1.
The values of x and y in the given figure are:
Correct Answer
A. X = 10; y = 14
2.
For any positive integer a and 3, there exist unique integers q and r such that a = 3q + r, where r must satisfy :
Correct Answer
A. 0 ≤ r < 3
Explanation
The statement "0 ≤ r < 3" is the correct answer because it satisfies the given conditions that r must be an integer and it must be between 0 and 3. This means that r can be either 0, 1, or 2, which are the only possible remainders when a is divided by 3.
3.
The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is:
Correct Answer
A. 66
Explanation
The highest common factor (HCF) of two numbers is the largest number that divides both numbers without leaving a remainder. The least common multiple (LCM) of two numbers is the smallest multiple that is divisible by both numbers. In this case, the HCF is 33 and the LCM is 264. When the first number is divided by 2, the quotient is 33. This means that the first number is 66 (33 multiplied by 2). Since the LCM is 264, the other number must be 264 divided by the first number, which is 66. Therefore, the other number is 66.
4.
What will be the least possible number of the planks, if three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length?
Correct Answer
A. 5
Explanation
To divide the three pieces of timber into planks of the same length, we need to find the greatest common divisor (GCD) of the lengths of the timber. The GCD of 42, 49, and 63 is 7. Therefore, the planks will have a length of 7 meters. To find the least possible number of planks, we divide the total length of the timber (154 meters) by the length of each plank (7 meters), which gives us 22 planks. However, since we want the least possible number of planks, we need to round up to the next whole number, which is 23. Therefore, the least possible number of planks is 23 divided by 3 (since we have 3 pieces of timber), which equals 7.
5.
The decimal expansion of the rational number will terminate after
Correct Answer
A. 2
Explanation
The decimal expansion of a rational number will terminate after a certain number of decimal places if and only if the denominator of the rational number can be expressed as a power of 10. In this case, the denominator is 2, which can be expressed as 10^1. Therefore, the decimal expansion of the rational number will terminate after 1 decimal place, which is the correct answer.
6.
1. The decimal expansion of number has:
Correct Answer
A. A terminating decimal
Explanation
The question asks about the decimal expansion of a number. A terminating decimal is a decimal that ends or terminates after a certain number of decimal places. This means that the number can be expressed as a fraction with a denominator that is a power of 10. In contrast, a non-terminating decimal either repeats a pattern of digits or does not repeat at all. The correct answer states that the decimal expansion of the number is a terminating decimal, indicating that it ends after a certain number of decimal places.
7.
L.C.M. of 23 × 32 and 22 × 33 is :
Correct Answer
A. 23
Explanation
The L.C.M. (Least Common Multiple) is the smallest multiple that two or more numbers have in common. In this case, we need to find the L.C.M. of (23 × 32) and (22 × 33). To find the L.C.M., we need to consider the highest power of each prime factor that appears in the given numbers. Both numbers have 2 as a common factor, and the highest power of 2 is 2^2. However, 23 is a prime number and does not have any common prime factors with the other number. Therefore, the L.C.M. is 23.
8.
If A = 2n + 13, B = n + 7, where n is a natural number then HCF of A and B is:
Correct Answer
A. 2
Explanation
The HCF (Highest Common Factor) is the largest number that divides both A and B without leaving a remainder. In this case, A = 2n + 13 and B = n + 7. To find the HCF, we need to find the common factors of A and B. By simplifying the equations, we can see that A = 2(n + 6) + 1 and B = n + 6 + 1. It is clear that both A and B have a common factor of 2. Therefore, the HCF of A and B is 2.
9.
The product of rational and irrational no is
Correct Answer
A. Rational
Explanation
The product of a rational number and an irrational number is always rational. This is because when we multiply a rational number (which can be expressed as a fraction) with an irrational number, the result can still be expressed as a fraction. Therefore, the product is rational.
10.
If two positive integers a and b are written as a = x^{3}y^{2} and b = xy^{3}; x, y are prime numbers, then HCF (a, b) is
Correct Answer
A. XY
Explanation
In this question, we are given that a = x^3y^2 and b = xy^3, where x and y are prime numbers. The highest common factor (HCF) of two numbers is the largest number that divides both of them. In this case, we can see that both a and b have factors of x and y. However, a has an additional factor of y^2, which is not present in b. Therefore, the largest number that divides both a and b is xy. Hence, the correct answer is XY.