Mathematics - Trigonometric Functions, Identities and Equation
Exam Duration: 45 Mins Total Questions : 30
\(cos2\theta cos2\phi +{ sin }^{ 2 }\left( \theta -\phi \right) \) is equal to
- (a)
\(sin2\left( \theta +\phi \right) \)
- (b)
\(cos2\left( \theta +\phi \right) \)
- (c)
\(sin2\left( \theta -\phi \right) \)
- (d)
\(cos2\left( \theta -\phi \right) \)
The value of (1+tan1o)(1+tan44o)(1+tan2o)(1+tan43o)...... (1+tan22o)(1+tan23o) is
- (a)
1
- (b)
222
- (c)
38
- (d)
211
The maximum and minimum values of cos2x - 6sinxcosx + 3sin2x + 2 are
- (a)
\(4-\sqrt { 10 } ,4+\sqrt { 10 } \)
- (b)
\(2-\sqrt { 10 } ,2+\sqrt { 10 } \)
- (c)
\(3-\sqrt { 5 } ,4+\sqrt { 5 } \)
- (d)
\(None\quad of\quad the\quad above\)
If \(\alpha ={ sin }^{ 8 }\theta +{ cos }^{ 14 }\theta ,\) then which of the following is true?
- (a)
\(\alpha >1\)
- (b)
\(\alpha \le 1\)
- (c)
\(\alpha =0\)
- (d)
\(\alpha <0\)
If [sin x] + [√2 cos x] = - 3, x ∈ [0, 2π] ([.] denotes the greatest integer function), then x belongs to
- (a)
\(\left(\pi,{5\pi\over 4}\right)\)
- (b)
\(\left[\pi,{5\pi\over 4}\right]\)
- (c)
\(\left({5\pi\over 4},2\pi\right)\)
- (d)
\(\left[{5\pi\over 4},2\pi\right]\)
If sin x + cos x + tan x + cot x + see x + cosec x = 7 and sin 2x = a - b,√c, then a - b + 2c is
- (a)
0
- (b)
14
- (c)
28
- (d)
42
If 2 tan2 x - 5 see x is equal to 1 for exactly 7 distinct values of x ∈ \(\left[0,{n\pi\over 2}\right], n\epsilon\ N\) then the greatest value of n
- (a)
6
- (b)
12
- (c)
13
- (d)
15
The number of solutions of the equation sin5x-cos5x\(={1\over cos\ x}-{1\over sin\ x}(sin\ x\neq cos\ x)\)is
- (a)
0
- (b)
1
- (c)
infinite
- (d)
none of these
The equation (cos p -1) x2 + (cos p) x + sin p = 0, where x is a variable, has real roots. Then the interval of p may be any one of the followings
- (a)
(0, 2π)
- (b)
(-π, 0)
- (c)
\(\left(-{\pi\over 2},{\pi\over 2}\right)\)
- (d)
(0, π)
The number of solutions of the equation \(sin\left(\pi\ x\over 2\sqrt3\right)=x^2-2\sqrt3x+4\)
- (a)
forms an empty set
- (b)
is only one
- (c)
is only two
- (d)
is greater than two
Values of x and y satisfying the equation sin7 y = |x3 - X2 - 9x + 91 + Ix3 - X2 - 4x + 4|+ sec22y + cos2 y are
- (a)
x = 1, Y = nπ ,n ∈ I
- (b)
x = 1, Y =2nπ+\(\pi\over 2\) ,n ∈ I
- (c)
x = 1, Y =2nπ ,n ∈ I
- (d)
none of these
The number of solutions of the equation \(1 + sin\ x\ sin^2\left( x\over 2\right)=0\) in [-π, π]
- (a)
zero
- (b)
1
- (c)
2
- (d)
3
2 sin x cos 2x = sin x, if
- (a)
\(x=n\pi+{\pi\over 6}(n\epsilon I)\)
- (b)
\(x=n\pi-{\pi\over 6}(n\epsilon I)\)
- (c)
\(x=n\pi(n\epsilon I)\)
- (d)
\(x=n\pi+{\pi\over 2}(n\epsilon I)\)
solution (x, y) of the system of equations \(x-y={1\over 3}\)and \(cos^2(\pi\ x)-sin^2(\pi\ y)={1\over 2} \)is given by
- (a)
\(\left\{{7\over 6},{5\over 6}\right\}\)
- (b)
\(\left\{{2\over 3},{1\over 3}\right\}\)
- (c)
\(\left\{-{5\over 6},-{7\over 6}\right\}\)
- (d)
\(\left\{{13\over 3},{11\over 3}\right\}\)
When ever the terms on the two sides of the equation are of different nature, then equations are known as Non standard form, some of them are in the form of an ordinary equation but can not be solved by standard procedures.
Non standard problems require high degree of logic, they also require the use of graphs, inverse properties of functions, in equalities.
The number of solutions of the equation \(2cos\left(x\over 2\right)=3^x+3^{-x}\)is
- (a)
1
- (b)
2
- (c)
3
- (d)
none of these
When ever the terms on the two sides of the equation are of different nature, then equations are known as Non standard form, some of them are in the form of an ordinary equation but can not be solved by standard procedures.
Non standard problems require high degree of logic, they also require the use of graphs, inverse properties of functions, in equalities .
The number of real solutions of the equation sin (ex) = 5X + 5-x is
- (a)
0
- (b)
1
- (c)
2
- (d)
infinitely many
The circular measures of two angles of a triangle are \({1\over 2}and {1\over 3}\) find the third angle in English system.
- (a)
130015'20''
- (b)
132015'20''
- (c)
132016'22''
- (d)
122015'44''
Find the distance from the eye at which a coin of diameter 2cm should be held so as just to conceal the full moon whose angular diameter is 31'
- (a)
2.18m
- (b)
3.15m
- (c)
2.22m
- (d)
4.13m
Convert 2400 into radian
- (a)
2π/3
- (b)
4π/3
- (c)
7π/3
- (d)
π/3
Find x from the equation
cosec(900 +θ)+xcosθcot(900+θ)=sin(900+θ)
- (a)
cotθ
- (b)
tanθ
- (c)
-tanθ
- (d)
-cotθ
The value of 3 tan6100-27tan4100+33 tan2100 equals
- (a)
0
- (b)
-1
- (c)
1
- (d)
None of these
If 0 < x < π and cosx+sinx=1/2 then tan x is
- (a)
\((1-\sqrt7)\over4\)
- (b)
\((4-\sqrt7)\over3\)
- (c)
\(-(4-\sqrt7)\over3\)
- (d)
\((1+\sqrt7)\over4\)
Solve the equation
2(cosx+cos2x)+sin2x(1+2cosx)=2sinx
- (a)
\(nπ+(-1)^n\left(-{\pi\over 2}\right), n\in I\)
- (b)
2nπ+π or 2nπ± n∈I
- (c)
nπ, n∈I
- (d)
Both (a) and (b)
if sin \(\theta\) + cos \(\theta\) = 1 , then the value of sin 2\(\theta\) is equal to
- (a)
1
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
0
- (d)
-1
Lex x be a real number different from Zero.
Statement -I : \({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \) cannot be less than 2.
Statement - II : \({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \) is not equal to cos\(\theta\)
- (a)
If both statement-I and statement -II are trur and statement -II is the correct explantion of statement -I
- (b)
If both statement -I and statement II are true but statment-II is not the correct explanation of statement-I
- (c)
If statement -I is true but statement -II is false
- (d)
if statement -I is false and statement -II is true
Statement - I: If tan \(\left( \frac { \pi }{ 2 } sin\theta \right) \) =cot \(\left( \frac { \pi }{ 2 } cos\theta \right) \) , then sin \(\theta\) + cos \(\theta\) = \(\pm\) \(\sqrt { 2 } \)
Statement -II : -\(\sqrt { 2 } \) \(\le\) sin \(\theta\)+ cos \(\theta\) \(\le\) \(\sqrt { 2 } \)
- (a)
If both statement-I and statement -II are trur and statement -II is the correct explantion of statement -I
- (b)
If both statement -I and statement II are true but statment-II is not the correct explanation of statement-I
- (c)
If statement -I is true but statement -II is false
- (d)
If statement -I is true but statement -II is false
Which of the following statements is/are true?
Statement-I: The general solution of the equation
\(5cos^2\theta + 7sin^2\theta - 6 = 0 \)is \(n\pi+pm{\pi\over 4}, n\in Z\)
Statement-II: Find the general solution of the equation
sinx - 3sin2x + sin3x = cosx - 3cos2x + cos 3x is nπ + \({\pi\over 8},n\in Z\)
- (a)
Only Statement-I
- (b)
Only Staternent-Il
- (c)
Both Staternent-I and Staternent-Il
- (d)
Neither Statement-I nor Staternent-Il
Which of the following statements is INCORRECT?
- (a)
The value of tan22°30' is \(1\over \sqrt2-1\)
- (b)
If cosα + cosl3 = 0 = sinα + sinβ, then cos2α + cos2β = -2cos(α + β)
- (c)
If sinθ + cosθ = 1, then the general value of θ is \({n\pi\over2 }.n\in Z\)
- (d)
If 2sin2θ = 3cosθ, where 0 < θ < 2π, then the value θ is \({\pi\over3}\ or\ {5\pi\over 3}\)
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