Question
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
- 5
- 6
- 2
- 3
Hint:
The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. It also goes by the name quadratic equations. Here we have to find the number of integral solutions of λ.
The correct answer is: 3
A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form:
ax² + bx + c = 0
where a, b, and c are constant terms and x is the unknown variable.
Now we have given
are the roots of equation
where
, such that
and
.
The equation is:
4x2−16x+λ=0
First finding the discriminant, we get:
D=b2-4ac
Applying it, we get:
D=162−16λ>0
λ<16...........................(i)
Now, we have sum of roots as: ![alpha plus beta equals 16 over 4 equals 4](data:image/png;base64,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)
we have product of the roots ![alpha cross times beta equals lambda over 4](data:image/png;base64,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)
Now as per the condition, we have:
and
, multiplying both, we get:
![2 less than alpha beta less than 6
2 less than lambda over 4 less than 6
S i m p l i f y i n g space i t comma space w e space g e t colon
8 less than lambda less than 24....................... left parenthesis i i right parenthesis
C o m b i n i n g space i space a n d space i i comma space w e space g e t colon
8 less than lambda less than 16](data:image/png;base64,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)
D=b2-4ac
Applying it, we get:
D=162−16λ>0
λ<16...........................(i)
Now, we have sum of roots as:
we have product of the roots
Now as per the condition, we have:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
Related Questions to study
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.
The equation of the directrix of the conic
is
The equation of the directrix of the conic
is
The conic with length of latus rectum 6 and eccentricity is
The conic with length of latus rectum 6 and eccentricity is
For the circle
centre and radius are
For the circle
centre and radius are
The polar equation of the circle of radius 5 and touching the initial line at the pole is
The polar equation of the circle of radius 5 and touching the initial line at the pole is
The circle with centre at and radius 2 is
The circle with centre at and radius 2 is
Statement-I : If ![fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text then end text bold italic A plus bold italic B plus bold italic C equals bold 0](data:image/png;base64,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)
Statement-II :If ![fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text then end text A plus B minus C equals 0](data:image/png;base64,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)
Which of the above statements is true
Statement-I : If ![fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text then end text bold italic A plus bold italic B plus bold italic C equals bold 0](data:image/png;base64,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)
Statement-II :If ![fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text then end text A plus B minus C equals 0](data:image/png;base64,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)
Which of the above statements is true