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12th Maths Weekly Test -1 (Theory Of Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (Theory Of Equations)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

A zero of x³ + 64 is

0
4
4i
-4

A polynomial equation in x of degree n always has

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

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+2x+3 has
one negative and two imaginary zeros

one positive and two imaginary zeros

three real zeros

no solution
The number of real numbers in [0,2π] satisfying sin⁴x-2sin²x+1 is
2

4

1

∞

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

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

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

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

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

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

0
3
-11
-3

Part - B

Generic 2 - Answer All Question

Construct a cubic equation with roots 1,2, and 3

Solve the equation x³−9x²+14x+24=0 if it is given that two of its roots are in the
ratio 3:2.

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\(\delta\).

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 \(\frac { q-q' }{ p'-p } \).
Solve the equation 3x³-16x²+23x-6=0 if the product of two roots is 1.

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away.

Part - C

Generic 3 - Answer All Question

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

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.

Find the condition that the roots of x³+ax²+bx+c = 0 are in the ratio p:q:r.

Form a polynomial equation with integer coefficients with \(\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } } \) as a root.
Solve: 2x+2^x-1+2^x-2=7x+7^x-1+7^x-2

Part - D

Generic 5 - Answer All Question

If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abe≠ 0) is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \) are H.P.

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

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+62x⁴-126x³+65x²+127x-140=0, find all roots.
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.

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