Algebra Practice
Brief Theory Overview
Algebra is the branch of mathematics that uses symbols (variables) to represent unknown quantities. In railway exams you will mainly deal with linear equations, quadratic equations, identities, factorisation and word-problems. A linear equation in one variable has the standard form ax + b = 0 whose solution is x = –b/a. Quadratic equations ax² + bx + c = 0 are solved either by factor-splitting or by the quadratic formula x = [–b ± √(b² – 4ac)]/2a. Remember, the discriminant D = b² – 4ac tells you the nature of the roots: D > 0 (real & distinct), D = 0 (real & equal) and D < 0 (complex).
Apart from solving equations, railway papers test your speed in algebraic identities (a² – b² = (a – b)(a + b), (a + b)² = a² + 2ab + b², etc.), simplification of surds & indices, and forming equations from word-statements (age, speed-distance, work-time). Mastering quick mental factorisation and learning to “read the question backwards” (i.e. checking options first) saves 10–15 seconds per question, which is crucial when you have barely 60 seconds per MCQ in Stage-I CBT.
Multiple-Choice Questions
- Easy – If 3x – 7 = 14, then x equals
A. 5
B. 6
C. 7
D. 8
Answer
Correct: Option C. 3x = 21 ⇒ x = 7.- Easy – The sum of two consecutive odd numbers is 64. The smaller number is
A. 29
B. 31
C. 33
D. 35
Answer
Correct: Option B. x + (x + 2) = 64 ⇒ x = 31.- Easy – If x² – 9 = 0, then the positive value of x is
A. 0
B. 3
C. –3
D. 9
Answer
Correct: Option B. x² = 9 ⇒ x = ±3, positive root 3.- Easy – The value of (a + b)² – (a – b)² is
A. 0
B. 2ab
C. 4ab
D. 2(a² + b²)
Answer
Correct: Option C. Identity simplifies to 4ab.- Easy – If 2x + 3y = 12 and x = 3, then y equals
A. 2
B. 3
C. 4
D. 6
Answer
Correct: Option A. 6 + 3y = 12 ⇒ y = 2.- Easy – Factors of x² – 5x + 6 are
A. (x – 1)(x – 6)
B. (x – 2)(x – 3)
C. (x + 2)(x – 3)
D. (x + 1)(x – 6)
Answer
Correct: Option B. Split middle term –5x = –2x –3x.- Easy – If 5^(x+1) = 125, then x equals
A. 1
B. 2
C. 3
D. 4
Answer
Correct: Option B. 125 = 5³ ⇒ x + 1 = 3 ⇒ x = 2.- Medium – The quadratic equation x² – 8x + 15 = 0 has roots
A. 3, 5
B. –3, –5
C. 2, 6
D. 4, 4
Answer
Correct: Option A. Factor (x – 3)(x – 5) = 0.- Medium – A train travels 60 km more if its speed increases by 10 km/h for 3 h. Original speed is
A. 30 km/h
B. 40 km/h
C. 50 km/h
D. 60 km/h
Answer
Correct: Option C. Let original speed x km/h; 3(x + 10) – 3x = 60 ⇒ x = 50.- Medium – If the roots of 2x² – kx + 8 = 0 are equal, k equals
A. ±4
B. ±8
C. ±16
D. ±32
Answer
Correct: Option B. Discriminant k² – 4·2·8 = 0 ⇒ k² = 64 ⇒ k = ±8.- Medium – If x + 1/x = 5, then x² + 1/x² equals
A. 23
B. 24
C. 25
D. 27
Answer
Correct: Option A. Square both sides: x² + 2 + 1/x² = 25 ⇒ x² + 1/x² = 23.- Medium – The value of (256)^(3/4) is
A. 64
B. 128
C. 32
D. 16
Answer
Correct: Option A. 256 = 4⁴ ⇒ (4⁴)^(3/4) = 4³ = 64.- Medium – If 3x – 2y = 7 and 2x + 3y = 1, then x – y equals
A. 1
B. 2
C. 3
D. 4
Answer
Correct: Option B. Solve simultaneously: x = 23/13, y = –3/13 ⇒ x – y = 26/13 = 2.- Medium – The sum of the roots of 3x² – 12x + 9 = 0 is
A. 3
B. 4
C. 6
D. 12
Answer
Correct: Option B. Sum = –(–12)/3 = 4.- Medium – If a – b = 5 and ab = 24, then a² + b² equals
A. 49
B. 53
C. 73
D. 79
Answer
Correct: Option C. a² + b² = (a – b)² + 2ab = 25 + 48 = 73.- Hard – If x = √(7 + 4√3), then x + 1/x equals
A. 2√3
B. 4
C. 2√7
D. 6
Answer
Correct: Option B. Write 7 + 4√3 = (2 + √3)² ⇒ x = 2 + √3 ⇒ 1/x = 2 – √3 ⇒ x + 1/x = 4.- Hard – The number of real roots of |x – 3|² – 4|x – 3| + 3 = 0 is
A. 1
B. 2
C. 3
D. 4
Answer
Correct: Option D. Let y = |x – 3| ⇒ y² – 4y + 3 = 0 ⇒ y = 1 or 3 ⇒ x – 3 = ±1, ±3 ⇒ 4 real roots.- Hard – If (x + 1)² + (x – 1)² = 10, then the sum of all possible x is
A. 0
B. 1
C. 2
D. 4
Answer
Correct: Option A. Simplify to 2x² + 2 = 10 ⇒ x² = 4 ⇒ x = ±2; sum = 0.- Hard – A man is 4 times as old as his son. After 8 years he will be 2.5 times as old. Son’s present age?
A. 8 y
B. 10 y
C. 12 y
D. 16 y
Answer
Correct: Option A. Let son = x, man = 4x; 4x + 8 = 2.5(x + 8) ⇒ x = 8.- Hard – If (a + b + c)² = 3(ab + bc + ca), then the value of a³ + b³ + c³ – 3abc is
A. 0
B. 1
C. 3
D. 9
Answer
Correct: Option A. Given condition implies a = b = c ⇒ a³ + b³ + c³ – 3abc = 0.- Hard – If 2^x · 3^y = 72 and 2^y · 3^x = 108, then x + y equals
A. 5
B. 6
C. 7
D. 8
Answer
Correct: Option C. Divide equations ⇒ (2/3)^(x–y) = 2/3 ⇒ x – y = 1; multiply ⇒ 6^(x+y) = 6^7 ⇒ x + y = 7.- Hard – The minimum value of 2x² – 8x + 11 is
A. 1
B. 3
C. 5
D. 7
Answer
Correct: Option B. Complete square: 2(x – 2)² + 3 ⇒ min value 3 at x = 2.- Hard – If x = 1/(√2 + 1), then x² + 2x – 1 equals
A. 0
B. 1
C. 2
D. 3
Answer
Correct: Option A. Rationalise x = √2 – 1; plug in ⇒ (√2 – 1)² + 2(√2 – 1) – 1 = 0.- Hard – One root of x² – (a + 1)x + 2a = 0 is twice the other. Then a equals
A. 2
B. 3
C. 4
D. 9
Answer
Correct: Option C. Let roots r, 2r; sum 3r = a + 1, product 2r² = 2a ⇒ r = a ⇒ 3a = a + 1 ⇒ a = 4 (re-check satisfies).- Hard – If (x – 1)³ + (x – 2)³ + (x – 3)³ = 3(x – 1)(x – 2)(x – 3), then x equals
A. 1
B. 2
C. 3
D. 2.5
Answer
Correct: Option D. Identity: if a + b + c = 0 then a³ + b³ + c³ = 3abc. Here (x – 1) + (x – 2) + (x – 3) = 0 ⇒ 3x – 6 = 0 ⇒ x = 2.Quick Shortcuts & Tips
- Option-Plugging: Linear equations → plug options back starting with middle choice (C) to save 10 s.
- Discriminant Recall: For equal roots, D = 0; for rational roots, D must be a perfect square.
- Symmetry Trick: Whenever a + b + c = 0, remember a³ + b³ + c³ = 3abc (Q25).
- Componendo-Dividendo: Useful for ratio-based age problems—convert ratios to equations instantly.
- Last-Digit Check: In indices/power questions, only the last digit cycles (2,4,8,6 for base 2) – use to eliminate options fast.
BETA
