EQUATION, INEQUATION AND EXPRESSION
1) If x is a Real number such that x(x²+1), (-1/2)x², 6 are three consecutive terms of an AP then the next two consecutive terms of AP are
a) 14, 6 b) -2,-10 c) 14, 22 d) none
2) The number of real solutions of x - 1/(x²-4) = 2 - 1/(x²-4) is
a) 0 b) 1 c) 2 d) infinite
3) The number of values of a for which (a²- 3a+ 2)x²+ (a²- 5a +6)x + a² -4=0 is an identity in x is
a) 0 b) 2 c) 1 d) 3
4) The number of values of the pair (a,b) for which a(x +1)²+ b(x²- 3x -2)+ x +1=0 is an identity in x is
a) 0 b) 1 c) 2 d) infinite
5) The number of values of the triple (a,b,c) for which
a cos 2x + b sin²x + c =0 is satisfied by all real x is
a) 0 b) 2 c) 3 d) infibite
6) The polynomial (ax²+ bx+ c)(ax²- dx- c), ac ≠ 0, has
a) four real zeros b) at least 2 real zeros c) atmost two real zero d) no real zeros
7) Let f(x)= ax³+ 5x²- bx +1. If f(x) when divided by 2x + 1 leaves 5 as remainder, and f'(x) is divisible by 3x -1 then
a) a= 26, b=10
b) a= 24, b=12
c) a= 26, b=12 d) none
8) ₓ3ⁿ₊ ᵧ3ⁿ is divisible by x+ y if
a) n is any integer ≥ 0
b) n is an odd positive integer
c) n is an even positive integer
d) n is a rational number
9) If x,y are rational numbers such that
x+ y + (x -2y)√2= 2x - y + (x - y -1)√6 then
a) x and y cannot be determined
b) x=2, y=1 c) x=5, y=1 d) none
10) The number of real solutions of the equation
2ˣ⁾² + (√2 +1)ˣ = (5+ 2√2)ˣ⁾² is
a) one b) two c) four d) infinite
11) The number of real solutions of the equation eˣ= x is
a) 1 b) 2 c) 0 d) none
12) The sum of the real roots of the equation x²+ |x| - 6=0 is
a) 4 b) 0 c) -1 d) none
13) The solution of the equation 2x - 2[x] = 1, where [x]= the greatest integer less than or equal to x, are
a) x= n + 1/2, n ∈ N
b) x= n - 1/2, n ∈ N
c) x= n + 1/2, n ∈ Z
d) x= n + 1, n ∈ Z
14) The number of real solutions of the equation sin(eˣ)= 5ˣ + 5⁻ˣ is
a) 0 b) 1 c) 2 d) infinitily many
15) The number of real solutions 1+ |eˣ -1|= eˣ(eˣ -2) is
a) 0 b) 1 c) 2 d) 4
16) The equation 2 sin²(x/2) cos²x = x + 1/x, 0< x ≤π/2 has
a) one real solution
b) no real solution
c) infinity any real solutions d) none
17) If y≠ 0 then the number of values of the pair (x,y) such that
x + y + x/y = 1/2 and (x + y)x/y = -1/2, is
a) 1 b) 2 c) 0 d) none
18) The number of real solutions of the equation log₀.₅x = |x| is
a) 1 b) 2 c) 0 d) none
19) The equation √(x +1) - √(x -1)= √(4x -1) has
a) no solution b) one solution c) two solution d) more than two solutions
20) The number of solutions of the equation |x|= cosx is
a) one b) two c) three d) zero
21) The product of all solutions of the equation (x -2)² - 3|x -2|+ 2= 0 is
a) 2 b) -4 c) 0 d) none
22) If 0< x <1000 and [x/2] + [x/3] + [x/5]= 31x/30, where [x] is the greatest integer less than or equal to x, the number of possible values of x is
a) 34 b) 32 c) 33 d) none
23) The solution set of (x)²+ (x +1)⅖= 25, where (x) is the least integer greater than or equal to x, is
a) (2,4) b) (-5,-4) U (2,3] c) [-4,-3)U [3, 4) d) none
24) If èˣ⁺¹ = 6ˡᵒᵍ₂3 then x is
a) 3 b) 2 c) log₃2 d) log₂3
25) If (√2)ˣ + (√3)ˣ= (√13)ˣ⁾² then the number of values of x is
a) 2 b) 4 c) 1 d) none
26) The number of real solutions of the equation (6- x)/(x²-4) = 2+ x/(x +2) is
a) two b) one c) zero d) none
27) The number of real solutions of
√(x²- 4x +3) + √(x²-9)= √(4x²- 14x +6) is
a) one b) two c) three d) none
28) If [x]= the greatest integer less than or equals to x, and (x)= the least integer greater than or equal to x, and [x]²+ (x)²> 25 then x belongs to
a) [3,4] b) (-∞,-4] c) [4, ∞) d) (-∞, -4] U [4, ∞)
29) Let R= the set of real numbers , Z= the set of integers , N= the set of natural numbers. If S be the solution set of the equation (x)²+ [x]²= (x -1)⅖+ [x +1]², where (x)= the least integer greater than or equal to x and [x]= the greatest integer less than or equal to x, then
a) S= R b) S= R - Z c) S= R - N d) none
30) If [x]²= [x +2], where [x]= the greatest integer less than or equal to x, then x must be such that
a) x=2,-1 b) x ∈ [2,3) c) x ∈ [-1,0) d) none
31) The solution set of |(x +1)/x| + |x +1|= (x +1)²/|x| is
a) {x : x ≥ 0}
b) {x : x > 0} U {-1}
c) {-1,1}
d) {x: x ≥ 1 or x ≤ -1}
32) The number of solutions of |[x] - 2x |= 4, where [x] is the greatest integer ≤ x, is
a) 2 b) 4 c) 1 d) infinite
33) The set of real values of x satisfying |x -1|≤ 3 and |x -1|≥ 1 is
a) [2,4] b) (-∞,2] U [4, + ∞) c) [-2,0] U [2, 4] d) none
34) the set of real values of satisfying | | x -1| -1|≤ 1 is
a) [-1,3] b) [0,2] c) [-1,1] d) none
35) If x y Z (the set of integers) such that x²- 3x < 4 then the number of possible values of x is
a) 3 b) 4 c) 6 d) none
36) If x is an integer satisfying x²- 6x + 5≤ 0 and x²- 2x > 0 then the number of possible values of x is
a) 3 b) 4 c) 2 d) infinite
37) The solution set of the equation log₁/₃ (x²+ x +1)+ 1 > 0 is
a) (-∞,-2) U (1, +∞) b) [-1,2] c) (-2,1) d) (-∞, +∞)
38) 5ˣ + (2√3)²ˣ ≥ 13ˣ then the solution set for x is
a) [2, +∞) b) {2} c) (-∞,2] d) {0,2]
39) If 3ˣ⁾² + 2ˣ > 25 then the solution set is
a) R b) (2, +∞) c) (4, +∞) d) none
40) If sinˣα + cosˣα ≥ 1, 0 < α< π/2, then
a) x ∈ [2,+∞) b) x ∈ (-∞,2] c) x ∈ [-1,1] d) none
41) The solutions set of x²+ 2 ≤ 3x ≤ 2x²- 5 is
a) φ b) [1,2] c) (-∞,-1] U [5/2, +∞) d) none
42) The solution set of (x²- 3x +4)/(x +1) > 1, x ∈ R, is
a) (3, +∞) b) (-1,1) U (3, +∞) c) [-1,1] U [3, +∞) d) none
43) The number of integral solutions of (x +2)/(x²+1) > 1/2 is
a) 4 b) 5 c) 3 d) none
44) If a, b,c are non-zero , unequal rational numbers then the roots of the equation abc²x² + (3a²+ b²)cx - 6a²- ab + 2b²= 0 are
a) rational b) imaginary c) irrational d) none
45) If l, m are real and l ≠ m then the roots of the equation
(l - m)x² - 5(l + m)x -2(l - m)= 0 are
a) real and equal
b) non real Complex
c) real and unequal d) none
46) If a, b, c, d are four consecutive terms of an increasing AP then the roots of the equation (x - a)(x - c) + 2(x - b)(x - d)=0 are
a) real and distinct b) nonreal Complex c) real and equal d) integers
47) If a,b,c are three distinct positive real numbers then the number of real roots of ax²+ 2b |x| - c =0 is
a) 4 b) 2 c) 0 d) none
48) The equation x²- 6x + 8 + λ(x²- 4x +3)= 0, λ ∈ R, has
a) real and unequal roots for all λ
b) real roots for λ< 0 only
c) real roots for λ > 0 only
d) real and unequal roots for λ = 0 only
49) If cosθ, sinφ, sinθ are in GP then roots of x²+ 2 cotφ. x +1=0 are always
a) equal b) real c) imaginary d) greater than 1
50) The roots of ax²+ bx + c=0, where a≠ 0 and co-efficients are real, are nonreal Complex and a+ c < b. Then
a) 4a + c >2b b) 4a + c < 2b c) 4a+ c =2b d) none
51) The equation (a +2)x²+ (a -3)x = 2a -1, a≠ -2 has roots rational for
a) all Rational values of a except a= -2
b) all real values of a except a= -2
c) rational values of a> 1/2 d) none
52) If a. 3ᵗᵃⁿˣ + a. 3⁻ᵗᵃⁿˣ -2= 0 has real solutions, x ≠ π/2, 0≤ x <π, then the set of possible values of the parameter a is
a) [-1,1] b) [-1,0] c) (0,1] d) (0,+∞)
53) If a> 1, roots of the equation (1- a)x²+ 3ax -1= 0 are
a) one positive root and one negative root
b) both negative
c) both roots positive
d) both nonreal Complex
54) If a∈R, b ∈ R then the equation x²- abx - a²= 0 has
a) one positive root and one negative root
b) both roots positive
c) both root negative
d) non-real roots
55) If the roots of the equation x²- 2ax + a²+ a -3=0 are less than 3 then
a) a< 2 b) 2≤a≤ 3 b) 3< a ≤4 d) a> 4
56) if α, β are the roots of x²- 3x + a =0, a∈ R and α < 1< β then
a) a ∈ (-∞,2) b) a ∈ (-∞, 9/4] c) a ∈ (2, 9/4] d) none
57) If α, β be the roots of 4x² - 16x + λ= 0, λ∈ R such that 1< α< 2 and 2< β< 3 then the number of integral solution of λ is
a) 5 b) 6 c) 2 d) 3
58) The number of integral values of a for which x¹- (a -1)x +3=0 has both roots positive and x²+ 3x + 6 - a =0 has both roots negative is
a) 0 b) 1 c) 2 d) infinite
59) If X denotes the set of real numbers p for which the equation x²= p(x + p) has its greater than p then X is equal to
a) (-2,-1/2) b) (-1/2,1/4) c) null set d) (-5,0)
60) If cos⁴x + sin²x - p =0, p ∈ R has real solutions then
a) p≤1 b) 3/4≤ p ≤ 1 c) p≥ 3/4 d) none
61) If one root of the equation (k²+1)x²+ 13x + 4k=0 is reciprocal of the other than k has the value
a) -2+√3 b) 2-√3 c) 1 d) none
62) If the ratio of the roots of λx²+ μx + v =0 is equal to the ratio of the roots of x²+ x +1=0 then , λ, μ, v are in
a) AP b) GP c) HP d) none
63) p,q,r and s are integers. If the AM of the roots of x²- px + q²=0 and GM of the roots of x²- rx + s²=0 are equal then
a) q is an odd integer
b) r is an even integer
c) p is an even integer
d) s is an odd integer
64) If α, β are roots of the equation (x - a)(x - b)= c, c≠ 0, then the roots of the equation (x - α)(x - β)+ c =0 are
a) a,c b) b,c c) a,b d) a+ c, b+ c
65) If the roots of 4x²+ 5k = (5k +1)x differ by unity then the negative value of k is
a) -3 b) -1/5 c) -3/5 d) none
66) The harmonic mean of the the roots of the equation
(5+ √2)x² - (4+ √5)x + 8 + 2√5= 0 is
a) 2 b) 4 c) 6 d) 8
67) If the product of the roots of the equation x²- 5x + ₄log₂λ= 0 is 8 then λ is
a) ±2√2 b) 2√2 c) 4 d) none
68) If the roots of a₁x²+ b₁x + c₁ = 0 are α₁, β₁ and those of a₂x² + b₂x + c₂ =0 are α₂β₂ such that α₁α₂ = β₁β₂ = 1 then
a) a₁/a₂ = b₁/b₂ = c₁c₂
b) a₁/c ₂ = b₁/b₂ = c₁a₂
c) a₁a₂ = b₁/b₂ = c₁c₂ d) none
69) If α, β are the roots of ax²+ c = bx then the equation (a + cy)²= b²y in y has the roots
a) α⁻¹, β⁻¹ b) α², β² c) αβ⁻¹, α⁻¹β d) α⁻², β⁻²
70) If the roots of ax²- bx - c=0 change by the same quantity then the expression in a, b, c that does not change is
a) (b²- 4ac)/a²
b) (b - 4c)/a
c) (b² + 4ac)/a² d) none
71) If α, β are the roots of x²- px + q=0 then the product of the roots of the quadratic equation whose roots are α²- β² and α³- β³ is
a) p(p²- q)² b) p(p²- q)(p²- 4q)
c) p(p²- 4q)(p²+ q) d) none
72) If the sum of the roots of the quadratic equation ax²+ bx + c=0 is equal to the sum of the squares of their reciprocals than b²/ac + bc/a² is equal to
a) 2 b) -2 c) 1 d) -1
73) If the absolute value of the difference of roots of the equation x²+ px +1=0 exceeds √(3p) then
a) p<-1 or p> 4 b) p> 4 c) -1< p< 4 d) 0≤ p < 4
74) If α, β are roots of x²+ px + q=0 and γ, δ are the roots of x²+ px - r=0 then (α - γ)(α - δ) is equal to
a) q+ r b) q- r c) -(q+ r) d) -(p + q+ r)
75) If α, β roots of 375x²- 25x - 2=0 and sₙ = αⁿ + βⁿ then limₙ₋∞ ⁿᵣ₌₁∑ sᵣ is
a) 7/116 b) 1/12 c) 29/358 d) none
76) The quadratic equation whose roots are the AM and HM of the roots of the equation x²+ 7x -1= 0 is
a) 14x²+ 14x - 45= 0
b) 45x²- 14x + 14=0
c) 14x²+ 45x - 14=0 d) none
77) Let α ≠ β and α²+ 3 = 5α while β²= 5β -3. The quadratic equation whose roots are α/β and β/α is
a) 3x²- 31x +3=0
b) 3x²- 19x +3=0
c) 3x²+ 19x +3=0 d) none
78) If a, b are rational and b is not perfect square then the quadratic equation with rational co-efficients whose one root is 1/(a + √b) is
a) x²- 2ax + (a²- b)= 0
b) (a²- b)x²- 2ax +1=0
c) (a²- b)x²- 2bx +1=0 d) none
79) If 1/(4- 3i) is a roots of ax²+ bx +1=0, where a,b are real, then
a) a= 25, b= -8 b) a=25, b=8 c) a=5, b=4 d) none
80) If α, β be the roots of the equation x(1+ x²)+ x²(6+ x)+ 2=0 then the the value of α⁻¹ + β⁻¹ + γ⁻¹ is
a) -3 b) 1/2 c) -1/2 d) none
81) If the roots of x³- 12x²+ 39x -28=0 are in AP then their common difference is
a) ±1 b) ±2 c) ±3 d) ±4
82) The roots of the equation x³+ 14x²- 84x - 216=0 are in
a) AP b) GP c) HP d) none
83) If z₀ = α + iβ, i= √-1, then the roots of the cubic equation
x³- 2(1+ α)x²+ (4α + α²+ β²)x - 2(α²+ β²)=0 are
a) 2, z₀,z₀ b) 1, z₀, -z₀ c) 2, z₀, -z₀ d) 2, -z₀, z₀
84) If 3 and 1+ √2 are two roots of a cubic equation with rational co-efficients then the equation is
a) x³ - 5x²+ 9x - 9=0
b) x³ - 3x² - 4x + 12 =0
c) x³ - 5x²+ 7x + 3=0 d) none
85) Let a,b,c be real numbers and a≠ 0. If α is a root of a²x²+ bx + c=0, β is a root of a²x² - bx - c =0 and 0<α<β then the equation a²x²+ 2bx + 2c =0 has a root γ that always satisfies
a) γ= (1/2) (α+β) b) γ= α+β/2
c) γ= α d) α < γ< β
86) Let a,b,C be three real numbers such that 2a + 3b + 6c = 0. Then the equation ax²+ bx + c= 0
a) imaginary roots
b) at least one root in (0,1)
c) atleast one root in (-1,0)
d) both roots in (1,2)
87) If the equation 2x²- 7x +1=0 and ax²+ bx +2=0 have a common root then
a) a= 2, b= -7 b) a= -7/2, b= 1
c) a= 4, b= -14 d) none
88) The quadric equation x¹+ (a²-2)x - 2a²=0 and x²- 3x +2=0 have
a) no common root for all x ∈ R
b) exactly one common root for all x ∈R
c) two common roots for some a ∈ R d) none
89) If the equation ax²+ bx + c=0 and cx²+ bx + a=0, a≠ c have a negative common root then the value of a - b + c is
a) 0 b) 2 c) 1 d) none
90) If the equation x²+ ix + a=0, x²- 2x + ia =0, a≠ 0 have a common root then
a) a is real b) a= 1/2 + i c) a= 1/2 - i
d) the other root is also common
91) If x²- 2r pᵣx + r= 0, r= 1,2,3 are three quadratic equations of which each pair has exactly one root common then the number of solutions of the triplet (p₁, p₂, p₃) is
a) 2 b) 1 c) 9 d) 27
92) If (λ²+ λ -2)x² + (λ +2)x < 1 for all x ∈ R then λ belongs to the interval
a) (-2,1) b) (-2,2/5) c) (2/5,1) d) none
93) The least integral value of k for which (k -2)x²+ 8x + k + 4 > 0 for all x ∈R, is
a) 5 b) 4 c) 3 d) none
94) The set of possible values of x such that 5ˣ + (2√3)²ˣ - 169 is always positive is
a) [3, +∞) b) [2,+∞) c) (2,+∞) d) none
95) If all real values of x obtained from the equation 4ˣ - (a -3)2ˣ + a - 4 =0 are non-positive then
a) a ∈ (4,5) b) a ∈ (0,4) c) a ∈ (4,∞) d) none
96) The set of possible values of λ for which
x² -(λ² - 5λ +5)x + (2λ² - 3λ -4)= 0 has roots whose sum and product are both less than 1 is
a) (-1, 5/2) b) (1,4) c) [1,5/2] d) (1,5/2)
97) If log₁₀x + log₁₀y ≥ 2 then the smallest possible value of x+ y is
a) 10 b) 30 c) 20 d) none
98) If f(x)= (x²-1)/(x²+1) for every real number x then the minimum value of f
a) does not exist because f is unbounded.
b) is not attained even through f is bounded.
c) is equals to 1
d) is equals to -1
99) If ax²+ bx + 6= 0 does not have two distinct real roots, where a ∈R, b ∈R, then the least value of 3a + b is
a) 4 b) -1 c) 1 d) -2
100) If ab = 2a + 3b, a>0, b > 0 then the minimum value of ab is
a) 12 b) 24 c) 1/4 d) none
101) If x²+ px +1 is a factor of the expression ax³+ bx + c then
a) a²+ c²= - ab
b) a²- c²= - ab
c) a² - c²= ab d) none
102) If x²-1 is a factor of x⁴+ ax³+ 3x - b then
a) a=3, b=-1 b) a=-3, b=1 c) a=3, b= 1 d) none
103) The number of values of k for which
{x²- (k - 2)x + k²}{x²+ kx + (2k -1)} is a perfect square is
a) 1 b) 2 c) 0 d) none
104) If x + λy - 2 and x - μy +1 are factors of the expression 6x²- xy - y² - 6x + 8y - 12 then
a) λ= 1/3, μ= 1/2
b) λ= 2, μ= 3
c) λ= -1/3, μ= -1/2 d) none
105) If x - y and y - 2x are two factors of expression x³- 3x²y + λxy²+ μy³ then
a) λ= 11, μ= -3
b) λ= 3, μ= -11
c) λ= 11/4, μ= -3/4 d) none
106) If x + y and y + 3x are two factors of the expression λx³ - μx²y + xy²+ y³ then the third factor is
a) y+ 3x b) y - 3x c) y= x d) none
107) If x, y, z are real and distinct then
f(x,y)= x²+ 4y²+ 9z²- 6yz - 3zx - 2xy is always
a) non negative b) non positive c) zero d) none
108) If x²+ y²+ z² = 1 then the value of xy+ yz+ zx lies in the interval
a) [1/2,2] b) [-1,2] c) [-1/2,1] d) [-1,1/2]
109) If a∈ R, b∈ R then the factors of the expression a(x²- y²) - bxy are
a) real and different b) real and identical c) complex d) none
110) If a,b,c are in HP then the expression a(b - c)x²+ b(c - a)x + c(a - b)
a) has real and distinct factors
b) is a perfect square
c) has no real factor d) none
111) three number of positive integral values of k for which
(16x²+ 12x +39) + k(9x² - 2x +11) is a perfect square is
a) 2 b) 0 c) 1 d) none
112) If (x -1)² is a factor of x⁴+ ax³+ bx²+ cx -1 then the other factor is
a) x -3 b) x +1 c) x +2 d) none
113) If x²- bx + c= 0 has equal integeral roots then
a) b and c are integers
b) b and c are even integers
c) b is an even integer and c a perfect square of a positive integer
d) none
114) Let A, G and H be the AM, GM and HM of two positive numbers a and b. The quadratic equation whose roots are A and H is
a) Ax² - (A²+ G²)x + AG²= 0
b) Ax² - (A²+ H²)x + AH²= 0
c) Hx² - (H²+ G²)x + HG²= 0 d) none
115) Let A, G and H are the AM, GM and HM respectively of two unequal positive integers . Then the equation Ax² - |G|x - H = 0 has
a) both roots as fractions
b) atleast one root which is a negative fraction
c) exactly one positive root
d) at least one root which is an integer
116) Let x²- px + q=0, where p∈ R, q ∈R, have the the roots α, β such that α + 2β =0 then
a) 2p²+ q= 0 b) 2q²+ p=0 c) q< 0 d) none
117) The cubic equation whose roots are the AM, GM and HM of the roots of x²- 2px + q²= 0 is
a) (x - p)(x - q)(x - p - q)= 0
b) (x - p)(x - |q|)(px - q²)= 0
c) x³- (p + |q|+ q²/p)x²+ (p |q| + q²+ |q|³/p)x - |q|³= 0
d) none
118) If x²+ ax + b =0 and x²+ bx + a=0, a ≠ b, have a common root α then
a) a+ b =1 b) α + 1=0 c) α = 1 d) a+ b +1=0
119) The line y + 14=0 cuts the curve whose equation is x(x²+ x +1) + y =0 at
a) three real points
b) one real point
c) at least one real point
d) no real point
120) If a,b,c are in GP, where a, c are positive, then the equation ax²+ bx + c=0 has
a) real roots
b) imaginary roots
c) ratio of roots= 1: ω where ω is non real cube root of unity
d) ratio of roots= b : ac
121) If α β are the roots of the equation x²+ x +3=0 then the equation 3x²+ 5x +3=0 has a root
a) α/β b) β/α c) α/β + β/α d) none
122) If α, β are the roots of x²- 2ax + b²= 0 and γ, δ are the roots of x²- 2bx + a²= 0 then
a) AM of α, β = GM of γ, δ
b) GM of α, β = AM of γ, δ
c) α, β, γ, δ are in AP
d) α, β, γ, δ are in GP
123) If the roots of the equation ax²- 4x + a²= 0 are imaginary and the sum of the roots is equal to their products then a is
a) -2 b) 4 c) 2 d) none
124) If x, y are three consecutive terms of a GP, where x> 0 and the common ratio is r, then inquality z + 3x > 4y holds for
a) r ∈ (- ∞,1) b) r= 24/5 c) r< (3, ∞) d) r= 1/2
125) The equation ||x -1|+ a|= 4 can have real solutions for x belongs if a belongs to the interval
a) (-∞,4) b) (-∞,-4] c) (4,+∞) d) [-4,4]
126) The equation |x +1| |x -1|= a²- 2a -3 can have real solutions for x if
a) (- ∞,1] U [3,+∞) b) [1- √5, 1+ √5]
c) [1-√5, -1] U [3,1+√5] d) none
127) The common roots of the equation x³+ 2x²+ 2x +1=0 and 1+ x¹³⁰ + x¹⁹⁸⁸= 0 (where ω is a nonreal cube root of unity)
a) ω b) ω² c) -1 d) ω - ω²
128) If α is a root of the equation 2x(2x +1)= 1 then the other root is
a) 3α³- 4α b) -2α(α +1) c) 4α³- 3α d) none
129) For the equation 2x²+ 6 √2 x +1=0
a) roots are rational
b) if one root is p + √q then the other is - p + √q
c) roots are irrational
d) if one root is p + √q then the other is p - √q
130) If α, β are the real roots of x²+ px + q=0 and α⁴, β⁴ are the roots of x²- rx + s=0 then the equation x²- 4qx + 2q² - r=0 has always
a) two real roots
b) two negative roots
c) two positive roots
d) one positive root and one negative root
131) The equation ₓ{(3/4)(log₂x)²} + log₂x - 5/4 ₌ √₂ has
a) at least one negative solution
b) exactly one irrational solution
c) exactly three real solutions
d) two nonreal Complex roots
132) If a,b,c are rational and no two of them are equal then the equations
(b - c)x² + (c - a)x + a - b = 0
And a(b - c)x²+ b(c - a)x + c(a - b)= 0
a) have rational roots
b) will be such that at least one has rational roots
c) have exactly one root common
d) have at least one root common
133) The equation x²+ b²= 1 - 2bx and x²+ a²= 1- 2ax have one and only one root common, then
a) a- b= 2 b) a - b +2= 0 c) |a - b|= 2 d) none
134) If px²+ qx + r=0 has no real roots and p,q,r are real such that p+ r > 0 then
a) p - q + r< 0 b) p - q + r> 0 c) p+ r = q d) all of these
135) Let p and q be roots of the equation x²- 2x + A =0 and let r and s be the roots of the equation x²- 18x + B=0 if p< q < r < s are in arithmetic progression then
a) A= -83, B= -3 b) A= -3, B= 77 c) q= 3, r= 7 d) p+ q+ r+ s= 20
136) The quadratic equation x²- 2x + λ= 0, λ ≠ 0,
a) cannot have a real root if λ< -1
b) can have a rational roots if λ is a perfect square
c) cannot have an integer root if n½- 1 < λ < n²+ 2n where n=0,1,2,3,...
d) none
137) A quadratic equation whose roots are (γ/α)² and (β/α)², where α, β, γ are the roots of x³+ 27=0 is
a) x²- x +1=0 b) x²+ 3x +9=0 c) x²+ x +1=0 d) x²- 3x +9 =0
138) The graph of the curve x²- 3x - y -2
a) between the lines x= 1 and x= 3/2
b) between the lines x= 1 and x= 2
c) strictly below the line 4y =1 d) none
139) a(x²- y²)+λ {x(y +1)+ 1} can be resolved into linear rational factors . Then
a) λ= 1 b) λ= 4a²/(a -1) (a≠ 1)
c) λ= 0, a= 1 d) none
140) x²- 4 is a factor of f(x)= (a₁x² + b₁x + c₁) (a₂x² + b₂x + c₂) if
a) b₁= 0, c₁+ 4a₁= 0
b) b₂ =0, c₂+ 4a₂= 0
c) 4a₁ + 2b₁ + c₁ =0, 4a₂ + c₂ = 2b₂
d) 4a₁ + c₁ = 2b₁, 4a₂ + 2b₂ + c₂ = 0
141) ax² + by² + cz² + 2ayz+ 2bzx+ 2cxy can be resolved into linear factors if a,b,c are such that
a) a= b = c b) ab+ bc + ca= 1
c) a+b +c = 0 d) none
142) If a,b are real roots of x²+ px +1=0 and c,d are the real roots of x²+ qx +1=0 then (a - c)(b - c)(a + d)(b + d) is divisible by
a) a+b +c+d b) a+b - c - d c) a - b+ c - d d) a - dpb - c - d
143) If x∈ [2,4] then for the expression x² - 6x +5
a) the least value =-4
b) the greatest value= 4
c) the least value= 3
d) the greatest value= -3
144) If 0< a <5, 0< b < 5 and (x²+5)/2 = x - 2 cos(a + bx) is satisfied for at least one real x then the greatest value of a+ b is
a) π b) π/2 c) 3π d) 4π
145) Let f(x)= x²(x +2)+ x +3. Then
a) f(-3- k)< 0 and f(-2+ k)> 0 for all k > 0
b) f(-3- k)> 0 and f(-2+ k)< 0 for all k > 0
c) f(x)=0 has a root α such that [α]+ 3 =0, where [α] is the greatest integer less than or equal to α.
d) f(x)= 0 has exactly one root α such that (α)+ 2= 0, where (α) is the smallest integer greater than or equal to α.
1c 2a 3c 4a 5d 6b 7c 8a 9b 10a 11c 12b 13c 14a 15b 16b 17b 18a 19a 20b 21c 22c 23b 24d 25c 26b 27a 28d 29b 30d 31b 32b 33c 34a 35b 36a 37c 38c 39c 40b 41a 42b 43c 44a 45c 46a 47b 48a 49b 50b 51a 52c 53c 54a 55a 56a 57d 58c 59c 60b 61b 62b 63c 64c 65b 66b 67b 68b 69d 70c 71b 72a 73b 74c 75b 76c 77b 78b 79a 80c 81c 82b 83a 84d 85d 86b 87c 88b 89a 90c 91a 92b 93a 94c 95a 96d 97c 98d 99d 100b 101c 102b 103a 104a 105c 106b 107a 108c 109a 110b 111c 112b 113ac 114ac 115bc 116ac 117bc 118cd 119b 120bc 121ab 122ab 123c 124abcd 125ab 126ac 127ab 128bc 129bc 130ad 131bc 132ac 133abc 134b 135bcd 136ac 137c 138c 139bc 140abcd 141ac 142ab 143ad 144c 145acd
TRIGONOMETRICAL FUNCTIONS AND IDENTITIES
1) If tanθ= a - 1/4a then secθ - tanθ is equal to
a) -2a, 1/2a b) -1/2a, 2a c) 2a d) 1/2a, 2a
2) sec²θ = 4xy/(x + y)², where x ∈ R, is true if and only if
a) x+ y≠ 0 b) x= y, x ≠ 0 c) x=y d) x≠ 0, y≠ 0
3) sin²θ= (x + y)²/4xy, where x ∈R, y ∈ R, gives real θ if and only if
a) x+ y=0 b) x=y c) |x|= |y|≠ 0 d) none
4) coseecθ = (x²- y²)/(x²+ y²), where x ∈R, y ∈ R, gives real θ if and only if
a) x=y ≠0 b) |x|= |y|≠ 0 c) x + y =0, x≠0 d) none
5) If sinθ + cosecθ=2 then the value of sin⁸θ + cosec⁸θ is equal to
a) 2 b) 2⁸ c) 2⁴ d) none
6) If x= r sinθ cosβφ, y= r sinθ sinφ and z= r cosθ then the value of x²+ y²+ z² is independent of
a) θ, φ b) r, θ c) r, φ d) r
7) Let p= a cosθ - b sinθ. Then for all real θ
a) p> √(a²+ b²) b) p< - √(a²+ b²) c) - √(a²+ b²) ≤ p ≤ √(a²+ b²) d) none
8) If 0°<θ <180° then √[2+ √{2+ √(2+....+√2(1+ cosθ))}] there being n number of 2's is equal to
a) 2 cos(θ/2ⁿ)
b) 2 cos(θ/2ⁿ⁻¹)
c) 2 cos(θ/2ⁿ⁺¹) d) none
9) The value of tan(π/16) + 2 tan(π/8) + 4 is equal to
a) cot(π/8) b) cot(π/16) c) cot(π/16) -4 d) none
10) The value of sin78° - sin66° - sin42°+ sin6° is
a) 1/2 b) -1/2 c) -1 d) none
11) The value of √3 cosec20° - sec20° is equal to
a) 2 b) 4 c) 2{sin20°/sin40°} d) 4. Sin20°/sin40°
12) The maximum value of 1+ sin(π/4+ θ) + 2 cos(π/4 - θ) for real values of θ is
a) 3 b) 5 c) 4 d) none
13) The minimum value of cos2θ + cosθ for real values of θ is
a) -9/8 b) 0 c) -2 d) none
14) The value of cosec10° - √3 sec10° is equal to
a) 1/2 b) 2 c) 4 d) 8
15) The least value of cos²θ - 6 sinθ cosθ + 3 sin²θ +2 is
a) 4+ √10 b) 4 - √10 c) 0 d) none
16) If cos⁴θ sec² α, 1/2 and sin⁴θ cosec² α are in AP then
cos⁸θ sec⁶α, 1/2 and sin⁸θ cosec⁶α are in
a) AP b) GP c) HP d) none
17) If tan(π/9) , x and tan(5π/18) are in AP and tan(π/9), y and tan(7π/18) are also in AP, then
a) 2x = y b) x> y c) x= y d) none
18) If cos(x - y), cosx and cosx(x + y) are in HP then | cosx. sec(y/2)| equals
a) 1 b) 2 c) √2 d) none
19) If 2sinα. cosβ. sinγ = sinβ sin(α+γ) then tanα, tanβ and tanγ are in
a) AP b) GP c) HP d) none
20) If tanα = √a, where a is a rational number which is not a perfect squares, then which of the following is a rational number?
a) sin2α b) tan2α c) cos2α d) none
21) Let f(θ)= cotθ/(1+ cotθ) and α + β=5π/4. Then the value of f(α). f(β) is
a) 2 b) -1/2 c) 1/2 d) none
22) If tan(α/2) and tan(β/2) are the roots of the equation 8x²- 26x +15=0 then cos(α + β) is equal to
a) -627/725 b) 627/725 c) -1 d) none
23) If sinα + sinβ = a and cosα - cosβ = b then tan{(α - β)/2} is equal to
a) -a/b b) -b/a c) √(a²+ b²) d) none
24) If 0<β < α < π/4, cos(α + β)= 3/5 and cos(α - β)= 4/5 then sin2α is equal to
a) 1 b) 0 c) 2 d) none
25) If cosα = (1/2)(x + 1/x), cosβ =(1/2)(y + 1/y) then cos(α - β) is equal to
a) x/y + y/x b) xy + 1/xy c) (1/2)(x/y + y/x) d) none
26) If sin α /(1+ sin α + cos α )= λ, then (1+ sin α - cos α )/(1+ sin α ) is equal to
a) 1/λ b) λ c) 1- λ d) 1+ λ
27) If |tanA|< 1 and |A| is acute then [{√(1+ sin2A) + √(1- sin2A)}/{√(1+ sin2A) - √(1- sin2A)}] is equal to
a) tanA b) - tanA c) cotA d) - cotA
28) tanθ tan(π/3 + θ) tan(π/3 - θ) is equal to
a) tan2θ b) tan3θ c) tan³θ d) none
29) The set of all possible values of α in [-π,π] such that √{(1- sin α)/(1+ sin α)} is equal to secα - tanα is
a) [0,π/2) b) [0,π/2) U (π/2,π)
c) [-π,0] d) (-π/2, π/2)
30) For all real values of θ, cotθ - 2 cot2θ is equal to
a) tan2θ b) tanθ c) - cot3θ d) none
31) Let a= cosA + cosB - cos(A+ B) and
b= 4 sin(A/2) sin(B/2) cos{(A+ B)/2}. Then a- b is equal to
a) 1 b) 0 c) -1 d) none
32) If tanθ + tan(θ+ π/3)+ tan(θ- π/3)= k tan3θ then k is equal to
a) 1 b) 3 c) 1/3 d) none
33) If a secα - c tanα = d and b secα + d tanα = c then
a) a²+ c²= b²+ d²
b) a²+ d²= b²+ c²
c) a²+ b²= c²+ d²
d) ab= cd
34) If cos20°- sin20°= p then cos40° is equal to
a) -p √(2- p²)
b) p √(2- p²)
c) p + √(2- p²) d) none
35) If 3 sinθ + 4 cosθ =5 then the value of 4 sinθ - 3 cosθ is
a) 0 b) 5 c) 1 d) none
36) If cos2x + 2cosx = 1 then sin²x(2- cos²x) is equal to
a) 1 b) -1 c) -√5 d) √5
37) If 0< φ < π/2, x = ∞ₙ₌₀∑ cos²ⁿφ, y= ∞ₙ₌₀∑ sin²ⁿφ and z= ∞ₙ₌₀∑ cos²ⁿφ. sin²ⁿφ then
a) xyz= xz+ y b) xyz= xy + z c) xyz= x+ z+ y d) xyz= yz+ x
38) Let n be an odd integer. If sin nθ = ∞ᵣ₌₀∑ bᵣ sinʳθ for all real θ then
a) b₀= 1, b₁ =3
b) b₀= 0, b₁ =n
c) b₀= -1, b₁ =n
d) b₀= 0, b₁ =n² - 3n -3
39) If cos5θ = a cos⁵θ + b cos³θ+ c cosθ then c is equal to
a) -5 b) 1 c) 5 d) none
40) If sin³x sin3x = ⁿₘ₌₀∑ cₘ cosmx is an identity in x, where cₘ's are constants then the value of n is
a) 4 b) 6 c) 9 d) none
41) The value of
sin(π/14) sin(3π/14) sin(5π/14) sin(7π/14) sin(9π/14) sin(11π/14) sin(13π/14) is equal to
a) 1 b) 1/16 c) 1/64 d) none
42) The numerical value of sin(π/18) sin(5π/18) sin(7π/18) is equal to
a) 1 b) 1/8 c) 1/4 d) 1/2
43) The value of tan63° - cot63° is equal to
a) 2 √(10-2√5)/(√5+1)
b) 2 √(10 + 2√5)/(√5+1)
c) (√5 -1)√(10-2√5)/4 d) none
44) The value of cos 9° - sin9° is
a) - √(5- √5)/2
b) (5 + √5)/4
c) (1/2) √(5- √5) d) none
45) The value of 2tan(π/10) + 3 sec(π/10) - 4 cos(π/10) is
a) 0 b) √5 c) 1 d) none
46) The value of tan20° + 2 tan50° - tan70° is
a) 1 b) 0 c) tan50° d) none
47) If α, β, γ and δ be four angles of a cyclic quadrilateral then the value of cosα + cosβ + cosγ + cosδ is
a) 1 b) 0 c) -1 d) none
48) If 4nα =π then cotα cot2α cot3α .....cot(2n -1)α is equal to
a) 1 b) -1 c) ∞ d) none
49) The value of cos12° cos24° cos36° cos48° cos72° cos84° is
a) 1/64 b) 1/32 c) 1/16 d) 1/128
50) The value of cos(π/11) + cos(3π/11)+ cos(5π/11) + cos(7π/11) + cos(9π/11) is
a) 0 b) 1 c) 1/2 d) none
51) ⁿ⁻¹ᵣ₌₁∑ cos²(rπ/n) is equal to
a) n/2 b) (n -1)/2 c) n/2 -1 d) none
52) The value of sin(π/n) + sin(3π/n) + sin(5π/n)+ ....to n terms is equal to
a) 1 b) 0 c) n/2 d) none
53) The sum of the real roots of cos⁶x + sin⁴x=1 in the interval -π≤ x ≤ π is equal to
a) 0 b) π c) - π d) none
54) The number of real solutions of the equation cos⁷x + sin⁴x =1 in the interval [-π, π] is
a) 2 b) 3 c) 5 d) none
55) If the solutions for θ from the equation sin²θ - 2sinθ +λ =0 lie in ∪ n ∈ z (2nπ - π/6, conjugate of (2n +1)π+ π/6) then the set of possible values of λ is
a) (-5/4,1] b) (-∞,1] c) (-5/4, + ∞] d) [1]
56) If ABCD is a convex quadrilateral such that 4 secA + 5=0, then the Quadratic equation whose roots are tanA and cosecA is
a) 12x²- 29x +15=0
b) 12x²- 11x -15=0
c) 12x² + 11x -15=0 d) none
57) If ABCD is a cyclic quadrilateral such that 12tanA - 5=0 and 5 cosB +3=0 then the quadratic equation whose roots are cosC, tanD is
a) 39x²- 16x - 48=0
b) 39x² + 88x + 48=0
c) 39x²- 88x + 48=0 d) none
58) The number of real solutions of the equation sin(eˣ) = 2ˣ + 2⁻ˣ is
a) 1 b) 0 c) 2 d) infinite
59) The equation (cos p -1) x²+ (cosp)x+ sin p =0 in x has real roots. Then the set of values of p is
a) [0,2π] b) [-π,0] c) [-π/2, π/2] d) [0,π]
60) If eˢᶦⁿˣ - e⁻ˢᶦⁿˣ - 4=0 then the number of real values of x is
a) 0 b) 2 c) 1 d) infinite
61) If sinθ = p, |p|≤ 1, then the quadratic equation whose roots are tan(α/2) and cot(α/2) is
a) px²+ 2x + p=0 b) px² - x + p=0 c) px²- 2x + p=0 d) none
62) If secα and cosecσ are the roots of x²- px + q=0 then
a) p² = q(q -2) b) p² = q(q + 2) c) p² + q²= 2q d) none
63) The number of values of x in the interval [0,5π] satisfying the equation 3 sin²x - 7 sinx +2=0 is
a) 0 b) 5 c) 6 d) 10
64) If x= α , β satisfies both the equations cos²x + a cosx + b =0 and sin²x + p sinx + q= 0 then the relation between a,b, p and q is
a) 1+ b + a²= p²- q -1
b) a²+ b²= p²+ q²
c) b + q = a²+ p²- 2 d) none
65) If 0≤ a ≤ 3, 0 ≤b ≤ 3 and the equation x²+ 4 + 3 cos(ax + b)= 2x has atleast one solution then the value of a+ b is
a) 0 b) π/2 c) π d) none
66) The equation cosθ = x + p/x , x ∈R, has a real solutions for θ. Then
a) p= 1/2 b) p≤ 1/4 c) p≥ 1/4 d) none
67) if f(x)= (sin3x)/(sinx), x ≠ nπ, then the range of values of f(x) for real values of x is
a) [-1,3] b) (-∞, -1] c) (3, +∞) d) [-1,3)
68) The set of values of k ∈R such that the equation cos2θ + cosθ + k =0 admits of a solution for θ is
a) [0,9/8] b) [0, +∞) c) [-2,0] d) none
69) The set of values of λ ∈R such that tan²θ + secθ= λ holds for some θ is
a) (-∞,1] b) (-∞,-1] c) φ d) [-1, +∞)
70) If tanA + tanB + tanC = tanA tanB tanC then
a) A, B, C must be angles of a triangle
b) the sum of any two of A, B, C is equal to the third
c) A+ B+ C must be an integral multiple of π
d) none
71) If x= sin(α -β ) sin(γ-δ), y= sin(β-γ) sin(α-δ) and, z= sin(γ-α) sin(β-δ) then
a) x+ y+ z=0 b) x+ y - z=0 c) - x+ y+ z=0 d) x³+ y³+ z³=3xyz
72) sin(15π/32) sin(7π/16)sin(3π/8) is equal to
a) 1/(8√2 cos(15π/32))
b) 1/(8 sin(π/32))
c) (1/4√2) cosec(π/16)
d) (1/8√2) cosec(π/32)
73) Which of the following is a rational number?
a) sin15 b) cos15 c) sin15 cos15 d) sin15 cos75
74) If a= 1/(5cosx + 12 sinx) then for all real x
a) the least positive value of a is 1/13
b) the greatest negative value of a is - 1/13
c) a ≤ 1/13 d) -1/13≤ a ≤ 1/13
75) Let y= sin²x + cos⁴x . Then for all real x
a) The maximum value of y is 2
b) the minimum value of y is 3/4
c) y ≤ 1 d) y≥ 1/4
76) Let y = sinx sin(60+x) sin(60- x). Then for all real x
a) the minimum value of y is -1/4
b) the maximum value of y is 1
c) y≤ 1/4 d) y≥ -1
77) Let fₙ(θ)= tan(θ/2) (1+ secθ)(1+ sec2θ)(1+ sec4θ) ....(1+ sec2ⁿθ). Then
a) f₂(π/16)= 1 b) f₃(π/32)= 1 c) f₄(π/64)= 1 d) f₅(π/128)= 1
78) Let 0 ≤ θ≤ π/2 and x = X cosθ+ Y sinθ, y= X sinθ - Y cosθ such that x²+ 4xy + y²= aX²+ bY², where a,b are constants. Then
a) a=-1, b =3 b) θ=π/4 c) a=3, b=-1 d) θ=π/3
79) If 7 cosx - 24 sinx = λ cos(x + α), 0< α < π/2, be true for all x ∈ to R then
a) λ= 25 b) α = sin⁻¹(24/25) c) λ= -25 d) α= cos⁻¹(7/25)
80) If A+ B =π/3 and cosA + cosB =1 then
a) cos(A - B)= 1/3
b) |cosA - cosB|= √(2/3)
c) cos(A - B)= -1/3
d) |cosA - cosB|= 1/2√3
81) If tanθ = a ≠ 0, tan2θ= b ≠ 0 and tanθ + tan2θ = tan3θ then
a) a= b b) ab= 1 c) a+ b =0 d) b= 2a
82) cos³x sin2x = ⁿₘ₌₁∑aₘ sin mx is an identity in x. Then
a) a₃ = 3/8, a₂= 0
b) n=6, a₁= 1/2
c) n=5, a₁= 1/4
d) ∑aₘ = 3/4
83) If 1+ cos(x - y)= 0 then
a) cosx - cos y=0
b) cosx + cos y=0
c) sinx + sin y=0
d) cosx + siny=1
84) If A≥ 0, B≥ 0, A + B =π/3 and y= tanA tanB then
a) the maximum value of y is 3
b) the minimum value of y is 1/3
c) the maximum value of y is 1/3
d) the minimum value of y is 0
85) cos(2π/7) + cos(4π/7)+ cos(6π/7) is equal to
a) an integer
b) a positive rational number
c) a negative rational number
d) an irrational number
1a 2b 3c 4d 5a 6a 7c 8a 9b 10b 11b 12c 13a 14c 15b 16a 17a 18c 19c 20c 21c 22a 23b 24a 25c 26b 27c 28b 29d 30b 31a 32b 33c 34b 35a 36a 37b 38b 39c 40b 41c 42b 43a 44c 45a 46b 47b 48a 49a 50c 51c 52b 53a 54b 55a 56b 57a 58b 59d 60a 61c 62b 63c 64c 65c 66b 67d 68a 69d 70c 71ad 72ad 73c 74ab 75bc 76ac 77abcd 78bc 79abd 80bc 81c 82acd 83bc 84cd 85c
