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12th Maths Monthly Test - 1 ( Theory of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Theory of Equations)-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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

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

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

a≥0
a>0
a<0
a≤0

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 polynomial x³-kx²+kx²+9x has three real roots if and only if, k satisfies
|k|≤6

k=0

|k|>6

|k|≥6

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

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 quadratic equation whose roots are ∝ and β is

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

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 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

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

0
3
-11
-3

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

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

\(\frac{1}{3}\) ≤k≤
k≥5
k≤0
none

The polynomial x³+2x+3 has

one negative and two imaginary zeros
one positive and two imaginary zeros
three real zeros
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

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

no solution
one solution
two solution
more than one solution

Part - B

Generic 2 - Answer All Question

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.

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

Solve: \(8x^{ \frac { 3 }{ 2n } }-8x^{ \frac { -3 }{ 2n } }\)=63

Find the monic polynomial equation of minimum degree with real coefficients having 2-\(\sqrt{3}\)i as a root.
Solve: (2x-1)(x+3)(x-2)(2x+3)+20=0

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)
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 \(\frac { q-q' }{ p'-p } \).

Show that the equation 2x²−6x+7=0 cannot be satisfied by any real values of x.

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4=0, form a cubic equation whose roots are
\(\frac { 1 }{ \alpha } ,\frac { 1 }{ \beta } ,\frac { 1 }{ \gamma } \)

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

Solve the equation 6x⁴-5x³-38x²-5x+6=0 if it is known that \(\frac{1}{3}\) is a solution.

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x⁹-4x⁸+4x⁷-3x⁶+2x⁵+x³+7x²+7x+2=0

Construct a cubic equation with roots 1,1, and −2

Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

Show that the equation x⁹-5x⁵+4x⁴+2x²+1=0 has atleast 6 imaginary solutions.

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

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a-1=0 assumes the least value.

Part - C

Generic 3 - Answer All Question

Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² =353702.

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.

If p is real, discuss the nature of the roots of the equation 4x²+4px+p+2=0 in terms of p.

Solve the equation 2x³+11x²−9x−18=0.
Find the condition that the roots of x³+ax²+bx+c = 0 are in the ratio p:q:r.

Solve the equation x³-5x²-4x+20=0

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

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

Solve the equation x⁴-9x²+20=0.
Solve:(x-1)⁴+(x-5)⁴=82

Find the number .of real solu,tlons of sin (e^x) -5^x + 5^-x

Solve the equation 7x³-43x²=43x-7

Part - D

Generic 5 - Answer All Question

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

Solve the following equation: x⁴-10x³+26x²-10x+1=0
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.

If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc =0.

Form the equation whose roots are the squares of the roots of the cubic equation x³+ax²+bx+c = 0.

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 the n. Prove that a,b,c,d are in G.P and ad=bc

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