Option (A): \(14+(-13)\)

\(|14|=14,\;|-13|=13\)

Here, \(|14|\) has a bigger absolute value than \(|-13|\).

Since  \(14\) has a positive sign, so the sum will also have a positive sign.

\(14+(-13)=\) a positive integer

Hence, option (A) is incorrect.

Option (B): \(-12+10\)

\(|-12|=12,\;|10|=10\)

Here, \(|-12|\) has a bigger absolute value than \(|10|\).

Since \(-12\) has a negative sign, so the sum will also have a negative sign.

\(-12+10=\) a negative integer

Hence, option (B) is correct.

Option (C): \(-10+12\)

\(|-10|=10,\;|12|=12\)

Here, \(|12|\) has a bigger absolute value than \(|-10|\).

Since \(12\) has a positive sign,  so the sum will also have a positive sign.

\(-10+12=\) a positive integer

Hence, option (C) is incorrect.

Option (D): \(7+(-5)\)

\(|7|=7,\;|-5|=5\)

Here, \(|7|\) has a bigger absolute value than \(|-5|\).

Since  \(7\) has a positive sign, so the sum will also have a positive sign.

\(7+(-5)=\) a positive integer

Hence, option (D) is incorrect.

Option (A): 

\(-6-(-7)\)

Here, Negative – Negative

If the absolute value of the second integer is greater than the absolute value of the first integer, we get a positive integer as the answer.

\(\because\;|-6|=6,\;|-7|=7\) and \(7>6\)

\(\therefore\;-6-(-7)=\) Positive integer

Hence, option (A) is correct.

Option (B): 

\(6-7\)

Here,  Positive – Positive

If the second integer is greater than the first, we get a negative integer as the answer.

\(\therefore\;6-7=\) Negative integer

Hence, option (B) is incorrect.

Option (C): 

\(-7-(-6)\)

Here,  Negative – Negative

If the absolute value of the first integer is greater than the absolute value of the second integer, we get a negative integer as the answer.

\(\because\;|-7|=7,\;|-6|=6\) and \(7>6\)

\(\therefore\;-7-(-6)=\) Negative integer

Hence, option (C) is incorrect.

Option (D): 

\(-6-7\)

Here,  Negative – Positive

If the first integer is negative and the second integer is positive, we get a negative integer as the answer.

\(-6-7=\) Negative integer

Hence, option (D) is incorrect.

Option (A): \(-3×4\)

Here, \(-3\) is a negative integer and \(4\) is a positive integer.

When we multiply two integers with different signs, we get a negative integer as the answer.

Thus, \(-3×4=\) a negative integer

Hence, option (A) is correct.

Option (B): \(-3×(-4)\)

Here, \(-3\) is a negative integer and \(-4\) is also a negative integer.

When we multiply two integers with the same sign, we get a positive integer as the answer.

Thus, \(-3×(-4)=\) a positive integer

Hence, option (B) is incorrect.

Option (C): \(3×4\)

Here, \(3\) is a positive integer and \(4\) is also a positive integer.

When we multiply two integers with the same sign, we get a positive integer as the answer.

Thus, \(3×4=\) a positive integer

Hence, option (C) is incorrect.

Option (D): \(5×20\)

Here, \(5\) is a positive integer and \(20\) is also a positive integer.

When we multiply two integers with the same sign, we get a positive integer as the answer.

Thus, \(5×20=\) a positive integer

Hence, option (D) is incorrect.

Option (A): \(12\div(-4)\)

Here, \(12\) is a positive integer and \(-4\) is a negative integer.

When dividend and divisor are of different signs, we get a negative integer as the answer.

Thus, \(12\div(-4)=\) a negative integer

Hence, option (A) is incorrect.

Option (B): \(-15\div5\)

Here, \(-15\) is a negative integer and \(5\) is a positive integer.

When dividend and divisor are of different signs, we get a negative integer as the answer.

Thus, \(-15\div5=\) a negative integer

Hence, option (B) is incorrect.

Option (C): \(-12\div(-4)\)

Here, \(-12\) is a negative integer and \(-4\) is also a negative integer.

When dividend and divisor are of the same sign, we get a positive integer as the answer.

Thus, \(-12\div(-4)=\) a positive integer

Hence, option (C) is correct.

Option (D): \(-12\div4\)

Here, \(-12\) is a negative integer and \(4\) is a positive integer.

When dividend and divisor are of different signs, we get a negative integer as the answer.

Thus, \(-12\div4=\) a negative integer

Hence, option (D) is incorrect.