Mathematics · Quantitative Aptitude

Algebra and Arithmetic

406 Questions

Algebra and arithmetic questions cover fundamental mathematical operations, inequalities, and binomial products. They assess core quantitative reasoning skills required for various aptitude tests. Solving these problems strengthens the understanding of number systems and algebraic identities.

Binomial productsLinear inequalitiesLeast common multipleQuadratic equationsInteger properties

Algebra and Arithmetic Questions

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

For a set of positive numbers, consider the following statements:
1. If each number is reduced by $2$, then the geometric mean of the set may not always exists.
2. If each number is increased by $2$, then the geometric mean of the set is increased by $2$.
Which of the above statements is/are correct?

  1. $1$ only
  2. $2$ only
  3. Both $1$ and $2$
  4. Neither $1$ nor $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

1. Consider the two numbers $1$ and $4$, geometric mean of $1$ and $4$ is $\sqrt {1 \times 4} = 2$.
When each number is reduced by $2$, the numbers become $-1$ and $2$ whose geometric mean does not exist.
2. Now consider two numbers $2$ and $7$. Their geometric mean is $\sqrt {14}$. The new numbers are $4$ and $9$ whose geometric mean is $\sqrt {4\times 9} = 6$ which is not equal to $2\sqrt {14}$.
Thus only statement $1$ is true.

Multiple choice

Which of the following is NOT a property of the expected value?

  1. It is a linear operator

  2. It is a constant

  3. It is always positive

  4. It is always negative
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The expected value is not always positive. It can be negative or zero, depending on the distribution of the random variable.

Multiple choice

What is the negation of the statement "∀x ∈ R, x^2 ≥ 0"?

  1. ∃x ∈ R, x^2 < 0

  2. ∀x ∈ R, x^2 > 0

  3. ∃x ∈ R, x^2 ≤ 0

  4. ∀x ∈ R, x^2 ≠ 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of a universal statement is an existential statement with the opposite quantifier and the opposite truth value.

Multiple choice

What is the negation of the statement "∃x ∈ R, x^2 = 2"?

  1. ∀x ∈ R, x^2 ≠ 2

  2. ∃x ∈ R, x^2 > 2

  3. ∀x ∈ R, x^2 < 2

  4. ∃x ∈ R, x^2 ≤ 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of an existential statement is a universal statement with the opposite quantifier and the opposite truth value.

Multiple choice

If x is a variable, what does the inequality x > 0 imply?

  1. x is a positive number.

  2. x is a negative number.

  3. x is zero.

  4. x can be any real number.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inequality x > 0 implies that the variable x is greater than zero. This means that x must be a positive number.

Multiple choice

If x is a variable, what does the inequality x ≤ 0 imply?

  1. x is a positive number.

  2. x is a negative number.

  3. x is zero.

  4. x can be any real number.
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The inequality x ≤ 0 implies that the variable x is less than or equal to zero. This means that x can be either a negative number or zero.

Multiple choice

The set of all real numbers between 0 and 1 is:

  1. Open.

  2. Closed.

  3. Both open and closed.

  4. Neither open nor closed.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of all real numbers between 0 and 1 is open because it does not contain any of its boundary points.

Multiple choice

What is the result of adding zero to any number?

  1. The number itself

  2. Zero

  3. The number divided by two

  4. The number multiplied by two
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Adding zero to any number does not change the value of the number. It remains the same.

Multiple choice

What is the result of subtracting zero from any number?

  1. The number itself

  2. Zero

  3. The number divided by two

  4. The number multiplied by two
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Subtracting zero from any number does not change the value of the number. It remains the same.

Multiple choice

What is the result of multiplying any number by zero?

  1. The number itself

  2. Zero

  3. The number divided by two

  4. The number multiplied by two
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiplying any number by zero always results in zero. This is because zero represents the absence of quantity, so multiplying any number by zero means there is no quantity to multiply.

Multiple choice

What is the result of dividing any number by zero?

  1. The number itself

  2. Zero

  3. Infinity

  4. Undefined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Division by zero is undefined. This is because dividing a number by zero means trying to find how many times zero goes into that number, which is impossible. Therefore, division by zero is undefined.

Multiple choice

In the division of two numbers, if one of the numbers is zero, what is the quotient?

  1. The other number

  2. Zero

  3. Infinity

  4. Undefined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Division by zero is undefined. Therefore, if one of the numbers in a division operation is zero, the quotient is undefined.

Multiple choice

What is the initial condition for Eulerian numbers?

  1. $A(0, 0) = 1$
  2. $A(1, 0) = 1$
  3. $A(0, 1) = 1$
  4. $A(1, 1) = 1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The initial condition for Eulerian numbers is $A(0, 0) = 1$.

Multiple choice

Brahmagupta's identity states that $a^2 + b^2 = c^2 + d^2$ if and only if $ab = cd$. This identity is also known as:

  1. Brahmagupta's Formula

  2. Brahmagupta's Theorem

  3. Brahmagupta's Identity

  4. Brahmagupta's Conjecture
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Brahmagupta's identity is a mathematical equation that relates the squares of two pairs of numbers. It is named after the Indian mathematician Brahmagupta, who discovered it in the 7th century.