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Quantitative Aptitude — Algebra
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Quantitative Aptitude — Algebra

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27 mins
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35
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Quantitative Aptitude — Algebra — Questions with Answers Open after you finish the quiz — all 35 questions, with answers and explanations.
Q1. Simplify the expression: 3(x + 2) - 2(x - 1)
  1. x + 8
  2. x + 4
  3. x + 5
  4. x + 7
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Answer: A. x + 8
Given expression: 3(x + 2) - 2(x - 1) Apply distributive property: = (3 * x) + (3 * 2) - (2 * x) - (2 * -1) = 3x + 6 - 2x + 2 Combine like terms: = (3x - 2x) + (6 + 2) = x + 8
Q2. Solve the linear equation: 5x - 7 = 3x + 9
  1. x = 6
  2. x = 8
  3. x = 7
  4. x = 5
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Answer: B. x = 8
Given equation: 5x - 7 = 3x + 9 Subtract 3x from both sides: 5x - 3x - 7 = 9 2x - 7 = 9 Add 7 to both sides: 2x = 9 + 7 2x = 16 Divide by 2: x = 16 / 2 x = 8
Q3. The sum of two numbers is 45. If one number is twice the other, find the smaller number.
  1. 10
  2. 20
  3. 15
  4. 30
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Answer: C. 15
Let the smaller number be x. Then the larger number is 2x. According to the problem, their sum is 45: x + 2x = 45 3x = 45 x = 45 / 3 x = 15 The smaller number is 15.
Q4. Simplify: (a^5 * a^3) / a^4
  1. a^2
  2. a^3
  3. a^5
  4. a^4
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Answer: D. a^4
Using the exponent rules: (a^m * a^n) = a^(m+n) (a^m / a^n) = a^(m-n) First, simplify the numerator: a^5 * a^3 = a^(5+3) = a^8 Now, divide: a^8 / a^4 = a^(8-4) = a^4
Q5. Factorize the expression: x^2 - 16
  1. (x-4)^2
  2. (x-4)(x+4)
  3. (x+4)^2
  4. x(x-16)
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Answer: B. (x-4)(x+4)
The expression x^2 - 16 is in the form of a^2 - b^2, which is a difference of squares. Here, a = x and b = 4 (since 4^2 = 16). The formula for difference of squares is a^2 - b^2 = (a - b)(a + b). So, x^2 - 16 = (x - 4)(x + 4).
Q6. Solve the following system of linear equations for x and y: 1) x + y = 7 2) x - y = 3
  1. x=4, y=3
  2. x=3, y=4
  3. x=5, y=2
  4. x=2, y=5
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Answer: C. x=5, y=2
Given equations: 1) x + y = 7 2) x - y = 3 Add equation (1) and equation (2): (x + y) + (x - y) = 7 + 3 2x = 10 x = 10 / 2 x = 5 Substitute x = 5 into equation (1): 5 + y = 7 y = 7 - 5 y = 2 So, x = 5 and y = 2.
Q7. If p = 3 and q = -2, find the value of 2p - 3q.
  1. 0
  2. 6
  3. 10
  4. 12
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Answer: D. 12
Given p = 3 and q = -2. Substitute these values into the expression 2p - 3q: 2(3) - 3(-2) = 6 - (-6) = 6 + 6 = 12
Q8. Expand the expression: (2x + 3)^2
  1. 4x^2 + 12x + 9
  2. 4x^2 + 9
  3. 2x^2 + 12x + 9
  4. 4x^2 + 6x + 9
Show answer
Answer: A. 4x^2 + 12x + 9
Use the algebraic identity (a + b)^2 = a^2 + 2ab + b^2. Here, a = 2x and b = 3. So, (2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2 = 4x^2 + 12x + 9
Q9. Simplify the expression: (x/3) + (x/4)
  1. x/7
  2. x/12
  3. 7x/12
  4. 5x/12
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Answer: C. 7x/12
To add fractions, find a common denominator. The least common multiple (LCM) of 3 and 4 is 12. Convert each fraction to have a denominator of 12: x/3 = (x * 4) / (3 * 4) = 4x/12 x/4 = (x * 3) / (4 * 3) = 3x/12 Now add the fractions: 4x/12 + 3x/12 = (4x + 3x) / 12 = 7x/12
Q10. The present age of a father is three times the age of his son. If the sum of their ages is 48 years, what is the son's age?
  1. 10 years
  2. 16 years
  3. 18 years
  4. 12 years
Show answer
Answer: D. 12 years
Let the son's present age be x years. Then the father's present age is 3x years. The sum of their ages is 48 years: x + 3x = 48 4x = 48 x = 48 / 4 x = 12 So, the son's age is 12 years.
Q11. Simplify: 3x + 5y - x + 2y
  1. 2x + 7y
  2. 4x + 7y
  3. 2x + 3y
  4. 4x + 3y
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Answer: A. 2x + 7y
Combine like terms: (3x - x) + (5y + 2y) = 2x + 7y.
Q12. If 2x + 7 = 15, what is the value of x?
  1. 3
  2. 4
  3. 5
  4. 6
Show answer
Answer: B. 4
Subtract 7 from both sides: 2x = 15 - 7 = 8. Then divide by 2: x = 8 ÷ 2 = 4.
Q13. Factorize the expression: x² + 5x + 6
  1. (x + 1)(x + 6)
  2. (x + 2)(x + 3)
  3. (x + 2)(x + 4)
  4. (x + 5)(x + 1)
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Answer: B. (x + 2)(x + 3)
Find two numbers that multiply to 6 and add to 5, which are 2 and 3. So, (x + 2)(x + 3).
Q14. If a = 3 and b = 2, what is the value of a² + b²?
  1. 10
  2. 11
  3. 12
  4. 13
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Answer: D. 13
Substitute a = 3 and b = 2 into the expression: 3² + 2² = 9 + 4 = 13.
Q15. If x + y = 7 and x - y = 3, what are the values of x and y?
  1. x = 5, y = 2
  2. x = 2, y = 5
  3. x = 4, y = 3
  4. x = 3, y = 4
Show answer
Answer: A. x = 5, y = 2
Add the two equations: (x + y) + (x - y) = 7 + 3 ⇒ 2x = 10 ⇒ x = 5. Substitute x = 5 into x + y = 7 ⇒ 5 + y = 7 ⇒ y = 2.
Q16. Expand (2a + 3b)²
  1. 4a² + 9b²
  2. 4a² + 12ab + 9b²
  3. 2a² + 6ab + 3b²
  4. 4a² + 6ab + 9b²
Show answer
Answer: B. 4a² + 12ab + 9b²
Use the identity (x + y)² = x² + 2xy + y². Here x = 2a, y = 3b. So (2a)² + 2(2a)(3b) + (3b)² = 4a² + 12ab + 9b².
Q17. The sum of two consecutive integers is 45. What is the smaller integer?
  1. 21
  2. 22
  3. 23
  4. 24
Show answer
Answer: B. 22
Let the integers be x and x + 1. Then x + (x + 1) = 45 ⇒ 2x + 1 = 45 ⇒ 2x = 44 ⇒ x = 22. The smaller integer is 22.
Q18. Simplify: (a³)⁴ ÷ a⁶
  1. a⁵
  2. a⁶
  3. a⁷
  4. a⁸
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Answer: B. a⁶
(a³)⁴ = a^(3×4) = a¹². Then a¹² ÷ a⁶ = a^(12-6) = a⁶.
Q19. If x = 9, what is the value of √x + 5?
  1. 8
  2. 7
  3. 6
  4. 5
Show answer
Answer: A. 8
Substitute x = 9 into the expression: √9 + 5 = 3 + 5 = 8.
Q20. Subtract (2x² + 3x - 1) from (5x² + 7x + 4).
  1. 3x² + 4x + 5
  2. 3x² + 4x + 3
  3. 7x² + 10x + 3
  4. 3x² - 4x + 5
Show answer
Answer: A. 3x² + 4x + 5
(5x² + 7x + 4) - (2x² + 3x - 1) = 5x² - 2x² + 7x - 3x + 4 - (-1) = 3x² + 4x + 5.
Q21. Simplify: (5a)^2 + b^2 - (3a)^2
  1. 16a^2 + b^2
  2. 4a^2 + b^2
  3. 16a^2 - b^2
  4. 4a^2 - b^2
Show answer
Answer: A. 16a^2 + b^2
Expand each term: (5a)^2 = 25a^2 and (3a)^2 = 9a^2. So expression = 25a^2 + b^2 - 9a^2 = 16a^2 + b^2.
Q22. If (3x/a + y/b) = 7 and (x/a - y/b) = 1, find the values of x and y respectively.
  1. 2a, b
  2. 2a, 2b
  3. 3a, b
  4. 3a, 2b
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Answer: A. 2a, b
Add both equations: (3x/a + y/b) + (x/a - y/b) = 7 + 1, which gives 4x/a = 8, so x/a = 2, giving x = 2a. Substitute x/a = 2 into the second equation: 2 - y/b = 1, so y/b = 1, giving y = b.
Q23. Factors of x^2 + 9x + 18 are:
  1. (x+6)(x+3)
  2. (x-6)(x-3)
  3. (x+6)(x-3)
  4. (x-6)(x+3)
Show answer
Answer: A. (x+6)(x+3)
Split the middle term: 9x = 6x + 3x. So x^2 + 6x + 3x + 18 = x(x+6) + 3(x+6) = (x+6)(x+3).
Q24. If 25x^2 = 625^2 - 375^2, then find the value of x.
  1. 100
  2. 120
  3. 150
  4. 180
Show answer
Answer: A. 100
Use identity a^2 - b^2 = (a+b)(a-b). So 25x^2 = (625+375)(625-375) = 1000 x 250 = 250000. Then x^2 = 250000/25 = 10000, giving x = 100.
Q25. On dividing 10x^3 - 20x^2 + 8x - 5 by 2x - 4, the remainder is:
  1. 3
  2. 5
  3. 11
  4. -5
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Answer: C. 11
Use Remainder Theorem. Put 2x - 4 = 0, so x = 2. Substitute in polynomial: 10(2)^3 - 20(2)^2 + 8(2) - 5 = 10(8) - 20(4) + 16 - 5 = 80 - 80 + 16 - 5 = 11.
Q26. Given p = -1, q = 2, r = 0 and s = 1/2, find the value of 3(p^2 + q^2 + r^3).
  1. 15
  2. -15
  3. 12
  4. 18
Show answer
Answer: A. 15
Substitute values: 3((-1)^2 + 2^2 + 0^3) = 3(1 + 4 + 0) = 3 x 5 = 15.
Q27. If a^3 + b^3 + c^3 - 3abc = 0, then find the value of (a^3 + b^3 + c^3) / (abc).
  1. 0
  2. 1
  3. 3
  4. -3
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Answer: C. 3
Given a^3 + b^3 + c^3 - 3abc = 0, it implies a^3 + b^3 + c^3 = 3abc. Therefore, (a^3 + b^3 + c^3) / (abc) = 3abc / abc = 3 (assuming abc is not zero).
Q28. Given that (p^2 + q^2) = 45, then find the value of (p + q)^2 + (p - q)^2.
  1. 45
  2. 90
  3. 135
  4. 180
Show answer
Answer: B. 90
Use identity: (p+q)^2 + (p-q)^2 = 2(p^2 + q^2). So value = 2 x 45 = 90.
Q29. If x = 2, y = 3, z = 4, then find the value of (x^3 + y^3 + z^3 - 3xyz) / (x^2 + y^2 + z^2 - xy - yz - zx).
  1. 9
  2. -9
  3. 6
  4. -6
Show answer
Answer: A. 9
Use identity: x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx). So the expression simplifies to (x+y+z). Substitute values: x+y+z = 2+3+4 = 9.
Q30. If (a^2 + 1/a^2) = 7, then find the value of (5a + 5/a).
  1. plus or minus 15
  2. plus or minus 20
  3. plus or minus 25
  4. plus or minus 30
Show answer
Answer: A. plus or minus 15
(a + 1/a)^2 = a^2 + 1/a^2 + 2 = 7 + 2 = 9, so a + 1/a = plus or minus 3. Then 5(a + 1/a) = 5a + 5/a = plus or minus 15.
Q31. If (x - 1/x) = 2, find the value of (x^3 - 1/x^3).
  1. 8
  2. 14
  3. 10
  4. 12
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Answer: B. 14
Use identity: x^3 - 1/x^3 = (x - 1/x)^3 + 3(x - 1/x) = (2)^3 + 3 x (2) = 8 + 6 = 14.
Q32. If 3p(p+q+r) = 150, 3q(p+q+r) = 90 and 3r(p+q+r) = 60, then find the value of (p + 2q + 3r).
  1. 17
  2. 20
  3. 25
  4. 30
Show answer
Answer: A. 17
Add all three equations: 3(p+q+r)(p+q+r) = 150 + 90 + 60 = 300. So 3(p+q+r)^2 = 300, giving (p+q+r)^2 = 100, and p+q+r = 10 (assuming positive sum). Then, 3p(10) = 150 => p = 5. 3q(10) = 90 => q = 3. 3r(10) = 60 => r = 2. So, p + 2q + 3r = 5 + 2(3) + 3(2) = 5 + 6 + 6 = 17.
Q33. If 2.5a = 0.05b, then find the value of (b - a)/(b + a).
  1. 49/51
  2. 51/49
  3. 49/50
  4. 50/49
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Answer: A. 49/51
From 2.5a = 0.05b, we get a/b = 0.05/2.5 = 5/250 = 1/50. Now (b-a)/(b+a) = (1 - a/b)/(1 + a/b) = (1 - 1/50)/(1 + 1/50) = (49/50)/(51/50) = 49/51.
Q34. If (x + y + 2) = 0, then find the value of (x^3 + y^3 + 8 - 6xy).
  1. -1
  2. 1
  3. 0
  4. 2
Show answer
Answer: C. 0
Use identity: a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca). Take c = 2: x^3 + y^3 + 2^3 - 3xy(2) = x^3 + y^3 + 8 - 6xy. The expression is (x+y+2) x (...). Since (x+y+2) = 0, the entire expression equals 0.
Q35. If (p + q)^2 - pq = 0, then find the value of (p^3 - q^3)/(p - q).
  1. 0
  2. 1
  3. 2
  4. 3
Show answer
Answer: A. 0
We know (p^3 - q^3)/(p - q) = p^2 + pq + q^2. Now, p^2 + pq + q^2 = p^2 + 2pq + q^2 - pq = (p+q)^2 - pq. Given (p+q)^2 - pq = 0. So, the value of the expression is 0.
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