Identities MCQ Quiz - Objective Question with Answer for Identities - Download Free PDF
Last updated on Jul 17, 2025
Latest Identities MCQ Objective Questions
Identities Question 1:
Simplify the following.
Answer (Detailed Solution Below)
Identities Question 1 Detailed Solution
Given:
Formula used:
a3 + b3 = (a + b)(a2 - ab + b2)
Calculation:
⇒ (0.013 + 0.0033)/(5 × 0.01 × 5 × 0.01 - 5 × 0.01 × 5 × 0.003 + 5 × 0.003 × 5 × 0.003)
⇒ (0.013 + 0.0033)/(25 × 0.01 × 0.01 - 25 × 0.01 × 0.003 + 25 × 0.003 × 0.003)
⇒ (0.013 + 0.0033)/25(0.012 - 0.01 × 0.003 + 0.0032)
⇒ (0.01 + 0.003)/25
⇒ 13/25 × 10- 3
∴ The value is 13/25 × 10-3
Identities Question 2:
If
Answer (Detailed Solution Below)
Identities Question 2 Detailed Solution
Concept used:
If
Calculation:
Now,
⇒
⇒
⇒ 6 × (-√32)
⇒ - 24√2
∴ The correct answer is - 24√2
Mistake Points Please note that
if 0 < x < 1
then
1/x > 1
so, 1/x4 > 1
and 0 < x4 < 1
so,
x4 - 1/x4 < 0
so the answer will be negative.
Identities Question 3:
The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.
Answer (Detailed Solution Below)
Identities Question 3 Detailed Solution
Given:
The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144.
Calculation:
Let the numbers are x and y
(x - y)2 = 324
So, x - y = 18, xy = 144
(x + y)2 = (18)2 + 4× 144
⇒ 900
⇒ x + y = 30
Then, x is (30 + 18) / 2 = 24 and y = 6
So , x2 - y2 = 242 - 62
⇒ 576 - 36 = 540
∴ The correct option is 2
Identities Question 4:
Solve:
Answer (Detailed Solution Below)
Identities Question 4 Detailed Solution
Given:
Solve: (0.1 × 0.1 × 0.1+0.02 × 0.02 × 0.02)/(0.2 × 0.2 × 0.2+0.04 × 0.04 × 0.04)
Formula Used:
Basic arithmetic operations and exponentiation.
Calculation:
(0.1 × 0.1 × 0.1 + 0.02 × 0.02 × 0.02) / (0.2 × 0.2 × 0.2 + 0.04 × 0.04 × 0.04)
⇒ (0.13 + 0.023) / (0.23 + 0.043)
⇒ (0.001 + 0.000008) / (0.008 + 0.000064)
⇒ 0.001008 / 0.008064
⇒ 0.125
The correct answer is option 2.
Identities Question 5:
Answer (Detailed Solution Below)
Identities Question 5 Detailed Solution
Given:
If x + (1/x) = 4, find the value of x3 + (1/x3).
Formula used:
(x + (1/x))3 = x3 + (1/x3) + 3(x + (1/x))
Calculation:
x + (1/x) = 4
⇒ (x + (1/x))3 = 43
⇒ x3 + (1/x3) + 3(x + (1/x)) = 64
⇒ x3 + (1/x3) + 3 × 4 = 64
⇒ x3 + (1/x3) + 12 = 64
⇒ x3 + (1/x3) = 64 - 12
⇒ x3 + (1/x3) = 52
∴ The correct answer is option (2).
Top Identities MCQ Objective Questions
If x −
Answer (Detailed Solution Below)
Identities Question 6 Detailed Solution
Download Solution PDFGiven:
x - 1/x = 3
Concept used:
a3 - b3 = (a - b)3 + 3ab(a - b)
Calculation:
x3 - 1/x3 = (x - 1/x)3 + 3 × x × 1/x × (x - 1/x)
⇒ (x - 1/x)3 + 3(x - 1/x)
⇒ (3)3 + 3 × (3)
⇒ 27 + 9 = 36
∴ The value of x3 - 1/x3 is 36.
Alternate Method If x - 1/x = a, then x3 - 1/x3 = a3 + 3a
Here a = 3
x - 1/x3 = 33 + 3 × 3
= 27 + 9
= 36
If
Answer (Detailed Solution Below)
Identities Question 7 Detailed Solution
Download Solution PDFGiven:
x - (1/x) = (- 6)
Formula used:
If x - (1/x) = P, then
x + (1/x) = √(P2 + 4)
If x + (1/x) = P, then
x3 + (1/x3) = (P3 - 3P)
x5 - (1/x5) = {x3 + (1/x3)} × {x2 - 1/x2} + {x - (1/x)}
Calculation:
x - (1/x) = (- 6)
x + (1/x) = √{(- 6)2 + 4} = √40 = 2√10
So, x2 - 1/x2 = (x + 1/x) (x - 1/x) = 2√10 × (-6) = -12√10
and x3 + (1/x3) = (√40)3 - 3√40
⇒ 40√40 - 3√40 = 37 × 2√10 = 74√10
Now,
x5 - (1/x5) = {x3 + (1/x3)} × {x2 - 1/x2} + {x - (1/x)}
⇒ {74√10 × (-12√10)} + (- 6)
⇒ - 74 × 12 × (√10 × √10) - 6
⇒ (- 8880) - 6 = - 8886
∴ The correct answer is - 8886.
If p – 1/p = √7, then find the value of p3 – 1/p3.
Answer (Detailed Solution Below)
Identities Question 8 Detailed Solution
Download Solution PDFGiven:
p – 1/p = √7
Formula:
P3 – 1/p3 = (p – 1/p)3 + 3(p – 1/p)
Calculation:
P3 – 1/p3 = (p – 1/p)3 + 3 (p – 1/p)
⇒ p3 – 1/p3 = (√7)3 + 3√7
⇒ p3 – 1/p3 = 7√7 + 3√7
⇒ p3 – 1/p3 = 10√7
Shortcut Trick x - 1/x = a, then x3 - 1/x3 = a3 + 3a
Here, a = √7 ( put the value in required eqn )
⇒p3 – 1/p3 = (√7)3 + 3 × √7 = 7√7 + 3√7
⇒p3 – 1/p3 = 10√7.
Hence; option 4) is correct.
If x = √10 + 3 then find the value of
Answer (Detailed Solution Below)
Identities Question 9 Detailed Solution
Download Solution PDFGiven:
x = √10 + 3
Formula used:
a2 - b2 = (a + b)(a - b)
a3 - b3 = (a - b)(a2 + ab + b2)
Calculation:
⇒ 1/x = √10 - 3
Squaring both side of (1),
∴ The required value is 234.
Shortcut TrickGiven:
x = √10 + 3
Formula used:
⇒
Calculation:
x = √10 + 3
⇒ 1/x = √10 - 3
⇒
⇒
⇒
∴ The required value is 234.
If a + b + c = 14, ab + bc + ca = 47 and abc = 15 then find the value of a3 + b3 +c3.
Answer (Detailed Solution Below)
Identities Question 10 Detailed Solution
Download Solution PDFGiven:
a + b + c = 14, ab + bc + ca = 47 and abc = 15
Concept used:
a³ + b³ + c³ - 3abc = (a + b + c) × [(a + b + c)² - 3(ab + bc + ca)]
Calculations:
a³ + b³ + c³ - 3abc = 14 × [(14)² - 3 × 47]
⇒ a³ + b³ + c³ – 3 × 15 = 14(196 – 141)
⇒ a³ + b³ + c³ = 14(55) + 45
⇒ 770 + 45
⇒ 815
∴ The correct choice is option 1.
If
Answer (Detailed Solution Below)
Identities Question 11 Detailed Solution
Download Solution PDFGiven:
Formula used:
(a + 1/a) = P ; then
(a2 + 1/a2) = P2 - 2
(a3 + 1/a3) = P3 - 3P
Calculation:
a + (1/a) = 7
⇒ (a2 + 1/a2) = (7)2 - 2 = 49 - 2 = 47
⇒ (a3 + 1/a3) = (7)3 - (3 × 7) = 343 - 21 = 322
a5 + (1/a5) = (a2 + 1/a2) × (a3 + 1/a3) - (a + 1/a)
⇒ 47 × 322 - 7
⇒ 15134 - 7 = 15127
∴ The correct answer is 15127.
The sum of values of x satisfying x2/3 + x1/3 = 2 is:
Answer (Detailed Solution Below)
Identities Question 12 Detailed Solution
Download Solution PDFFormula used:
(a + b)3 = a3 + b3 + 3ab(a + b)
Calculation:
⇒ x2/3 + x1/3 = 2
⇒ (x2/3 + x1/3)3 = 23
⇒ x2 + x + 3x(x2/3 + x1/3) = 8
⇒ x2 + 7x - 8 = 0
⇒ x2 + 8x - x - 8 = 0
⇒ x (x + 8) - 1 (x + 8) = 0
⇒ x = - 8 or x = 1
∴ Sum of values of x = -8 + 1 = - 7.If (a + b + c) = 19 and (a2 + b2 + c2) = 155, find the value of (a - b)2 + (b - c)2 + (c - a)2.
Answer (Detailed Solution Below)
Identities Question 13 Detailed Solution
Download Solution PDFGiven:
(a + b + c) = 19
(a2 + b2 + c2) = 155
Formula used:
a2 + b2 + c2 - (ab + bc + ca) = (1/2) × [(a - b)2 + (b - c)2 + (c - a)2]
Calculation:
a + b + c = 19
Squaring both sides
⇒ (a + b + c)2 = (19)2
⇒ a2 + b2 + c2 + 2 × (ab + bc + ca) = 361
⇒ 155 + 2 × (ab + bc + ca) = 361
⇒ 2 × (ab + bc + ca) = (361 - 155)
⇒ (ab + bc + ca) = 206/2 = 103
Now,
a2 + b2 + c2 - (ab + bc + ca) = (1/2) × [(a - b)2 + (b - c)2 + (c - a)2]
⇒ 2 × (155 - 103) = (a - b)2 + (b - c)2 + (c - a)2
⇒ (a - b)2 + (b - c)2 + (c - a)2 = 104
∴ The correct answer is 104.
If a + b + c = 0, then (a3 + b3 + c3)2 = ?
Answer (Detailed Solution Below)
Identities Question 14 Detailed Solution
Download Solution PDFFormula used:
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
Calculation:
a + b + c = 0
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
⇒ a3 + b3 + c3 - 3abc = 0 × (a2 + b2 + c2 - ab - bc - ca) = 0
⇒ a3 + b3 + c3 - 3abc = 0
⇒ a3 + b3 + c3 = 3abc
Now, (a3 + b3 + c3)2 = (3abc)2 = 9a2b2c2
If
Answer (Detailed Solution Below)
Identities Question 15 Detailed Solution
Download Solution PDFGiven:
x2 + (1/x2) = 7
Formula used:
x2 + (1/x2) = P
then x + (1/x) = √(P + 2)
and x - (1/x) = √(P - 2)
⇒ x2 - (1/x2) = {x + (1/x)} × {x - (1/x)}
Calculation:
x2 + (1/x2) = 7
⇒ x + (1/x) = √(7 + 2) = √9
⇒ x + (1/x) = 3
⇒ x - (1/x) = -√(7 - 2)
⇒ x - (1/x) = - √5 {0 < x < 1}
x2 - (1/x2) = {x + (1/x)} × {x - (1/x)}
⇒ 3 × (- √5)
∴ The correct answer is - 3√5.
Mistake Points
Please note that
0 < x < 1
so
1/x > 1
so
x + 1/x > 1
and
x - 1/x < 0 (because 0 < x < 1 and 1/x > 1 so x - 1/x < 0)
so
(x - 1/x)(x + 1/x) < 0