= 1.
3) If cotx + tanx = m, secx - cosx =n, eliminate x.
4) If c cos²x + 3c cosx sin²x = m, c sin³x + 3c cos²x sinx = n, then show that (m+ n)²⁾³ + (m - n)²⁾³ = 2c²⁾³.
5) If cosx+ sinx =√2 cosx , then show that cosx - sinx = √2 sinx.
6) If sinx + 5 cosx = 5, show that 5 sinx - 3 cosx = ± 3.
7) If a cosx + b sinx = p, a sinx - b cosx = q, show that a²+ b²= p²+ q².
8) If a cosx - b sinx = c, show that a sinx + b cosx = ±√(a²+ b²- c²).
9) If a sinx + b cosx = c, then show that (a - b tanx)/(b + a tanx)= ± √(a²+ b²- c²)/c.
10) If tan²(1- e²). Show that secx + tan³x coséx = (2- e²)³⁾².
11) If ax/cosθ + by/sinθ = (a²- b²), and ax sinθ/cos²θ - by cosθ/sin²θ = 0, show that (ax)²⁾³ + (by)²⁾³ = (a²- b²)²⁾³.
12) If sinθ = (m²- n²)/(m²+ n²), determine the values of tanθ, secθ, and cosecθ.
13) If tanθ = 2x(x +1)/(2x +1), determine sinθ and cosθ.
14) If cosθ = 2x/(1+ x²), find the values of tanθ and cosecθ.
15) If secx = p + 1/4p, then secx + tanx = 2p or 1/2p.
16) If secθ + tanθ = p, obtain the values of secθ, tanθ, sinθ in terms of p.
17) If cosx/cosy , sinx/siny= b, then (a²- b²) sin² y = a²- 1
18) If tanθ = p/q, show that (p sinθ - q cosθ)/(p sinθ + q cosθ) = (p²- q²)/(p²+ q²).
19) Is the equation sec²θ = 4xy/(x + y)² possible for real values of x and y ?
If not, then find out a relation between x and y so that it may be possible. x= y but x≠ 0
20) If m²+ n²+ 2mn cosθ = 1, and p²+ q²+ 2pq cosθ = 1, and mp + nq + (mq+ np) cosθ = 0, show that m²+ n²= cosec²θ.
1b 2)e) √(2- a²), 1- (1/2) (a²-1)² f) T g) T hb ib jb
EXERCISE - 4
1) The value of sin⁶θ+ cos⁶θ+ 3 sin²θ cos²θ is
a) 0 b) 1 c) 2 d) 3
2) The least value of 2 sin²θ+ 3 cos²θ is
a) 1 b) 2 c) 3 d) 5
3) The greatest value of sin⁴θ + cos⁴θ is
a) 1/2 b) 1 c) 2 d) 3
4) The value of sin²θ cos²θ(sec²θ + cosec²θ) is
a) 2 b) 4 c) 1 d) 0
5) If sinθ + cosecθ= 2, then sin²θ+ cosec²θ is equal to
a) 1 b) 4 c) 2 d) none
6) For how many values of x between 0 and 2π is the Equation 2 cosec2x citx - cot²x = valid?
a) 0 b) 2 c) 1 d) none
7) Incorrect statement is
a) sinθ= -1/5 b) cosθ= 1 c) secθ =1/2 d) tanθ= 20
8) sec²θ= 4xy/(x + y)² is true if and only if
a) x + y ≠ 0 b) x=y, x≠ 0 c) x= y d) x≠ 0, y ≠ 0
9) If x= a cos²θ sinθ and y= a sin²θ cosθ, then (x²+ y²)³/x²y² is independent of θ. T/F
10) The inequality ₂sin²θ + ₂cos²θ≥ 2√2 holds for all real θ T/F
11) The equation sinθ = x + 1/x holds true for all real θ. T/F
12) The least value of tan²θ + cot²θ is _____.
13) The value of sinθ cosθ(tanθ + cotθ) is ______
14) If for real x, the equation x + 1/x = 2 cosθ holds, then cosθ= ____
15) If secθ - cotθ = q, then the value of cosecθ = _____
1b 2b 3b 4c 5c 6d 7c 8b 9t 10t 11f 12)2 13) 1 14) cosθ= ±1 15)
RAW- A
1) If ⁴⁻ˣP₂ =
6 Then the value of x is
a) 1 b) 2 c) 3 d) 4
2) If ¹⁰Pᵣ = 5040 then the value of ʳP₂ is
a) 4 b) 12 c) 16 d) 24
3) The number of arrangements of two letters of the word BANANA in which two N's do not appear adjacently is
a) 40 b) 60 c) 80 d) 10
4) The number of 4 digit number, all digits occurring once is
a) ¹⁰O₃ b) ¹⁰P₄ c) 9 x ⁹P₃ d) ¹⁰C₄
5) The total number of permutation of n things taken r at a time in which 3 particular things never occur is
a) n -3 b) ⁿCᵣ c) ⁿ⁻³Pᵣ₋₃ d) ⁿ⁻³Pᵣ
6) Which one of the following relations is correct ?
a) ⁿPᵣ = ⁿ⁻¹Pᵣ + ⁿ⁻¹Pᵣ₋₁
b) ⁿPᵣ+ ⁿPᵣ₊₁
c) ⁿPᵣ= ⁿ⁻¹Pᵣ + r. ⁿ⁻¹Pᵣ₋₁
d) ⁿPᵣ + ⁿPᵣ₋₁ = ⁿ⁺¹Pᵣ
7) In how many ways can 5 persons be arranged in a circular way ?
a) 4! b) 24 c) (1/5) ⁶P₃ d) all of these
8) The number of ways in which 5 '+' signs and 3 '×' signs can be arrarin a row is
a) 56 b) 72 c) 65 d) 81
9) How many ways are there to arrange the letters of the word GARDEN with the vowels in alphabetical order ?
a) 360 b)120 c) 240 d) 480
10) The number of ways in which 29 persons can be seated around a table if there are 9 chairs is
a) ²⁹P₉/2 b) ²⁰P₉/2 c) ²⁹P₉/9 d) none
1a 2b 3a 4c 5d 6c 7d 8a 9a 10b
RAW - B
1) The number of ways in which 6 different balls can be put in two boxes of different sizes so that no boxe is empty
a) 64 b) 60 c) 62 d) 30
2) A polygon has 54 diagonals. Then the number of its sides are
a) 9 b) 12 c) 15 d) 13
3) The number of 6 digit numbers , whose digits are all odd (1, 3, 5, 7, 9) is
a) 6⁵ b) ⁶C₅ c) 5⁶ d) 6!/2!
4) If ⁿP₁ + ⁿP₂=9 then n is
a) 2 b) 6 c) 3 d) none
5) If ⁿ⁺¹P₃ = 2. ⁿP₃, then n is
a) 5 b) (1/2) ¹⁰P₁ c) 5!/4! d) all of these
6) If ⁿP₃= 24, then ⁿP₂ is
a) 20 b) 12 c) 15 d) none
7) In how many ways can 6 men stand in a row, if two of them. A and B always stand together?
a) 240 b) 480 c) 640 d) none
8) How many 4 digit numbers can be formed with the numbers 1, 2, 3, 4, (taken only one at a time)?
a) 2⁴ b) 4² c) 24 d) none
9) A 6 digit number is formed by y1,2,....9 and repetition is not allowed. If odd postions are occupied by only odd digits, the number of such arrangements is
a) 1440 b) 340240 c) 1240 d) none
10) All the letters of the word SUNRISE are taken together and arranged in such a way that no two vowels occur together. The number of arrangements is
a) 2160 b) 720 c) 4120 d) none
11) All the letters of the word KOLKATA are taken together and arranged in such a way that two Ks do not come together. The number of arrangement is
a) 720 b) 900 c) 810 d) none
12) There are four gentlemen, 3 ladies, and three children. They have to take their seats on 2 rows-- 5 on the left and 5 on the right. Ladies are allowed to sit only on the left row and the gentlemen only in the right on row, whereas children may sit on any row. Then the number of possible sitting in arrangement is
a) 41250 b) 43200 c) 44120 d) 44210
13) A number less than 600 is to be formed by using 1, 2, 3, 4,5, repetition is not allowed. The number of ways this can be done is
a) 85 b) 80 c) 60 d) none
14) One 5 digit number is to be formed by using 1, 2, 3 and 4. All digit should be used. The number of ways this can be done is
a) 60 b) 120 c) 240 d) 480
15) The total number of factors of 1800, excluding itself is
a) 35 b) 34 c) 33 d) 30
16) 12 books are to be kept on a table. If 2 particular books are kept side by side, the total number of such arrangements is
a) 12x 12! b) 12 x 11! c) 11 x 12! d) 10 x 11!
17) 3- digit numbers are written with the help of 4, 5, 6, 7. If repitation is not allowed the sum of such numbers is
a) 7376 b) 7326 c) 14652 d) 2442
18) The number of positive terms in the sequence {xₙ}= 96/Pₙ - ⁿ⁺³P₃/Pₙ₊₁, n ∈ N is
a) 6 b) 8 c) 7 d) 9
19) The value of ¹⁰₌₁ᵣ∑rP(r,r) are
a) P(11,11) b) P(11,11) -1 c) P(11,11)+1 d) none
20)
21) ∞ᵣ₌₁∑ (r²+1) r! = 100 × 101!, then m equal to
a) 100 b) 101 c) 102 d) none
1c 2b 3c 4c 5d 6b 7a 8c 9a 10b 11b 12b 13a 14c 15b 16d 17c 18c 19b 20c 21a
RAW- C
1) How many 3-digit number of distint digit can be formed from the digits 2, 3, 7, 8, 9. 60
2) 3 persons enter a railway carriage , where there are 5 vacant seats. In how many ways can they seat themselves ? 60
3) How many numbers are there between 100 and 1000 in which all the digits are distinct . 648
4) How many 9 digit numbers of different digits can be formed ? 3265920
5) How many 3 digit numbers can be formed without using the digit 0, 2, 3, 4, 5 and 6. 64
6) How many odd numbers less than 1000 can be formed? By using the digits 0, 2, 5, 7 when the repetition of digit is allowed ? 32
7) How many numbers are there between hundred 100 and 1000 such that atleast one of their digits is 7 ? 252
8) How many numbers are there between 100 and 1000 that have one of their digit as 7 ? 255
9) In how many ways can 5 persons sit in a car, 2 including the driver in the front seat and 3 in the back seat , if two particular persons do not know driving ? 72
10) Given 5 flags of different colours. How many different signals can be generated by hosting the flags on a vertical pole (one below the other) if each signal requires the use of two flags ? 20
RAW- D
1) How many words can be made from the letters of the word COSTING so that the power
a) are always together. 1440
b) are never together. 3600
c) may occupally only odd positions ? 1440
2) Out of the letters A, B, C, p, q and r, how many different words can be formed if
a) the words always begin with a capital letter. 360
b) the words always begin with a capital letter and end also with a capital letter. 144
3) How many even numbers greater than 300 can be formed with the digits 1,2,3,4 and 5, no repetition being allowed? 111
4) firnd the sum of the numberr that can be formed by using all the five digits of 12345 only once. 3999960
5) In how many ways can the letters of the word CONSTANT be arranged keeping the two vowels together always ? 2520
6) How many different 9-digit numbers can be formed from the number 223355888 by arranging its digits so that the odd digits occupying even positions? 60
7) a) How many numbers of 5 significant digits can be formed using the digit 5, 6, 7, 8, 0 ? 96
b) with no digit be repeated in any number. 42
c) How many of them are divided by 5 ? 30
d) How many of them are divisible by 4. 20
8) Find the rank of the word MAKE when its letters are arranged as in a dictionary. 120960
9) Find how many arrangement can be formed with the letters of word JUSTAPOSED, the vowels always coming together. 7560
10) in how many ways can be laters of the word FOOTBALL be arranged so that the two O's do not come together ? 120
11) How many different words can be formed taking all the letters of the word PEOPLE in which two P's will and come together ? 96
12) How many number of 5 digit can be formed with the digit 0, 2, 5, 6, 7 without taking any of these digits more than once ? 340
13) How many numbers less than 10000 can be formed with the digits 3, 5, 7, 9 repetition being allowed? 1290
14) How many numbers each lying between 9 and 10000 can be formed with the digits 0, 1, 2, 3, 6, 7, 8 (digits can be repeated)? 64864800
15) Find the number of words with the letters of the word CONTAMINATION.
RAW- E
1) In how many ways can three sport prizes be given to 20 boys when a boy may receive any number of prizes ? 8000
2) In how many ways 8 books on Mathematics and 7 books on English can be placed on a shelf so that books on the same subject remain together ? 2x 8! x 7!
3) 7 Candidates are to be examined, 2 in Mathematics and the remaining in different subjects. In how many ways can they be seated in a row so that the examinees in Mathematics may not sitt together ?
4) Each of the 6 squares in the strip shown in figure given below
is to be coloured with any one of ten different colours so that no two adjacent squares have the same colour. Find the number of ways are colouring the strip . 590490
5) The adjacent figure is to be colouring using four different colours. In how many ways can this be done if no two adjacent triangle have the same colour ? 320
6) A man has 6 friends. In how many ways can he invite 1 or more of his friends at a dinner ? 63
7) a) There are 8 true -false statements in a question paper. How many sequence of answers are possible ? 256
b) if there are 10 true-false statements , then what will be the number if a student can answer one or more questions. 310-1
8) In how many ways can 8 books be arranged on a shelf if
a) any arrangement is possible ?
b) 3 particular books must always stand together ?
c) two particular books must occupy the ends ?
9) In how many ways can 10 B. SC students and 7 B. Com students may sit together? (10!) x P(11,7)
10) The first name of a person consists of 8 letters in which one letter occurs more than once, while the other letters are different. If the number of Permutations of the letters of his name taken all at a time be 6720. Find the number of the like letter occurs. 3
11) Two different books have three volumes of each and 3 other different books have 2 volumes for each. In how many different ways can be 12 volumes be arranged on a shelf, so that the volumes of the same book may remain side by side? 34560
RAW- F
1) How many three digits even numbers can be formed from 1,2,3,4,5 and 6 if the digits can be repeated? 108
2) Find the total number of ways of answering six multiples-choice questions, each question having four choices. 4⁶
3) There are four letters between Delhi and Patna. In how many different ways can a man go from Delhi to Patna and return, if for returning
a) any one of the routes is taken? 16
b) the same route is taken? 4
c) the same route is not taken? 12
4) Find the value of
a) 25!/23!. 600
b) LCM [4!, 5!, 6!]. 720
c) 5.6.7.8.9.10 in factorial form. 10!/4!
d) 1.3.5........(2n -1). 2n!/(2ⁿ. n!)
e) (2×3)! And show that it is not equal to 2! × 3!.
5) Find the value of n if
a) (n +1)!= 12 × (n -1)! 3
b) (n +3)!= 56(n +1)!. 5
c) (n +1)!= 90(n -1)!. 9
d) 1/8! + 1/9! = n/10!. 100
6) Show that 2.6.10.14 to n factors= (n +1)(n +2)(n +3).....to n factors.
7) Prove that in equation (n!)²≤ nⁿ. n! ≤ (2n)! for all positive integers n.
8) (2n +1)!/n!= 2ⁿ(1.3.5.....(2n -1)(2n +1))
9) Find the value of n such that
a) ⁿP₅ = 42 ⁿP₃, n > 4. 10
b) ⁿP₄/ⁿ⁻¹P₄ = 5/3, n > 4. 10
c) 5.⁴P₃ = 6. ⁵Pₙ₋₁. 8,3
d) ⁿP₄ = 20. ⁿP₂. 7
e) ⁿP₄ = 2. ⁵P₃. 5
f) ¹⁵Pₙ = 2730. 3
g) ⁵⁶Pₙ₊₆ : ⁵⁴Pₙ₊₃ = 30800:1. 10,41
h) ²ⁿP₃ = 100. ⁿP₂. 13
i) 16. ⁿP₃ = 13. ⁿ⁺¹P₃. 15
j) ²ⁿ⁻¹Pₙ : ²ⁿ⁺¹Pₙ₋₁ = 2217. 10
k) ⁿP₄ : ⁿP₅ = 12. 6
l) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5. 4
m) ¹¹Pₙ = ¹²Pₙ₋₁. 9
n) ¹⁵Pₙ₋₁ : ¹⁶Pₙ₋₂ = 3:4. 14
o) ⁿ⁻¹P₃ : ⁿ⁺¹P₃ = 5:12. 8
10) Prove the following:
a) P(n,n)= 2P(n, n -2), i.e., ⁿPₙ = 2. ⁿPₙ₋₂.
b) ⁿPₙ = ⁿPₙ₋₁.
c) ⁿPₓ = ⁿ⁻¹Pₓ + x. ⁿ⁻¹Pₓ₋₁.
d) ⁿPₙ = n ⁿ⁻¹Pₓ₋₁.
e) ²ⁿPₙ = 2ⁿ[1.3.5.....(2n -1)]
f) ⁿPₓ = nⁿ⁻¹Pₓ = (nm- x +1) ⁿPₓ₋₁.
11) If ⁿ⁺ˣP₂ = 110 and ⁿ⁻ˣP₂ = 20, find n and x. 8,3
12) Let ⁿPₓ denote the number of Permutation of n different things taken x at a time then show that 1+ 1. ¹P₁ + 2. ²P₂ + 3. ³P₃ + .....+ n. ⁿPₙ = ⁿ⁺¹Pₙ₊₁.
13) If ⁹P₅ + 5 ⁹P₄ = 10 ¹⁰Pₓ, find x. 5
14) Show that ⁹P₃ + 3 ⁹P₂ = ¹⁰P₃.
RAW- G
1) How many
a) four digit number. 3024
b) 3-digit numbers can be formed by using the digit 1 to 9 if repetition of digit is not allowed. 504
2) How many numbers lying between 100 and 1000 can be formed with the digit 0, 1, 2, 3, 4 and 5 if the repetition of the digits is not allowed ?
How many natural number less than 1000 can be formed by the same numbers and same process. 131
3) How many three digits even numbers can be made using the digits 1,2,3,4, 6 and 7 if no digit is repeated ? 60
4) How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4,5,6,7 and 8 when no digit is repeated in any such numbers? 12
5) How many words can be formed the letters of the word SUNDAY? How many of these begin with D ? 120
6) How many words beginning with C the and ending with Y can be formed by using the letters of the word COURTESY? 72
7) How many permutations can be formed by the letters of the word VOWELS, when
a) each word begin with O and ends with L ? 24
b) all consonants come together? 6
8) Find how many arrangements can be made with the letters of the word MATHEMATICS? In how many of them are the vowels together ?
In how many of them do the constants come together ? How many of them starts with M ?
9) How many arrangements can be made out of the letters of the words
a) INDIA. 60
b) ENGINEERING.
c) COMMERCE. 5040
d) CHANDIGARH. 907200
e) INTERMEDIATE. 19958400
f) MONDAY. 720
g) ORDINATE. 4032
h) CATS. 24
i) JAIPUR. 720
j) ORIENTAL (with A and E occupying odd place)? 8640
10) Find the number of arrangements that can be made out of the words
a) ROOT. 12
b) BANANA. 60
c) PINE APPLE. 30240
d) INSTITUTE. 30240
e) CIVILIZATION. 19958400
RAW - H
1) How many different signals can be given with five different flags by hoisting any number of them ? 325
2) In how many ways can five persr occupy 3 vacant seats ? 60
3) In how many ways can 10 people line up at a ticket window of a cinema hall ? 10!
4) In how many will can 5 children stand in a queue ? 120
5) In how many ways can 4 different books one each in chemistry, physical, biology, and mathematics be arranged in a shell? 24
6) Ram wants to arrange three economics, two hisr, and four language books on a shelf. If the books on the same subject are different, determine the number of all possible arrangements? 9!
7) 7 students are contesting the election for the president ship of the students union. In how many ways can their names be listed on the ballot papers ? 5049
8) 10 students are participating in a race . In how many ways can the first three prizes be won ? 720
9) There are three different things to be worn in four fingers with atmost one in each finger. In how many ways can this be done ? 24
10) If there are 6 periods in each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period ? 3600
11) In how many ways can n books on n subjects be arranged in a cupboard so that no two books on particular subjects are together ? (n -2)(n -1)!
12) There are six English, four Sanskrit and 5 Hindi books. In how many ways can they be arranged on a shelf so as to keep all the books of the same language together? 6x720x24x120
13) In how many ways can 9 examination papers be arranged so that the best and the worst papers are never together ? 282240
14) In how many ways can 6 rings of different types be worn in four fingers ? 4096
15) if 20 persons were invited for a party, in how many ways can they and the host be seated in a circular table ? In how many of these ways will 2 particular persons be seaten on either side of the host? 2x 18!
16) In how many ways can a party of men and 4 women be seated at a circular table so that no two women are adjacent? 144
17) A round table conference is to be held between delegates of 20 countries. In how many ways can they be seated if two particular delegates may with to seat together ? 2 x 18!
18) In how many ways can 8 persons be seated at a round table so that all shall not have the same neighbour in any two arrangement ? 7!/2
19) In how many different ways can 20 different pearls be arranged to form a necklace? 19!/2
20) In how many different ways can a garland of 16 different flowers be made? 15/2
RAW - I
1) In how ways 3 boys and 5 girls can be arranged in a row so that two boys sit together? 14400
2) What is the value of n when ⁿP₂ = 12? 4
3) How many numbers between 100 and 1000 can be formed with the digit 0,2, 3, 4, 0, 8 and 9, each digit occuring only once? 100
4) How many words can be formed with the letters of the word MONDAY ? How many of these words begin with M but do not end with Y ? 720, 96
5) How many 4-digit numbers can be formed with the numbers 1, 2, 3, 4 (taken only one at a time) ? 12
6) How many four digit numbers can be formed with the digits 1,2,5,6,7 no digit being repeated? How many of them are divisible by 5? 120, 24
7) How many words can be formed with the letters of the word STATIR? 50400
8) Find the value of ⁸P₈. 40320
9) In how many ways the letters of the word BALLOON be arranged so that two L's do not come together? 360
10) Show that ²ⁿPₙ = 2ⁿ[1.3.5.....(2n -1)]
11) ²ⁿ⁻¹Pₙ : ²ⁿ⁺¹Pₙ₋₁ = 5:3, find n. 4
12) Show that ⁿPₓ = ⁿ⁻¹Pₓ + x ⁿ⁻¹Pₓ₋₁
ARITHMETIC PROGRESSION
R. N- 1
1) Find the 35th and n-th term of the progression -2, 1, 4, 7,...... 3n -5
2) In an AP the 11th term is 22 and 25th term is 65. Find the AP. -7,-4,-1,2..., 80
3) Can (-447) be a term of the AP: 8, 5, 2....? No
4) The pth term of an AP is 3p - 5. Find the common difference and the 15th term of the AP . 3, 40
5) If pth term of an AP is q and qth term is p, then find (p+ q)th term. 0
6) Insert 5 Arithmetic mean between (-5) and 7. -3,-1,1,3,5
7) There are n arithmetic means between 5 and 35. If the ratio of second and the last mean is 1:4, find the value of n. 17
8) Find the middle term of the series: 5+ 8+ 11+...+ 89. 47
9) Find the sum:
a) 35 + 32+ 29+...+ (-25). 105
b) 1+ 4+ 6+ 9+ 11+ 14 .... up to 21 terms. 551
10) Find the sum of 4+7+10+13+....+100. 1716
11) The 3rd terms of an AP is 1/5 and 5th term is 1/3; show that the sum of the 15th of the series' is 8.
12) The sum of the first n terms of an AP 3n²+ 5n; find the value of the 10th term of the series. Which term of the series is 152 ? 62, 25
13) The first term of an AP is 22 and the common difference is (-4). If the sum of n terms of this progression is 64, find the value of n. 4 or 8
14) The sums of the first P, q and r are number of terms of a series in AP are x, y, z respectively. Find the value of x(q-r)/p + y(r - p)/q + z(p - q)/r. 0
15) The ratio of the sum of the first n terms of two different AP's is (2n +1):(2n -1). Find the ratio of their 8th terms. 31:29
16) The sum of 4 integers in AP is 24 and their product is 945. Find the integers. 3,5,7,9 or 9,7,5,3
17) if a,b,c are in AP, then prove that a²(b + c), b²(c + a), c²(a+ b) will also in AP.
18) If (b - c)², (c - a)², (a - b)² are in AP, then prove that 1/(b - c), 1/(c - a), 1/(a - b) will be in AP.
19) If a₁, a₂, a₃,....., aₙ are in AP, then show that, 1/(a₁a₂) + 1/(a₂a₃) + 1/(a₃a₄) + .....+ 1/aₙ₋₁ aₙ) = (n -1)/(a₁aₙ).
20) The monthly salary of an office employee increases annually in AP. If he was drawing Rs200 per month during 11th year and Rs380 per month during 29th year, find out his initial monthly salary and rate of annual increment. Find out his salary at the time of retirement on complation of 32 years of service. Rs410
21) A person lends Rs9000 to his friend on condition that he will charge no intrest but he has to return the amount in monthly installment which will increase Rs20 per month . In How many months will the loan be refunded if the first installment be Rs 640 which will be paid one month after receiving the loan ? What is the amount of the last installment ? 12, Rs860
22) If a² is the arithmetic mean between xy and (1/2) (x²+ y²) then show that a is the AM between x and y. x=y= a
23) The first term of an AP is 2, last term is 29 and the sum of the terms is 155. Find the number of terms of the series. 10
24) If the sum of the first n terms of an AP is n², find the common difference. 2
25) If Sₙ = nP + n(n -1)Q/2 where Sₙ denotes the sum of n terms of an AP, then the common differences is
a) P+ Q b) 2P+ 3Q c) 2Q d) Q. d
R. N- 2
1) Find the indicating terms of the following series:
a) 21st and n terms of 5, 8, 11,..... 65, (3n +2)
b) 25th and p-th terms of 16+ 11+ 6+...... -104, 21- 5p
c) (n + 1)th term of 1/n , (n+1)/n, (2n +1)/n, .....n+ 1/n
2) The 15-th term of an AP is 49 and common difference is 4; find the 25th term. 89
3) The first term of an AP is 10 and 16th term is 0; find the 31st term. -10
4) The 12th term and the 21st term of an AP are respectively 82 and 145. Find the 32nd term. 222
5) The ratio of third and the 5th term of an AP is 2:3. Find the ratio of 7th and 9th terms. 4:5
6) In the AP 7, 12, 17,.......which term is 162 ? 32th
7) Can 103 be any term of the AP 5, 9, 13,.....? No
8) The n-th term of an AP is 4n -1. Find the common difference and 10th term of the AP. 4,39
9) The p-th term of an AP is q and q-th term is p; find the n-th and (p + q - n)-th terms. p+ q - n, n
10) The m-th terms of an AP is 1/n and n-th term is 1/m, find its mn-th term. 1
11) If p times the p-th term of an AP be q times the q-th term, find (p+ q)-th term. 0
12) If 9th term of an AP is 0, show that the 29th term is double and the 19th term.
13) Prove that in any AP the sum of the equidistant terms from the beginning and end is constant.
14) The p-th, q-th and r-th terms of an AP are respectively a, b and c. Show that a(q - r) + b(r - p) + c(p - q)=0.
15) Insert 10 arithmetic means between 2 and 57. 7,12,17,22,27,32,37,42,74,52
16) There are n arithmetic means between 4 and 40. If the ratio of first mean and last mean is 2:9, then find the value of n. 8
17) Find the middle terms of the following series:
a) 2 +4 +6 +......198. 100
b) 1+ 5 + 9+ ....+333. 165, 169
18) Find the sum of the following series:
a) 2 +5 +8+.... up to 25 terms. 950
b) 49 +44 +39+.... up 21 terms. -21
c) 1/2+ 3/2+ 5/2+...... up to n terms. n²/2
d) 1+ 4 + 7+......+ 37. 246
e) 9 + 7 + 5+.....+ (- 25). -144
f) (a - b)²+ (a²+ b²)+ (a+ b)²+......(a²+ 20ab + b²). 12(a²+ 9ab + b²)
g) 3 + 4 + 8+ 9 +13 +14 +.....up to 20 terms. 520
h) 1 - 3 + 5 - 7+ 9 - 11+.....up to 101 terms. 101
19) Find the sum of each of the following series without using the summation formula of AP
a) 1 + 3 + 5+.....up to n terms. n²
b) 5+ 8 + 11+..... up to 51 terms. 4080
20) The 5th and 11th terms of an AP are 41 and 20 respectively. What is its first term ? Find the sum of its first 11 terms. 55, 425/2
21) The 5th and 12th terms of an AP are 30 and 665 respectively. Find the sum of its first 20 terms. 1150
R. N- 3
1) The sum of first 9 terms of an AP is 171 and the sum of first 24 terms is 996. Find the sum of the first 16 term of the AP. 472
2) The 12th term of an AP 35 and the sum of first 16 terms is 392. Find the sum of its first 28 terms . 1190
3) The n-th term of an AP is (2n + 1). Find the sum of first n terms of the AP. n²+ 2n
4) The sum of the first n terms of an AP is n²+ 3n. Which term of it is 162 ? 80th
5) The sum of the first n terms of a series is pn²+ qn. Prove that the series is an AP .
6) if the sum of the first p terms of an AP is 4p²+ p, find the common difference of AP. 8
7) Show that the sum of n terms of the sequence (4, 12, 20, 28,...) is the square of an even number.
8) At least how much should be added with the sum of n terms of the series 8+16+24+ 32+.... to make the result a perfect square? 1
9) The middle term of an AP consisting of 11 terms is 12, find the sum of 11 terms of that series. 132
10) The first term of an AP is 9, last term is 96 and sum of the terms is 1575. Find the number of terms and common difference of the AP. 30,3
11) If the n-th term of an AP is p, then show that the sum of first (2n -1) terms is (2n -1)p.
12) If the sum of first P terms of an AP is equals to to the sum of the first Q terms , then show that the sum of the first (P+ Q) term is zero.
13) The first, second and last terms of an AP are a, b, c respectively. Show that the sum of the terms of the series is {(a+ c)(b + c - 2a)}/2(b - a)
14) The sum of the first n terms of an AP is q, and the n-th term is p; show that the first term is (2q - pn)/n.
15) The p-th term of an AP is 1/q and q-th term is 1/p. Show that the sum of pq number of terms is (pq+1)/2.
16) The p-th and q-th terms of an AP are respectively and b. Show that the sum of the first (p + q) terms is (1/2) (p+ q) {a + b + (a - b)/(p- q)}
17) The sum of the first p-terms of an AP and the sum of the first q terms of the same AP is p. Find the sum of the first (p + q) terms of the AP. -(p+ q)
18) How many terms of the series (9+ 12+ 15+....) must be added to get the sum 306? 12
19) How many terms of the series 27+24+21+18+..... must be added to get the sum 126?. 7 or 12
20) If Sₙ be the sum of first n terms of an AP, then show that, S₃ₙ = 3(S₂ₙ - Sₙ).
21) If S₁ + S₂, S₃ be the sum of n terms of three AP 's, the first term of each AP being 1 and the respective common difference being 1,2,3, show that S₁ + S₃ = 2S₂.
R. N - 4
1) How many numbers are there between 20 and 200 which are multiple of 6 ? Find the sum of thost numbers. 30, 3330
2) Find the sum of all the multiples of 13 between 750 and 1000. 16549
3) Calculate the sum of the integers from 1 to 100, which are not divisible by 3 or 5. 2632
4) The sum of n terms of two AP s are in the ratio (n +1): (n +3). Find the ratio of their 6-th terms. 6:7
5) The sum of the three numbers in AP is 15 and their product is 80, find the numbers. 2,5,8 or 8,5,2
6) Divide 36 in to three parts such that the parts are in AP and the sum and the squares of the parts is 450. 9,12,15
7) The sum of three number in AP is 12 and sum of their cubes is 288, find the numbers. 2,4,6 or 6,4,2
8) Four numbers are in AP. If the sum of the extremes be 10 and the product of the mean be 24, find the numbers. 2,4,6,8 or 8,6,4,2
9) 4 integers are in AP, if the sum is zero and product is 9, then find the integers . -3,-1,1,3 or 3,1,-1,-3
10) Five numbers are in AP. If their sum be 30 and the sum of their squares be 220, find the numbers. 2,4,6,8,10 or 10,8,6,4,2
11) The digits of a 3 digited number are in AP and the sum of the digits is 15. If 594 be added to this number the digits will be reversed . Find the original number. 258
12) If a,b,c are in AP, then show that
a) 1/bc , 1/ca, 1/ab are in AP
b) a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP
c) 1/(√b + √c), 1/(√c + √a), 1/√a + √b) are in AP
13) If a/(b + c), b/(c + a), c/(a+ b) are in AP and a+ b + c ≠ 0, then show that 1/(b + c) , 1/(c + a), 1/(a+ b) are in AP
14) If (b + c - a)/a, (c + a - b)/b, (a + b - c)/c are in AP and a+ b + c ≠ 0, then show that 1/a, 1/b, 1/c are in AP.
15) if a²(b + c), b²(c + a), c²(a + b) are in AP, and ab+ bc+ ca≠ 0, show that a,b,c are in AP.
16) If (a+ b)/(1- ab) , b, (b + c)/(1- bc) are in the AP, show that 1/a, b, 1/c are in AP.
17) If (b²+ c²- a²)/2bc , (c²+ a²- b²)/2ca, (a²+ b²- c²)/2ab are in AP, then show that a(b + c - a), b(c + a - b), c(a+ b - c) are in AP.
18) If a², b², c² are in AP, then show that 1/(b + c), 1/(c + a), 1/(a+ b) will be in AP.
19) The 3 sides of a right angler triangle are in AP. Show that the ratios of the sides are 3:4:5.
20) The three sides of a right angled triangle are in AP, if the length of the shortest side is 15cm, find the length of the longest side. 25cm
21) if a is the Arithmetic mean between b and c and 1/c is the Arithmetic mean between 1/a and 1/b, show that b²= ac.
R. N - 5
1) if 1/{2(y - a)} is the arithmetic mean between 1/(y - x) and 1/(y - z), show that, (x - a)(z - a)= (y - a)².
2) If (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the arithmetic mean between a and b, find the value of n. 0
3) The sum of first 50 terms of an AP is 200 and the sum of the next 50 terms 50 terms is 2700; find the common difference and the first term. 1, -41/2
4) if the last number of 30 consecutive odd integers be 127, find the sum of the numbers. 2940
5) Find the least value of n for which the sum of the first n terms of the series 12 + 20+ 28+..... is greater than 1020. 16
6) Which term of the series 19 + 91/5+ 87/5+.... is the first negative term ? Find also the least number of terms for which the sum of the series is negative. 25th
7) The sum of the first three of an AP consisting of n terms is p and the sum of the last three terms is q. Show that the sum of all terms is n(p +q)/6.
8) If the sum of first n terms of an AP is Sₙ; then show that,
Sₙ₊₃ - 3Sₙ₊₂ + 3Sₙ₊₁ - Sₙ = 0.
9) If the sum of p terms of an AP is the sum of q terms as p²: q², then show that p-th term= (2p -1): (2q -1).
10) If a₁ , a₂, a₃ ,.....a₂ₙ₊₁ are in AP, then show that,
1/a₁a₃ + 1/a₃a₅ +....+ 1/a₂ₙ₋₁a₂ₙ₊₁ = n/a₁a₂ₙ₊₁.
11) If a₁, a₂, a₃,...., aₙ are in AP where aᵢ > 0, for all values of i, show that 1/(√a₁ + √a₂) + 1/(√a₂ + √a₃)+ .....+ 1/(√aₙ₋₁ + √aₙ) = (n -1)/((√a₁ + √aₙ).
12) An object falling from rest covers 16 meters in the first second, 48 metres in the second, 80 metres in the third second and 112 metres in the 4th second. What is the distance covered by the object and the 10th second and in 10 seconds ? 304,1600,
13) A man has to travel 162 miles. He travels 30 miles on the first day, 27 miles on the second day, 24 miles on the third day and so on. How many days does he take for the journey ? 9 days
14) In order to pay off a debt of Rs 650 by monthly installment, a person pays Rs20 as the first installment and increases each subsequent instalments by Rs10 than the proceeding one. In how many months will the debt be cleared off ? 10 months
15) The interior angles of a polygon are in AP having common difference 5°. If the least angle be 120°, find the number of sides of the polygon. 9
16) A class consists of a number of students whose ages are in AP, the common difference being 4 months. If the youngest student is just 8 years old and the sum of the ages of all the students is 168 years, find the number of students in the class. 16
17) A man arranges to pay off a debt of Rs36000 by 40 annual installments which form an AP. When 30 of the instalment are paid, he dies leaving one third of the debt unpaid . How much did he pay in the first installment ? Rs510
18) A sets out from a place and travels at the rate of 5 miles an hour. B. sets out 9/2 hours after A and travels in the same direction, 3 miles in the first hour, 7/2 miles in the second hour, 4 miles in 3rd hour and so on. Find in how many hours B will overtake A. 15 hours
19) The cost of sinking a 600 metre deep tubbewell is 25 paise for the first metre and an additional 4 paise for every subsequent metres. Find the cost of sinking the 500th metre and the entire tubewell. Rs20.21, Rs7338
20) To verify the cash balance, the auditor of a bank employes on assistant to count Rs4500. He counted at the rate of Rs150 per minute for the first 10 minutes , then he begins to count at the rate of Rs2 less in every minute than he did in the previous minute. Find how much time he will take to count Rs4500 ? 34
21) Two posts are offered to a man. In the first one, the initial salary is Rs1200 per month and increases annually by Rs80; in the second one, the initial salary is Rs850 per month and increases annually by Rs120. If the man has to retire after 20 years, which post will be financially acceptable to him ? Second post
RAW- 1
1) The three angles of a right angled triangle are in AP. Find the angles . 30,60,90
2) The sum of the first five terms of an AP is 20, find its third term. 4
3) The 10th term of an AP is 44 and common difference is 4; find the first term of the series. 8
4) The 15th term of an AP is 33 and the 32nd term is 67; find the first term and the common difference. 5, 2
5) The first two terms of an AP are 5 and 8 and the last term is 80; find the number of terms. 26
6) 3k+1, 7k, and 10k +8 are in AP, what will be the common difference ? 35
7) The third term of an AP is twice that of its first term and the fourth term is 15; find the common difference of the AP. 3
8) Find one arithmetic mean between (-5) and 15. 5
9) Find the arithmetic mean between (x - y)² and (x + y)². x²+ y²
10) 2, x, 8, y are in AP; find the values of x and y. 5,11
11) In an AP the first term is (-5), the last term is 25 and the number of terms is 10. What is the sum of the series ? 100
12) A polygon has 25 sides, the lengths of which are in AP. If perimeter of the polygon be 1100cm and the length of the largest side is 10 times that of the smallest, find the length of the smallest side. 8cm
13) The sum of how many terms of the AP 21+ 18+ 15+ 12+....will be zero? 15
14) Find the sum of the first hundred natural numbers . 5050
15) If each term of a series in AP be multiplied by 3, would the series so obtained be again in AP. Yes
16) Let Sₙ denotes the sum of first n terms of an AP. If S₂ₙ = 3Sₙ then find the values of S₃ₙ/Sₙ.
17) On a straight road, there are 100 stones, placed one after another, the distance between any two consecutive stones being 1 metre . Also, there is a basket, 1 metre behind the first stone. Starting from the basket, a man walks to collect the stone one by one and put them into the basket. What distance to would he have to walk. 10100
GEOMETRIC PROGRESSION
EXERCISE - 1
1) Find the ninth and the n-th term of the series 4, - 8,16, - 32,... 1024, (-2)ⁿ⁺¹
2) The 4-th and the 7-th term of a GP are -135 and 3645 respectively. Find the progression. What is the 6-th term of the GP? -1215.
3) Which term of the GP √6, 2√3, 2√6, 4√3,....is 64√3? 12
4) if x, y, k be the p-th, q-th, and r-th terms respectively both of an AP and of a GP, then show that xʸ⁻ᵏ. yᵏ⁻ˣ. zˣ⁻ʸ=1.
5) Insert three geometric means between 2 and 162. 6,18,54 or -6,18,-54
6) Find the sum :
a) 1 - 1/2 + 1/2²+..... up to 20 terms. (2/3)(1- 2⁻²⁰
b) 5+ 555 + 55555+....up to n terms. (50/891) (10²ⁿ -1) - 5n/9.
7) Find the sum to n terms of the GP series: 1+ 1/2+ 1/2²+.... 2 - 1/2ⁿ⁻¹
8) If the 6th and the 9th terms of a GP are 64 and 512, find the sum of first 9 terms of the GP. 1022
9) If the sum of first n terms of a GP is p and the sum of the first 2n terms is 3p, then show that the sum of first 3n terms is 7p.
10) If in a GP first term is a, n-th term is b, and product of first n term is P, then prove that P²= (ab)ⁿ.
11) If S be the sum, P the product and R the sum of the reciprocals of n terms of a GP., prove that P²= (S/R)ⁿ.
12) Find three numbers in GP whose sum is 35 and product is 1000. 20,10,5
13 The sum of three numbers in AP is 15. If 1, 4 and 19 are added to them respectively, the results are in GP. Find the numbers. 26,5,16 or 2,5,8
14) The ratio of AM and GM of two numbers is m: n; show that the ratio of the number is m + √(m²- n²): m - √(m²- n²).
15) If one Arithmetic mean A and two Geometric mean p,q ne inserted between two given quantities then prove, that, p²/q + q²/p = 2A.
16) Show that the arithmetic mean of two positive quantities can never be less than their geometric mean.
17) Find the sum : 1+ (1+ 3) + (1+ 3+ 3²)+.....up to n terms. (1/4)(3ⁿ⁺¹ - 2n -3)
18) Find the n-th term and the sum up to n-th term of the series:
1+3+7+15+...... 2ⁿ⁺¹ - n -2
19) Find the sum: 1+ 3/2+ 5/2²+ 7/2³+.... up to n terms. 6- (2n +3)/3ⁿ⁻¹ .
20) Prove that in a GP, the product of any two terms equidistant from the beginning and the end is constant.
21) In a GP, first term is 7, last time is 448 and sum of the terms is 889; find the common ratio and number of terms of the GP. 2, 7
EXERCISE - B
1) A, B and C have together Rs5700 and the amount of money possessed by them form a GP. If B had Rs150 more, the amounts would form an AP. Find the amounts they possess. 2700,1800,1200
2) A bouncing tennis ball rebounds each time to a height equal to one half the height of the previous bounce . If it is dropped from a height of 16 m, find the total distance it has travelled when it hits the ground for the 10th time . 47+15/16
3) The second term of a GP is b and the common ratio is r. If the product of the first three terms of the GP is 64, find b. 4
4) The n-th term of a series is 4.3ⁿ⁻¹. show that the series is a GP.
5) If the (p + q)th term of a GP is m and (p - q )th terms is n, then find its p-th term. ±√(mn)
6) If a, b, c are in GP and a, 2b, 4c are in AP, then find the common ratio of the GP. 1/2
7) Find 10th term and n-th term of 16,8,4,2,..... 1/32, 2⁵⁻ⁿ
8) The 8th and 2p-th terms of 6, -12,24, -48,......(p is an integer). -768, -3.2²ᵖ
9) The 10th and (2n +1)-th terms of the series 4, 2√2, 2, √2,.... √2/8, 2²⁻ⁿ
10) The fourth term of a GP is 32 and the common ratio is (-1/2), find the 10th term. 1/2
11) The first term of a GP is 64 and 4th term is 216; find the 7th term. 729
12) the second term of a GP is 9/16 and the fifth term is 1/6; find the 7th term. 2/27
13) The 9th term of a GP is 36 and 21-st term is 972; find the 5-th term. 12
14) The p-th term of a GP is q and the q-th term is p; find the (2p - q)-th term. q²/p
15) Which term of the series 4, -8, 16,-32,.... 1024? 9
16) Can 640 be any term of the GP 5, -10, 20,....? No
17) If a,b,c are in AP and x, y, z are in GP., then show that, xᵇ⁻ᶜ, yᶜ⁻ᵃ, zᵃ⁻ᵇ = 1
18) If p,q,r are in AP then, show that the p-th, q-th and r-th terms of any GP are in GP.
19) If p-th, q-th and r-th terms of a GP are a, b and c respectively, then show that aᑫ⁻ʳ. bʳ⁻ᵖ. cᵖ⁻ᑫ = 1.
20) aᵖ = bᑫ = cʳ and a, b, c are in GP then show that p⁻¹, q⁻¹, r⁻¹ will be in AP.
21) If a¹⁾ˣ = b¹⁾ʸ = c¹⁾ᶻ and a, b and c are in GP, then show that x, y, z will be in AP.
EXERCISE - 3
1) Insert two Geometric means between 5 and 135. 15,45
2) if 5, x, y, z , 405 are in GP, then find the value of x, y, z. ±15,45,±135
3) Find the sum of the following series:
a) 1- 3+ 9 - 27+....to 9 terms. 4921
b) 0. 3+ 0.03+ 0.003 + .....to n terms. (1/3)(1- 1/10ⁿ)
c) 1/8+ 1/4+ 1/2+.....+64. 127+7/8
d) 2+ √2+1+....to 12 terms. (63/32)(2+ √2)
e) (a+b)/(a- b) + 1+ (a - b)/(a+ b)+.... n-th term. (a+ b)²/2b(a- b)[1- {(a - b)/(a+ b)}ⁿ]
f) 1/2 - 1/3 + 1/4 - 1/9 + 1/8 - 1/27+....up to 2n terms. 1/2 - 1/2ⁿ + 1/(2.3ⁿ)
g) .9+ .99+ .999+......up to n terms. n - (1/9)(1- 1/10ⁿ)
h) 11+ 103+ 1005+.... to n-th term. (10/9)(10ⁿ -1)+ n²
i) 4 + 44 + 444+....up to n terms. (40/81)(10ⁿ -1) - 4n/9
j) 0.6+ 0.66+ 0.666+.... upto n terms. 2n/3 - (2/27)(1- 10ⁿ)
k) (2+ 1/2)²+ (2²+ 1/2²)²+ (2³+ 1/2³)²+....to n-th term. (4/3) 2²ⁿ - 1/(3.2²ⁿ)+ 2n -1
4) How many terms of the series. 3+ (-6)+ 12+ (-24)+ .... must be added from the first so that the sum maybe -1023 ? 10
5) if the 4th term of a GP is 24 and 7th term is 192, find the sum of its first 10 terms. 3069
6) in a GP, the sum of first and second terms is 16 and the sum of the fourth and 5th terms is 432. Find the sum of first 6 terms of the GP. 1456
7) The sum of the first three terms of a GP is 84 and the sum of the first 6 terms is 756. Find the sum of the first 8 terms of the GP. 3060
8) The ratio of the sum of first three terms and first 6 terms of a GP is 125 :152. Find the common ratio of the GP. 3/5
9) If S₁, S₂, S₃ be respectively the sum of first n, 2n, and 3n terms of a GP, then show that: S₁(S₃ - S₂) = (S₂ - S₁)²
10) If S be the sum, P the product and R the sum of the reciprocals of the 5 terms , in a GP, then show that P½= (S/R)⁵.
11) The sum of 2n terms of a series in GP is 2R and the sum of the reciprocals of those terms is R; find the continued product of the terms.
12) Find three numbers in GP whose sum is 21 and product is 64. 1,4,16 or 16,4,1
13) The sum of thhree consecutive numbers in GP is 21 and the sum of their square is 189. Find the numbers . 12,6,3 or 3,6,12
14) Divide 26 into 3 parts such that the parts form a GP and the sum of the products taking two parts at a time is 156. 2,6,18 or 18,6,2
15) Sum of the four numbers in GP is 180; sum of their two extreme members is 108; find the numbers . 12,24,48,96 or 96,48,24,12
16) The sum of three positive numbers of AP is 21. If 4, 5 and 8 are added respectively to the numbers , the results are in GP. Find the numbers. 4,7,10
17) The first, 8th and 22-nd terms of an AP are 3 consecutive terms of a GP. Find the common ratio of the GP . 2
18) A number consists of 3 digits which are GP and the product of the digits is 216. If 495 be added to the number, the digits will be reversed. Find the original number. 469
19) The Arithmetic mean of two positive numbers is 15, and their Geometric mean is 9. Find the numbers. 3 and 27
20) If AM of two numbers be twice their GM, show that the numbers are as (2+ √3): (2- √3).
21) If 2 quantities are in the ratio (3+ 2√2): (3- 2√2) then show that their Arithmetic mean is thrice of their geometric mean.
EXERCISE - 4
1) The difference of two positive numbers is 54 and the difference of their arithmetic and geometric means is 9. Find the numbers. 72,18
2) If G be the geometric mean and p and q be two arithmetic means between two given quantities, then show that G²= (2p- q)(2q - p).
3) If a,b,c are in GP and x,y are Arithmetic mean of a, b and c respectively, show that
a) a/x + c/y = 2
b) 1/x + 1/y = 2/b
4) If a, b,c are in AP and b,c,a are in GP, prove that 1/c, 1/a , 1/b are in AP.
5) If a,x,y,b are in AP and c², x, y, d³ are in GP, show that, a+ b = cd(c + d).
6) Let A be the arithmetic mean and G be the positive geometric mean of two unequal positive quantities , show that A> G > G²/A.
7) If a,b,c are in GP, then show that
a) a²+ b², ab+ bc, b²+ c² are in GP
b) (a²+ ab+ b²)/(bc+ ca+ ab)= (a+ b)/(b+ c).
c) a²b²c²(1/a³ + 1/b³+ 1/c³)= a³+ b³+ c³.
d) 1/(a+ b) , 1/2b, 1/(b + c) are in AP
8) If a,b,c,d are in GP then show that
a) a²+ b², b²+ c², c²+ d² are in GP
b) (a - b)², (b - c)², (c - d)² are in GP
c) (b - c)²+ (c - a)²+ (d - b)²= (a - d)²
d) (a²+ ac + c²)(b²+ bd + d²)= (ab + bc + CD)².
e) a²+ b²+ c², ab + bc + cd, b²+ c²+ d² are in GP
9) Find the sum: (1+ (1+4)+(1+4+4²)+.....up to n terms. (4/9)(4ⁿ -1) - n/3
10) Find the term and sum up to n-th term:
a) 3+5+9+17+.... 2ⁿ+ 1, 2ⁿ⁺¹+ n -2
b) 1+4+13+40+.... (1/2) (3ⁿ -1), (3/4) (3ⁿ -1) - n/2
11) Find the sum up to n-th term:
a) 1+ 2a + 3a²+ 4a³+..... (1- aⁿ)/(1- a)² - naⁿ/(1- a)
b) 1+ 2/3 + 3/3²+ 4/3³+.... (9/4) - (2n +3)/4.3ⁿ⁻¹
12) If the number of terms of a GP be odd, show that the product of the first and the last terms is equal to the square of the middle term.
13) The first and last terms of a GP are respectively 36 and 972 and the sum of the terms is 1440. Find the common ratio and number of terms of the GP. 3,4
14) Three numbers are in GP whose product is 216. If 4 be added to the first term and 6 to the second term, then the resulting numbers and the third number are in AP. Obtain the numbers in GP. 2,6,18 or 18,6,2
15) The sum of three numbers in GP is 70. If the two extremes be multiplied each by 4 and the mean by 5, the product are in AP. Find the numbers. 10,20,40 or 40,20,10
16) The sum of three numbers in GP is 14. If the first two numbers are each increased by 1 and the third number decreased by 1, the resulting numbers are in AP. Find the numbers in GP. 2,4,8 or 8,4,2
17) Insert two numbers between 6 and 16 such that the first three numbers are in AP and the last three numbers are in GP. 9 and 12
18) Insert two numbers between 4 and 12 such that the first three numbers are in GP and the last three numbers are in AP. 6 and 9
19) The first terms of an AP and of a GP are each 4 and their third terms are same. If the second term of the AP exceeds that of the GP by 8, find AP and the GP. 4,20,36. 4,12,36
20) If a,b,c are in AP and a,d, d are in GP, then show that a, A'- b, d - c are in GP
21) If a,b,c are three unequal numbers such that a,b,c are in AP and b - c, c - b, a are in GP, then show that a:b: c = 1:2:3.
EXERCISE - 5
1) If Sₙ = 1+ 1/2 + 1/2²+......+1/2ⁿ⁻¹, find Sₙ and the least value of n for which 2 - Sₙ < 1/100. 2 - 1/2ⁿ⁻¹, 8
2) If (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the geometric mean between a and b, then find the value of n. -1/2
3) Solve for x:
a) 1+ a + a²+ a³+....aˣ = (1+ a)(1+ a²)(1+ a⁴)(1+ a⁸)(1+ a¹⁶). 31
b) 2+ 4+8+.....+ x= 1022. 512
4) If x,y,z are in GP and a= x - 1/x, b = y - 1/y, c= z - 1/z, then show that r + 1/r = (a+ c)/b, where r is the common ratio of the GP.
5) A man borrowed Rs8190 without intrest and repaid the sum in 12 installments. If the amount of each installment be double that the just preceding instalment, find the amount of the first and last installments . Rs2, Rs4096
6) A man borrows Rs19682 without interest and agree to repay the money in 9 monthly installment, each installment being thrice the preceding one. After the 7th installment has been paid, he wants to repay the balance in lump. How much has he to pay now. Rs17496
7) When a certain tennis ball is dropped from a height it bounces to a height of three-fifth the distance from which it fell. If the ball is dropped from a height of 125 metres, how far has it travelled when it hits the ground for the 6-th time? 470+21/25
8) The product of first three terms of a GP is 27/8; find the middle term. 3/2
9) The first term of a GP is 5 and 4-th term is 320; find the common ratio. 4
10) The second term of a GP is - 24 and 5th term is 81; find its first term. 16
11) The first two terms of a GP are 9 and 3. Which term of the series is 1/243 ? 8th
12) A GP has first term 6 and last term 96. If each term is twice its preceding term, find the number of terms. 5
13) 2x, 2x +1 and 2x +3 are in GP; find the common ratio. 2
14) If 3a +1, 6a - 4 and 3a - 2 are in GP. Find the value of a. 1
15) The third term of a GP is equal to the square of the first term and 4th term is 32, find the common ratio. 2
16) The sum of first two terms of a GP is 16 and sum of next two terms is 144. Find the common ratio of the series. ±3
17) if the r-th term of the series is 4. 3ʳ⁻², show that the series is a GP
18) the sum of first n terms of a series is 4(3ⁿ -1). Show that the series is a GP.
19) Find the geometric mean between 3/4 and 16/27. ±2/3
20) If 3 positive numbers a,b,c are in GP, then show that a+ c > 2b.
21) if x, y, z are such that (x²+ y²)(y²+ z²)= (xy+ yz)², then prove that x,y,z are in GP.
EXERCISE - 6
1) If p,q,r are in GP, find the value of (q - p)/(q - r) + (q + p)/(q+ r).
2) Can any terms of a GP be 0 ? No
3) Is there any two different real numbers such that AM and GM are equal ? No
4) What type of series will be formed by the reciprocals of the term of a GP ? GM
5) Can three different quantities be in AP and GP at the same time ? No
6) The third term of a GP is 4; the product of the first term is
a) 4² b) 4³ c) 4⁴ d) 4⁵. d
7) The sum of the first n terms of a GP is denoted by Sₙ. If S₂ₙ = 4Sₙ, then what is the value of S₃ₙ/Sₙ ? 13
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