A zero of x³ + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

A polynomial equation in x of degree n always has

(a)

n distinct roots

(b)

n real roots

(c)

n complex roots

(d)

at most one root

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

(a)

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

(b)

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

(c)

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

(d)

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

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

(a)

a ≥ 0

(b)

a > 0

(c)

a < 0

(d)

a ≤ 0

The polynomial x³ + 2x + 3 has

(a)

one negative and two imaginary zeros

(b)

one positive and two imaginary zeros

(c)

three real zeros

(d)

no zeros

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

(a)

0

(b)

n

(c)

< n

(d)

r

The quadratic equation whose roots are ∝ and β is ___________

(a)

(x - ∝)(x -β) = 0

(b)

(x - ∝)(x + β) = 0

(c)

∝ + β = \(\frac{b}{a}\)

(d)

∝ β = \(\frac{-c}{a}\)

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

(a)

\(\frac{1}{3}\) ≤ k ≤

(b)

k ≥ 5

(c)

k ≤ 0

(d)

none

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

(a)

one of the roots is 2

(b)

one of the roots is 1

(c)

one of the roots is -1

(d)

one of the roots is -2

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

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

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

(a)

b²-4ac = 0

(b)

b² - 4ac <0

(c)

b² - 4ac >0

(d)

b² - 4ac ≥ 0

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

(a)

0, 2

(b)

0, 1

(c)

0, 3

(d)

0, √3

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

(a)

0

(b)

3

(c)

-11

(d)

-3

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

(a)

c>0

(b)

c<0

(c)

c=0

(d)

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.

(a)

no

(b)

1

(c)

2

(d)

infinite

Find a polynomial equation of minimum degree with rational coefficients, having 2 +√3 i as a root.

Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

Obtain the condition that the roots of x³+ px²+ qx + r = 0 are in A.P.

Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

Discuss the nature of the roots of the following polynomials:
x²⁰¹⁸+1947x¹⁹⁵⁰+15x⁸+26x⁶+2019

Determine the number of positive and negative roots of the equation x⁹- 5x⁸-14x⁷= 0.

Find the exact number of real zeros and imaginary of the polynomial x⁹+9x⁷+7x⁵+5x³+3x.

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

Find the sum of the squares of the roots of ax⁴+ bx³+ cx²+ dx + e = 0. \(a \neq 0\)

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

Form a polynomial equation with integer coefficients with \(\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } } \) as a root.

Prove that a line cannot intersect a circle at more than two points.

Solve the equations
x⁴+ 3x³- 3x - 1 = 0

Find the number of real solutions of sin (e^x) -5^x + 5^-x

Solve: (x-1)⁴+(x-5)⁴ = 82

Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

Solve the equation (x-2) (x-7) (x-3) (x+2)+19 = 0

Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504

Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴ + 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

Solve: (x - 4)(x - 7)(x - 2)(x + 1) = 16

If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².