Maths-

General

Easy

Question

If $alpha comma beta$ be that roots $4 x squared minus 16 x plus lambda equals 0$ where $lambda element of R$ , such that $1 less than alpha less than 2$ and $2 less than beta less than 3$ then the number of integral solutions of λ is

5
6
2
3

hint 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 $alpha comma beta$ are the roots of equation $4 x squared minus 16 x plus lambda equals 0$ where $lambda element of R$ , such that $1 less than alpha less than 2$ and $2 less than beta less than 3$ .
The equation is:
$4 x^{2} - 16 x + λ = 0$
$First finding the discriminant, we get:$
D=b²-4ac
Applying it, we get:
D=16²−16λ>0
λ<16...........................(i)
Now, we have sum of roots as: $alpha plus beta equals 16 over 4 equals 4$
we have product of the roots $alpha cross times beta equals lambda over 4$
Now as per the condition, we have:
$1 less than alpha less than 2$ and $2 less than beta less than 3$ , multiplying both, we get:
$2 less than alpha beta less than 6 2 less than lambda over 4 less than 6 S i m p l i f y i n g space i t comma space w e space g e t colon 8 less than lambda less than 24....................... left parenthesis i i right parenthesis C o m b i n i n g space i space a n d space i i comma space w e space g e t colon 8 less than lambda less than 16$

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 $8 less than lambda less than 16$

If α,β then the equation $x squared minus 3 x plus 1 equals 0$ with roots $fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction$ 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, $x to the power of 2 end exponent minus x minus 1 equals 0$ will be the equation for the roots $fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction$ .

If α,β then the equation $x squared minus 3 x plus 1 equals 0$ with roots $fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction$ will be

Maths-General

General

maths-

The equation of the directrix of the conic $2 equals r left parenthesis 3 plus C o s invisible function application theta right parenthesis$ is

maths-General

General

maths-

The conic with length of latus rectum 6 and eccentricity is

maths-General

General

maths-

For the circle $r equals 6 C o s space theta$ centre and radius are

maths-General

General

Maths-

The polar equation of the circle of radius 5 and touching the initial line at the pole is

Maths-General

General

maths-

The circle with centre at and radius 2 is

maths-General

General

Maths-

Statement-I : If $fraction numerator x squared plus 3 x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis squared end fraction text then end text bold italic A plus bold italic B plus bold italic C equals bold 0$
Statement-II :If $fraction numerator x squared plus 2 x plus 3 over denominator x cubed end fraction equals A over x plus B over x squared plus C over x cubed text then end text A plus B minus C equals 0$
Which of the above statements is true

Maths-General

General

Maths-

If x, y are rational number such that $x plus y plus left parenthesis x minus 2 y right parenthesis square root of 2 equals 2 x minus y plus left parenthesis x minus y minus 1 right parenthesis square root of 6$ Then

Maths-General

General

maths-

A class contains 4 boys and g girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being sent every week. During, the picnic, the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of g is -

maths-General

General

maths-

There are three piles of identical yellow, black and green balls and each pile contains at least 20 balls. The number of ways of selecting 20 balls if the number of black balls to be selected is twice the number of yellow balls, is -

maths-General

General

maths-

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a³ + b³ + c³ must be divisible by -

maths-General

General

maths-

For x $element of$ R, let [x] denote the greatest integer $less or equal than$ x, then value of $open square brackets negative fraction numerator 1 over denominator 3 end fraction close square brackets$ + $open square brackets negative fraction numerator 1 over denominator 3 end fraction minus fraction numerator 1 over denominator 100 end fraction close square brackets$ + $open square brackets negative fraction numerator 1 over denominator 3 end fraction minus fraction numerator 2 over denominator 100 end fraction close square brackets$ +…+ $open square brackets negative fraction numerator 1 over denominator 3 end fraction minus fraction numerator 99 over denominator 100 end fraction close square brackets$ is -

maths-General

General

maths-

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 $less or equal than$ x $less or equal than$ 5, 12 $less or equal than$ y $less or equal than$ 18, z $less or equal than$ –1

maths-General

General

maths-

Ravish write letters to his five friends and address the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes -

maths-General

General

maths-

In how many ways can we get a sum of at most 17 by throwing six distinct dice -

maths-General

Have a query? Ask a Tutor!!

Can’t find what you’re looking for?

If be that roots where , such that and then the number of integral solutions of λ is

562 3

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

Related Questions to study

If α,β then the equation with roots will be

If α,β then the equation with roots will be

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 Statement-II :If Which of the above statements is true

Statement-I : If Statement-II :If Which of the above statements is true

If x, y are rational number such that Then

If x, y are rational number such that Then

There are three piles of identical yellow, black and green balls and each pile contains at least 20 balls. The number of ways of selecting 20 balls if the number of black balls to be selected is twice the number of yellow balls, is -

There are three piles of identical yellow, black and green balls and each pile contains at least 20 balls. The number of ways of selecting 20 balls if the number of black balls to be selected is twice the number of yellow balls, is -

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a3 + b3 + c3 must be divisible by -

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a3 + b3 + c3 must be divisible by -

For x R, let [x] denote the greatest integer x, then value of++ +…+is -

For x R, let [x] denote the greatest integer x, then value of++ +…+is -

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 x 5, 12 y 18, z –1

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 x 5, 12 y 18, z –1

Ravish write letters to his five friends and address the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes -

Ravish write letters to his five friends and address the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes -

In how many ways can we get a sum of at most 17 by throwing six distinct dice -

In how many ways can we get a sum of at most 17 by throwing six distinct dice -

With Turito Academy.

With Turito Foundation.

Get an Expert Advice From Turito.

Turito Academy

With Turito Academy.

Test Prep

With Turito Foundation.

Courses

Quick Links

If $alpha comma beta$ be that roots $4 x squared minus 16 x plus lambda equals 0$ where $lambda element of R$ , such that $1 less than alpha less than 2$ and $2 less than beta less than 3$ then the number of integral solutions of λ is

5
6
2
3

If α,β then the equation $x squared minus 3 x plus 1 equals 0$ with roots $fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction$ will be

If α,β then the equation $x squared minus 3 x plus 1 equals 0$ with roots $fraction numerator 1 over denominator alpha minus 2 end fraction comma fraction numerator 1 over denominator beta minus 2 end fraction$ will be

The equation of the directrix of the conic $2 equals r left parenthesis 3 plus C o s invisible function application theta right parenthesis$ is

The equation of the directrix of the conic $2 equals r left parenthesis 3 plus C o s invisible function application theta right parenthesis$ is

For the circle $r equals 6 C o s space theta$ centre and radius are

For the circle $r equals 6 C o s space theta$ centre and radius are

If x, y are rational number such that $x plus y plus left parenthesis x minus 2 y right parenthesis square root of 2 equals 2 x minus y plus left parenthesis x minus y minus 1 right parenthesis square root of 6$ Then

If x, y are rational number such that $x plus y plus left parenthesis x minus 2 y right parenthesis square root of 2 equals 2 x minus y plus left parenthesis x minus y minus 1 right parenthesis square root of 6$ Then

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a³ + b³ + c³ must be divisible by -

If a, b, c are the three natural numbers such that a + b + c is divisible by 6, then a³ + b³ + c³ must be divisible by -

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 $less or equal than$ x $less or equal than$ 5, 12 $less or equal than$ y $less or equal than$ 18, z $less or equal than$ –1

The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 $less or equal than$ x $less or equal than$ 5, 12 $less or equal than$ y $less or equal than$ 18, z $less or equal than$ –1