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 = _____
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