Scroll:Algebra new >> Root of the equation >> ps (4149)

Written Instructions:

For each Multiple Choice Question (MCQ), four options are given. One of them is the correct answer. Make your choice (1,2,3 or 4). Write your answers in the brackets provided..

For each Short Answer Question(SAQ) and Long Answer Question(LAQ), write your answers in the blanks provided.

Leave your answers in the simplest form or correct to two decimal places.


 

1)  

 Show that the root of the equation x2 + 2(a + b) x+2 (a2+b2)= 0 are unreal.




Answer:_______________




2)  

 

{}


Answer:_______________




3)  

 Ffind the value of the first 27 terms of the geometric series.


Answer:_______________




4)  

 Find the sum of the first 20 terms of the geometric series.

 



Answer:_______________




5)  

 Factorize each of the following polynomials.

 (4x2 - x - 6)


Answer:_______________




6)  

 If the root of the equation (a2 + b2)x2 - 2(ac + bd)x + c2 + d2 = 0,

where ad - bc   ≠   0, are equal, prove that ab   ≠    cd  





Answer:_______________




7)  

 Show that the root of the equation x2 + 2(a + b) x+2 (a2+b2)= 0 are unreal.




Answer:_______________




8)  

 

{}


Answer:_______________




9)  

 Ffind the value of the first 27 terms of the geometric series.


Answer:_______________




10)  

 Find the sum of the first 20 terms of the geometric series.

 



Answer:_______________




 

1)  

 Show that the root of the equation x2 + 2(a + b) x+2 (a2+b2)= 0 are unreal.



Answer: 2


SOLUTION 1 :

Let the given equation be Ax2 + Bx + C = 0

Thus, A = 1, B = 2(a + b), C = 2(a2 + b2)

Now, the discriminant of Ax2 + Bx + C = 0 is

B2 - 4AC = [ 2(a + b)]2 - 4 x 1 x 2 (a2 + b2)

    = 4(a + b)2 - 8(a2 + b2)

    = 4 (a2 + 2ab + b2) - 8a2 - 8b2

    = 4a2 + 8ab + 4b2 - 8a2 - 8b2

    = -4a2 + 8ab - 4b2

    = -4 (a2 - 2ab + b2)

    = -4 (a - b)2 < 0

Hence the roots of the given equation are unreal.



2)  

 

Answer: x-1

Answer: x-2

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3)  

 Ffind the value of the first 27 terms of the geometric series.

Answer: 16



4)  

 Find the sum of the first 20 terms of the geometric series.

  Answer: 154




5)  

 Factorize each of the following polynomials.

Answer: x-1 (4x2 - x - 6)



6)  

 If the root of the equation (a2 + b2)x2 - 2(ac + bd)x + c2 + d2 = 0,

where ad - bc   ≠   0, are equal, prove that ab   ≠    cd  



Answer: 2



SOLUTION 1 :

(a2 + b2)x2 - 2(ac + bd)x + c2 + d2 = 0

comparing with Ax2 + Bx + C = 0

we get A = a2 + b2, B = -2(ac + bd) , C = c2 + d2

given that the equation has equal roots.

in this  â–º=0

B2 - AC=0

[-2(ac + bd)]2 - 4[(a2 + b2) (c2 + d2)] = 0

4(a2c2 + 2abcd + b2d2) - 4(a2c2 + a2d2 +cb2c2 +cb2d2) = 0

a2c2 + 2abcd + b2d2 - a2c2 - a2d2 -b2c2 - b2d2= 0

2abcd - a2d2 - b2c2 = 0

(ad - bc)2 = 0

ad - bc = 0

ad = bc

ab = cd

Hence proved.

 



7)  

 Show that the root of the equation x2 + 2(a + b) x+2 (a2+b2)= 0 are unreal.



Answer: 2


SOLUTION 1 :

Let the given equation be Ax2 + Bx + C = 0

Thus, A = 1, B = 2(a + b), C = 2(a2 + b2)

Now, the discriminant of Ax2 + Bx + C = 0 is

B2 - 4AC = [ 2(a + b)]2 - 4 x 1 x 2 (a2 + b2)

    = 4(a + b)2 - 8(a2 + b2)

    = 4 (a2 + 2ab + b2) - 8a2 - 8b2

    = 4a2 + 8ab + 4b2 - 8a2 - 8b2

    = -4a2 + 8ab - 4b2

    = -4 (a2 - 2ab + b2)

    = -4 (a - b)2 < 0

Hence the roots of the given equation are unreal.



8)  

 

Answer: x-1

Answer: x-2

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9)  

 Ffind the value of the first 27 terms of the geometric series.

Answer: 16



10)  

 Find the sum of the first 20 terms of the geometric series.

  Answer: 154