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The value of \sin ^{-1}\left ( \frac{12}{13} \right )-\sin ^{-1}\left ( \frac{3}{5} \right )  is equal to : 
 

  • Option 1)

    \frac{\pi }{2}-\cos ^{-1}\left ( \frac{9}{65} \right )

  • Option 2)

    \pi -\sin ^{-1}\left ( \frac{63}{65} \right )

  • Option 3)

    \frac{\pi }{2}-\sin ^{-1}\left ( \frac{56}{65} \right )

     

  • Option 4)

    \pi -\cos ^{-1}\left ( \frac{33}{65} \right )

 
  Option 1) Option 2) Option 3)   Option 4)

The angles A,B and C of a triangle ABC are in A.P.  and a:b=1:\sqrt3.

If c = 4 cm , then the area ( in sq. cm) of this triangle is: 

  • Option 1)

    \frac{2}{\sqrt3}

  • Option 2)

    4\sqrt3

  • Option 3)

    2\sqrt3

  • Option 4)

    \frac{4}{\sqrt3}

 
In  , A,B,C are in A.P. =>  Now, it is given that a:b= Now, Area of                                                                                         Option 1) Option 2) Option 3) Option 4)

Let S be the set of all \alpha \epsilon R such that the equation, \cos 2x+\alpha \sin x=2\alpha -7 has a solution. Then S is equal to:

 

  • Option 1)

    R

  • Option 2)

    [1,4]

  • Option 3)

    [3,7]

  • Option 4)

    [2,6]

 
=>  =>  =>  For atleast one solution, Option 1) R Option 2) [1,4] Option 3) [3,7] Option 4) [2,6]

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45^{\circ} from a point A on the plane. Let B be the point 30 m vertically above point A.  If the angle of elevation of distance (in m) of the foot of the tower from the point A is:

 

  • Option 1)

    15(3+\sqrt{3})

     

     

     

     

  • Option 2)

    15(5-\sqrt{3})

  • Option 3)

    15(3-\sqrt{3})

  • Option 4)

    15(1+\sqrt{3})

from (1) and (2) or Option 1)        Option 2)Option 3)Option 4)

The derivative of \tan ^{-1}\left ( \frac{\sin x-\cos x}{\sin x+\cos x} \right ),

with respect to \frac{x}{2}, where \left ( c\epsilon \left ( 0,\frac{\pi }{2} \right ) \right ) is :

 

  • Option 1)

    1

     

     

  • Option 2)

    \frac{2}{3}

  • Option 3)

    \frac{1}{2}

  • Option 4)

    2

 
Derivation of             Now,  Option 1) 1     Option 2) Option 3) Option 4) 2

The number of solutions of the equation 1+\sin ^{4}x=\cos ^{2}3x,   x\equiv \left [ -\frac{5\pi }{2},\frac{5\pi }{2} \right ] is : 

 

 

  • Option 1)

    4

  • Option 2)

    3

  • Option 3)

    5

  • Option 4)

    7

 
  Trigonometric Identities - - wherein They are true for all real values of      Results from General Solution - -                                                       Rewrite the trignometric equation  at  This value is also true for  Option 1) Option 2) Option 3) Option 4)

The equation y=\sin x\sin \left ( x+2 \right )-\sin ^{2}\left ( x+1 \right ) represents a straight line lying in : 

 


 

  • Option 1)

    first, third and fourth quadrants 

  • Option 2)

    third and fourth quadrants only

  • Option 3)

    first, second and fourth quadrants

  • Option 4)

    second and third quadrants only

 
               Option 1) first, third and fourth quadrants  Option 2) third and fourth quadrants only Option 3) first, second and fourth quadrants Option 4) second and third quadrants only

If \cos^{-1}x-\cos^{-1}\frac{y}{2}=\alpha , where -1\leq x\leq 1,

-2\leq y\leq 2 , x\leq \frac{y}{2}, then for all x,y,4x^{2}-4xy\cos \alpha +y^{2}

is equal to : 

  • Option 1)

    4\sin ^{2}\alpha

  • Option 2)

    2\sin ^{2}\alpha

  • Option 3)

    4\sin ^{2}\alpha-2x^{2}y^{2}

  • Option 4)

    4\cos ^{2}\alpha+2x^{2}y^{2}

 
                         where , ,                           Squaring both the sides So, option (1) is correct. Option 1) Option 2) Option 3) Option 4)

ABC is a triangular park with AB = AC = 100 metres.A vertical tower is situated at the midpoint of BC. if the angles of elevation of the top of the tower at A and B are  \cot^{-1}(2\sqrt2) respectively , then the height of the tower (in meters) is : 

  • Option 1)

    \frac{100}{3\sqrt3}

  • Option 2)

    10\sqrt5

  • Option 3)

    20

  • Option 4)

    25

  AM = y ; MD = h ( tower height ) ............................(1)   =>   =>  =>  from (1) .............................(2) ..........................................(3) correct option(3)    Option 1)Option 2)Option 3)20Option 4)25

The value of   \sin 10 ^{o}\sin 30 ^{o}\sin 50 ^{o}\sin 70 ^{o}  is :

  • Option 1)

    \frac{1}{16}

  • Option 2)

    \frac{1}{32}

  • Option 3)

    \frac{1}{18}

  • Option 4)

    \frac{1}{36}

 
Option 1) Option 2) Option 3) Option 4)

Two poles are standing on a horizantal ground are of heights 5\:m\:\:and\:\:10\:m respectively . The line joining their tops makes an angle of 15^{0} with the ground . Then the distance ( in m ) between the poles, is :

  • Option 1)

    5(2+\sqrt{3})

  • Option 2)

    5(\sqrt{3}+1)

  • Option 3)

    \frac{5}{2}(2+\sqrt{3})

  • Option 4)

    10(\sqrt{3}-1)

    Option 1)Option 2)Option 3)Option 4)

Let S=\left \{ \theta \epsilon \left [ -2\pi ,2\pi \right ] :2cos^{2}\theta +3sin\theta =0\right \}

Then the sum of the elements of S is :

  • Option 1)

     \frac{13\pi }{6}          

  • Option 2)

    \frac{5\pi }{3}

  • Option 3)

    2\pi

  • Option 4)

    \pi

 
Given equation      OR Hence solution is  Option 1)             Option 2) Option 3) Option 4)

The value of cos^{2}10^{\circ}-cos10^{\circ}\; cos50^{\circ}+cos^{2}50^{\circ} is:

  • Option 1)

       \frac{3}{4}+cos20^{\circ}

  • Option 2)

    3/4

  • Option 3)

    \frac{3}{2}\left ( 1+cos\; 20^{\circ} \right )

  • Option 4)

    3/2

Option 1)   Option 2)Option 3)Option 4)

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is : 

  • Option 1)

  • Option 2)

  • Option 3)

  • Option 4)

 
Let be side of      is smallest angle Three angles are,  Given,  Use sine rule.                                                               Option 1) Option 2) Option 3) Option 4)

\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}}dx is equal to :

(Where c is a constant of integration.)
 

  • Option 1)

    2x+\sin x+\sin 2x+c

  • Option 2)

    2x+\sin x+2\sin 2x+c

  • Option 3)

    x+2\sin x+\sin 2x+c

     

  • Option 4)

    x+2\sin x+2\sin 2x+c

 
Option 1) Option 2) Option 3)   Option 4)

If 2y=\left ( \cot ^{-1}\left ( \frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x} \right ) \right )^{2},x\; \epsilon \left ( 0,\frac{\pi}{2} \right ) then \frac{dy}{dx} is equal to :


 

  • Option 1)

    \frac{\pi }{6}-x

  • Option 2)

    \frac{\pi }{3}-x

  • Option 3)

    x-\frac{\pi }{6}

     

  • Option 4)

    2x-\frac{\pi }{3}

 
then , Option 1) Option 2) Option 3)   Option 4)

If \cos(\alpha+\beta )=\frac{3}{5},\sin(\alpha-\beta )=\frac{5}{13} and 0<\alpha,\beta <\frac{\pi}{4} then \tan(2\alpha) is equal to :
 

  • Option 1)

    \frac{21}{16}

  • Option 2)

    \frac{63}{16}

  • Option 3)

    \frac{33}{52}

     

  • Option 4)

    \frac{63}{52}

 
                                                                                                                          Option 1) Option 2) Option 3)   Option 4)

If \alpha = \cos ^{-1}\left ( \frac{3}{5} \right ),\beta = \tan ^{-1}\left ( \frac{1}{3} \right ), where 0<\alpha ,\beta <\frac{\pi}{2}, then \alpha -\beta is equal to :
 

  • Option 1)

    \sin ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )
     

  • Option 2)

    \tan ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )

  • Option 3)

    \cos ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )

     

  • Option 4)

    \tan ^{-1}\left ( \frac{9}{14} \right )

 

\alpha=\cos^{-1}\frac{3}{5}

\beta =\tan^{-1}\left ( \frac{1}{3} \right )\; \; \; 0<\alpha ,\beta <\frac{\pi}{2}

Then

\cos\: \alpha=\frac{3}{5}, \tan \alpha =\frac{4}{3},\alpha =\tan ^{-1}\left ( \frac{4}{3} \right ),

\tan \beta =\frac{1}{3}                        

\alpha -\beta =\cos^{-1}\left ( \frac{3}{5} \right )-\tan^{-1}\left ( \frac{1}{3} \right )

               =\tan^{-1} \frac{4}{3}-\tan^{-1} \frac{1}{3}=\tan^{-1}\left ( \frac{\frac{4}{3}-\frac{1}{3}}{1+\frac{4}{3}\times\frac{1}{3}} \right )

               

               =\tan^{-1}\left ( \frac{1}{1+\frac{4}{9}}\right )=\tan^{-1}\left ( \frac{9}{13} \right )

               \tan^{-1}\left ( \frac{9}{13} \right )

No option matched.


Option 1)

\sin ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )
 

Option 2)

\tan ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )

Option 3)

\cos ^{-1}\left ( \frac{9}{5\sqrt{10}} \right )

 

Option 4)

\tan ^{-1}\left ( \frac{9}{14} \right )

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Given \frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}  for a \Delta ABC with usual notation. 

If \frac{\cos A}{\alpha }=\frac{\cos B}{\beta }=\frac{\cos C}{\gamma }, then the ordered triad (\alpha ,\beta ,\gamma ) has a value :

  • Option 1)

    (5,12,13)

  • Option 2)

    (7,19,25)

  • Option 3)

    (19,7,25)

  • Option 4)

    (3,4,5)

  Cosine Rule - Cosine Rule - null Using Cosine Formula, Option 1)Option 2)Option 3)Option 4)

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y.

If x^{2}-c^{2}=y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

  • Option 1)

     

    \frac{y}{\sqrt3}

  • Option 2)

     

    \frac{c}{\sqrt3}

  • Option 3)

     

    \frac{c}{3}

  • Option 4)

     

    \frac{3}{2} y

  Allied Angles - - wherein The trigonometric ratios for angles in all the four quadrants.     Trigonometric Ratios of Special Angles - - wherein These are the values of trigonometric ratios for standard angles.      Let a,b,c be 3 sides of trangle a+b=x ab=y Circumradius ,     Option 1)  Option 2)  Option 3)  Option 4) 
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