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12th Maths Weekly Test -4 (Theory Of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -4 (Theory Of Equations)-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

A zero of x³ + 64 is

0
4
4i
-4

If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

mn
m+n
mⁿ
n^m

A polynomial equation in x of degree n always has

n distinct roots
n real roots
n imaginary roots
at most one root

If α,β and γ are the roots of x³+px²+qx+r, then \(\Sigma \frac { 1 }{ \alpha } \) is

-\(\frac { q }{ r } \)
\(\frac { p }{ r } \)
\(\frac { q }{ r } \)
-\(\frac { q }{ p } \)

According to the rational root theorem, which number is not possible rational root of 4x⁷+2x⁴-10x³-5?

-1
\(\frac { 5 }{ 4 } \)
\(\frac { 4 }{ 5 } \)
5

The polynomial x³-kx²+kx²+9x has three real roots if and only if, k satisfies

|k|≤6
k=0
|k|>6
|k|≥6

The number of real numbers in [0,2π] satisfying sin⁴x-2sin²x+1 is

2
4
1
∞

If x³+12x²+10ax+1999 definitely has a positive root, if and only if

a≥0
a>0
a<0
a≤0

The polynomial x³+2x+3 has

one negative and two imaginary zeros
one positive and two imaginary zeros
three real zeros
no solution

The number of positive roots of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

0
n
<n
r

If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax²+bx+c=0 then the other root is

-p+√q
p-iq
p-√q
-p-√q

The quadratic equation whose roots are ∝ and β is

(x - ∝)(x -β) =0
(x - ∝)(x + β) =0
∝+β=\(\frac{b}{a}\)
∝.β=\(\frac{-c}{a}\)

Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

n
n -1
n+1
(n-r)

If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then

\(\frac{1}{3}\) ≤k≤
k≥5
k≤0
none

Let a > 0, b > 0, c >0. h n both th root of th quatlon ax²+b+C= 0 are

real and negative
real and positive
rational numb rs
none

The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has

no solution
one solution
two solution
more than one solution

lf the root of the equation x³ +bx²+cx-1=0 form an lncreasing G.P, then

one of the roots is 2
one of the rots is 1
one of the rots is -1
one of the rots is -2

For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has

one solution
two solution
at least two solution
no solution

If the equation ax2+ bx+c=0(a>0) has two roots ∝ and β such that ∝<- 2 and β > 2, then

b2-4ac=0
b² - 4ac <0
b² - 4ac >0
b² - 4ac≥0

If (2+√3)x²-2x+1+(2-√3)x²-2x-1=\(\frac { 2 }{ 2-\sqrt { 3 } } \) then x=

0,2
0,1
0,3
0, √3

If ∝, β, ૪ are the roots of the equation x³-3x+11=0, then ∝+β+૪ is __________.

0
3
-11
-3

If ∝, β,૪ are the roots of 9x3-7x+6=0, then ∝ β ૪ is __________

\(\frac{-7}{9}\)
\(\frac{7}{9}\)
0
\(\frac{-2}{3}\)

If x^{2 -} hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

0
1
4
3

If ax² + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

c>0
c<0
c=0
c≥0

If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

no
1
2
infinite

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