THEORY OF QUADRATIC EQUATIONS
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1) Two non integer roots of the equation
(x²+ 3x)² - (x²+ 3x) -6=0 are
a) (1/2) (-3+ √11), (1/2) (-3- √11)
b) (1/2) (-3+ √7), (1/2) (-3- √7)
c) (1/2) (-3+ √21), (1/2) (-3- √21) d) none
2) Two non integer roots of
{(3x -1)/(2x +3)}⁴ - 5{(3x -1)/(2x +3)}⅖ +4= 0 are
a) -5/7,-2/5 b) -2/4,7/5 c) 5/7,7/5 d) -2/5, 3/5
3) Sum of the roots of the equation
4ˣ - 3. 2ˣ⁺³ + 128=0 are
a) 5 b) 6 c) 7 d) 8
4) The only value of x satisfying the equation is
6√{x/(x +4)} - 2√{(x +4)/x} = 11 where x ∈ R
a) 4/35 b) -4/35 c) 16/3 d) none
5) The number of real values of x satisfying the equation
2(x²+ 1/x²) - 9(x + 1/x) + 14= 0
a) 1 b) 2 c) 3 d) 4
6) The non integer roots of x⁴- 3x³- 2x²+ 3x +1= 0 are
a) (1/2)(3+ √13), (1/2)(3 - √13)
b) (1/2)(3- √13), (1/2)(-3 - √13)
c) (1/2)(3+ √17), (1/2)(3 - √17) d) none
7) The number of real solution of
1/(x +1) + 1/(x +5) = 1/(x +2) + 1/(x +4) is
a) 0 b) 1 c) 2 d) 3
8) Number of real solutions of
(x -1)(x +1)(2x +1)+2x -3)= 15 is
a) 0 b) 2 c) 3 d) 4
9) The number of solutions of the equation
√[2x √(2x +4)]= 4 is
a) 0 b) 1 c) 2 d) 4
10) The number of solutions of
√(3x²+ x +5)= x -3 is
a) 0 b) 1 c) 2 d) 4
11) The number of solutions of
√(4- x) + √(x +9)= 5 is
a) 0 b) 1 c) 2 d) 3
12) The number of real solutions of
√(x²-4x +3) + √(x²-9)= √(4x²- 14x +6) is
a) 0 b) 1 c) 2 d) 4
13) The value of a for which one root of the equation
(a²- 5a +3)x² + (3a -1)x +2=0 is twice as large as other, is
a) -2/3 b) 1/3 c) -1/3 d) 2/3
14) Eange of the function f(x)= (x²+ x +2)/(x²+ x +1), x ∈ R is
a) (1, ∞) b) (1,3/2) c) (1,7/3] d) 1,7/5]
15) If f(x)= x²+ 2bx + 2c² and g(x)= - x²- 2cx + b² are such that minimum f(x)> maximum g(x), then relation between b and c, is
a) no relation b) 0< c<b/2 c) |c|< |b|/√2 d) |c|> √2 |b|
16) if a, b are the roots of x²+ px +1= 0, and c, d are the roots of x²+ qx +1= 0, the value of
E= (a - c)(b - c)(a + d)(b + d) is
a) p²- q² b) q²- p² c) q² + p² d) none
17) If 4ˣ - 3ˣ⁻¹⁾² = 3ˣ⁺¹⁾² - 2²ˣ⁻¹ , then the value of x is
a) 5/2 b) 2 c) 3/2 d) 1
18) For a> 0, a≠ 1, the number of values of x satisfying the equation
2logₓa + logₐₓa + 3 logₐ²ₓ a= 0 is
a) 2 b) 3 c) 4 d) infinite
19) The number of solutions of
√(x+1 - √(x -1)= 1 (x ∈R)
a) 1 b) 2 c) 4 d) infinite
20) If a, b, c are real and a≠ b, then the roots of the equation
2(a - b)x² - 11(a+ b + c)x -3(a - b)= 0 are
a) real and equal
b) real and unequal
c) purely imaginary d) none
21) Let a> 0, b> 0 and c> 0. Then both the roots of the equation
2ax²+ 3bx + 5c= 0
a) are negative
b) have real parts
c) have positive real parts d) none
22) If a, b, c are real, then both the roots of the equation
(x - b)(x - c)+ (x - c)(x - a)+ (x - a)(x - b)= 0 are always
a) positive b) negative c) real d) none
23) The equation
2x - 3/(x -2) = 4 - 3/(x -2) has
a) no root b) one root c) two equal roots d) none
24) If a, b,c are positive real numbers which are in GP , then the equation ax²+ 2bx + c= 0 and dx² + 2ex + f= 0 have common root if a/d, b/e, c/f are in
a) AP b) GP c) HP d) none
25) If P(x)= ax²+ bx + c and Q(x)= - ax²+ dx + c, where ac ≠ 0, then P(x) Q(x)= 0 has
a) no real root
b) exactly two real roots
c) atleast two distinct real roots d) none
26) If the product of the roots of the equation
x²+ 5kx + 2e⁴ˡⁿᵏ -1=0 is 31, then sum of the root is
a) -10 b) 5 c) -8 d) none
27) The number of real roots of
(7+ 4 √3)|ˣ|⁻⁸ + (7- 4 √3)|ˣ|⁻⁸ = 14 is
a) 0 b) 2 c) 4 d) none
28) Sum of all the values of x satisfying the equation
log₁₇log₁₁(√(x +11) + √x)= 0 is
a) 25 b) 36 c) 171 d) 0
29) let α, β be the roots of the equation (x - a)(x - b)= c with 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
30) If p,q are roots of x²+ px + q= 0, then
a) p=1 b) p= 1 or 0 c) p= -2 d) p= -2 or 0
31) The equation √(x +1) - √(x -1)= √(4x -1), (x ∈R)
a) no solution b) one solution c) two solution d) more than two solutions
32) The sum of all the real roots of the equation
|x -2|²+ |x -2| - 2= 0 is
a) 7 b) 4 c) 1 d) none
33) Let p and q be the roots of x²- 2x + A= 0 and r and s be the roots of x²- 18x + B= 0. If p< q < r< s are in AP, then ordered pair (A, B) is equal to
a) (-3,77) b) (77,-3) c) (-3,-77) d) none
34) In a triangle PQR, angle R= π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c=0 where a≠ 0, then
a) a+ b= c b) b+ c= a c) a+ c= b d) b= c
35) If α, β (α> β) are the roots of the equation x²+ bx + c= 0 c< 0< b, then
a) 0< α < β b) 0< α < β<|α| c) α < β<0 d) α < 0< |α|< β
36) For the equation 3x²+ px + 3= 0 , p> 0, if one of the roots is square of the other, than p is equals to
a) 1/3 b) 1 c) 3 d) 2/3
37) If the roots are the equation x² - 2ax + a²- 3 = 0 are real and less than 3, then
a) a<2 b) 2≤ a≤3 c) 3<a ≤4 d) a> 4
38) If b> a, then the equation (x - a)+x - b) -1= 0 has
a) both roots in [a, b]
b) both roots in (- ∞, a)
c) both roots in (b, ∞)
d) one root in (-∞,a) and other in (b, ∞).
39) Let α, β be the roots of x² - x + p = 0 and γ, δ be the roots of x² - 4x + q = 0 . If α, β, γ, δ are in GP then the integral value of p and q respectively, are
a) -2,- 32 b) -2,3 c) -6,3 d) -6, -32
40) If a, b, c are not all equal and α and β be the roots of the equation ax² + bx + c = 0, then value of (1+ α+ α²)(1+ β+ β²) is
a) 0 b) positive c) negative d) non negative
41) If a,b,c are in AP and if the equations
(b - c)x²+ (c - a)x + (a - b)= 0 and
2(c + a)x²+ (b + c)x = 0 have a common root, then
a) a², b², c² are in AP
b) a², c², b² are in AP
c) c², a², b² are in AP d) none
42) Value of
x= √[6+ √{6+ √{6+....up to
a) 3 b) 2 c) 1 d) none
43) two complex numbers α and β are such that α + β = 2 and α⁴+ β⁴= 272, then the quadratic equation whose roots are α and β is
a) x²-2x -16= 0
b) x²-2x + 12= 0
c) x²-2x -8= 0 d) none
44) The equation (cos p -1)x² + (cos p)x + sin p = 0 in variable x has real roots, if p belongs to the interval
a) 0,2π) b) (-π,0) c) (-π/2,π/2) d) (0,π)
45) If the roots of the equation
1/(x + a) + 1/(x + b) = 1/c are equal in magnitude but opposite in sign, then their product is
a) (1/2) (a²+ b²) b) - (1/2) (a²+ b²) c) ab/2 d) -ab/2
46) If the quadratic equations x²-11x + a= 0 and x²-14x + 2a= 0 have common root, then the values of a are
a) 0, 24 b) 0,-24 c) 1,-1 d) -2,1
47) If α, β are the roots of the equation ax² + bx + c = 0, then the value of α³+ β³ is
a) (3abc+ b³)/a³
b) (a³+ b³)/3abc
c) (3abc- b³)/a³
d) - (3abc+ b³)/a³
48) If the sum of the roots of the quadratic equations ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then
a) ab², ca², bc² are in AP
b) ab², bc², ca² are in AP
c) ab², bc², ca² are in AP d) none
49) If the ratio of the roots of the equation x² + bx + c = 0 is the same as that of the ratio of the roots of x² + qx + r = 0, then
a) br²= qc² b) cq²= rb² c) q²c²= b²r² d) Bpbq= rc
50) If a, b are the non zero distinct roots of x² + ax + b = 0, then the least value of x² + ax + b is
a) 2/3 b) 9/4 c) -9/4 d) 1
51) If a+ b+ c= 0, then the quadratic equation 3ax² + 2bx + c = 0 has
a) at least one root in [0,1]
b) one root in [2,3] and other is [-2, -1]
c) imaginary roots d) none
52) If a< b < c< d, then the equation 3(x - a)(x - c)+ 5(x - b)(x - d)=0
a) real and distinct rootes
b) real and equal roots
c) purely imaginary roots d) none
53) For real x, the function (x - a)(x - c)/(x - b) will assume all real values provided
a) a< b < c b) b< c < a c) c< a < cpb d) none
54) Let a,b,c ∈R 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 + β) d) α <γ< β
55) Suppose p,q,r,s ∈R and α, β be the roots of x²+ px + q= 0 and α⁴, β⁴ be the roots of x²- rx + s= 0, then the equation x²- 4qx + 2q² - r= 0 has always
a) two imaginary roots
b) two positive roots
c) two negative roots
d) one positive and one negative root
56) The equation
x⁽³/⁴⁾⁽ˡᵒᵍ₂ˣ⁾^²⁺ ˡᵒᵍ₂ˣ ⁻ ⁵/⁴= √2 has
a) exactly two real roots
b) no real root
c) one irrational root d) none
57) Let f(x) be a quadratic expression which is positive for all x, if g(x)= f(x)+ f'(x) then for all real x,
a) g(x)< 0 b) g(x)> 0 c)g(x) = 0 d) g(x)≥ 0
58) If α, β are the roots of ax²+ bx + c= 0, then the quadric equation whose roots are 2α+3 and 2β+3 is
a) 4ax² - 3bx + c= 0
b) 6a¹x² - 4abx + 6c= 0
c) ax² +2(b- 3a)x + 9a+ 2b= 0 d) none
59) If α, β are the roots of the equation ax²+ bx + c= 0, then the equation whose roots are α³, β³ is
a) a³y²+ (b³- 3abc)y+ c³= 0
b) a³y²+ (3abc - b³)y- c³= 0
c) a²y²+ 2aby+ c³= 0 d) none
60) If sinα and cosα are the roots of 25x²+ 5x -12= 0, then value of sin2α is
a) 12/25 b) -12/25 c) -24/25 d) 4/5
61) Let P(x) be a polynomial with integral coefficients . If there exist two integers a and b such that P(a) - P(b)= 1, then
a) both a and b must be even
b) both a and b must be odd
c) a and b are two consecutive integers d) none
62) Let am b, c be non zero real such that
¹₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx
²₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx
Then the quadratic equation has ac¹+ bx + c= 0 has
a) no root in (0,2)
b) at least one root in (1,2)
c) a double root +0,2) d) none
63) If a, b, c are distinct real numbers, then the expression
f(x)= a²{(x - b)(x - c)}/{(a- b)(a - c)} + b²{(x - c)(x - a)}/{(b- c)(b - a)} + c²{(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to
a) x²- (a + b + c)x + abc
b) x² + x - abc c) x² d) none
64) The number of real solutions of the equation
27¹⁾ˣ + 12¹⁾ˣ = 2(8¹⁾ˣ) is
a) 0 b) 1 c) infinite d) none
65) If 0< a < b< c < d, then the quadratic equations ax²+ {1+ a(b + c)}x + abc - d = 0 has
a) real and distinct roots out of which one lies between c and d
b) real and distinct roots out of which one lies between a and b
c) real and distinct roots out of which one lies between b and c
d) nonreal roots
SAP-2
1) If α, β are the roots are the roots of x²+ px + q= 0 and γ, δ are the roots of x²+ rx + s= 0, then the value of (α- γ)(α-δ)(β -γ)(β -δ) is
a) (r - p)² - (q- s)²
b) (r - p)² + (q- s)²
c) (r - p)² - (q- s)² - 2rp(r - p)(q - s) d) none
2) The number of real solutions of x²+ 5|x| + 4=0 is
a) 4 b) 2 c) 1 d) 0
3) The number of real solutions of x²- 3|x|+2=0 is
a) 4 b) 2 c) 1 d) 0
4) If 2+ i √5 is a root of x²- px + q= 0 where p and q are real, then the ordered pair (p,q) is equal to
a) (4,9) b) (9,4) c) (3,3) d) (2,3)
5) If the quadratic equation 2x²+ ax + b= 0 and 2x²+ bx + a = 0, (a≠ b) have a common root, the value of a+ b is
a) -3 b) -2 c) -1 d) 0
6) If a,b,c, d and p are distinct real numbers such that
(a²+ b²+ c²)p²+ 2(ab + bc+ cd)p + (b²+ c²+ d²)≤ 0, then a, b, c, d
a) are in AP
b) are in GP
c) are in HP d) none
7) The number of real roots of of the equation sin(eˣ)= 5ˣ + 5⁻ˣ is
a) 0 b) 1 c) 2 d) infinitely many
8) The value of a for which the equation x³+ ax +1=0 and x⁴+ ax²+ 1= 0 have common root is
a) 2 b) -2 c) 0 d) none
9) The roots of the equation |x²- x -6|= x +2 are
a) -2, 1, 4 b) 0, 2, 4 c) 0, 1, 4 d) -2, 2, 4
10) The number of real roots of the equation |x²| - 3|x|+ 2= 0
a) 4 b) 1 c) 2 d) infinite
11) If α, β are the roots of x²- P(x +1) - C = 0 then value of
(α+1)(β +1) is
a) C -1 b) 4+ C c) 1+ C d) 1- C
12) Let f(x) be quadratic expression such that f(x)< 0 , x∈ R. If g(x)= f(x)+ f'(x)+ f"(x) then for x∈ R.
a) g(x)< 0 g(x)≤ 0 c) g(x)> 0 d) g(x)≥ 0
13) If x+1 is a factor of x⁴+ (p -3)x³ - (3p -5)x²+ (2p+9)x + 12, then value of p is
a) -2 b) 2 c) 1 d) -1
14) The equation ₓ(3/4)(log₅x)²+ log₅x - 5/4 = √5 has
a) only one real solution
b) exactly two real solution
c) one irrational solution
d) infinite number of solutions
15) Let f and g be two real valued functions and S={x | f(x)=0} and T={x | f(x)=0}, then S∩T represent the set of roots of
a) f(x) g(x)= 0
b) f(x)²+ g(x)² = 0
c) f(x) + g(x)= 0
d) f(x)/g(x)= 0
16) The number of real roots of the equation (x +3)²+ (x +1)²+ (x -5)²+ (x -6)²= 0 is
a) 2 b) 1 c) 0 d) none
17) If a,b,c are in AP and if (b - c)x²+ (c - a)x + (a - b)= 0 and 2(c + a)x²+ (b + c)x = 0 have a common root, then
a) a², b², c² are in AP
b) a², b², c² are in GP
c) a², b², c² are in HP d) none
18) If a,b,c are positive real numbers , then the number of positive real roots of the equation ax²+ bx + c= 0 is
a) 0 b) 1 c) 2 d) infinite
19) If the roots of the equation x²+ p²= 8x + 6p are real, then p belongs to the interval
a) [2,8] b) [-8,2] c) [- 2,8] d) [-8, -2]
20) If sum of the roots of the equation (a+ 1)x²+ (2a +3)x + (3a+4)= 0 is -2, then the product of the roots is
a) 1 b) -1 c) 2 d) -2
21) If 3- 4i is a root of x²- px + q=0 where p, q ∈R, then value (2p - q)/(p+ q) is
a) -12/31 b) -13/31 c) -15/31 d) none
22) If x= 1+ i is a root of x³- ix + 1 - i = 0, then the equation whose roots are the remaining two roots of x³- ix + 1 - i = 0 is
a) x²+(1+ i)x + 1+ i= 0
b) x²- (1+ i)x + 1+ i= 0
c) x²+ 2(1+ i)x - 2= 0 d) none
23) If α and β be the roots of the equation x²+ px - 1/2p²= 0, where p∈ E. Then the minimum possible value of α⁴+ β⁴ is
a) 2 b) 2√2 c) 2+√2 d) none
24) The equation √{x +3 - 4√(x -1)}+ √{x +8 - 6√(x -1)}= 0 has
a) no solution
b) exactly one solution
c) exactly two solutions
d) more than two solutions
25) The equation | x- x² - 1|= |2x - 3 - x²| has
a) no solution b) exactly one solution
c) exactly two solutions
d) more than two solutions
26) If sinα, cosα are the roots of the equation ax²+ bx+ c= 0, (a≠ 0), then
a) a²- b²+ 2ac=0
b) a²+ b² - 2ac=0
c) (a - c)²= b²+ c² d) none
27) If x ∈R, and k= (x²- x+ 1)/(x²+ x+ 1), then
a) 1/3≤k ≤ 3 b) k≥ 5 c) k≤ 0 d) none
28) If the quadratic equation x²+ bx+ ca= 0 and x²+ cx+ ab = 0 have a common root and b≠ c, then their other roots will satisfy the equation
a) x²+ (b+ c)x+ bc= 0
b) x²+ ax+ bc= 0
c) x²+ ax+ bc= 0 d) none
29) If the inequality (mx²+ 3x+ 4)/(x²+ 2x+ 2) < 5 is satisfied for all x∈ R, then
a) m< 5 b) m > 5 c) m < 71/24 d) m > 71/24
30) If a, b, c are distinct real numbers, then the equation expression
{(x - b)(x - c)}/{(a- b)(a - c)} + {(x - c)(x - a)}/{(b- c)(b - a)} + {(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to
a) 1 b) x c) x² d) none
31) If ax²+ bx+ c, a, b, c ∈ R, a≠ 0 has no real zeros and a- b + c< 0, then value of ac is
a) positive b) zero c) negative d) non negative
32) The number of real roots of the equation eˢᶦⁿˣ + e⁻ˢᶦⁿˣ = 4 is
a) zero b) one
c) more than one but finitely many
d) infinitely many
33) If α , β are the roots of the equation ax²+ bx+ c=0 and α + h, β+ h are the roots of the equation Ax²+ 2Bx+ C=0, then
a) (b²- ac)/(B²- AC) = a/A
b) (b²- ac)= (B²- AC)
c) h = (bA - aB)/Aa
d) h = (Ac + aC)/(Aa+ Bb)
34) The quadratic equation x²+ 7x= 14(q²+1), where q is an integer has
a) real and distinct roots
b) integral roots
c) imaginary roots d) none
35) Let a,b,c ∈R and a> 0. If the quadratic equation ax²+ bx+ c=0 has two real roots α and β such that α < -1 and β > 1, then value of c/a + |b/a| is
a) less than 2 b) less than 1 c) less than 0 d) less than -1
36) Let a, b, c ∈R and a≠ 0 be such that (a+ c)²< b², then the quadratic equation ax²+ bx + c= 0 has
a) imaginary roots
b) real roots
c) two real roots lying between (-1,1) d) none
37) The integral values of a for which the equation (x - a)(x - 10) +1= 0 has integral roots are
a) -1,3 b) 2,3 c) 12,8 d) -8,-12
38) The number of real solution of 4ˣ⁺¹·⁵ + 9ˣ⁺⁰·⁵ = (10)(6ˣ) is
a) zero b) one c) two d) infinite
39) The number of real solution of ₂sin²x + ₅(₂cos²x)= 7 is
a) zero b) 1 c) finitely many d) infinitely many
40) The number of values of k for which the equation x²- 2x + k= 0 has two distinct roots lying in the interval (0,1) is
a) 0 b) 1 c) 2 d) infinitely many
41) If roots of the equation ax² + bx + c = 0 are real and are of the form α/(α -1), (α+1)/α, then value of (a+ b + c)² is
a) 4ac - b² b) b²- 4ac c) c²+ a²- 2b² d) none
42) Let x be an integer and x²+ x +1 is divisible by 3. When x is divided by 3, it leaves remainder
a) 0 b) 1 c) 2 d) any of (a),(b),(c)
43) If α, β are the roots of the equation x²+ ax+ b=0, then maximum value of - x²+ ax+ b + (1/4) (α - β)² is
a) (1/4) (a²- 4b) b) (1/4) (b²- 4a) c) a²/2 d) none
44) If both the roots of the equation x²+ bx+ c =0 lie in the interval (0,1), then
a) b = -1 , c= 2
b) b > -2 , c< 1
c) b = -5 , c< 2 d) none
45) Let a, b, c ∈R be such that a+ b+ c< 0, a - b+ c< 0 and c> 0. If α and β are the roots of the equation ax²+ bx + c =0, then the value of [α] + [β] is
a) 2 b) 1 c) -1 d) 0
46) If roots of the equation x²- 2mx + m²-1= 0 lie in the interval (-2,4), then
a) - 1< m < 3 b) 1< m < 5 c) 1< m < 3 d) - 1< m < 5
47) The value of √[8 + 2√{8+ 2√(8+ 2√(8+.....is
a) 10 b) 6 d) 8 d) none
48) The number of solutions of the equation
Sin(πx/2√3)= x²- 2√3 x + 4 is
a) 1 b) 2 c) 0 d) infinite
(Hint: sin(πx/2√3)= (x- √3)² + 1≥ 1 => Sin(πx/2√3)≥ 1. But Sin(πx/2√3)≤ 1. Thus Sin(πx/2√3)= 1 . So, (x - √3)²+ 1= 1= x= √3 check x=√3 satisfies the given equation)
49) The number of solutions of |x +2|= 2(3- x) is
a) 1 b) 2 c) 3 d) 0
50) If α, β are the roots of the equation ax²+ bx + c=0, then the equation ax²+ bx(x -1)+ c(x -1)²= 0 has roots
a) α /(1- α), β/(1- β)
b) (1-α)/α, (1-β)/β
c) α /(1+ α), β/(1+ β) d) none
51) Two non integer roots of
(x²- 5x)² - 7(x²- 5x)+ 6= 0 are
a) (1/2) (5+ √29), (1/2) (5- √29)
b) (1/2) (-5+ √29), (1/2) (-5+ √29)
c) (1/2) (-5+ √14), (1/2) (-5- √41) d) none
52) The number of real roots of
{(x-1)/(x +1)}⁴ - 13{(x-1)/(x +1)}²+ 36=0 is
a) 0 b) 2 c) 3 d) 4
53) The number of negative roots of
9ˣ⁺² - 6(3ˣ⁺¹)+1= 0 is
a) 0 b) 1 c) 2 d) 4
54) The number of real roots of 81{(2x-5)/(3x +1)}⁴ - 45{(2x-5)/(3x +1)}²+ 4=0 , x≠ 1/3 is
a) 1 b) 2 c) 3 d) 4
55) The number of irrational roots of
(x²+ 3x +2)² - 8(x²+ 3x) -4=0 is
a) 0 b) 2 c) 3 d) 4
56) The number of roots of the equation
√{x/(x -3)} + √{(x -3)/x}= 5/2, x≠ 0, x≠ 3 is
a) 0 b) 2 c) 3 d) 4
57) The number of irrational roots of the equation
4(x - 1/x)²+ 8(x + 1/x)= 29 is
a) 0 b) 2 c) 4 d) infinite
58) Irrational roots of the equation
2x⁴+ 9x²+ 8x²+ 9x +2= 0 are
a) -2-√3, 2 +√3
b) 2-√3, 2 +√3
c) -2 +√3, -2 -√3 d) none
59) Sum of the roots of the equation
4(x - 1/x)² - 4(x - 1/x)+ 1=0 is
a) 5 b) 1 c) -5/2 d) -1
60) The number of irrational roots of the equation
(x -1)(x -2)(3x -2)(3x +1)= 21 is
a) 0 b) 2 c) 3 d) 4
61) Product of roots of the equation
x - √(3x -6)=2 is
a) 10 b) 5 c) 7 d) 24
62) Product of roots of the equation
2√(2x +1)= 2x -1 is
a) -3 b) -5 c) 5 d) 3
63) Product of roots of the equation
√(13 - x²)= x +5 is
a) -6 b) 7 c) 6 d) -7
64) The number of roots of the equation
√(x²-4) - (x -2)= √(x²-5x +6) is
a) 0 b) 1 c) 2 d) 3
65) The product of the roots of the equation
√(x²-4x +3) + √(x²- 7x + 12) = 3 √(x -3) is
a) 15 b) -15 c) 20 d) -20
66) If a≠ b and difference between the roots of x²+ ax + b= 0 is equal to difference between the roots of x²+ bx + a = 0, then
a) a+ b +4= 0 b) a+ b - 4= 0 c) a- b -4= 0 d) a - b +4= 0
67) α ≠ β but α²= 5α -3 and β²= 5β -3, then the equation whose roots are α/β and β/α is
a) 3x²+ 19x + 3= 0 b) x² - 5x + 3= 0 c) 3x²- 19x + 3= 0 d) none
68) If α ∈ (0,π/2), then the expression √(x²+ x) + tan²α/√(x²+ x) is always greater than or equal to
a) 2 tanα b) 2 c) 1 d) sec²α
69) If a, b ∈R, and the equation
x²+ (a - b)x - a - b + 1= 0 has real roots for all b ∈R, then a lies in the interval
a) (1, ∞) b) (0,∞) c) (-∞,1) d) (-1,1)
70) If α and β (α < β) are the roots of the equation x²+ bx + c = 0, where c< 0 < b, then
a) 0<α< β b) α <0<β <|α| c) α < β < 0 d) none
1d 2d 3a 4a 5b 6b 7a 8b 9d 10a 11d 12a 13b 14c 15b 16c 17d 18a 19c 20b 21b 22d 23c 24d 25b 26a 27a 28c 29c 30a 31a 32a 33c 34a 35d 36c 37c 38c 39d 40a 41b 42b 43c 44d 45c 46a 47d 48a 49a 50c 51a 52d 53b 54c 55d 56b 57b 58c 59b 60d 61a 62a 63c 64d 65a 66a 67c 68a 69a 70b
∈ ∞
α β γδ
TRIGONOMETRICAL RATIOS AND IDENTITIES
SAP- 1
1) 2(sin⁶x + cos⁶x) - 3(sin⁴x+ cos⁴x)+ 1=0
2) 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π- x)] is equal to
a) 0 b) 1 c) 3 d) sin4x + sin6x e) none
3) sin⁶x + cos⁶x + 3sin²x cos²x = 1
4) 3(sinx - cosx)⁴ + 6(sinx + cosx)² + 4(sin⁶x + cos⁶x) is independent of x.
5) (sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2 sin²x cos²x).
6) (3+ cos4x) cos2x= 4(cos⁸x - sin⁸x).
7) If sinx+ cosx= a, then find the values of|sinx - cosx| and cos⁴x + sin⁴x.
8) If sinx + cosecx = 2, then sin²x + cosec²x is equal to 2. T/F
9) f(x)= cos²x + sec²x≥ 2. T/F
Or minimum value of f(x) is 2.
10) 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
11) Let A= sin⁸x + cos¹⁴x, then for all real x
a) A≥ 1 b) 0< A ≤1 c) 1/2< A ≤ 3/2 d) none
12) If x, y are acute, sinx= 1/2, cos y= 1/3, then (x + y) belong to
a) (π/3,π/2) b) (π/2,2π/3) c) (2π/3,5π/6) d) (5π/6,π)
13) (tanx + cot x)²= sec²x + cosec²x = sec²x cosec²x.
14) (1+ tan x tan y)² + (tanx - tan y)² = sec²x sec²y.
15) (secx - tan x)/(sec x + tan x)= 1- 2 secx tanx + 2 tan²x.
16) 1/(secx - tan x) - 1/cosx = 1/cosx - 1/(secx + tanx).
17) (secx + tan x -1)(secx - tanx +1) - 2 tan x= 0
18) If (secx + tanx)(sec y + tan y)(sec z + tan z)= (secx - tan x)(sec y - tan y)(sec z - tan z) show that each of the side is equal to ±1.
19) If (1+ sinx)(1+ sin y)(1+ sin z)= (1- sin x)(1- sin y)(1- sin z), show that each side is equal to ± cosx cos y cos z.
20) Let f(x)= sinx (sinx + sin3x). Then f(x).
a) ≥ 0 only when x≥ 0
b) ≤ 0 for all real x
c) ≥0 for all real x
d) ≤ 0 only when x ≤ 0
21) The maximum value of (cosx₁). (cosx₂)......(cosxₙ), under the restriction 0≤ x₁, x₂, .....xₙ≤ π/2 and (cotx₁).(Cotx₂).....(cotxₙ)= 1 is
a) 1/2ⁿ⁾² b) 1/2ⁿ c) 1/2n d) 1
22) √{(1- sinx)/(1+ sinx)}= secx - tan x.
23) √{(1+ cosx)/(1 - cosx)}= cosecx + cotx.
24) If sinx + sin²x= 1, then show that cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x -1= 0
25) If sinx+ sin²x + sin³x = 1, then cos⁶x - 4cos⁴x + 8cos²x = _____.
26) sec⁴x (1- sin⁴x) - 2 tan²x = 1.
27) tan²x - sin²x = sin⁴x sec²x= tan²x sin²x.
28) (cotx + tant)/(cot y + tanx)= cotx tan y.
29) (sinx + cosx)(tanx + cotx)= secx + cosecx
30) (cosx cosecx - sinx secx)/(cosx + sinx)= cosecx - secx.
31) (1+ cotx - cosecx)(1+ tanx + secx)= 2
32) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1
33) (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.
34) cot²x(secx -1)/(1+ sinx) = sec²x. (1- sinx)/(1+ secx).
35) (secx +1- tanx)/(tanx - secx +1)= (1+ cosx)/sinx.
36) cosx/(1- tanx) + sinx/(1- cotx)= sinx + cosx.
37) tₙ= sinⁿx + cosⁿx, then (t₃ - t₅)/t₁ = (t₅ - t₇)/t₃.
38) tanx/(1- cotx) + cotx/(1- tanx)= secx cosecx +1.
39) (sinx + cosecx)²+ (cosx + secx)²= tan²x + cot²x +7.
40) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - cosecx/sec²x.
41) (secx - cosecx)(1+ tanx + cotx)= tanx secx - cotx cosecx.
42) {2sinx tanx(1- tanx)+ 2 sinx sec²x}/(1+ tanx)²= 2sinx/(1+ tanx).
43) (tanx + cosec y)²+ (cot y - secx)²= 2 tanx cot y(cosecx + sec y).
44) {(1+ sinx - cosx)/(1+ sinx + cosx)}²= (1- cosx)/(1+ cosx).
45) If 2sinx/(1+ cosx + sinx)= y, then (1- cosx + sinx)/(1+ sinx) is also y.
46) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)}. sin²x cos²x = (1- sin²x cos²x)/(2+ sin²x cos²x).
47) (cosecx - secx)(cotx - tanx)= (cosecx + secx)(secx cosecx -2).
48) If tanx+ sinx = m and tanx - sinx = n, then show that m²- n² = 4√(mn).
49) Eliminate x from the relations
a secx = 1- b tan x and a² sec²x = 5+ b² tan²x.
50) If cosecx - sinx = m, secx - cosx = n, eliminate x.
51) If cosecx - sinx = a³, secx - cosx= b³, then a²b²(a² + b²)= 1.
52) If cotx + tanx = a, secx - cosx = b eliminate x.
53) If c cos³x + 3c cosx sin²x = m, c sin³x + 3c cos²x sinx = n, then show that (m + n)²⁾³ + (m - n)²⁾³= 2c²⁾³.
54) If cosx + sinx= √2 cosx, show that cosx - sinx =√2 sinx.
55) If 3 sinx + 5 cosx = 5, show that 5 sinx - 3 cosx = ±3.
56) If a cosx + b sin x = p, a sinx - b cosx = q, show that a² + b² = p² + q².
57) If a cosx - b sin x = c, show that a sinx + b cosx = ±√(a² + b² + c²).
58) If a sinx + b cosx = c, then show that (a - b tanx)/(b + a tanx)= ±√(a² + b² + c²)/c.
59) If tan²x = (1- e²), show that secx + tan³x cosecx = (2- e²)³⁾².
60) 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.
61) If sinθ = (m² - n²)/(m²+ n²), determine the values of tanθ, secθ, cosecθ.
62) If tanθ = 2x(x+1)/(2x +1), determine sinθ and cosθ.
63) If cosθ = 2x/(1+ x²), find the values of tanθ and cosecθ.
64) If secx = p + 1/4p, then secx + tanx = 2p or 1/p
65) If secθ + tanθ = p, obtain the values of secθ, tanθ, sinθ in terms of p.
66) If cosx/cos y = a, sinx/sin y = b, then (a² - b²)sin²y= a² -1
67) If tanθ = p/q, show that (p sinθ - q cosθ)/(p sinθ + q cosθ) = (p² - q²)/(p² + q²).
68) 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.
69) If m² + m'² + 2mm' cosθ = 1,
n² + n'² + 2nn' cosθ = 1 and mn + m'n' + (mn' + m'n) cosθ = 0 show that m² + n² = cosec²θ.
SAP-2
1) The value of sin⁶θ + cos⁶θ + 3 sin²θ cos²θ is
a) 0 b) 1 c) 2 d) 3
2) The least value of 2sin²θ+ 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
2cosec2x cotx - cot²x = 1 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
TRUE OR FALSE
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.
10) The inequality ₂sin²θ + ₂cos²θ≥ 2√2 holds for all real θ.
11) The equation sinθ = x + 1/x holds true for all real θ.
FILL IN THE BLANK
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 cosecθ - cotθ = q, then the value of cosecθ = _____
Trigonometry full. 12 mix
SAP-1
1) cos(540° - θ) - sin(630- θ) is equal to
a) 0 b) 2 cosθ c) 2 sinθ d) sinθ - cosθ
2) 2sec²x - sec⁴x -2 cosec²x + cosec⁴x = 15/4 if tanx is equal to
a) 1/√2 b) 1/2 c) 1/2√2 d) 1/4
3) If 2 sinθ/(1+ cosθ + sinθ)= x, then cosθ/(1+ sinθ) is equal to
a) 1/x b) x c) 1+ x d) 1- x
4) If cosx = (2 cos y -1)/(2- cos y) (0< x, y<π), x+ y=π then tan(x/2) is
a) ⁴√3 b) √3 c) 3 d) 3²
5) If tan25°= x, then (tan155° - tan 115°)/(1+ tan155° tan115°) is equal to
a) (1- x²)/2x b) (1+ x²)/2x c) (1+ x²)/(1- x²) d) (1- x²)/(1+ x²)
6) If sinx + cos y= a and cosx + sin y= b, then tan{(x - y)/2} is equal to
a) a+ b b) a- b c) (a+ b)/(a - b) d) (a - b)/(a+ b)
7) The value of the determinant
Sin²13 sin²77 tan135
Sin²77 tan135 sin²13
Tan135 sin²13 sin²77 is equal to
a) -1 b) 0 c) 1 d) 2
8) If A= 130° and x= sinA+ cosA, then
a) apx> 0 b) x < 0 c) x = 0 d) x ≥ 0
9) If tan²36° + k(sin18°+ cos36°)= 5, then the value of k is
a) 2 b) 2√5 c) 4 d) 4√5
10) sin3θ/cos2θ< 0 if θ lies in
a) +13π/48, 14π/48)
b) (14π/48,18π/48)
c) 18π/48,23π/48)
d) any of these intervals
11) If cosx + cos y= a, sin x + sin y= b and k is the arithmetic mean between x and y then sin2k + cos2k is equal to
a) (a+ b)²/(a²+ b²)
b) (a - b)²/(a²+ b²)
c) (a²- b²)/(a²+ b²). d) none
12) If (sinx)/a= (cosx)/b= (tanx)/c= k, then bc + 1/ck + ak/(1+ bk) is
a) k(a + 1/a) b) (1/k)(a + 1/a) c) 1/k² d) a/k
13) sin²x + cos²(x + y)+ 2 sinx sin y cos(x + y) is independent of
a) x b) y c) both x and y d) none
14) If uₙ = sin nθ secⁿθ, vₙ = cos nθ secⁿθ, n≠ 1, θ≠ pπ, n, p∈I,
then (vₙ - vₙ₋₁)/uₙ₋₁ + 1uₙ/nvₙ= 0 for
a) all values of n
b) finite numbers of values of n
c) infinite number of values of n
d) no values of n
15) If 1/cosx cos y + tanx tan y= tan z, 0< x, y< π then 1- tan²z< 0 for
a) all values of x and y
b) no values of x and y
c) finite number of values of x and y
d) infinite number of values of x and y.
16) tan203° + tan22°+ tan203°+ tan22°=
a) -1 b) 0 c) 1 d) 2
17) If sin32°= k and cosx = 1- 2k²; α, β are the values of x between 0° and 360° with α< β, then
a) α+β= 180 b) β - α = 200 c) β= 4α + 40 d) β = 5α - 20
18) The minimum value of 27 tan²θ+ 3 cot²θ is
a) 9 b) 18 c) 27 d) 30
19) The value of sin12 sin48 sin54 is
a) sin30 b) sin²30 c) sin³30 d) cos³30
20) tan⁶(π/9) - 33 tan⁴(π/9)+ 27 tan²(π/9)=
a) tan(π/3) b) tan²(π/3) c) tan(π/6) d) tan²(π/6)
21) if 3 sin y = sin(2x + y), then tan(x + y)- 2 tan x is
a) independent of x
b) independent of y
c) independent of both x and y
d) independent of none of them
22) Let n be a fixed positive integer such that sin(π/2n)+ cos(π/2n)=√n/2 then
a) n< 4 b) n> 8 c) n= 6 d) none
23) If A= sin²θ + cos⁴θ, then for all values of θ
a) 1≤ A ≤ 2 b) 3/4 ≤ A ≤ 1 c) 13/16≤ A ≤ 1 d) 3/4≤ A ≤ 13/16
24) If a= cosφ cosψ + sinφ sinψ cosδ
b= cosφ sinψ - sinφcosψ cosδ and
c= sinφ sinδ, then a²+ b²+ c²=
a) -1 b) 0 c) 1 d) none
25) In a triangle ABC, BP is drawn perpendicular to BC to meet CA in P, such that CA= AP, then BP/AB=
a) 2 sinA b) 2 sinB c) 2 sinC d) none
26) If x+y = z, then cos²x + cos² y + cos²z - 2 cosx cos y cos z is equal to
a) cos²z b) sin²z c) 0 d) 1
27) If sin2θ= k, then the value of tan³θ/(1+ tan²θ) + cot³θ/(1+ cot²θ) is equal to
a) (1- k²)/k b) (2- k²)/2 c) k²+1 d) 2- k²
28) If sin²A = x, then sinA sin2A sin3A sin4A is a polynomial in x, the sum of whose coefficients is
a) 0 b) 40 c) 168 d) 336
29) If sinA/sinB =√3/2 and cosA/cosB=√5/2, 0< A, B <π/2, then tanA+ tanB is equal to
a) √3/√5 b) √5/√3 c) 1 d) (√3+√5)/√5
30) If cosθ= cosx cos y, then tan{(θ+ x)/2} tan{(θ- x)/2} is equal to
a) tan²(x/2) b) tan²(y/2) c) tan²(θ/2) d) cot²(y/2)
31) If α, β, γ, δ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of 4 sin(α/2)+ 3 sin(β/2)+ 2 sin(γ/2) + sin(δ/2) is equal to
a) 2√(1- k) b) 2√(1+ k) c) 2√k d) none
32) The equation a sinx + b cosx = c, where|c|> √(a²+ b²) has
a) a unique solution
b) infinite number of solutions
c) no solution d) none
33) If cotx equals to the integral solution of the inequality 4x²- 16x +15< 0 and sin y equals to the slope of the bisector of the first quadrant, then sin(x + y) sin(x - y) is equal to
a) -3/5 b) -4/5 c) 2/√5 d) 3
34) The value of cos(2π/7) + cos(4π/7)+ cos(6π/7) + cos(7π/7) is
a) 1 b) -1 c) 1/2 d) -3/2
35) The greatest value of f(x)= 2 sinx + sin2x on [0,3π/2], is given by
a) 9/2 b) 5/2 c) 3√3/2 d) 3/2
36) If x= a cos³θ sin²θ, y= a sin³θ cos²θ and (x²+ y²)ᵖ/(xy)ᑫ, (p,q ∈N) is independent of θ, then
a) 4p= 5q b) 4q= 5p c) p+ q= 9 d) pq= 20
37) If (a - b) sin(x + y)= (a+ b) sin(x - y) and a tan(x/2) - b tan(y/2)= c, then the value of sin y is equal to
a) 2ab/(a²- b²- c²) b) 2bc/(a²- b²- c²) c) 2bc/(a²- b²+ c²) d) 2ab/(a²- b²+ c²)
38) If (cosx - cos y)/(cosx - cos z)=( sin²y cos z)/(sin²z cos y) then cosx is equal to
a) (cos y - cos z)/(1+ cos y cos z)
b) (cos y - cos z)/(1- cos y cos z)
c) (cos y + cos z)/(1+ cos y cos z) d) none
39) If 0< x, y<π and cosx+ cos y - cos(x + y)= 3/2 then sinx + cos y is equal to
a) 0 b) 1 c) (√3+1)/2 d) √3
40) If sinθ, cosθ, tanθ are in GP, then cos⁹θ+ cos⁶θ+ 3 cos⁵θ-1 is equal to
a) -1 b) 0 c) 1 d) none
41) If sinα, sinβ , sinγ are in AP and cosα, cosβ, cosγ are in GP then (cos²α+ cos²γ - 4 cosα cosγ)/(1- sinα sinγ).
a) -2 b) -1 c) 0 d) 2
42) If cosα + cosβ = a and sinα + sinβ = b and α - β = 2θ, then cos3θ/cosθ=
a) a²+ b²-2 b) a²+ b²-3 c) 3- a²- b² d) (a²+ b²)/4
43) If cosA= 3/4, then the value of 16 cos²(A/2) - 32 sin(A/2) sin(5A/2) is
a) -4 b) -3 c) 3 d) 4
44) If D= 1 cosθ 1
-sinθ 1 -cosθ
-1 sinθ 1
Then D lies in the interval
a) [0,4] b) [2,4] c) [2-√2, 2+√2] d) [-2,2]
45) The value of θ lying between θ= 0 and θ= π/2 and satisfying the equation of determinant
1+ sin²θ cos²θ 4 sin4θ
sin²θ 1+ cos²θ 4sin4θ =0
sin²θ cos²θ 1+ 4sin4θ
is
a) 3π/24 b) 5π/24 c) 11π/24 d) π/24
46) If x= sin³p/cos²p , y= cos³p/sin²p; and sin p+ cos p= 1/2, then x + y=
a) 75/18 b) 44/9 c) 79/18 d) 48/9
47) If sinθ+ cosecθ= 2, then sinⁿθ+ cosecⁿθ=
a) 2ⁿ b) 2⁻ⁿ c) 2 d) 2n
48) If a cosA - b sinA = c, then a sinA + b cosA is equal to
a) ±√(a²+ b²- c²) b) ±√(apb²+ c²- a²) c) ±√(c²+ a²- b²) d) ±√(a²+ b² + c²)
49) The general solution of the trigonometrical equation sinx + cosx = 1 is
a) x= 2nπ, n∈ I
b) x= 2nπ +π/2, n∈ I
c) px= nπ + (-1)ⁿπ/4, -π/4, n∈ I d) none
50) The value of x between 0 and 2π which satisfy the equation sinx √(8cos²x)= 1 are in AP. The common difference of the AP is
a) π/8 b) π/4 c) 3π/8 d) 5π/8
51) Number of solutions of the equations tanx + secx = 2 cosx lying in the interval [0,2π] is
a) 0123
52) The number of all possible triplets (a₁, a₂, a₃) such that a₁+ a₂ cos2x + a₃ sin²x= 0 for all x is
a) 0 b) 1 c) 3 d) infinite
53) If tan(cotx)= cot(tanx), then the value of sin2x is
a) π/4
b) 4/(2n -1)π, n∈I - {-1,0}
c) 2/π
d) 4/(2n +1)π, n∈I , n> 7
54) sinx + 2 sin2x = 3+ sin3x, 0≤ x≤ 2π has
a) 2 solutions in I quadrant
b) one solutions in II quadrant
c) no solutions in any quadrant
d) one solutions in each quadrant
55) Let f(x)= (sin²θ)x² + (cos²θ)x + cos²θ, f(x)= 0 has no real roots, then cos²θ can be
a) 3/4 b) 1/4 c) 1/8 d) 1/16
56) The equation (cosp - 1)x² + (cos p)x + sin p= 0 where x is a variable, has real roots if p lies in the interval
a) (0,2π) b) (-π,0) c) (-π/2,π/2) d) (0,π)
57) The set of values of x for which
(tan3x - tan2x)/(1+ tan3x tan2x)= 1 is
a) φ b) {π/4} c) {nπ+π/4, n= 1,2,3...}
d) {2nπ +π/4, n= 1,2,3....}
58) The general solution of the equation
(1+ sinx+....+(-1)ⁿ sinⁿx +...)/(1+ sinx+ ....+ sinⁿx+....)= (1- cos2x)/(1+ cos2x), x≠(2x +1)π/2, n∈ U is
a) (-1)ⁿ(π/3)+ nπ
b) (-1)ⁿ(π/6)+ nπ
c) (-1)ⁿ⁺¹(π/6)+ nπ
d) (-1)ⁿ⁻¹(π/3)+ nπ, (n∈I)
59) The number of solutions of the equation
sin⁵x - cos⁵x = 1/cosx - 1/sinx (dinx≠ cosx) is
a) 0 b) 1 c) infinite d) none
60) General solution of the equation log₂ sinx - log₂cosx - log₂(1- tanx) - log₂(1+ tanx)+ 1=0 is
a) (2n+1)π/8 b) (16n+1)π/8 c) (2n+1)π/4 d) none
61) The smallest positive root of the equation tanx - x = 0 lies in
a) I quadrant b) II quadrant c) III quadrant IV quadrant
62) If sin⁴x + cos⁴y +2= 4 sinx cos y, 0≤x, y ≤π/2 then sinx + cos y=
a) - 2 b) 0 c) 2 d) none
63) tan(pπ/4)= cot(qπ/4) if
a) p+ q = 0 b) p+ q= 2n +1 c) p+ q = 2n d) p+ q= 2(2n+1)
64) If sinx = cos y, √6 sin y = tan z and 2 sin z = √3 cosx; u, v, w denotes respectively sin⅖x, sin²y , sin²z then the value of the triplet(u,v,w) is
a) (1,00) b) (0,1,0) c) (1/2,1/2,3/4) d) (1/2,3/4,1/2)
65) A solution (x,y) of the system of equation x - y = 1/3 and cos²(πx) - sin²(πy)= 1/2 is given by
a) (2/3,1/3) b) (7/6,1/6) c) (13/6,11/6) d) (1/6,5/6)
66) If x+ y + z= π, tanx tan y = 2, tanx + tany + tan z= 6, then the value of z is
a) nπ+π/4, n∈ I b) nπ + tan⁻¹2, n ∈ I c) nπ + tan⁻¹3, n ∈ I d) none
67) cos2x - 3 cosx +1= 1/{(cot2x - cotx) sin(x -π)} holds.
a) if cosx = 0 b) if cosx = 1 c) if cosx= 2/5 d) for no real value of x.
68) cos(x - y) - 2 sinx + 2 sin y= 3 if
a) sinx= sin y
b) x+ y= 2nπ, (x - y)= (2km-1)π/2
c) x= 2kπ -π/2, y= 2nπ +π/2
d) cos(x - y)= -1 (n,k∈ I )
69) The equation 8 cosx cos2x cos4x = sin6x/sinx has a solution given by
a) x= nπ b) x= nπ+π/4 c) x= (2n +1)π/14 d) x= (2n +1)π/7 (n ∈ I )
70) cos3θ/(2 cos2θ -1)= 1/2 if
a) θ= nπ+π/3 b) θ= 2nπ± π/3 c) θ= 2nπ± π/6 d) θ= nπ+π/6
71) If 6 cos2θ + 2 cos²(θ/2)+ 2 sin²θ= 0, -π<θ < π, then θ is equal to
a) π/3 b) π/3, cos⁻¹(3/5) c) cos⁻¹(3/5) d) π/3, π - cos⁻¹(3/5)
72) The number of integral values of a for which the equation cos2x + a sinx = 2a -7 possesses solution is
a) 2 b) 3 c) 4 d) 5
73) The least difference between the roots of the equation 4 cosx (2- 3 sin²x)+ (cos2x +1)= 0 (0≤ x≤π/2) is
a) π/6 b) π/4 c) π/3 d) π/2
74) The equation cos⁴x - (a+ 2) cos²x - (a +3)= 0 possesses a solution
a) a>-3 b) a< -2 c) -3≤ a ≤ -2 d) a is any positive integer
75) The solution of |cosx|= cosx -2 sinx is
a) x= nπ b) x= nπ+π/4 c) x= nπ+(-1)ⁿ(π/4) d) x= (2n +1)π+π/4
1a 2a 3d 4a 5a 6d 7b 8a 9c 10a 11d 12b 13a 14d 15a 16c 17c 18b 19c 20b 21c 22c 23b 24c 25c 26d 27b 28a 29d 30b 31b 32c 33b 34d 35c 36a 37b 38c 39c 40b 41a 42b 43c 44c 45c 46c 47c 48a 49c 50b 51c 52d 53b 54c 55a 56d 57a 58b 59a 60b 61c 62c 63d 64a 65c 66c 67d 68c 69c 70b 71d 72d 73a 74c 75d
SAP-2
1) If 2 tanx + cot y = tan y, then the value of tan(y - x) is
a) tanx b) cotx c) tan y d) cot y
2) If cos(x - y)= a cos(x + y), then cotx cot y is equal to
a) (a-1)/(a+1) b) (a+1)/(a-1) c) (a-1) d) (a+1)
3) sin²A + sin²(A+ B)+ 2 sinA cosB sin(B-A) is equal to
a) sin²A b) sin²B c) cos²A d) cos²B
4) If 3 sin2θ/(5+ 4 cos2θ)= 1, then the value of tanθ is equal to
a) 1 b) 1/3 c) 3 d) none
5) If x= a sec³θ tanθ, y= b tan³θ secθ, then sin²θ is equal to
a) x/a - y/b b) x/a + y/b c) xy/ab d) ay/bx
6) cotθ - cot3θ is equal to
a) 2 sinθ sin3θ
b) 2 cosθ cos3θ
c) 2 cosθ cosec3θ
d) 2 sinθ cosec3θ
7) If 0<x, y < 2π, the number of solutions of the system of equations sinx sin y = 3/4 and cosx cos y = 1/4 is
a) 0 b) 1 c) 2 d) infinite
8) If A and B be acute positive angles satisfying 3 sin²A+ 2 sin²B= 1, 3 sin2A - 2 sin2B = 0 then
a) B=π/4 - A/2 b) A =π/4 - 2B c) B=π/2 - A/4 d) A =π/4 - B/2
9) If tan x, tan y , tan zare the roots of the equation x³- px² - r= 0, then the value of (1+ tan²x)(1+ tan²y)(1+ tan²z) is equal to
a) (p - r)² b) 1+ (p -r)² c) 1- (p -r)² d) none
10) If A, B, C are the angles of triangle such that angle A is obtuse then
a) tanA tanB< 1
b) tanB tanC < 1
c) tanA tanC< 1
d) tanA tanB tanC< 1
11) If tan(x/2)= cosecx - sinx, then cos²(x/2) is equal to
a) sin18 b) cos36 c) sin36 d) cos18
12) If a sin²x + b cos²x = a cos²y + b sin²y= 1 and a tanx= b tany(a≠ b) then
a) a+ b= 2ab b) a- b= 2ab c) a - b+ 2ab = 0 d) a + b+ 2ab = 0
13) The acute angle of a rhombus whose side is a mean proportional between its diagonals is
a) 15 b) 20 c) 30 d) 80
14) Given the height h and the angle bisector l drawn from the vertex of the right angle of a triangle, then cosine of an acute angle of the triangle is given by
a) (h+ √(l²- h²))/√2h
b) (h - √(l²- h²))/√2h
c) h/l d) (h - √(l²- h²))/√2l
15) If 2 sin²(x +π/4)+ √3 cos2x > 0, then
a) cos(2x -π/6)> -1/2
b) sin(2x -π/6)< -1/2
c) sin(2x -π/6)> -1/2
d) cos(2x -π/6)< -1/2
16) The equation sin⁴x + cos⁴x = a has a real solution if
a) 0< a≤ 1 b) 1/2≤ a≤1 c) 1/4≤ a≤1/2 c) -1≤ a≤1
17) x= ∞ₙ₌₀∑ cos²ⁿx, y= ∞ₙ₌₀∑ sin²ⁿx, z= x= ∞ₙ₌₀∑ cos²ⁿx sin²ⁿ2x, |cosx|< 1, |sinx|< 1 then x+ y+ z is equal to
a) xy b) yz c) zx d) xyz
18) For n∈ I, the line x= nπ+ π/2 does not intersect the graph of
a) cot(x+π) b) cos(x+π) c) sinx d) tanx
19) The least positive value of x satisfying (sin²2x + 4 sin⁴x - 4 sin²x cos²x)/(4- sin²2x - 4 sin²x)= 1/9 is
a) π/3 b) π/6 c) 2π/3 d) 5π/6
20) In a triangle ABC right angled at C, sin²A/sin²B - cos²A/cos²B is equal to
a) (a²- b²)c²/a²b²
b) (a⁴ + b⁴)/a²b²
c) (b²- c²)a²/b²c²
d) (c²- a²)b²/c²a²
21) If (tanx)/2= (tany)/3= (tanz)/5 and x+ y+z=π, then the value of tan²x + tan²y + tan²z is
a) 38/3 b) 38 c) 114 d) none
22) If the angles A, B, C of a triangle are in AP such that sin(2A + B)= 1/2 then sin(B+ 2C)=
a) -1/2 b) 1/2 c) √3/2 d) 1/√2
23) cos7.5°=
a) √{(2+√2+√6)/8}
b) √{(4+√2+√6)/8}
c)(2√2+√3+1)/2√2
d) √{(4+√2+√6)/4}
24) If tanx+ tan y= a, cotx + cot y= b, x - y = m (≠0) then
a) ab<4 ab=4 c) ab>4 d) ab= 0
25) If x/y= cosA/cosB then (x tanA + y tanB)/(x + y)=
a) (sinA+ cosB)/(cosA+ sinB)
b) (sinA+ sinB)/(cosAcosB)
c) tan{(A+ B)/2}
d) cot{(A- B)/2}
26) If x= a(cosθ+ θ sinθ), y= a(sinθ - θ cosθ) then aθ=
a) x+ y= a b) √(x²+ y²- a²) c) √(x²& y²+a²) d) x - y + a
27) (1+ cos(π/8))(1+ cos(3π/8))(1+ cos(5π/8))(1+ cos(7π/8))=
a) 1/2 b) cos(π/8) c) 1/8 d) (1+√2)/2√2
28) If 2 cosx + 2 cos3x= cos y, 2 sinx + 2 sin3x= sin y then the value of cos2x is
a) -7/8 b) 1/8 c) -1/8 d) 7/8
29) (cosA)/3= (cosB)/4= 1/5, -π/2< 0, 0<B<π/2, then 3sunA + 4 sinB=
a) 0 b) -1 c) 24/5 d) 1
30) The value of log₃tan1° + log₃tan2°+.....+ log₃tan89° is
a) 3 b) 1 c) 2 d) 0
31) If x= X cosθ - Y sinθ, y= X sinθ+ Y cosθ and ax²+ 2bxy + cy²= AX²+ 2HXY+ BY², then
a) H= 0 if θ= 0 b) H= 0 if θ= π/2 c) A+ B= a+ c d) H= c - a if θ= π/4
32) If tan²((π/2) - θ)/sec²θ. Cot²θ/sec((π/2- θ)). sin((π/2- θ)/sin⁴θ= cotⁿθ then n=
a) 2 b) 4 c) 6 d) 8
33) If sin5θ= a sin⁵θ+ b sin³θ+ c sinθ+ d, then
a) a+ b+ c=0
b) a+ b+ c + d=0
c) 5a+ 3b - 4c=0
d) a- 3c+ d=0
34) The number of solutions of
Sinθ+ 2Sin2θ+ 3Sin3θ+ 4Sin4θ= 10, 0<θ<π is
a) 0 b) 1 c) 2 d) 4
35) If tanA - tanB= x and cotB - cotA= y, then the value of cot(A - B) is
a) (x - y)/xy b) 1/x²+ 1/y² c) (x+y)/xy d) xy
36) cos3θ/cos³θ + sin3θ/Sin³θ is equal to
a) 3cos2θcosecθ
b) 3cot2θ/Sec2θ
c) 12 cot2θcosec2θ
d) 12 tan2θSec2θ
37) (x tanθ + y cotθ) (x cotθ+ y tanθ) - 4xy cos²θ=
a) x²+ y² b) 4xy c) (x + y)² d) none
38) cos11- cos2x is
a) a positive integer
b) a negative integer
c) a positive rational number
d) a negative rational number
39) If sinA, cosA and tanA are in GP., then cot⁶A - cot²=
a) -1 b) 0 c) 1 d) none
40) If tanA tanB, tanC satisfy the equation 3tan³θ - 4 tan²θ+ 3 tanθ +1=0, then A+ B+ C=
a) 0 b) π/2 c) 3π/4 d) 2π
41) if x sinθ + ysin2θ+ z sinSin3θ = sin4θ, (θ≠ nπ) then 8 cos³θ - 4z cos²θ - 2(y +2) cosθ is equal to
a) x - y b) x - z c) y - z d) none
42) The number of values of sinx satisfying sin5x= 5 sinx is
a) 0 b) 1 c) 2 d) 3
43) If sinx, sin y are the roots of the equation
a sin²θ+ b sinθ + + c= 0 and sinx + 2 sin y= 1 then a²+ 2b²+ 3ab + ac=
a) -1 b) 0 c) 1 d) a+ b+ c
44) If sin(θ/2)= a, cos(θ/2)= b, then
(1+ sinθ)(3 sinθ + 4cosθ+5)=
a) (a+ b)²(a+ 3b)²
b) (a+ b)²(3a+ b)²
c) (a- b)²(a- 3b)²
d) (a- b)²(3a- b)²
45) if cosx - Sinx = 1/2, then tan2x=
a) √7/3 b) √7/4 c) 3/√7 d) 2/√7
46) Which of the following gives the least value of A
a) cos2A= sin3A
b) cos3A= sin7A
c) tanA= cot3A
d) cotA= tan2A
47) If A, B, C are acute positive angles such that A+ B + C=π and cotA cotB cotC = k, then
a) k≥3 b) k≤ 1/3√3 c) k≤√3 d) k≤ 1/3√3
48) If sinA= sinB and cosA= cosB; A≠ B, then
a) tan{(A- B)/2}= 0
b) cos(A+ B)= 1
c) tan{(A+ B)/2}= 0
d) sin(A- B)= 1/2
49) cos22+ cos78+ cos 80=
a) 4 sin11 sin39 sin40
b) 1+ 4 cos11 cos39 cos40
c) 1+ 4 sin11 sin39 sin40
d) 4 cos11 cos39 cos40
50) tanx + (1/2) tan(x/2)+ (1/2²) tan(x/2²) + .....+ (1/2ⁿ⁻¹) Tan(x/2ⁿ⁻¹) is equal to
a) 1/2ⁿ cot(x/2ⁿ) - 2 cot2x
b) (1/2ⁿ⁻¹) cot(x/2ⁿ⁻¹) - 2 cot2x
c) tan{(2ⁿ -1)x/2ⁿ⁻¹}
d) 2 cot2x - (1/2ⁿ⁻¹) cot(x/2ⁿ⁻¹)
51) If 4nx=π, then the value of tanx tan2x tan3x.....tan(2n -1)x is
a) -1 b) 0 c) 1 d) none
52) The value of
(3+ cot76 cot16)/(Cot76+ cot16) is
a) cot44 b) cot46 c) tan2 d) cot92
53) If x cosθ= y cos(θ+ 2π/3)= z cos(θ+ 4π/3) then xy+ yz + zx=
a) cos²θ b) sin²θ c) 1 d) 0
54) If A> 0, B> 0 and A+ B =π/3 then the maximum value of tanA tanB is
a) 1/√3 b) 1/3 c) √3 d) 3
55) If tanθ, 2 tanθ+2, 3 tanθ+ 3 are in GP, then the value of
(7- 5 cotθ)/(9- 4√(sec²θ-1)) is
a) 12/5 b) -33/28 c) 33/100 d) 12/13
56) If sinθ+ cosθ= a and cosθ- sinθ= b , then sinθ(sinθ - cosθ)+ sin²θ(sin²- cos²θ)+ sin³θ(sin³θ - cos³θ)+ ....is equal to
a) (1- ab)/(1+ ab)
b) (1- a²/(3- a²)
c) (1- ab)/(1+ ab) + (1- a²/(3- a²)
d) (1+ ab)/(1- ab) + (a²-1)/(3- a²)
57) If x> 0 and the determinant
x sinθ cosθ
- sinθ x. 1 = 0 then
cosθ 1. x
x<√2 b) x=√2 c) x>√2 d) none
58) If x₁, x₂, x₃,.....xₙ are in AP whose common difference is θ, then the value of sinθ(secx₁. secx₂+ secx₂ secx₃+....secxₙ₋₁ secxₙ) is
a) sin nθ/(cosx₁ cosxₙ)
b) sin(n -1)θ/cosx₁ cosxₙ
c) sin nθ cosx₁ cosxₙ
d) cos(n -1)θ/sinx₁sinxₙ
59) If xₙ₊₁ = √(1/2) (1+ xₙ), then cos[√(1- x₀²)/(x₁x₂x₃....to infinite)] (-1< x₀< 1) is equal to
a) -1 b) 1 c) x₀ d) 1/x₀
60) If (1+ √(1+ x) tanθ= (1- √(1- x)) then x=
a) sinθ b) sin2θ c) sin4θ d) cos4θ
61) If f(θ)= sinθ(sinθ+ sin3θ), then f(θ)
a) ≥ only when θ≥ 0
b) ≤ 0 for all real θ
c) ≥ 0 for all real θ
d) ≤ 0 only when θ≤ 0
62) In a right angled triangle, the hypotenuse is 2√2 times the length of the perpendicular drawn from the opposite vertex in its hypotenuse then the other two angles are
a) π/3 , π/6 b) π/4,π/4 c) π/8, 3π/8 d) π/2, 5π/12
63) √cos2x + √(1+ sin2x)= √(sinx + cosx) if
a) sinx+ cosx= 1
b) x=2nπ
c) x= nπ+π/4
d) sinx - cosx= 0
64) If cot(π/3) cos(2πx)=√3, the general solution of the equation
a) 2nπ± π/3 b) n±1/3 c) n±1/6 d) n±1/2
65) 2 cos²x + 4 cosx = 3 sin²x if
a) cosx= (-2+√14)/5
b) cosx= (-2+√19)/5
c) sinx= (-2+√14)/5
d) sinx= (-2+√19)/5
66) sinx + 2 sin2x= 3+ sin3x
a) if sinx + cos2x= 0
b) if sin2x -1=0
c) If cosx= 0
d) for no real value of x
67) 6 tan²x - 2 cos²x= cos2x if
a) cos2x= -1 b) cos2x= 1 c) cos3x = -1/2 d) cos2x = 1/2
68) The greatest value of cosθ for which cos5θ= 0 is
a) 0 b) (1+√5)/4 c) √{(5+√5)/8} d) √{(√5+1)/4}
69) If tanpθ= tan qθ, then the values of θ form an AP with common difference
a) π/(p+ q) b) π/p c) π/q d) π/(p - q)
70) The number of pairs (x, y) satisfying the equation sinx + sin y= sin(x + y) and|x|+ |y|= 1 is
a) 2 b) 4 c) 6 d) infinite
71) The equation ˣ₀∫ (t²- 8t +13) dy= x sin(a/x) has a solution if sin(a/6)=
a) 0 b) 1 c) 3 d) 6
72) The smallest positive root of the equation √sin(1- x)= √cosx is
a) 1/2+ π/4 b) 1/2+ 3π/4 c) 1/2+ 5π/4 d) 1/2+ 7π/4
73) The sum of the roots of the equation
a) 4 cos³x - 4 cos²x - cos(π+ x) - 1= 0 in the interval [0,315] is pπ, where p is equal to
a) 2500 b) 2550 c) 2600 d) 2651
74) A solution (x,y) of x²+ 2x sinxy +1= 0 is
a) (1,0) b) (1,7π/2) c) (-1,7π/2) d) (-1,0)
75) eˢᶦⁿˣ - e⁻ˢᶦⁿˣ= 4 for
a) all real values of x
b) some x∈ [0,π/2]
c) some x ∈ (-π/2,π/2)
d) some x ∈ (-π/2,π/2)
1d 2b 3b 4c 5d 6c 7c 8a 9b 10b 11b 12a 13c 14d 15a 16b 17d 18d 19b 20a 21a 22a 23b 24c 25c 26b 27c 28a 29a 30d 31c 32d 33c 34a 35c 36c 37c 38d 39c 40b 41b 42b 43b 44a 45c 46b 47b 48a 49c 50b 51c 52a 53d 54b 55c 56c 57d 58b 59c 60c 61c 62c 63b 64c 65b 66d 67d 68c 69d 70c 71b 72d 73b 74b 74d
θ
