1) α is the real cube root of 2 and β, γ are its imaginary cube roots,
then (xβ + yγ + zα)/(xγ + yα + zβ) is equal to
a) ³√2 b) ³√2 ω½ c) ω² d) none
2) The modulus amplitude form of the complex number -1 - I is
a) -√2(cos(π/4) + i sin(π/4))
b) √2(cos(3π/4) + i sin(3π/4))
c) √2(cos(-3π/4) + i sin(-3π/4)) d) none
3) The amplitude of √12 + 6{(1- i)/(1+ i)} is
a) π/3 b) 2π/3 c) -π/3 d) -2π/3
4) If x√2 = 1+ √-1, then the value of x⁶+ x⁴+ x² +1 is
a) 0 b) 4 c) -4 d) none
5) The value of √i + √-i is/are
a) √2 b) ±√2 c) ±√2 i d) ±√2, ±√2 i
6) If y= √(z² + 6x +8), then (1- iy)¹⁾² is equal to
a) ±(1/√2){i √(x +4) - √(x +2)}
b) ±(i/√2){√(x +4) - √(x +2)}
c) ±(1/√2){√(x +4) - i √(x +2)} d) none
7) The square roots of a² + 1/a² - 4i(a - 1/a) - 6 are
a) ±(a - i/a -2)
b) ±(a - 1/a -2i)
c) ±(a - 1/a +2i) d) none
8) If ³√(x + iy)= a + ib, where a,b, x, y are real, then x/a + y/b is equal to
a) 4(b² - a²) b) 4(a² + b²) c) 4(a² - b²) d) none
9) If n is a +ve integer, not a multiple of 3, the {(1+ √-3)/2}ⁿ + {(-1- √-3)/2}ⁿ is equal to
a) -1 b) 2 c) 0 d) none
10) If x + iy moves on the line 3x + 4y +5=0, then the least value of |x + it| is
a) 0 unit b) 1/5 unit c) 1 unit d) none
11) If z₁ = √3 i and z₂= -1+ √3 i, then amp(z₁z₂) is equal to
a) 7π/6 b) -5π/6 c) 5π/6 d) none
12) If {(2- i)x + i}/(1+ i) + {(1+ 2i)y + i}/(1- i) = -1/2 + 5i/2, where x and y are real, then x - y is equal to
a) 1 b) -1 c) 6 d) 8
13) If {(1+ i)x - 2i}/(3+ i) + {(2- 3i)y + i}/(3- i)= i, where x and y are real, then 4x + 9y is equal to
a) 10 b) -10 c) 3 d) -3
14) The modulus of {(1- i)/(3+ i)} + 4i/5 is
a) √5 unit b) √11/5 unit c) √5/5 unit d) none
15) The least positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is
a) 2 b) 4 c) 8 d) 16
16) If (√3 + i)¹⁰⁰ = 2¹⁰¹ (a + ib), then a is equal to
a) 4 b) -4 c) 1/4 d) -1/4
17) If (a + 1)²/(2a - i) = p + iq, then p² + q² is equal to
a) (a² +1)²/(4a² -1) b) (a²+1)²/(2a² -1) c) (a² +1)²/(4a² +1) d) none
18) The complex numbers z= x + iy which satisfy the equation|(z - 5i)/(z + 5i)|= 1, lie on
a) x = -5 b) y= 6 c) the x-axis d) the y-axis
19) For any complex number z= x + it, if the imaginary part of (2z +1)/(iz + 1) is -2, then the locus of z is
a) a straight line
b) a circle
c) an ellipse d) none
20) If the complex number z= x + iy satisfies the condition |(z - k)/(z + ki)|= 1, where k is any real number, then the locus of z is
a) a straight line b) a circle c) an ellipse d) none
21) The complex number z= x + iy satisfying the condition amp{(z - i)/(z + i)}=π/4 lies on
a) a straight line b) a circle c) an ellipse d) none
22) The complex number z= x + iy satisfying the condition amp{(z -1)/(z +1)}=π/6 lies on
a) a straight line b) a circle c) an ellipse d) none
23) If the arguments of (z - a)(conjugate of z - b) and {(√3+ i)(1+ √3 i)}/(1+ i) are equal where z is a complex number and a, b are real numbers, then the locus of z is
a) a straight line b) a circle c) a parabola d) none
24) The locus of the complex number z= x + it satisfying the condition 'real part of 1/z = 1/4' is
a) a straight line at a distance of 4 unit from the imaginary axis
b) a circle with radius 1 unit
c) a circle with radius 2 unit
d) a straight line not passing through the origin.
25) The equation conjugate of b .z + b. conjugate of z= c, where b is a non-zero complex constant and C is a real, represents
a) a circle b) a straight line c) a parabola d) an ellipse
26) If z = x + it and if z. conjugate of z - (2+ 3i)z - (2- 3i). conjugate of z+ 9 = 0, then the locus of z in the complex plane is
a) a straight line b) an ellipse c) a circle d) none
27) If z= x + iy and w= (1- iz)/(z - i), then |w|= 1 implies that, in the complex plane
a) z lies on the imaginary axis
b) z lies on the real axis
c) z lies on the unit circle with centre at origin
d) none of these happens.
28) If x+ iy = 1/(2+ cosθ + i sinθ), where x, y, θ are real, then as θ varies, in the complex plane the point z= x + iy moves on
a) a straight line b) an ellipse c) a parabola d) a circle
29) Let i² = -1; then
a) i and - i each has exactly one square root.
b) i has two square root but - i does not have any.
c) neither i and - i each has any square root.
d) i and - i each has exactly two square root.
30) The expression {(1+ i)ⁿ/(1- i)ⁿ⁻² is equal to
a) iⁿ⁺¹ b) - iⁿ⁺¹ c) 1 d) -2iⁿ⁺¹
31) The argument of the complex number z= (1+ i √3)²/4i(1- i √3) is
a) π/6 b) π/4 c) π/2 d) none
32) Let z and w be two non-zero complex numbers such that |z|= |w| and amp z + amp w =π, then z is equal to
a) w b) - w c) conjugate of w d) - conjugate of w
33) If z and w are two complex numbers satisfying the equation|(z + w)/(z - w)|= 1, Then z/w is a number which is
a) positive real
b) negative real
c) zero d) none
34) If z₁ and z₂ are two complex numbers such that
|z₁ + z₂|= |z₁|+ |z₂|, then
a) arg z₁ = arg z₂
b) arg z₁ + z₂ = 0
c) arg (z₁z₂)= 0 d) none
35) If z₁ and z₂ are two complex numbers such that |z₁|²+ |z₂|²= |z₁ + z₂|², then
a) Re(z₁/z₂)= 0
b) Im(z₁/z₂)= 0
c) Re z₁z₂ = 0
d) Um(z₁z₂)= 0
36) If z₁ and z₂ are two complex numbers, then|z₁ + √(z₁² - z₂²)| + |z₁ - √(z₁² - z₂²)| is equal to
a) |z₁| b) |z₂| c) |z₁ + z₂| d) |z₁ + z₂|+ |z₁ - z₂|.
37) If a² + b² = 1, then (1+ a+ ib)/(1+ a - ib) is equal to
a) a+ ib b) a- ib c) b+ ia d) b- ia
38) If z is an imaginary number and z/(1+ z) is purely imaginary, then z
a) can be neither real or not purely imaginary
b) is real
c) is purely imaginary
d) satisfies none of these properties.
39) In a GP the first term and the common ratio are both (1.2) (√3+ i), then the absolute value of the nth term of the progression is
a) 2ⁿ b) 4ⁿ c) 1 d) none
40) The complex number z satisfying |z - 1|= |z -3|= |z - i| is
a) 2+ i b) 3/2 + i/2 c) 2+ 2i d) none
41) The value of the expression
1(2- ω)(2- ω²)+ 2(3- ω)(3- w²)+ ....+ (n -1)(n - ω)(n - ω²), where ω is an imaginary cube root of unity, is
a) {n(n +1)/2}²
b) n²(n+1)².4
c) n²(n+1)²/4 + n d) none
42) The number of solutions of the equation z² = conjugate of z, where z is a complex number, is
a) 2 b) 3 c) 4 d) none
43) The solution of the equation|z| - z = 1+ 2i is
a) 2 - 3i/2 b) 3/2 - 2i c) 3/2+ 2i d) -2+ 3i/2
44) The number of solutions of the equation z² + |z|²= 0, where z is a complex number, is
a) 1 b) 2 c) 3 d) none
45) If x= cosα + i sin α, y = cosβ + i sinβ, z= cosγ + i sinγ and x+ y+ z= 0, then 1/x + 1/y + 1/z is equal to
a) xyz b) 1 c) Re x + Re y + Re z d) 0
46) If x + 1/x = 2 cosθ then for any integer n, xⁿ - 1/xⁿ is equal to
a) 2 cosnθ b) 2i 2i sin nθ c) - 2i sin nθ d) none
47) 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
48) Given that the equation z² + (p + iq)z + r + i s= 0 where p,q,r and s are non-zero real numbers, has a real root, then
a) pqr= r² + p²s
b) pqs= s² + q²r
c) prs = q²+ r²p
d) qrs= p² + s²q
49) The equation z² + conjugate of z ² - 2|z|²+ z + conjugate of z represents
a) a straight line
b) a circle
c) an ellipse
d) a parabola
50) The complex numbers sinx + i cos2x and cosx - i sin2x are conjugate to each other for
a) x= 0 b) x= nπ c) x= (2n+1)π/2 d) no value of x.
51) The greatest value of the modulii of the complex numbers z satisfying the equation |z - 4/z|= 2 is
a) √5 -1 b) √5 c) √5+ 1 d) none
52) For any complex number z, the minimum value of |z| + |z -1| is
a) 0 b) 1/2 c) 1 d) 3/2
53) Let z₁ and z₂ be two complex numbers such that z₁/z₂ + z₂/z₁ = 1, then
a) z₁, z₂ and the origin are collinear
b) z₁, z₂ and the origin form a right angled triangle
c) z₁, z₂ and the origin form an equilateral triangle d) none
54) The maximum values of |z| when z satisfies the condition |z - 2/z|= 2, is
a) √3- 1 b) √3 c) √3 + 1 d) √2+ √3
55) If α is a root of the equation x²+ x +1= 0, then α³ᵐ + α³ⁿ⁺¹ + α³ᵖ⁺² (where m,n,p are three integers) is equal to
a) -3 b) 3 c) 1 d) 0
56) The area of the triangle in the complex plane formed by the points z, iz and z + iz is
a) |z|² b) (1/4 |z|² c) (1/2) |z|² d) z²/2
57) z₁, z₂, z₃, z₄ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z₁ - z₄ = z₂ - z₃ and amp (z₄ - z₁)/(z₂ + z₁) =π/2. Then the quadrilateral is
a) a square b) a rectangle c) a rhombus d) none
58) The complex numbers z₁, z₂, z₃ are respectively the vertices A,B, C of a parallelogram ABCD, then the fourth vertex D is
a) (1/2)(z₁ + z₂)
b) (1/2) (z₁ + z₂ + z₃)
c) (1/3)(z₁+ z₂ + z₃)
d) z₁ + z₃ - z₂.
59) Suppose z₁, z₂, z₃ are the vertices of an equilateral triangle circumscribing the circle |z|= 2. If z₁= 1+ √3 i and z₁z₂, z₃ are are in the anticlockwise sense, then z₂ is
a) 1- √3 i b) 2 c) 1- √3 I d) -2
60) In the complex plane the points -2+ i, 1+ 2i, 4+ 5i and 1+ 4i form
a) a square b) a rectangle c) a parallelogram d) none
61) The statement 'a + ib> c + id where a,b,c,d are real numbers and i²= -1' is
a) correct when a> c and b> d
b) correct when a> c and b= d
c) correct when a> c and b= d=0 d) never true
62) The statement 'mi > ni where m,n are real numbers and i² = -1' is
a) correct when m> n and m,n are positive rational numbers
b) correct when m> n and m,n are positive prime integers
c) correct when m> n =π
d) meaningless for all real numbers m,n.
63) The statement 'i² = -1' is equivalent to
a) i= √-1, taking positive square root
b) i= - √-1, taking positive square root
c) √-1= ± i
d) all the statement in above three options.
64) If arg z< 0, then arg(-z) - argz is equal to
a) π b) -π c) -π/2 d) π/2
65) If z² + z +1= 0 where z is complex number, then the value of (z + 1/z)²+ (z² + 1/z²)²+ (z³+ 1/z³)²+....+ (z⁶ + 1/z⁶)² is equal to
a) 6 b) 12 c) 18 d) 54
66) The value of ¹⁰ₖ₌₁∑ (sin(2kπ/11) + i cos(2kπ/11) is equal to
a) -1 b) - I c) I d) 1.
67) The value of ⁶ₖ₌₁∑(sin(2πk/7 - I cos(2πk/7)) is equal to
a) -1 b) - I c) I d) 1
68) Let n(∈ N) be a multiple of 5 and x= cos(2π/5)+ i sin(2π/5) then
1+ xⁿ+ x²ⁿ + x³ⁿ + x⁴ⁿ is equal to
a) 0 b) 5 c) -5 d) 5 i
69) Let α = cos(4π/3) + i sin(4π/3) then the value of {(1+ α)/2}³ⁿ is equal to
a) (-1)ⁿ b) (-i)ⁿ c) (-1)ⁿ/2³ⁿ d) 1/2³ⁿ.
70) Let zₙ = cos(2nπ/7) + i sin(2nπ/7), n = 0,1,2,....6 then z₁z₂z₃....z₆ is equal to
a) 0 b) 1 c) -1 d) - i
71) Let zₙ = cos(π/2ⁿ)+ i sin(π/2ⁿ) then z₁z₂z₃....to ∞ is equal to
a) 0 b) -1 c) 1 d) i
72) The product of values of (1+ i √3)³⁾⁴ is equal to
a) 80 -8i -8 d) 8
73) Let α, β are the roots of the equation x²- 2x cosθ +1= 0, then the equation whose roots are αⁿ and βⁿ, is
a) x² + 2x cos nθ + 1
b) x² - 2x cos nθ + 1
c) x² + 2x cos nθ - 1
d) x² - 2x cos nθ - 1
74) If |z|= 1 and w= (z -1)/(z +1), z≠ -1 then Re w is equal
a) 0 b) z/|z +1|² c) |z/(z +1)| z/|z +1|² d) √2/|z +1|²
75) If one root of ax²+ bx - 2c = 0 (a,b,c are real) is imaginary and 8a + 2b > c, then
a) a> 0 and c< 0
b) a> 0 and c> 0
c) a< 0, c > 0
d) a< 0 and c< 0
76) If x²+1= √3 x, then (x³ - 1/x³)ⁿ⁺¹ for any n ∈ N is equal to
a) ω b)0 c) ±1 d) ± i
77) Let z be a complex number and z= (1- t²)+ i √(1+ t²), where t is a real parameter, then the locus of z in the complex plane is
a) a straight line b) a parabola c) a hyperbola d) an ellipse
78) Let z be a complex number and z= 1- t + i √(t²+ t +2), where t is a real parameter then the locus of z is
a) a straight line b) a parabola c) a hyperbola d) an ellipse
79) If the three complex numbers z₁, z₂, z₃ are in AP then z₁, z₂, z₃ lie on
a) a straight line b) a circle c) a parabola d) an ellipse
80) Let z₁ and z₂ are two complex numbers such that |z₁|= |z₂|= 1, then |(z₁ - z₂)/(1- z₁. Conjugate of z ₂)| is equal to
a) 2 b) 1/2 c) 1 d) none
81) The modulus of the complex number {(2+ i√5)/(2- i √5)}¹⁰ + {(2- i√5)/(2+ i √5)}¹⁰ is equal to
a) 2 sin(10 cos⁻¹(2/3))
b) 2 sin(20 cos⁻¹(2/3))
c) 2 cos(20 cos⁻¹(2/3))
d) 2 cos(10 cos⁻¹(2/3))
82) Let C be the set of all complex numbers and A, B be two subsets of C x C defined by A={(z,w): |z|= |w| and z, w ∈ C}
B={(z,w): z²= w², z, w ∈ C} then
a) A= B b) A⊆B c) B⊆A d) none
83) If |z²- 1|= |z|²+ 1, then z lies on
a) the real axis b) the imaginary axis c) a circle d) an ellipse
84) If x+ iy= |6i -3i 1
4 3i -1
20 3 i | then
a) x=3, y= 1 b) x=1, y= 3 c) x=0, y= 3 d) x=0, y= 0
85) If z and w are two complex numbers such that |zw|= 1 and arg z - arg w=π/2, then conjugate of z. w is equal to
a) -1 b) i c) 1 d) - i
86) If z is a complex number such that iz³+ z²- z + i =0, then |z|=
a) 2 b) 1 c) √2 d) none
87) (√3/2 + i/2)¹⁷⁷ is equal to
a) i b) - I c) -1 d) √3/2 - i/2.
88) Let z and w be two complex numbers such that conjugate of z+ i. conjugate of w = 0 and arg zw =π, then arg z is equal to
a) π/4 b) π/2 c) 3π/4 d) 5π/4
89) If ω is an imaginary cube root of 1 then determinant
1 1+ ω² ω²
1- i -1 ω² -1
i -1+ ω -1 is equal to
a) 0 b) 1 c) I d) ω
90) The sum ¹⁹ₖ₌₁∑ (sin(kπ/5)+ i cos(kπ/5)) is
a) purely real and positive
b) purely imaginary
c) purely real and negative d) none
91) If w is a given complex number outside the circle with centre at origin and radius |a - 1| (a is real), then the points z, satisfying z. Conjugate of z - 2 conjugate w. z - w. Conjugate of w - 2w. Conjugate of w + 5(a -1)⅖= 0, lie on
a) a circle b) a parabola c) a straight line d) none
92) If Z and W represent diagonally opposite vertices of a square, then the other two vertices are given by the complex numbers
a) Z + iW and Z - iW
b) (1/2) (Z + W)± (i/2)(Z - W)
c) (1/2) (Z - W)± (i/2)(Z - W)
d) (1/2) (Z - W)± (i/2)(Z - W)
1c 2b 3c 4a 5d 6c 7b 8c 9a 10c 11b 12b 13c 14c 15c 16d 17c 18c 19a 20a 21b 22b 23b 24c 25a 26c 27b 28d 29d 30d 31c 32d 33c 34a 35a 36d 37a 38a 39c 40c 41b 42c 43b 44d 45d 46b 47b 48c 49d 50d 51c 52c 53c 54c 55d 56c 57b 58d 59d 60c 61c 62d 63c 64a 65b 66b 67c 68b 69c 70c 71b 72d 73b 74a 75a 76b 77b 78c 79a 80c 81c 82c 83b 84d 85d 86b 87b 88c 89a 90b 91a 92d
A. P, GP, HP
1) If for three real numbers a,b,c, the numbers (b + c)/a, (c + a)/b, (a+ b)/c are in AP and a+ b + c≠ 0, then a,b,c are in
a) AP b) HP c) GP d) none
2) If b(≠ 0) is the AM between a and c and ab+ cd+ ad= 3bc, then, between b and d, the number c is
a) the AM b) a GM c) the HM d) none
3) (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the AM between two positive unequal real numbers a and b if n=
a) 1 b) 0 c) 2 d) none
4) If a₁ = 2 and aₙ - aₙ₋₁ = 2n (n≥ 2), then aₙ =
a) n(n +1)/2 b) 2(n +1) c) n(n +1) d) none
5) For any positive integer n, (n +1)²+ (n +2)²+.......(2n)²=
a) 2n(2n +1)(4n +1)/6
b) [2n(2n +1)/2]²
c) (n/6) (2n +1)7n +1) d) none
6) The ratio of the sums of p AM's and Q AM's between two numbers is
a) q:p b) +p+2): (q +2) c) (p+1): (q+1) d) p:q
7) If for some non zero integer r, 7r is the GM between 3r+1) and (10r +8), then r is
a) 1 b) -1 c) 2 d) none
8) If 1/(a+ b), 1/2b, 1/(b + c) are in AP, then a,b,c are in
a) GP b) AP c) HP d) none
9) The ratio of HM to AM of two positive number is 24: 25. Then the ratio of the number is
a) 4:1 b) 3: 4 c) 4:3 d) 2:3
10) In a GP of alternatively positive and negative terms , each term is the AM of the next two terms . Then a common ratio of the GP is
a) 2 b) -2 c) 3 d) -3
11) In a GP of alternatively positive and negative terms , 9 terms each term is the HM of the next two terms. Then the common ratio of the GP is
a) 2/3 b) -3 c) (9+3√17)/4 d) none
12) (aⁿ⁻² + bⁿ⁻²)/(aⁿ⁻³ + bⁿ⁻³) is the HM between two positive unequal real numbers a and b if n is
a) 0 b) 3 c) 2 d) none
13) The AM , GM , HM between two unequal positive real numbers are in
a) GP b) AP c) HP d) none
14) Let p be the nth of (2n -1) AMs and Q be the nth of (2n -1) GMs between two unequal positive real numbers a and b. If r be the HM between a and b, then the roots of px²+ qx + r= 0 are
a) real and equal
b) real and unequal
c) imaginary d) none
15) If a₁, a₂, a₃, .....a₂ₙ are 2n AM's, b₁, b₂,.....b₂ₙ are 2n GM's and C be the HM between two unequal positive numbers, then ⁿᵢ₌₁∑ (aᵢ + a₂ₙ₋ᵢ₊₁)/(bᵢ b₂ₙ₋ᵢ₋₊₁)
a) 2nc b) 2n/c c) c/n d) ne
16) The AM of two unequal positive whole numbera is 2. If the greater number is increased by 1, their positive GM becomes the AM of the given numbers. Then the HM of these two numbers is
a) 2/3 b) 3/2 c) 1/2 d) none
17) For two unequal real numbers a and b of the same sign, a, p, b are in AP, a, q, b are in GP and a, r, b are in HP. If r= 25p. Then
a) 2q= p+ r b) q= 13p c) r= 5q d) |q|= 5|p|
18) If a,b,c,d be four unequal positive real numbers such that the the first three are in HP and last three terms and AP then .
a) the GM of first and the fourth = the GM of second and third
b) the GM of first and the third = the GM of second and fourth
c) the GM of first and second = the GM of third and fourth
d) none of the above is valid
19) If a,v,c are in GP (a>1, v >1, c> 1), then for any real number x (with x> 0, ≠1) then log ₐx, logᵥx, log꜀x are in
a) GP b) AP c) HP d) none
20) Three unequal positive real numbers a, b, c are such that a,b,c are in GP while log(5c/2a), log(7b/5c), log(2a/7b) are in AP. Then a, b, c are the lengths of the sides of
a) an isosceles triangle
b) an equilateral triangle
c) a scalene triangle d) none
21) {tₖ} is a sequence of positive integer such that t₁ = 1 and tₖ (k > 1)= 3ⁿ⁺¹ whenever, for some non-negative integer n, 1+ 2+ 2²+2³+....+2ⁿ< k ≤ 1+2²+2³+.....2ⁿ⁺¹. then t₁₁₅₅=
a) 3¹¹ b) 3¹⁰ c) 3⁹ d) none
22) If three distinct numbers x,y,z are in GP and are such that 4x, 12y, 11z are in AP , then the common ratio of the GP is
a) 3 b) 111 c) -2 d) or 2 or 2/11
23) If the first term and the third term of an AP are 7/3 and 11/5, then the largest value of tᵣ > 0 is
a) 35 b) 18 c) 36 d) 34
24) If x be the GM of two unequal positive real numbers a and b, then a+ x, 2x, b+ x are in
a) AP b) HP c) GP d) none
25) Given an AP, if S₅ = 25r (for some positive integer r) and if S'₇ = 98r when S'ₙ is the sum of the first n terms of an AP, each of whose term is twice the corresponding term of the given AP, then Sᵣ is
a) 35r b) 12r c) r³ d) 12r²
26) The first term and the common of a GP are respectively 1, 2. Then Sₙ, S₂ₙ - Sₙ, S₃ₙ - S₂ₙ are in
a) GP b) HP c) AP d) none
27) If b is the AM between a and c, d is the HM between c and e and c is the GM, between b and d, then, between a and e, the number c is the
a) AM b) GM c) HM d) none
28) The first term of an increasing AP is zero. then among the two GM's between the fifth and the 10th terms of the AP.
a) none is a term of the AP
b) one is the second term of the AP.
c) one is a term between the 5th and the 10th
d) one is the third of the AP.
29) The number of terms which are equal in the progressions 1, 3, 9, .....to 100th terms and 1, 3, 5, 7, 9,.....100th terms is
a) 3 b) 5 c) 2 d) 4
30) If xᵃ = yᵇ= kᶜ, where a, k, c are in GP, then logₖy=
a) logₓy b) logᵧx c) b/c d) c/a
31) If the HM between the numbers 1/3¹⁺²ˣ and 3²ˣ⁻¹ is 1/5, Then the set of all possible values of x is
a) { x ∈ R : 4x²-1= 0}
b) {3,1/3} c) {9,1/9} d) {-1/2}
32) The length of three edges of a rectangular parallelopiped are in GP. The volume and the total surface area of the parallekopiped are respectively 64 cubic metre and 112 square metre. The length of the smallest edge is
a) 4m b) 8m c) 3m d) 2m
33) If x be the AM and y, z be two GM's between two unequal positive real numbers , then the HM between these numbers is
a) 4x/yz b) x/yz c) yz/x d) y/zx
34) If x, y are two AM's and z and t are two GM 's between two unequal positive real numbers , then 3t/(x + y) is
a) The HM between the given numbers.
b) The GM between the given numbers
c) The AM between the given numbers
d) Thrice the HM between the given numbers
35) suppose you know the term supplier 3 natural number such that disting positive real number such that be a function such that a distinct and are in GP then for three distinct prime numbers the numbers are in AP not in AP but scene on consecutive terms of 180 not at all any three terms of an ap are in GP the minimum number of the terms 2 5 8 11 that ad up to a number exceding 15503215 the sum of all odd factors 590490 in a GP of alternatively positive and negative terms and 52 the first term is 48 in 12 common term of the AP 516 + 769 7275 is 8 term 7th term 9th term the lunch time of the producing 142 and 3282 3 consecutive terms of progression 15 the next term of the progression if their positive real numbers are in AP or in HP GP 18 let the three positive numbers be in HP then the numbersthe sum of the 16th and the 17 term of the sequence the sum of the 18th and the 19th term of the sequence the sum of the first world terms of the GPS 10 times the sum of the odd terms before the 12th term in a common ratio of 10 9 11 the copy centre in the product the coefficient of in the product the coefficient of in the product in the first time in the 20% series 0.30.006 0.009
ⁿ⁺¹ⁿ⁺¹ⁿⁿ ₁ₙₙ₋₁ₙ ²²²² ₁ₖ ²³ⁿ²³ⁿ⁺¹₁₁₅₅¹¹¹⁰⁹ ᵣ ³² ∈ ²ᵣₙₘₚₘₚₘₚ₂₉₃₈₄₇²²²²²²²₇₅₂₅ⁿⁿ⁻¹ⁿⁿ⁻¹₂₃₂₄₃₉₈₈₉₉₁₀₁₀₉¹¹³²₃²₃²₃¹²₃¹³₃¹³₃¹²¹⁰²⁵⁸¹¹²⁶²⁹¹⁵⁵¹⁵⁵³¹⁰¹⁶¹⁶¹⁷¹⁶¹⁶²⁷¹²⁷¹³ᵐₚ₌₁²ᵐₙ₌₁ⁿₚ₃₉₂₇₃ⁿ₌₁
PERMUTATION & COMBINATION
1) If 5 x ⁿP₄ = 6 x ⁿ⁻¹P₄ then the value of n is
a) 22 b) 24 c) 48 d) 50
2) ⁿ⁺¹P₄ : ⁿ⁻¹P₃ = 72: 5 then the value of n is
a) 8 b) 9 c) 10 d) 11
3) If ¹⁰Pᵣ = 5040 then r is equal to
a) 2 b) 3 c) 4 d) 5
4) If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁ then the values of n and r are respectively equal to
a) 2,5 b) 3,2 c) 6,3 d) 7,4
5) If ¹²Pᵣ = ¹¹P₆ + 6. ¹¹O₅ then the value of r is
a) 3 b) 4 c) 5 d) 6
6) If (n +2)!= 2550. n! then the value of n is
a) 47 b) 48 c) 49 d) 50
7) If ¹⁰Pᵣ = ⁹P₅ + 5. ⁹P₄ then the value of r is
a) 2 b) 3 c) 4 d) 5
8) If ⁴⁻ˣP₂ = 6 then the value of x is
a) 1 b) 2 c) 3 d) none
9) If ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5 then the value of n is
a) 4 b) 3 c) 2 d) none
10) ⁿCᵣ₊₁ + ⁿCᵣ₋₁ + 2 ⁿCᵣ is equal to
a) ⁿ⁺²Cᵣ b) ⁿ⁺²Cᵣ₊₁ c) ⁿ⁺¹Cᵣ d) ⁿ⁺¹Cᵣ₊₁
11) If ¹⁶Cᵣ = ¹⁶Cᵣ₊₂ then ʳC₄ is equal to
a) 21 b) 27 c) 35 d) 39
12) If 3 ˣ⁺¹C₂ + ²P₂. x = 4 ˣP₂, then the value of x is
a) 2 b) 3 c) 5 d) 7
13) If ²ⁿC₃ : ⁿC₂ = 44: 3 then n is equal to
a) 3 b) 4 c) 5 d) 6
14) If ⁿP₅ = 60. ⁿ⁻¹P₃ then n is equal to
a) 8 b) 10 c) 12 d) 14
15) The value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is
a) 24504 b) 44502 c) 42504 d) 45042
16) If ⁿC₄ = 21. ⁿ⁾²C₃ then the value of n is
a) 6 b) 10 c) 12 d) 14
17) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀∑⁴⁰⁺ʲC₁₀₊ⱼ is equal to
a) ⁷²C₃₁ b) ²⁷C₈ c) ¹⁹C₁₁ d) ⁵¹C₂₀
18) If ²⁸C₂ᵣ : ²⁴C₂ᵣ₋₄ = 225 : 11 then value of r is
a) 7 b) 9 c) 14 d) none
19) If ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800: 1, then the value of r is
a) 11 b) 21 c) 31 d) 41
20) If ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃, then
a) n< 6 b) n > 7 c) n <5 d) n > 6
21) If ⁿ⁺²C₈ : ⁿ⁻²C₄ = 171:2 then value of n is
a) 18 b) 19 c) 20 d) 21
22) The value of ⁴⁷C₄ + ⁵ᵣ₌₁∑⁵²⁻ʳC₃ is
a) 277025 b) 275027 c) 507227 d) 270725
23) If ⁿPᵣ = 504 and ⁿCᵣ = 84, then the value of n is equal to
a) 3 b) 6 c) 9 d) 12
24) The number of different algebraic expressions that can be made by combining the letters p,q,r,s and t in this order with the signs '+' and '-' taking all the letters together is
a) 21 b) 23 c) 31 d) 32
25) The number of different factors of 2160 is
a) 29 b) 39 c) 49 d) none
26) 8 different chocolates can be distributed equally between two boys in
a) 70 ways b) 35 ways c) 38 ways d) 19 ways
27) From a group of persons the number of ways of selecting 5 persons is equal to that 8 persons . The number of persons in the group is
a) 29 b) 25 c) 13 d) 11
28) A man has 5 oranges and 4 mangoes. How many different selections having at least one orange is possible ?
a) 25 b) 30 c) 35 d) 40
29) a man has 6 friends. The number of ways in which he can invite one or more than one of them to his house is
a) 6! b) 6!-1 c) 2⁶! d) none
30) A 5 digited number is divisible by 3 and it is formed by 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which such a number can be formed is
a) 126 b) 216 c) 621 d) 261
31) All the letters of the string AEPRAB are arranged in all possible ways. The number of such arrangements in which two vowels are not adjacent to each other is
a) 220 b) 115 c) 72 d) 65
32) The number of ways in which the letters of the string ANRTIPF can be arranged so that the vowels may appear in the odd places is
a) 1230 b) 1350 c) 1440 d) 1570
33) if there are 10 persons in a gathering and if each of them shakes hand with everyone else, then the number of hand shakes that takes place in the gathering is
a) 20 b) 45 c) 2¹⁰ d) 10²
34) The number of parallelograms that can be formed from a set of 4 parallel lines intersecting another set of 3 parallel lines is
a) 21 b) 20 c) 18 d) 16
35) The number of students to be selected at a time from a group of 16 students, so that the number of selections is the greatest is
a) 16 b) 14 c) 8 d) none
36) The number of different arrangements with the letters of the word ALGEBRA so that the two As are not together is
a) 1800 b) 2520 c) 720 d) none
37) The number of odd integer of 6 significant digits that can be formed with the digits 0, 1, 4, 5, 6, 7 without repetition of the digits is
a) 96 b) 108 c) 266 d) 288
38) The number of words that can be made by writing down the letters of world CALCUTTA such that each word starts and ends with a constant is
a) 7! b) 7!/2 c) 5.7!/2 d) 9.7!/2
39) The number of triangles that can be formed with 10 points as vertices, k of them being collinear, is 110. Then value of k is
a) 3 b) 5 c) 7 d) none of
40) The number of ways in which 5 '+' sign and 3 'X' sign can be arranged in a row is
a) 56 b) 65 c) 72 d) 81
41) The number of ways in a which 15 class XI students and 12 class XII students be arranged in a line so that no two class XII students may occupy connective positions is
a) 12! x 16!/4! b) 15! x 13!/4! c) 16! x 13!/4! d) 15! x 16!/4!
42) The number of strings of 3 letters that can be formed with the letters chosen from CALCUTTA is
a) 48 b) 62 c) 96 d) 102
43) The number of permutations of the letters of the word MADHUBANI where the arrangementa do not begin with M but end with I is
a) 16740 b) 17460 c) 14670 d) none
44) The number of ways in which committee of 5 perss may be formed out of 6 men and four women under the condition that at least one woman has to be selected necessarily is
a) 252 b) 246 c) 242 d) none
45) Given that balls of the same colour identical, the number of a ways in which 18 white balls and 19 red balls may be arranged in a row so that no two white balls may come together is
a) 180 b) 190 c) 200 d) 210
46) in an examination there are 3 multiple choice questions and each question has four choices. The number of ways in which one can fail to get all answers correct is
a) 12 b) 21 c) 36 d) 63
47) The number of diagonals that can be drawn by joining the vertices of an octagon is
a) 28 b) 20 c) 18 d) 16
48) Out of six given points 3 are collinear . The number of different straight lines that can be drawn by joining any two points from those 6 given points is
a) 12 b) 10 c) 9 d) none
49) The total number of selections of at least one red ball from 4 red balls and 3 blue balls. If the balls of the same colour are different, is
a) 95 b) 105 c) 120 d) 125
50) In an examination of 9 papers a candidate has to pass in more papers than the number of paper in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
a) 265 b) 255 c) 256 d) 625
51) The number of integers greater than 50000 that can be formed by using the digits 3, 5, 6, 6, 7 is
a) 54 b) 46 c) 32 d) none
52) The number of arrangements that can be formed from the letters of the word VIOLENT, so that the voeel may occupy only odd positions is
a) 576 b) 574 c) 572 d) none
53) in a group of 15 boys there are 7 boys scouts. The number of a ways in a which 12 boys can be selected from the groups so as to include at least 6 boys-scouts is
a) 125 b) 127 c) 252 d) 255
54) 15 distinct objects may be divided into 3 groups of 4,5 and 6 objects in
a) 230230 b) 320320 c) 360360 d) 630630
55) The number of different ways in which 1440 can be expressed as the product of two factors is
a) 18 b)) 16 c)) 14 d) none
56) The number of different rectangles (regarding every square as a rectangle as well) that are there on a chess board is
a) 1280 b) 1284 c) 1296 d) 1300
57) The number of arrangements which can be made out of the letters of the word ALGEBRA without changing the relative positions of the vowels and consonants is
a) 54 b) 64 c) 70 d) 72
58) The number of factors of 420 is
a) 22 b) 23 c) 24 d) none
59) a boat as a crew of 10 men of which 3 can row only one one side and 2 only on other. The number of ways the crew can be arranged in the boat is
a) 142000 b) 144000 c) 124000 d) none
60) There are 10 points in a plane of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through atleast three points of these points is
a) 117 b) 120 c) 122 d) 124
61) The number of 6 digited integers that can be made using that digit 3 and 4 and in which at least two digits are different, is
a) 60 b) 61 c) 62 d) none
62) The sum of the digit in unit place of all the 4 digited numbers formed with the help of 2,3,4,5, taken all at a time, is
a) 54 b) 108 c) 84 d) none
63) The number of different ways in which 15 distinct objects may be divided into 3 groups of 5 objects each, is
a) 216216 b) 126126 c) 216612 d) 126612
64) The number of different arrangements that can be made out of the letters of the word ALLAHABAD, such that the vowels may occupy the even positions only is
a) 70 b) 50 c) 60 d) 120
65) The number of ways in which 4 letters can be posted in 3 post boxes is
a) 256 b) 81 c) 12 d)) none
66) At an election a voter may vote for any number of candidates, not greater than the number to be selected. There are 10 candidates and 4 are to be elected. if a voter votes for at least one candidate, then the number of a ways in which he can vote is
a) 385 b) 1110 c) 5040 d) 6210
67) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order ?
a) 360 b) 240 c) 12⁰ d) 480
68) The number of ways of distributing 8 identical balls in distinct boxes so that none of the boxes is empty is
a) 38 b) 21 c) 5 d) ⁸C₃
69) The number of different solutions (x,y,z) of the equation x+ y+ z = 10, where each of x, y, z is a+ve integer, is
a) 36 b) ¹⁰C₃ - ¹⁰C₂ c) 10³ d) none
70) If ²ⁿC₁ + ²ⁿC₂ + ....+ ²ⁿCₙ₋₁ + (1/2) ²ⁿCₙ = 127, then n is equal to
a) 4 b) 5 c) 3 d) none
1b 2a 3c 4b 5d 6c 7d 8a 9a 10d 11c 12b 13d 14b 15c 16b 17d 18a 19d 20b 21b 22d 23b 24d 25b 26a 27c 28a 29d 30b 31c 32c 33b 34c 35c 36a 37d 38c 39b 40a 41d 42c 43d 44b 45b 46d 47b 48d 49c 50c 51d 52a 53c 54d 55a 56c 57d 58b 59b 60a 61c 62d 63b 64c 65b 66a 67a 68b 69a 70a
BINOMIAL THEOREM - 1 (TEST)
1) Write down the general term in the expansion of (a + x))ⁿ when n is a positive integer.
2) Write down the general term in the expansion of (x²+ 1/x))²ⁿ when n is a positive integer.
3) What is the sum of the indices of a and x in the 5th term in the expansion of (a + x)¹² ?
4) What is the number of terms in the expansion of (a + x)²⁰?
5) What is the number of terms in the expansion of (x + y)⁵(x - y)⁵?
6) How many different terms can be obtained by simplifying the expansion of (a + 2b)⁵⁰ (a - 2b)⁵⁰?
7) The rth term in the expansion of (x + 1/x)¹⁰ is independent of x; what is the value of r ?
8) If the coefficient of the 16th and 26th terms in expansion of (1+ x))ⁿ are equal, what is the value of n ?
9) if the coefficient of the (3r)th and (r +2)th terms in the expansion of (1+ x))²ⁿ are equal, then
a) n= 2r b) n= 3r c) n= 2r +1 d) none
10) if the coefficient of x⁶ in the expansion of (1+ x))ⁿ is 177100, what is the coefficient of xⁿ⁻⁶ in that expansion ?
11) Find the number of terms containing positive powers of x in the expansion of (2x - 1/3x²)¹².
1) ⁿCᵣ aⁿ⁻ʳxʳ
2) ²ⁿCᵣ. x⁴ⁿ⁻³ʳ 3) 12 4) 21 5) 6 6) 26 7) 6 8) 40 9) a 10) 177100 11) 4
3)
∞ ∈ ω
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