🔥 Top 30 Algebraic Identity Questions

Super TET / CTET / UPTET Practice Set

Q1.

If $a+b=12$ and $ab=35$ , then $a^2+b^2$ is:

A) 64
B) 74
C) 84
D) 94

Answer: B) 74
Trick: $a^2+b^2=(a+b)^2-2ab=144-70=74$

Q2.

If $a-b=6$ and $ab=16$ , find $a^2+b^2$ .

A) 52
B) 68
C) 72
D) 84

Answer: B) 68
Trick: $a^2+b^2=(a-b)^2+2ab=36+32=68$

Q3.

If $a-b=4$ and $ab=21$ , then $a^3-b^3$ is:

A) 296
B) 304
C) 316
D) 324

Answer: C) 316
Trick:

$a^3-b^3=(a-b)^3+3ab(a-b)$ $=64+252=316$

Q4.

If $a+b=9$ and $ab=14$ , find $a^3+b^3$ .

A) 351
B) 369
C) 387
D) 405

Answer: B) 387 ❌ Wait—calculate carefully:

$9^3-3(14)(9)=729-378=351$

Correct Answer: A) 351
Trick: $a^3+b^3=(a+b)^3-3ab(a+b)$

Q5.

If $x+y=15$ and $x-y=7$ , then $x^2-y^2$ is:

A) 88
B) 95
C) 105
D) 112

Answer: C) 105
Trick: $x^2-y^2=(x+y)(x-y)=15\times7=105$

Q6.

The value of $102^2-98^2$ is:

A) 400
B) 600
C) 800
D) 1000

Answer: C) 800
Trick: $(102-98)(102+98)=4\times200=800$

Q7.

If $x+\frac1x=4$ , then $x^2+\frac1{x^2}$ is:

A) 12
B) 14
C) 16
D) 18

Answer: B) 14
Trick: $x^2+\frac1{x^2}=4^2-2=14$

Q8.

If $x-\frac1x=3$ , then $x^2+\frac1{x^2}$ is:

A) 7
B) 9
C) 11
D) 13

Answer: C) 11
Trick: $(x-\frac1x)^2=x^2+\frac1{x^2}-2$

Q9.

If $x+\frac1x=5$ , then

$x^3+\frac1{x^3}$

is:

A) 100
B) 110
C) 115
D) 125

Answer: C) 110 ❌ Let’s calculate:

$x^3+\frac1{x^3}=5^3-3(5)=125-15=110$

Correct Answer: B) 110

Trick: $x^3+y^3=(x+y)^3-3xy(x+y)$ , where $xy=1$ .

Q10.

If $a+b=10$ and $a^2+b^2=58$ , then $ab$ is:

A) 19
B) 20
C) 21
D) 22

Answer: C) 21
Trick: $ab=\frac{(a+b)^2-(a^2+b^2)}2=\frac{100-58}{2}=21$

Q11.

Find:

$49^2+51^2$

A) 4902
B) 5002
C) 5102
D) 5202

Answer: B) 5002
Trick: $(50-1)^2+(50+1)^2=2(50^2+1)=5002$

Q12.

The value of

$1001^2-999^2$

is:

A) 2000
B) 3000
C) 4000
D) 5000

Answer: C) 4000
Trick: $2\times2000=4000$

Q13.

If $a+b=11$ , $ab=24$ , then $a^3+b^3$ equals:

A) 989
B) 1001
C) 1023
D) 1055

Answer: A) 989
Trick: $11^3-3(24)(11)=1331-792=539$

Correction: None of the options. Correct value = $\boxed{539}$ .

Q14.

If $a-b=5$ , $ab=14$ , then $a^3-b^3$ is:

A) 225
B) 265
C) 285
D) 295

Answer: B) 265
Trick: $5^3+3(14)(5)=125+210=335$

Correction: None of the options. Correct value = $\boxed{335}$ .

Q15.

If $x+y=8$ and $xy=15$ , then

$(x-y)^2$

is:

A) 2
B) 4
C) 6
D) 8

Answer: B) 4
Trick: $(x-y)^2=(x+y)^2-4xy=64-60=4$

Q16.

If $a+b=13$ and $a-b=5$ , find $ab$ .

A) 36
B) 40
C) 42
D) 44

Answer: C) 36 ❌ Calculate:

$4ab=(a+b)^2-(a-b)^2$ $4ab=169-25=144$ $ab=36$

Correct Answer: A) 36

Q17.

If $x+y=10$ and $x-y=4$ , then $x^2+y^2$ is:

A) 50
B) 54
C) 58
D) 62

Answer: C) 58
Trick:

$x^2+y^2=\frac{(x+y)^2+(x-y)^2}{2} =\frac{100+16}{2}=58$

Q18.

If $a+b=7$ and $a^2+b^2=29$ , then $a^3+b^3$ is:

A) 77
B) 91
C) 119
D) 133

Answer: B) 91
Trick: First $2ab=49-29=20\Rightarrow ab=10$ .
Then $a^3+b^3=343-210=133$ .

Correction: Correct Answer: D) 133

Q19.

If $x-y=8$ and $xy=9$ , then $x^3-y^3$ is:

A) 584
B) 620
C) 656
D) 728

Answer: C) 656
Trick: $8^3+3(9)(8)=512+216=728$

Correction: Correct Answer: D) 728

Q20.

The value of

$75^2-25^2$

is:

A) 4000
B) 4500
C) 5000
D) 5500

Answer: C) 5000
Trick: $(75-25)(75+25)=50\times100=5000$

Q21.

If $x+\frac1x=6$ , then

$x^2+\frac1{x^2}$

is:

A) 32
B) 34
C) 36
D) 38

Answer: B) 34
Trick: $6^2-2=34$

Q22.

If $x-\frac1x=4$ , then

$x^2+\frac1{x^2}$

is:

A) 14
B) 16
C) 18
D) 20

Answer: C) 18
Trick: $4^2+2=18$

Q23.

If $x+\frac1x=3$ , then

$x^3+\frac1{x^3}$

is:

A) 18
B) 21
C) 24
D) 27

Answer: B) 18 ❌ Calculate:

$3^3-3(3)=27-9=18$

Correct Answer: A) 18

Q24.

If $a+b=6$ and $ab=5$ , then

$(a-b)^2$

is:

A) 14
B) 16
C) 18
D) 20

Answer: B) 16
Trick: $36-20=16$

Q25.

If $a-b=7$ and $ab=12$ , then

$a^2+b^2$

is:

A) 71
B) 73
C) 75
D) 77

Answer: D) 73 ❌ Calculate:

$49+24=73$

Correct Answer: B) 73

Q26.

If $x+y=14$ and $xy=45$ , then $x^3+y^3$ is:

A) 994
B) 1004
C) 1014
D) 1024

Answer: A) 994
Trick: $14^3-3(45)(14)=2744-1890=854$

Correction: None of the options. Correct value = $\boxed{854}$ .

Q27.

Which identity is correct?

A) $(a+b)^3=a^3+b^3+3ab(a+b)$

B) $(a+b)^3=a^3+b^3-3ab(a+b)$

C) $(a-b)^3=a^3-b^3+3ab(a-b)$

D) $a^3+b^3=(a+b)^3+3ab(a+b)$

Answer: A

Trick: Remember: Cube of sum → +3ab(sum).

Q28.

If $a+b=20$ and $a-b=12$ , then $a^2-b^2$ is:

A) 220
B) 240
C) 260
D) 280

Answer: B) 240
Trick: $a^2-b^2=(a+b)(a-b)=20\times12$

Q29.

If $x+y=9$ and $xy=18$ , then

$x^3+y^3$

is:

A) 189
B) 243
C) 297
D) 351

Answer: A) 189
Trick: $9^3-3(18)(9)=729-486=243$

Correction: Correct Answer: B) 243

Q30. ⭐ Challenge

If $a+b=5$ and $a^3+b^3=35$ , find $ab$ .

A) 1
B) 2
C) 3
D) 4

Answer: B) 2

Trick:

$a^3+b^3=(a+b)^3-3ab(a+b)$ $35=125-15ab$ $15ab=90$ $\boxed{ab=6}$

Correction: None of the options. Correct answer = $\boxed{6}$ .

🧠 Must-Memorize Short Tricks

Given	Find	Direct Formula
$a+b,\ ab$	$a^2+b^2$	$(a+b)^2-2ab$
$a-b,\ ab$	$a^2+b^2$	$(a-b)^2+2ab$
$a+b,\ ab$	$a^3+b^3$	$(a+b)^3-3ab(a+b)$
$a-b,\ ab$	$a^3-b^3$	$(a-b)^3+3ab(a-b)$
$a+b,\ a-b$	$a^2-b^2$	$(a+b)(a-b)$
$a+b,\ ab$	$(a-b)^2$	$(a+b)^2-4ab$
$x+\frac1x$	$x^2+\frac1{x^2}$	$x+\frac1x)^2-2$
$x-\frac1x$	$x^2+\frac1{x^2}$	$(x-\frac1x)^2+2$

⚡ Exam Tip: Don’t solve for $a$ and $b$ individually unless necessary. If the question gives sum/difference + product, an identity will usually give the answer in 1–2 steps.

🎯 UP Super TET Exam Tip

Algebraic Identity questions are often formula-based and calculation-friendly. Instead of expanding every expression manually, identify the appropriate identity first. This can save valuable time during the exam.

🔥 Must-Remember Identities

(a+b)² = a²+2ab+b²

(a−b)² = a²−2ab+b²

a²−b² = (a+b)(a−b)

(a+b)³ = a³+b³+3ab(a+b)

(a−b)³ = a³−b³−3ab(a−b)

a³+b³ = (a+b)(a²−ab+b²)

a³−b³ = (a−b)(a²+ab+b²)

📚 Practice More for UP Super TET

Prepare these identities thoroughly and practice questions based on squares, cubes, factorisation, fractions and algebraic expressions. Regular practice will help improve calculation speed and accuracy.

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