Polynomials Test

Result

Polynomials Test - 32

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Weekly Quiz Competition

Question 1
1 / -0

What are the other two zeroes of the polynomial $$2x^4-6x^3+3x^2+3x-2$$ , if two of its zeroes are $$1$$ and $$2$$?

A
$$\sqrt2$$ , $$\dfrac{1}{\sqrt2}$$

B
$$\dfrac{1}{\sqrt2}$$,$$-\dfrac{1}{\sqrt2}$$

C
$$\dfrac{1}{\sqrt3}$$,$$-\dfrac{1}{\sqrt3}$$

D
$$-\dfrac{1}{\sqrt2}$$,$$\dfrac{1}{\sqrt3}$$

Solution
Question 2
1 / -0

Obtain all the zeros of polynomial $$x^4-6x^3+4x^2+6x-5$$, if two of its zeros are $$3+\sqrt4$$ and $$3-\sqrt4$$.

A
$$1$$ and $$4$$

B
$$1$$ and $$-1$$

C
$$-1$$ and $$4$$

D
none of these

Solution
Question 3
1 / -0

What are the other two zeros of the polynomial $$x^4-5x^3-10x^2+80x-96$$, if two positive consecutive primes are two of its zeros?

A
$$4$$ and $$8$$

B
$$-4$$ and $$8$$

C
$$4$$ and $$-4$$

D
None of these

Solution
Question 4
1 / -0

What are the zeros of the polynomial $$x^4-6x^3+7x^2+6x-8$$, if two of its zeros are $$a$$ and $$2a$$, where $$a= 2$$?

A
$$1, 2, 3, 4$$

B
$$-1, 2, 3, 4$$

C
$$-1, 1, 2, 4$$

D
$$-2, 1, 2, 4$$

Solution
Question 5
1 / -0

If two of the zeros of the polynomial $$x^4-12x^2+27$$ are $$3$$ and $$-3$$, then the sum of all the zeros is _____.

A
$$0$$

B
$$1$$

C
$$3$$

D
$$4$$

Solution
Question 6
1 / -0

Find the zeroes of the polynomial $$x^4+x^3-7x^2-x+6$$, if two of its zeroes are $$1$$ and $$-1$$

A
$$2$$ and $$3$$

B
$$2$$ and $$-2$$

C
$$3$$ and $$-3$$

D
$$2$$ and $$-3$$

Solution
Question 7
1 / -0

Obtain all the zeroes of the polynomial $$x^4+2x^3-13x^2-14x+24$$, if two of its zeroes are $$3$$ and $$-4$$.

A
$$1,-2,3,-4$$

B
$$1,2,-3,-4$$

C
$$-1,2,3,-4$$

D
$$1,-2,3,4$$

Solution
Question 8
1 / -0

The figure represents which polynomial

A
quadratic

B
cubic

C
linear

D
None of these

Solution

Given figure cuts the x - axis at two point.
So this is a quadratic equation.
$$\therefore$$ figure represents quadratic polynomial.
Option A is correct.
Question 9
1 / -0

The zeroes of the quadratic polynomial $$x^2 + 99x + 127$$ are

A
both positive

B
both negative

C
one positive and one negative

D
both equal

Solution

Product of zeroes = 127
The sign is positive it means both the zeroes have same sign.
Sum of zeroes = -99
The sign is negative and both have same sign hence the zeroes are both negative.
Question 10
1 / -0

If all three zeroes of a cubic polynomial $$x^3 + ax^2+ bx + c$$ are positive, then which of the following is correct about $$a,b$$ and $$c$$?

A
$$b$$ must be positive

B
$$a$$ must be positive

C
$$c$$ must be positive

D
$$b$$ must be negative

Solution

Let the roots of the equation $$x^3+ ax^2 + bx +c =0$$ be $$\alpha, \beta, \gamma$$

Then, sum of roots, $$\alpha + \beta + \gamma=- a$$.

Thus, $$a$$ is negative.

Product of roots, taken two at a time, $$\alpha.\beta + \alpha.\gamma + \gamma.\beta = b$$.

Thus $$b$$ is positive

Product of roots, $$\alpha.\beta.\gamma = -c$$.

Thus $$c$$ is negative.

Hence, $$a$$ and $$c$$ have negative sign, but $$b$$ has positive sign.