Home
Class 12
MATHS
find all the possible triplets (a(1), a(...

find all the possible triplets `(a_(1), a_(2), a_(3))` such that `a_(1)+a_(2) cos (2x)+a_(3) sin^(2) (x)=0` for all real x.

Text Solution

Verified by Experts

We have
`a_(1)+a_(2) cos (2x) +a_(3) sin^(2) x=0` for all real x
`:. a_(1)+a_(2) (1-2 sin^(2) x)+a_(3) sin^(2) x=0`
`rArr (a_(1)+a_(2))+(-2a_(2)+a_(3)) sin^(2) x=0`
Since this is true for all real x, we must have
`a_(!)+a_(2)=0` and `-2a_(2)+a_(3)=0`
`:. a_(2)=-a_(1)` and `a_(3)= -2a_(1)`
Thus, there exists infinite triplets `(a_(1), -a_(1), -2a_(1))`
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC EQUATIONS

    CENGAGE ENGLISH|Exercise Concept Application Exercise 4.1|12 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE ENGLISH|Exercise Concept Application Exercise 4.2|6 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE ENGLISH|Exercise Archives (Matrix Match Type)|1 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise ARCHIVES INTEGER TYPE|1 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE ENGLISH|Exercise SINGLE CORRECT ANSWER TYPE|38 Videos

Similar Questions

Explore conceptually related problems

The number of all possible 5-tuples (a_(1),a_(2),a_(3),a_(4),a_(5)) such that a_(1)+a_(2) sin x+a_(3) cos x + a_(4) sin 2x +a_(5) cos 2 x =0 hold for all x is

a_(1), a_(2),a_(3) in R - {0} and a_(1)+ a_(2)cos2x+ a_(3)sin^(2)x=0 " for all " x in R then (a) vectors veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = 4hati + 2hatj + hatk are perpendicular to each other (b)vectors veca=a_(1) hati+ a_(2) hatj + a_(3) hatk and vecb = hati + hatj + 2hatk are parallel to each each other (c)if vector veca=a_(1) hati+ a_(2) hatj + a_(3) hatk is of length sqrt6 units, then on of the ordered trippplet (a_(1), a_(2),a_(3)) = (1, -1,-2) (d)if 2a_(1) + 3 a_(2) + 6 a_(3) = 26 , " then " |veca hati + a_(2) hatj + a_(3) hatk | is 2sqrt6

If a_(1),a_(2),a_(3),a_(4),a_(5) are in HP, then a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5) is equal to

If a_(1),a_(2),a_(3),………. are in A.P. such that a_(1) + a_(5) + a_(10) + a_(15) + a_(20) + a_(24) = 225, then a_(1) + a_(2) + a_(3) + …… a_(23) + a_(24) =

If the coefficients of 4 consecutive terms in the expansion of (1+x)^(n) are a_(1),a_(2),a_(3),a_(4) respectively, then show that (a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))

a_(1),a_(2),a_(3),a_(4),a_(5), are first five terms of an A.P. such that a_(1) +a_(3) +a_(5) = -12 and a_(1) .a_(2) . a_(3) =8 . Find the first term and the common difference.

If a_(0), a_(1),a_(2),… are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending powers of x, prove that if E_(1) = a_(0) + a_(3) + a_(6) + …, E_(2) = a_(1) + a_(4) + a_(7) + … and E_(3) = a_(2) + a_(5) + a_(8) + ..." then " E_(1) = E_(2) = E_(3) = 3^(n-1)

It a_(1) , a_(2) , a_(3) a_(4) be in G.P. then prove that (a_(2)-a_(3))^(2) + (a_(3) - a_(1))^(2) + (a_(4) -a_(2))^(2) = (a_(1)-a_(4))^(2)

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

( 1 + x + x^(2))^(n) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(2n) x^(2n) , then a_(0) + a_(1) + a_(2) + a_(3) - a_(4) + … a_(2n) = .