# A-AS Level (CIE) Mathematics Paper-5: Specimen Questions with Answers 6 - 9 of 20

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## Question 6

### Question

MCQ▾If the system of equations has non-trivial solution, then …

### Choices

Choice (4) | |
---|---|

a. | |

b. | |

c. | |

d. |

### Answer

d.### Explanation

A nonhomogeneous system of linear equations has a unique non-trivial solution if and only if its determinant is non-zero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.

Writing given equations in determinant form.

Putting value of in given equations

## Question 7

### Question

MCQ▾If and then

### Choices

Choice (4) | |
---|---|

a. | 1 |

b. | |

c. | 2 |

d. |

### Answer

d.### Explanation

For any non-zero complex number , there exists a complex number such that , i.e.. , multiplicative inverse of .

and

Or

Comparing both sides we get,

and

Now,

[Using (1) ]

## Question 8

### Question

MCQ▾The remainder, when is divided by , is________

### Choices

Choice (4) | |
---|---|

a. | 10 |

b. | 6 |

c. | 8 |

d. | 9 |

### Answer

c.### Explanation

i.e.. , is divisible by

is divisible by for all odd

is divisible by and hence it is divisible by

Also, is divisible by

i.e.. ,

… Multiplying by

… put

Hence remainder is .

## Question 9

### Question

MCQ▾It be a complex number and . Then the value of lies in________

### Choices

Choice (4) | |
---|---|

a. | |

b. | |

c. | |

d. |

### Answer

d.### Explanation

**Complex Numbers**

When we combine a Real Number and an Imaginary Number, we get a Complex Number:

We know

Hence,

Now,

But

and

Least value of is

From (1) and (2)

Hence, the value of lies in