Polynomials MCQ test
POLYNOMIALS
MCQ test
Question of
Good Try!
You Got out of answers correct!
That's
1.If α, β, γ are the zeroes of polynomial 2x3+3x2−6x+2 then αβ+βγ+ γα=
Answer:2.If α, β, γ are the zeroes of polynomial ax3+bx2+cx+d then αβ+βγ+ γα=`c/a`
So, αβ+βγ+ γα=`-6/2=-3`
3.If one of the zero of the polynomial x2+kx−4 is 2 then k=
Answer:2 is the zero of p(x)=x2+kx−4
so,p(2)=22+2k−4=0
4+2k-4=0 ⇒ 2k =0 ⇒ k=0.
4.Zeroes of the cubic polynomial x3−4x are
Answer:x3−4x=0
x( x2−4)=0
x=0 or x2−4=0
x2=4 ⇒ x = `sqrt4`
∴ x = 2 or -2
The zeroes of cubic polynomial x3−4x are 0,2 and -2.
5.If product of zeroes of polynomial x2+4x−k is 5, then k=
Answer:Product of zeroes = ` (constant term)/(coefficient of x^2)`
So,`-k/1=5`⇒ k = -5
6.If sum of zeroes of polynomial ax2+bx+c is 0, then
Answer:Sum of zeroes = ` -(coefficient of x)/(coefficient of x^2)=-b/a=0 `
⇒ -b=0×a ⇒ b=0
7.The parabola y=x2 passes through
Answer:If x=2 ⇒ y=22 =4
So,the parabola y=x2 passes through (2,4)
8.If 6x2+x+1 is divided by 2x+1, then the remainder is
Answer:p(x)=6x2+x+1
2x+1=0 ⇒ x=`-1/2`
p(`-1/2`)=6(`-1/2`)2+(`-1/2`)+1
=`6 times 1/4 -1/2 +1=3/2-1/2+1=1+1=2`
So,the remainder is 2
9.The roots of quadratic polynomial 7x2−14x are
Answer:7x2−14x=7x(x-2)
7x(x-2)=0 ⇒ 7x=0 or x-2=0
∴ The roots of quadratic polynomial 7x2−14x are 0,2
10.The sum of coefficients x4 and x2 of a polynomial 2x5−3x4+2x3−3 is
Answer:x4 coefficient = -3
x2 coefficient = 0
So,The sum of coefficients x4 and x2=-3+0=-3
11.The graph of the linear polynomial is a
Answer:The graph of the linear polynomial is a line
12.If the divisor is x2 and quotient is -x while the remainder is -2, then the dividend is
Answer:Given g(x)=x2
q(x)=-x and r(x)=-2
division algorithm for polynomials is p (x) = q (x) . g (x) + r (x)
∴ dividend=(x2)(-x)+(-2)=-x3-2
13.A quadratic polynomial with zeroes are 5 and−3 is
Answer:p (x) = x2 – (sum of zeroes) x + product of zeroes
p (x) = x2 – (5-3) x + (5)(-3)=x2 – 2x – 15
14.The zero of p(x) = ax–b is
Answer:ax–b=0 ⇒ ax=b ⇒ x =`b/a`
15.The quadratic polynomial whose sum and product of zeroes are −`1/4` and `1/4` respectively is
Answer:p (x) = k[x2 – (sum of zeroes) x + product of zeroes]
p (x) = k[x2 – ( −`1/4`) x + `1/4`]=k`{4x2+x+1}/4`
If k=1 the polynomial is 4x2+x+1
16.The quadratic polynomial whose sum and product of zeroes are −√2 and √2 respectively is
Answer:p (x) = x2 – (sum of zeroes) x + product of zeroes
p (x) = x2 – (`-sqrt2`) x + `sqrt2`=x2 +`-sqrt2`x + `sqrt2`
17.A quadratic polynomial with zeroes are √2 and−√2 is
Answer:p (x) = x2 – (sum of zeroes) x + product of zeroes
p (x) = x2 – (`-sqrt2 + sqrt2`) x + (`-sqrt2 times sqrt2`)=x2 – 2
18.If α and β are zeroes of p(x) = 2x2–3x–1 then the value of 𝛼-1+𝛽-1=
Answer:α + β = `3/2`and αβ=`-1/2`
𝛼-1+𝛽-1=`1/α+1/β`
=`(α + β)/(αβ)=3/-1=-3`
19.If the graph of a polynomial does not intersect the x – axis, then that polynomial has ____
Answer:Graph of p (x) = ax2 + bx + c will neither touch nor intersect the x-axis then p(x) will not have any real ‘zero’.
20.For the polynomial 2x3–5x2–14x+8, the sum of the zeroes & products of zeroes are
Answer:sum of the zeroes = -(coefficient of x^2)/(coefficient of x^3 )=-((-5)/2)=5/2`
Product of zeroes =`-(constant term)/(coefficient of x^3)=-8/2=-4`
21.If α and β are the zeroes of the polynomial 3x2-2x+2, then the value of α+β+αβ is
Answer:α+β = `2/3` and αβ = `2/3` α+β+αβ = `2/3+2/3=4/3`
22.which of the following is a polynomial?
Answer:`sqrt5`x -3
23.If p(t) = t3-2 then p (-2) =
Answer:p(t) = t3-2
p(-2) = (-2)3-2 = -8 -2 = -10
24.A quadratic polynomial can have at most 2 zeroes
25.The degree of the constant polynomial is 0
26.If –1 is a zero of the polynomial f(x) =x2–7x–8 then other zero is
Answer:product of zeroes=`-8/1=-8`
If one zero is -1 then other zero is `(-8)/(-1)=8`
27.The value of p(x) = x3−3x−4 at x = −1 is
Answer:p(x) = x3−3x−4
p(-1) = (-1)3−3×(-1)−4=-1+3-4=-5+3=-2
28.The graph of the equation y = ax2+ bx+ c is an upward parabola, if
Answer:a>0
29.Sum of zeroes of a polynomial x2−5 is
Answer:Sum of zeroes = `-(coefficient of x)/(coefficient of x^2)=0/1=0`
30.Product of zeroes of polynomial 2m3−5m2+m−8 is
Answer:Product of zeroes = `-(constant term)/(coefficient of x^3) = -((-8)/2)=4`
31.Which of the following is a factor of the polynomial x3+x2-17x+15?
Answer:f(x)=x3+x2-17x+15
f(3)=33+32-17×3+15
= 27 + 9 -51+15=51-51=0
∴ x-3 is a factor of the polynomial x3+x2-17x+15.
32.If α, β are the zeroes of quadratic polynomial x2+x+1, then `1/α+1/β` is
Answer:(α+β)=`-1/1=-1`and αβ = `1/1=1`
`1/α+1/β = (α+β)/(αβ)=(-1)/(1)=-1`
33.If a polynomial of degree 7 is divided by a polynomial of degree 4, then the degree of the quotient is
Answer:If a polynomial of degree n is divided by a polynomial of degree m, then the degree of the quotient is n-m
∴ the answer is 7-4=3
34.The zeroes of the quadratic polynomial x2+24x+119 are
Answer:split the middle term 24x as the product of 119x2
x2+24x+119=x2+17x+7x+119
=x(x+17)+7(x+17)=(x+17)(x+7)
x+17=0 or x+7=0 ⇒ x=-17 or x=-7
So,the zeroes are -17 and -7
35.What is the degree of the polynomial 7u6`-3/2`u4+6u2-8?
Answer:The highest power of u in this polynomial is 6.So the degree of this polynomial is 6.
36.If α, β are the zeroes of quadratic polynomial ax2+bx+c=0,a≠0 then 1/α+1/β is
Answer:`1/α+1/β = (α+β)/(αβ)=((-b)/a)/(c/a)=-b/c`
37.The zeroes of the polynomial 2x3+ax2+bx-2 are -1 and -2 then (a,b) is
Answer:p(x)=2x3+ax2+bx-2
p(-1)=2×(-1)3+a×(-1)2+b×(-1)-2
=2×(-1)+a×(1)+b×(-1)-2=-2+a-b-2=0
⇒ a-b=4 ⇒ b = a-4 -----(1)
p(-2)=2×(-2)3+a×(-2)2+b×(-2)-2
=2×(-8)+a×(4)+b×(-2)-2=-16+4a-2b-2=0
⇒ 4a-2b=18 ⇒ 2a-b=9
from (1) 2a-a+4=9 ⇒ a =9-4=5 and b = 5-4 =1
Zeroes of a polynomial 2x−2/x=3 are
Answer:2x2-2=3x ⇒ 2x2-3x -2=0
2x2-4x +x -2= 2x(x-2)+1(x-2)=0
(x-2)(2x+1)=0 ⇒ x=2 or x=`-1/2`
∴ The zeroes are 2 and `-1/2`
38.which of the following is not a quadratic polynomial?
Answer:x2-2x=(-x)(3-x) and (x-2)(x+1)=(x-1)(x+3) are not quadratic polynomials.These are linear polynomials.
39.When f(x)=5x4-9x3-3x2+11x-18 is divided by (x-2) then the remainder is
Answer:f(x)=5x4-9x3-3x2+11x-18
let x-2=0 ⇒ x=2
f(2)=5 ×24-9 ×23-3 ×22+11 ×2-18
= 5 × 16 - 9 × 8 - 3 × 4 + 22 -18
= 80 - 72 - 12 + 22 -18 = 102 - 102 = 0
the remainder is 0
40.If 3 is a zero of the polynomial 2x2+kx–15 then other zero is
Answer:product of zeroes=`-15/2`
If one zero is 3 then other zero is `(-(15)/2)/3=-5/2`
41.If α, β are the zeroes of quadratic polynomial x2-5x+6 then (α-β)2 is
Answer:(α-β)2=(α+β)2- 4αβ
(α+β)=`5/1=5` and αβ=`6/1=6`
∴ (α-β)2= 52-4×6=25-24=1
42.If 0 is a zero of the polynomial ax2+bx+c then other zero is
Answer:Sum of zeroes=`-b/a`
If one zero is 0 then other zero is `-b/a`
43.If α, β, γ are the zeroes of polynomial ax3+bx2+cx+d then `1/α+1/β+ 1/γ`=
Answer:`1/α+1/β+ 1/γ=(α β + β γ + γ α)/(α β γ)`
α β + β γ + γ α=`c/a`and α β γ = `-d/a`
So,`1/α+1/β+ 1/γ=(c/a)/((-d)/a)=-c/d`
44.If zeroes of the polynomial p(x) =kx2–8x+1(k≠0) are multiplication inverses of one another then k is
Answer:If zeroes are multiplication inverses of one another then
product of zeroes = 1
So,`(constant term)/(coefficient of x^2)=1/k=1`⇒k=1
45.p(x)=x2-3x-28 and g(x)=x+4 then q(x)=---
Answer:p(x)=x2-3x-28
split the middle term -3x as the product of -28x2
we get x2-3x-28 = x2-7x + 4x-28
=x(x-7)+4(x-7)=(x-7)(x+4)
If g(x)=x+4 then q(x)=x-7
Post a Comment