A zero of x³ + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

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

(a)

2

(b)

4

(c)

1

(d)

∞

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

If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

(a)

n

(b)

n -1

(c)

n+1

(d)

(n-r)

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

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

(a)

\(\frac{-7}{9}\)

(b)

\(\frac{7}{9}\)

(c)

0

(d)

\(\frac{-2}{3}\)

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

6x⁶ - 35x⁵ + 56x⁴ -56x² + 35x - 6 = 0

(1)

4 Change of sign

p(x) = xⁿ.p\(\left( \frac { 1 }{ x } \right) \)

(2)

Reciprocal equation of type I

2x⁷-3x⁶-4x⁵+5x⁴+6x³-7x+8 = 0

(3)

even degree reciprocal equation of type II

9x⁹+2x⁵-x⁴-7x²+2 = 0

(4)

at least 6 imaginary roots

x⁹-5x⁸ -14x⁷= 0

(5)

4 Change of sign

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

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

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.

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2

If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

Find the condition that the roots of ax³+ bx²+ cx + d = 0 are in geometric progression. Assume a, b, c, d ≠ 0.

If the roots of x³+ px²+ qx + r = 0 are in H.P. prove that 9pqr = 27r²+2q³.

Solve: 2x+2^x-1+2^x-2 = 7x+7^x-1+7^x-2

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

Show that if p, q, r  are rational the roots of the equation x² − 2px + p² − q² + 2qr − r² = 0 are rational.

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