Advertisements
Advertisements
Question
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
Options
a2 + b2 + 2ac = 0
a2 – b2 + 2ac = 0
a2 + c2 + 2ab = 0
a2 – b2 – 2ac = 0
Advertisements
Solution
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation a2 – b2 + 2ac = 0.
Explanation:
Given that sin θ and cos θ are the roots of the equation ax2 – bx + c = 0
So sin θ + cos θ = `b/a` and sin θ cos θ = `c/a`
Using the identity (sinθ + cos θ)2 = sin2θ + cos2θ + 2 sin θ cos θ
We have `b^2/a^2 = 1 + (2c)/a`
or a2 – b2 + 2ac = 0
APPEARS IN
RELATED QUESTIONS
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
Prove that: \[\tan\frac{11\pi}{3} - 2\sin\frac{4\pi}{6} - \frac{3}{4} {cosec}^2 \frac{\pi}{4} + 4 \cos^2 \frac{17\pi}{6} = \frac{3 - 4\sqrt{3}}{2}\]
Prove that
Prove that:
\[\sec\left( \frac{3\pi}{2} - x \right)\sec\left( x - \frac{5\pi}{2} \right) + \tan\left( \frac{5\pi}{2} + x \right)\tan\left( x - \frac{3\pi}{2} \right) = - 1 .\]
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
If tan x = \[x - \frac{1}{4x}\], then sec x − tan x is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x2 + y2 + z2 is independent of
If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
Which of the following is correct?
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[\sqrt{3} \cos x + \sin x = 1\]
Solve the following equation:
sin x tan x – 1 = tan x – sin x
Solve the following equation:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0
Solve the following equation:
3sin2x – 5 sin x cos x + 8 cos2 x = 2
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
Write the values of x in [0, π] for which \[\sin 2x, \frac{1}{2}\]
and cos 2x are in A.P.
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
The number of solution in [0, π/2] of the equation \[\cos 3x \tan 5x = \sin 7x\] is
If \[\cot x - \tan x = \sec x\], then, x is equal to
The number of values of x in [0, 2π] that satisfy the equation \[\sin^2 x - \cos x = \frac{1}{4}\]
The solution of the equation \[\cos^2 x + \sin x + 1 = 0\] lies in the interval
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
Find the principal solution and general solution of the following:
tan θ = `- 1/sqrt(3)`
Solve the following equations:
2 cos2θ + 3 sin θ – 3 = θ
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Solve the following equations:
`sin theta + sqrt(3) cos theta` = 1
Solve the following equations:
cot θ + cosec θ = `sqrt(3)`
Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0
