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## Algebra Level 3 Test 2

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Question 1 |

Let S denotes the infinite sum 2 + 5x + 9x

^{2}+ 14x^{3}+ 20x^{4}+ â€¦..where IxI < 1 and the coefficient of x^{nâ€“1 }is Â½ n(n+3),(n = 1,2,â€¦â€¦â€¦). Then what will be the value of S(2 â€“ x)/(1-x) ^{3} | |

(2 â€“ x)/(1+x) ^{3} | |

(2 + x)/(1-x) ^{3} | |

(2 + x)/(1+x) ^{3} |

Question 1 Explanation:

Coefficient of x

S= 2 + 5x + 9x

Multiply by x

xS = 2x + 5x

Subtract second equation from first

S(1â€“ x) = 2 + 5x + 9x

Let p = S(1â€“ x)

P = 2 + 3x + 4x

Again multiply with x

Px = 2x +3x

On subtracting second from first

p(1 â€“x) = 2 + x + x

Now it is in G.P.

So the sum is 2 + x / (1+x)

S = (2-x)/ (1 â€“ x)

^{n}= 1/2(n - 1)(n + 4)S= 2 + 5x + 9x

^{2}+ 14x^{3}+ 20x^{4}+â€¦..Â Â (i)Multiply by x

xS = 2x + 5x

^{2}+ 9x^{3}+ â€¦â€¦â€¦.. (ii)Subtract second equation from first

S(1â€“ x) = 2 + 5x + 9x

^{2}+ 14x^{3}+ 20x^{4}Let p = S(1â€“ x)

P = 2 + 3x + 4x

^{2}+ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦Again multiply with x

Px = 2x +3x

^{2}+ â€¦â€¦â€¦â€¦â€¦.On subtracting second from first

p(1 â€“x) = 2 + x + x

^{2}Now it is in G.P.

So the sum is 2 + x / (1+x)

S = (2-x)/ (1 â€“ x)

^{3}Question 2 |

If x

^{2 }+ 5y^{2}+ z^{2}= 2y(2x + z), then which of the following statements is(are) necessarily true? A. x = 2y B. x = 2z C. 2x = z Only A | |

B and C | |

A and B | |

None of these |

Question 2 Explanation:

x

(x

(x â€“ 2y)

It can be true only if x = 2y and z = y

^{2}+ 5y^{2}+ z^{2}= 4yx + 2yz(x

^{2}+ 4y^{2}â€“ 4yx) + z^{2}+ y^{2}â€“ 2yz = 0(x â€“ 2y)

^{2}+ (z â€“ y)^{2 }= 0It can be true only if x = 2y and z = y

Question 3 |

Amol was asked to calculate the arithmetic mean of 10 positive integers, each of which had 2 digits. By mistake, he interchanged the 2 digits, say a and b, in one of these 10 integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b â€“ a equals

1 | |

2 | |

3 | |

None of these |

Question 3 Explanation:

Arithmetic mean is more by 1.8 times than what it should

have been, this means sum is more by 18.

So ba â€“ ab = 18,

This means b > Â a as sum has gone up, e.g. 31 â€“ 13 = 18

Hence, b â€“ a = 2

have been, this means sum is more by 18.

So ba â€“ ab = 18,

This means b > Â a as sum has gone up, e.g. 31 â€“ 13 = 18

Hence, b â€“ a = 2

Question 4 |

A child was asked to add first few natural numbers (i.e. 1 + 2 + 3 + â€¦) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong, the child discovered he had missed one number in the sequence during addition. The number he missed was

less than 10 | |

10 | |

15 | |

more than 15 |

Question 4 Explanation:

Sum of natural numbers = n(n+1)/2 = (n

So according to the question

(n

On solving it is clear that

1150 = n

n(n + 1)greater than equal to 1150

(n

Therefore for this the smallest value of x isÂ 20

^{2}+n) /2So according to the question

(n

^{2}+n) /2 â€“ x = 575On solving it is clear that

1150 = n

^{2}+ n â€“ 2xn(n + 1)greater than equal to 1150

(n

^{2}+n) greater than equal to 1150Therefore for this the smallest value of x isÂ 20

Question 5 |

The number of real roots of the equation p

^{2}/ (x) + q^{2}/ (x -1) = 1 where p and q are real numbers not equal to zero simultaneously, isNone | |

1 | |

2 | |

1 or 2 |

Question 5 Explanation:

p

p

This is a quadratic equation.

Hence, number of roots = 2 or 1 (1 in the case when both roots are equal).

^{2}/ (x) + q^{2}/ (x -1) = 1p

^{2}(x -1) + q^{2}(x) Â = x^{2}â€“ xThis is a quadratic equation.

Hence, number of roots = 2 or 1 (1 in the case when both roots are equal).

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