All 32 algebra questions the Punjab Police SI has asked since 2016: the eight identities, linear pairs by elimination, quadratics by sum and product, age tables, variation and logarithms.
Algebra in the Punjab Police SI paper is not about theory. Every one of the 32 algebra questions asked since 2016 is a sentence you have to turn into an equation, or an ugly expression that collapses the moment you spot the identity hiding in it.
That gives you only two skills to build. Translate: give the unknown a letter, write one equation per sentence, solve. Recognise: learn eight identities so well that (0.93³ − 0.7³)/(0.93² + 0.651 + 0.7²) is a two-second answer instead of a calculator job. This note takes both, part by part — identities, linear pairs, quadratics, ages, variation and the single logarithm question — with every SI question solved.
32
SI questions
2016–2023
18
linear and quadratic
the biggest slice
9
on identities
a³ ± b³ leads
4
on ages
one table solves all
The eight identities
Nine of the 32 questions are one of these in disguise.
Learn these by shape, not by letters. Whatever sits in the a-slot and the b-slot — a number, a decimal, a cube root — the identity still holds.
KEY TERM
a³ + b³ = (a + b)(a² − ab + b²)
This single line, read backwards, is the SI examiner's favourite trick. When you see a fraction whose denominator is a² − ab + b² and whose numerator is a³ + b³, the whole thing is just 1/(a + b). The sign rule is worth saying aloud: same sign outside, opposite sign in the middle. a³ + b³ takes (a + b) outside and −ab in the middle; a³ − b³ takes (a − b) outside and +ab in the middle.
a³ + b³→(a + b)(a² − ab + b²)→divide → 1/(a + b)
QUICK CHECK
Simplify the following : ((7)^{1/3} + (3)^{1/3})((7)^{2/3} − (21)^{1/3} + (3)^{2/3}) (SI 2 Jul 2023)
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Answer: 10
Put a = 7^(1/3) and b = 3^(1/3). Then a² = 7^(2/3), b² = 3^(2/3), and ab = 21^(1/3) — exactly the middle term. So the product is (a + b)(a² − ab + b²) = a³ + b³ = 7 + 3 = 10. Nothing is ever cubed by hand.
Read it as a and b with a = 6.28, b = 3.4. The numerator is a² − ab + b² and the denominator is a³ + b³ = (a + b)(a² − ab + b²). The bracket cancels and you are left with 1/(a + b) = 1/(6.28 + 3.4) = 1/9.68. Note that 3.4 × 6.28 is written out only so you recognise ab.
The hidden step is 0.651. Check it: 0.93 × 0.7 = 0.651, so the denominator is a² + ab + b² with a = 0.93 and b = 0.7. The numerator a³ − b³ = (a − b)(a² + ab + b²), so the answer is simply a − b = 0.93 − 0.7 = 0.23. Whenever a loose decimal appears in the middle of a denominator, multiply the two outer numbers — it is almost always ab.
THE TWO SQUARES YOU WILL USE MOST
a² + b² = (a + b)² − 2ab and a² + b² = (a − b)² + 2ab. A question almost never gives you a and b separately; it gives you their sum and their product, or their difference and their product. These two lines convert what you were given into what was asked, with no solving at all. The cube version works the same way: a³ − b³ = (a − b)³ + 3ab(a − b).
QUICK CHECK
If x = 2 + √5 and y = 2 − √5, then the value of x² + y² is : (SI 2 Jul 2023)
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Answer: 18
x + y = (2 + √5) + (2 − √5) = 4 and xy = (2)² − (√5)² = 4 − 5 = −1. Now x² + y² = (x + y)² − 2xy = 16 − 2(−1) = 16 + 2 = 18. Surds that are conjugates always give a clean sum and a clean product — never square them one at a time.
QUICK CHECK
If 3x − 2y = 10 and xy = 11, then the value of 27x³ − 8y³ is _______. (SI 2 Jul 2023)
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Answer: 2980
Write it as a cube difference with a = 3x and b = 2y, so a − b = 10 and ab = 6xy = 66. Then 27x³ − 8y³ = a³ − b³ = (a − b)³ + 3ab(a − b) = 10³ + 3(66)(10) = 1000 + 1980 = 2980. The whole question is one substitution; you never find x or y.
QUICK CHECK
If a = 0.25 and b = 0.75, then the value of the expression 0.5/(b² − a²) is ______. (SI 29 Jul 2023)
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Answer: 1
Do not square the decimals. b² − a² = (b + a)(b − a) = (0.75 + 0.25)(0.75 − 0.25) = 1 × 0.5 = 0.5. So the expression is 0.5 ÷ 0.5 = 1. The difference of two squares is the one identity that turns decimals into whole numbers.
QUICK CHECK
What is the remainder when (x^{7} + 1) n is divided by (x + 1)? (SI 14 Jul 2023)
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Answer: 0
By the remainder theorem, the remainder on dividing a polynomial p(x) by (x + 1) is p(−1). Here p(−1) = (−1)⁷ + 1 = −1 + 1 = 0. A remainder of zero also means (x + 1) is a factor — and indeed xⁿ + 1 is always divisible by x + 1 when n is odd.
QUICK CHECK
Simplify the expression given below. (x² + 3x − 54)/(x² − 36) + 9/(x + 6)² − 3/4 (SI 20 Jun 2023)
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Answer: (3/(x + 6) + 1/2)²
Factorise first. x² + 3x − 54 = (x + 9)(x − 6) and x² − 36 = (x + 6)(x − 6), so the first fraction is (x + 9)/(x + 6) = 1 + 3/(x + 6).
Now let t = 3/(x + 6). The second term 9/(x + 6)² is t². The whole expression becomes 1 + t + t² − 3/4 = t² + t + 1/4, which is the perfect square (t + 1/2)² — that is, (3/(x + 6) + 1/2)². Substituting a single letter for the repeated chunk is what makes the pattern visible.
QUICK CHECK
Simplify the following expression. (a^5 − a³b² − a²b³ + b^5)/(a^5 − 3a⁴b + 2a³b² + 2a²b³ − 3ab⁴ + b^5) (SI 29 Jun 2023)
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Answer: (a² + ab + b²)/(a² − 2ab + b²)
Group the numerator in pairs: a⁵ − a³b² − a²b³ + b⁵ = a³(a² − b²) − b³(a² − b²) = (a² − b²)(a³ − b³) = (a + b)(a − b)²(a² + ab + b²).
The denominator is (a + b)(a − b)⁴ — expand (a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴ and multiply by (a + b) to check it reproduces every term. Cancelling (a + b)(a − b)² leaves (a² + ab + b²)/(a − b)², and since (a − b)² = a² − 2ab + b² the answer is (a² + ab + b²)/(a² − 2ab + b²).
Turning sentences into equations
The largest group: fourteen questions.
Multiply each equation so that one unknown has the same coefficient in both, then subtract. Read the last line twice: the paper often asks for a combination, not for a single unknown.
FOUR STEPS THAT NEVER CHANGE
Name the unknown
↓
One sentence, one equation
↓
Eliminate
↓
Answer what was asked
QUICK CHECK
The cost of 5 pens and 2 pencils is ₹29 and the cost of 7 pens and 5 pencils is ₹45. Find the cost of 1 pen and 2 pencils. (SI 16 Oct 2022)
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Answer: ₹ 9
Let a pen cost p and a pencil c. Then 5p + 2c = 29 and 7p + 5c = 45. Multiply the first by 5 and the second by 2 to match the c terms: 25p + 10c = 145 and 14p + 10c = 90. Subtracting, 11p = 55, so p = 5 and then c = 2. The question asks for 1 pen and 2 pencils = 5 + 4 = ₹9 — not for p alone.
QUICK CHECK
In a group of cows and hens, the number of legs are 14 more than twice the number of heads. The number of cows is (SI 2016)
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Answer: 7
Let there be c cows and h hens. Heads = c + h; legs = 4c + 2h. The condition is 4c + 2h = 2(c + h) + 14. Expanding, 4c + 2h = 2c + 2h + 14, so 2c = 14 and c = 7. The hens cancel out completely — the number of hens is never determined, and does not need to be.
QUICK CHECK
Roy is attempting a question paper where he gets 3 marks for every correct answer and 1 mark is deducted for every wrong answer. After attempting 60 questions, Roy sees that his total marks is 100. Find the number of questions he answered incorrectly. (SI 16 Oct 2022)
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Answer: 20
Let c be correct and w wrong, with c + w = 60 and 3c − w = 100. Substituting w = 60 − c gives 3c − 60 + c = 100, so 4c = 160 and c = 40. Therefore w = 20. A faster route: if all 60 were right he would score 180; each wrong answer costs 3 + 1 = 4 marks, and he is 80 short, so 80 ÷ 4 = 20 wrong.
QUICK CHECK
The sum of two numbers is 100. The difference of their squares is 6000. Find the larger number. (SI 9 Jul 2023)
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Answer: 80
Never square anything here. a + b = 100 and a² − b² = 6000. Since a² − b² = (a + b)(a − b), we get 100(a − b) = 6000, so a − b = 60. Adding the two facts: 2a = 160, so the larger number is 80 (and the smaller is 20).
QUICK CHECK
A water tank is 2/3 full. When 30 litres are drawn from it, it is just 1/3 full. Half the capacity of the tank (in litre) is : (SI 2 Jul 2023)
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Answer: 45
Drawing 30 litres takes the tank from 2/3 full to 1/3 full, so those 30 litres are exactly 2/3 − 1/3 = 1/3 of the capacity. The full capacity is 90 litres, and the question asks for half of it, which is 45. Read the last line of a tank question carefully — it rarely asks for the capacity itself.
QUICK CHECK
When 2 is added to the numerator of a fraction and 5 is added to its denominator, the fraction becomes 1/2. When 2 is subtracted from both the numerator and the denominator, it becomes 1/3. Find the fraction. (SI 14 Jul 2023)
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Answer: 3/5
Let the fraction be n/d. “2 added to the numerator, 5 to the denominator gives 1/2” means 2(n + 2) = d + 5, so d = 2n − 1. “2 subtracted from both gives 1/3” means 3(n − 2) = d − 2, so d = 3n − 4. Setting them equal, 2n − 1 = 3n − 4, so n = 3 and d = 5 — the fraction is 3/5. Check both conditions: 5/10 = 1/2 ✓ and 1/3 ✓.
QUICK CHECK
The equation 19 (x + y) + 17 = 19 (− x + y) − 21 is true for x is equal to : (SI 18 Aug 2021)
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Answer: − 1
Expand both sides: 19x + 19y + 17 = −19x + 19y − 21. The 19y cancels on both sides, leaving 19x + 17 = −19x − 21. So 38x = −38 and x = −1. y disappears entirely, which is the point of the question — the equation holds for every y.
QUICK CHECK
The sum of two times a number and three times 52 is 342. What will be the sum of four times that number and two times 52? (SI 20 Jun 2023)
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Answer: 476
“Two times a number and three times 52 is 342” means 2n + 156 = 342, so 2n = 186 and n = 93. The question then asks for 4n + 2 × 52 = 372 + 104 = 476. Notice you could also double the first expression: 2(2n + 156) = 684, then adjust the 52 part — but solving for n is safer under exam pressure.
QUICK CHECK
A bonus was distributed to 12 workers instead of 16 workers. If each worker received $443 more, then find the total amount. (SI 18 Aug 2021)
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Answer: $21264
Let the total be T. Shared among 12, each gets T/12; among 16, each gets T/16. The difference is 443: T/12 − T/16 = 443. The LCM of 12 and 16 is 48, so (4T − 3T)/48 = 443, that is T/48 = 443 and T = 443 × 48 = $21,264. The whole question is one fraction subtraction — the trap is dividing 443 by something.
QUICK CHECK
If the length of a rectangle is either (x + 3y) units or 13 units, and its breadth is either (3x + y) units or 7 units, then find the value of y. (SI 24 Aug 2021)
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Answer: 4
“Either … or” here means the two descriptions are the same measurement: x + 3y = 13 and 3x + y = 7. Multiply the second by 3: 9x + 3y = 21. Subtracting the first, 8x = 8, so x = 1 and then y = 4. Check: length 1 + 12 = 13 ✓, breadth 3 + 4 = 7 ✓.
CHECK YOURSELF
If (3/5) × (x ÷ 6/5 − 2 ÷ 3) 5 = 3y and y = 4(4 ÷ 2/5 × 2 − 2 ÷ 5/3) ÷ 4/5, then which of the following is a whole number? (SI 14 Jul 2023)
Show answer
Work out y first, strictly left to right inside the bracket. 4 ÷ 2/5 = 10, and 10 × 2 = 20. Next 2 ÷ 5/3 = 6/5 = 1.2, so the bracket is 20 − 1.2 = 18.8. Then 4 × 18.8 = 75.2, and 75.2 ÷ 4/5 = 75.2 × 5/4 = 94. So y = 94.
Now x. The left side is (3/5) × (5x/6 − 2/3) × 5 = 3(5x/6 − 2/3), and this equals 3y = 282. So 5x/6 − 2/3 = 94, giving 5x/6 = 284/3 and x = 568/5 = 113.6.
Test the options: x + (y + 3)/5 = 113.6 + 97/5 = 113.6 + 19.4 = 133, a whole number. The other three give 359.6, 132.8 and 132.6.
Quadratics without the formula
Sum and product beat substitution every time.
For ax² + bx + c = 0 the roots add to −b/a and multiply to c/a. Most SI questions need only these two numbers, never the roots themselves.
QUICK CHECK
If q > 0, and the roots of x² − 17x + q = 0 are integers, then how many values of 'q' can be integers? (SI 19 Aug 2021)
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Answer: 8
The roots add to 17 and multiply to q. Since q > 0 and the roots are integers, both roots must be positive (two negatives could not add to +17, and a mixed pair would give q < 0). List the positive integer pairs adding to 17: (1, 16), (2, 15), (3, 14), (4, 13), (5, 12), (6, 11), (7, 10), (8, 9) — the next pair repeats, so there are 8 pairs and 8 values of q: 16, 30, 42, 52, 60, 66, 70, 72.
QUICK CHECK
Find the roots of the equation x² + 2x − 6 = 0. (SI 19 Aug 2021)
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Answer: √7 − 1, −√7 − 1
Complete the square rather than reaching for the formula: x² + 2x − 6 = 0 becomes (x + 1)² = 7, so x + 1 = ±√7 and x = −1 ± √7. The two roots are √7 − 1 and −√7 − 1. Sanity check with the sum: −b/a = −2, and (√7 − 1) + (−√7 − 1) = −2 ✓.
WHAT THE DISCRIMINANT TELLS YOU
D = b² − 4ac. If D > 0 there are two distinct real roots; if D = 0 the two roots are equal; if D < 0 there is no real root. If D is also a perfect square and a, b, c are integers, the roots are rational — which is exactly the test to apply when a question says “the roots are integers”.
Ages
Four questions, one table.
Give the present ages letters, then add or subtract the same number from every entry in a column. The condition in the question is always one row read across.
QUICK CHECK
A man is four times as old as his son. After 6 years, the man's age will be three times that of his son. The sum of their present ages is : (SI 31 Jul 2023)
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Answer: 60 years
Let the son be x now, so the man is 4x. After 6 years they are x + 6 and 4x + 6, and the condition is 4x + 6 = 3(x + 6). That gives 4x + 6 = 3x + 18, so x = 12. The son is 12, the man is 48, and the sum asked for is 60. Note the question asks for the sum, not for either age.
QUICK CHECK
Baarish is 8 years older than Radha. In five years, the sum of their ages will be 82. What are their present ages? (SI 14 Jul 2023)
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Answer: 32, 40
In five years each of them is 5 years older, so the pair gains 10 years in total. Their present ages therefore add to 82 − 10 = 72. If Radha is r, Baarish is r + 8, so 2r + 8 = 72 and r = 32. The ages are 32 and 40. Converting a future sum to a present sum first saves a whole equation.
QUICK CHECK
In 10 years, A will be twice as old as B was 10 years ago. If A is now 9 years older than B, the present age of B is ____. (SI 29 Jul 2023)
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Answer: 39 years
Two moments, so two shifts. If B is now b, then A is b + 9. Ten years from now A is b + 19; ten years ago B was b − 10. The condition b + 19 = 2(b − 10) gives b + 19 = 2b − 20, so b = 39. The trap is comparing both ages at the same moment — here they are ten years apart in opposite directions.
QUICK CHECK
A man's age is four times the sum of the ages of his two daughters. After six years, his age will be three times the sum of his daughters' ages. Find the man's present age. (SI 15 Jul 2023)
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Answer: 120 years
Let the daughters' ages add to s, so the man is 4s. After six years the man is 4s + 6, and the daughters' total grows by 6 + 6 = 12, becoming s + 12. The condition 4s + 6 = 3(s + 12) gives s = 30, so the man is 4 × 30 = 120 years. The step most candidates miss: with two daughters the sum rises by 12, not by 6.
THE SIX-PERSON AGE PUZZLE, ASKED THREE TIMES
The same puzzle about X, A, B, C, D and E appears in three different SI papers, each time asking for a different person. Solve it once, in this order:
X = √25 = 5 · C = 12X = 60 · A = 10% of C = 6 · B = X + 2.8 = 7.8 · D × A = 10B ⇒ D = 78/6 = 13 · E: D + 2 = 2E ⇒ E = 7.5
Always start from the one sentence that gives a bare number — here “X is the square root of 25” — and chain outwards. In the 15 July 2022 version B is defined as 30% more than A instead, which changes B but not C, so the answer to “What is C's age?” is still 60.
CHECK YOURSELF
There are six persons X, A, B, C, D and E. A's age is 10% of C's age. After 2 years, D's age will be 2 times E's age. Also, the product of D's age and A's age is 10 times B's age, and C's age is 12 times X's age. B's age is 2.8 years more than X's age, which is the square root of 25. What is A's age? (SI 20 Jun 2023)
Show answer
X = √25 = 5 and C = 12X = 60, so A = 10% of 60 = 6. The rest of the data (B = 7.8, D = 13) is not needed for this one.
CHECK YOURSELF
There are six persons X, A, B, C, D and E. A's age is 10% of C's age. After 2 years, D's age will be 2 times E's age. Also, the product of D's age and A's age is 10 times B's age, and C's age is 12 times X's age. B's age is 2.8 years more than X's age, which is the square root of 25. What is D's age? (SI 29 Jul 2023)
Show answer
Same chain: X = 5, C = 60, A = 6, B = X + 2.8 = 7.8. Then D × A = 10B gives 6D = 78, so D = 13.
CHECK YOURSELF
There are six people X, A, B, C, D, E. B’s age is 30% more than A’s age, which is 10% of C’s age. D’s age 2 years later will be 2 times the age of E. Also, the product of D’s age and A’s age is 10 times that B’s age and C’s age is 12 times that of X’s age. E’s age is 1.5 times that of X’s age, which is the square root of 25. What is C’s age? (SI 15 Jul 2023)
Show answer
This version defines B as 30% more than A, but the question only asks for C. X = √25 = 5 and C = 12X = 60 — one line, and the rest of the puzzle can be ignored. Read what is asked before you solve anything.
Variation and the one logarithm question
Small, but they come back.
KEY TERM
p varies inversely as q² means pq² = k
Inverse variation is a constant product, not a constant ratio. Find k from the pair you are given, then put the new value in. Direct variation is the mirror image: y = kx, a constant ratio. The SI paper has asked the inverse-square version twice with different numbers, so learn the two-step drill — find k, then substitute — rather than the answers.
p ∝ 1/q²→pq² = k→k from the given pair
QUICK CHECK
Given that q = 8 for p = 2, p varies inversely as the square of q. Find the value of p when q = 16. (SI 29 Jul 2023)
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Answer: 1/2
pq² = k. From p = 2 when q = 8: k = 2 × 64 = 128. At q = 16, p = 128 ÷ 256 = 1/2. Shortcut worth seeing: q doubled, so q² quadrupled, so p fell to a quarter — 2 ÷ 4 = 1/2.
CHECK YOURSELF
Given that q = 4 for p = 1 and p varies inversely as square of q. Which of the following options is equal to the value of p for q = 8? (SI 31 Jul 2023)
Show answer
Same drill: k = pq² = 1 × 16 = 16. At q = 8, p = 16 ÷ 64 = 1/4. Again q doubled and p fell to a quarter.
QUICK CHECK
If x² + y² = z², then 1/(log_{(z+x)} y) + 1/(log_{(z−x)} y) = ? (SI 29 Jun 2023)
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Answer: 2
Use the base-change rule 1/log_b a = log_a b. The expression becomes log_y(z + x) + log_y(z − x) = log_y[(z + x)(z − x)] = log_y(z² − x²).
But x² + y² = z², so z² − x² = y², and log_y(y²) = 2. Two logarithm facts carry this question: the reciprocal flips the base, and a sum of logs is the log of a product.
COMMON TRAPS
Answering x when the question asked for x + y, for a sum of ages, or for half the capacity. Underline the last line of the stem before you start.
Squaring surds or decimals one at a time. If you see a conjugate pair or a difference of squares, use the identity — it is always faster and always exact.
Forgetting that a loose decimal in a denominator is usually ab. 0.651 is 0.93 × 0.7.
In an ages question with two children, adding 6 to the total instead of 12. Each person ages separately.
Treating inverse variation as a ratio. pq² = k is a product; doubling q divides p by four, not by two.
QUICK BYTES
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
a² + b² = (a + b)² − 2ab
Roots: sum = −b/a, product = c/a
D = b² − 4ac decides how many real roots
Remainder of p(x) ÷ (x + 1) is p(−1)
Inverse variation: pqⁿ = k
1/log_b a = log_a b
REMEMBER
Same outside, opposite inside. a³ + b³ keeps the plus outside, (a + b), and flips to −ab in the middle. a³ − b³ keeps the minus outside, (a − b), and flips to +ab. Say it once before the exam and you will never mix up the four cube terms.
