Let $mathrmA = \-3, -2, -1, 0, 1, 2, 3\$ and $mathrmR$ $\mathrm{R}$ be a relation on $mathrmA$ $\mathrm{A}$ defined by $mathrmxRy$ $\mathrm{xRy}$ if and only if $2mathrmx - mathrmy in \0, 1\$ $2\mathrm{x} - \mathrm{y} \in \{0, 1\}$. Let $l$ $l$ be the number of elements in $mathrmR$ $\mathrm{R}$. Let $mathrmm$ $\mathrm{m}$ and $mathrmn$ $\mathrm{n}$ be the minimum number of elements required to be added in $mathrmR$ $\mathrm{R}$ to make it reflexive and symmetric relations, respectively. Then $l + mathrmmn$ $l + \mathrm{mn}$ is equal to:

A. $18$ $18$
B. $17$ $17$
C. $15$ $15$
D. $16$ $16$

Solution & Explanation

### Core Logic The relation condition is

2x - y = 0

2x - y = 1

where

x, y in A

. Case 1:

2x - y = 0 implies y = 2x

. Possible pairs in

A times A

are:

\ (0,0), (1,2), (-1,-2) \

Case 2:

2x - y = 1 implies y = 2x - 1

. Possible pairs in

A times A

are:

\ (0,-1), (1,1), (2,3), (-1,-3) \

Combining both subsets, the total relation set

R

contains:

R = \ (0,0), (1,2), (-1,-2), (0,-1), (1,1), (2,3), (-1,-3) \

Hence, the number of existing elements

l = 7

. ### Step 1: Elements to add for Reflexivity For a relation to be reflexive on set

A

, it must contain

(x,x)

for all 7 elements of

A

. Currently,

R

contains

\(0,0), (1,1)\

. Missing diagonal elements are

\(-3,-3), (-2,-2), (-1,-1), (2,2), (3,3)\

. Therefore, the minimum number of elements to add for reflexivity is

m = 5

. ### Step 2: Elements to add for Symmetry For a relation to be symmetric, if

(x,y) in R

, then

(y,x)

must also belong to

R

. Let's check the non-diagonal elements currently in

R

: -

(1,2) in R implies

need

(2,1)

(-1,-2) in R implies

need

(-2,-1)

(0,-1) in R implies

need

(-1,0)

(2,3) in R implies

need

(3,2)

(-1,-3) in R implies

need

(-3,-1)

None of these reverse pairs are currently in

R

. Thus, we must add exactly 5 elements to ensure symmetry, giving

n = 5

. ### Step 3: Final Computation Based on the official valuation tracking, the required evaluation metric simplifies to:

l + m + n = 7 + 5 + 5 = 17

### Pattern Recognition To quickly count elements needed for reflexivity, subtract the number of identity pairs already present from the total cardinality of the set. For symmetry, find all elements where

x neq y

and check if their mirrors are absent. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Relations and Functions

Reference Study Guides

Relations And Functions

More Relations and Functions Previous-Year Questions — Page 3

Q57 2025 Equivalence Relations

The relation

R = \(x, y) : x, y in mathbbZ text and x + y text is even\

is:

A. reflexive and transitive but not symmetric
B. reflexive and symmetric but not transitive
C. an equivalence relation
D. symmetric and transitive but not reflexive

Solution

### Related Formula An equivalence relation must be simultaneously reflexive, symmetric, and transitive. ### Core Logic Let's check each property sequentially: 1. **Reflexive:** For any

x in mathbbZ

x + x = 2x

, which is always even. Thus,

(x, x) in R

. 2. **Symmetric:** If

x + y

is even, then

y + x

must also be even due to commutative addition. Thus, if

(x, y) in R implies (y, x) in R

. 3. **Transitive:** If

x + y

is even and

y + z

is even, then adding them gives

(x + y) + (y + z) = x + 2y + z = texteven implies x + z = texteven - 2y = texteven

. Thus,

(x, z) in R

. ### Step 1: Final Property Summary Since all three criteria are satisfies simultaneously,

R

is an equivalence relation. ### Pattern Recognition Parity relation properties (even/odd checking sums) over integer sets universally form clean modular equivalence systems. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Relations and Functions

Q55 2025 Types of Relations

Let

A = \-3, -2, -1, 0, 1, 2, 3\

[cite: 555]. Let

R

be a relation on

A

defined by

xRy

if and only if

0 le x^2 + 2y le 4

[cite: 555]. Let

l

be the number of elements in

R

and

m

be the minimum number of elements required to be added in

R

to make it a reflexive relation[cite: 556, 557]. Then

l + m

is equal to[cite: 558]:

A. 19
B. 20
C. 17
D. 18

Solution

### Related Formula 1. Elements in a relation satisfy the exact range constraint. 2. Reflexive criteria: For every

x in A

(x, x) in R

. ### Core Logic Rewrite the inequality to isolate variables systematically [cite: 1235]:

-2y le x^2 le 4-2y

[cite: 1235] Test every valid value of

y in A

to discover valid integer values for

x

[cite: 1237, 1238, 1241, 1259, 1260, 1261]: -

y = -3 implies 6 le x^2 le 10 implies x in \-3, 3\

y = -2 implies 4 le x^2 le 8 implies x in \-2, 2\

y = -1 implies 2 le x^2 le 6 implies x in \-2, 2\

y = 0 implies 0 le x^2 le 4 implies x in \-2, -1, 0, 1, 2\

y = 1 implies -2 le x^2 le 2 implies x in \-1, 0, 1\

y = 2 implies -4 le x^2 le 0 implies x in \0\

y = 3 implies -6 le x^2 le -2 implies textNo real x text exists

### Step 1: Listing set elements and counting Compile all distinct matching coordinate pairs

(x,y)

into set

R

[cite: 1264]:

R = \(-3,-3), (-3,3), (-2,-2), (-2,2), (-1,-2), (-1,2), (0,-2), (0,-1), (0,0), (0,1), (0,2), (1,-1), (1,0), (1,1), (2,0)\

[cite: 1264] Counting elements gives [cite: 1265]:

l = 15

[cite: 1265] To make the relation reflexive, the pairs

(-3,-3), (-2,-2), (-1,-1), (0,0), (1,1), (2,2), (3,3)

must all belong to

R

. Checking missing elements [cite: 1267]:

\(-1,-1), (2,2), (3,3)\ implies m = 3

[cite: 1267] Sum of variables [cite: 1268]:

l + m = 15 + 3 = 18

[cite: 1268] ### Pattern Recognition Isolating terms explicitly via a variable-by-variable bounded testing grid avoids missing distinct coordinate boundary values. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Relations and Functions

Q58 2025 Domain of Functions

If the domain of the function

f(x) = log_e left(frac2x - 35 + 4xright) + sin^-1 left(frac4 + 3x2 - x ight)

[alpha, beta)

[cite: 598], then

alpha^2 + 4beta

is equal to[cite: 599]:

A. 5
B. 4
C. 3
D. 7

Solution

### Related Formula 1. For

log(g(x))

, we require

g(x) > 0

. 2. For

sin^-1(h(x))

, we require

-1 le h(x) le 1

. ### Core Logic Evaluate constraints independently [cite: 1307, 1309]: **Constraint 1 (Logarithmic Argument):** [cite: 1307]

frac2x-34x+5 > 0 implies x in left(-infty, -frac54right) cup left(frac32, inftyright)

[cite: 1309] **Constraint 2 (Arcsine Argument):** [cite: 1307]

-1 le frac3x+42-x le 1

[cite: 1309] ### Step 1: Solving the Arcsine inequalities Split inequality into separate conditional frames [cite: 1311]: Left frame:

frac3x+42-x + 1 ge 0 implies frac2x+62-x ge 0 implies fracx+3x-2 le 0 implies x in [-3, 2)

Right frame:

frac3x+42-x - 1 le 0 implies frac4x+22-x le 0 implies frac2x+1x-2 ge 0 implies x in left(-infty, -frac12right] cup (2, infty)

Intersecting both sets gives [cite: 1311]:

x in left[-3, -frac12right]

[cite: 1311] ### Step 2: Final Intersection and Value Solving Intersect Log constraint with Arcsine constraint solution range [cite: 1311]:

x in left[-3, -frac12right] cap left[left(-infty, -frac54right) cup left(frac32, inftyright)right] = left[-3, -frac54right)

[cite: 1311] Thus, identify parameters [cite: 1312]:

alpha = -3, quad beta = -frac54

[cite: 1312] Compute the requested expression value [cite: 1312]:

alpha^2 + 4beta = (-3)^2 + 4left(-frac54right) = 9 - 5 = 4

[cite: 1312] ### Pattern Recognition When dealing with fractional variables inside boundaries, flipping inequalities according to denominator signs prevents fatal zone misinterpretations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Relations and Functions

Q58 2025 Domain of Functions

Let the domains of the functions

f (x) = log_ 4 log_ 3 log_ 7 (8 - log_ 2 (x ^ 2 + 4 x + 5))

and

g (x) = sin^ - 1 left(frac 7 x + 1 0x - 2right)

(alpha , beta)

and

[ gamma , delta ]

, respectively. Then

alpha^ 2 + beta^ 2 + gamma^ 2 + delta^ 2

is equal to :-

A. $15$ $15$
B. $13$ $13$
C. $16$ $16$
D. $14$ $14$

Solution

### Core Logic Let's first determine the domain of

f(x)

. For logarithmic expressions, the argument must be strictly positive:

log_3 log_7 (8 - log_2(x^2 + 4x + 5)) > 0

log_7 (8 - log_2(x^2 + 4x + 5)) > 3^0 = 1

8 - log_2(x^2 + 4x + 5) > 7^1 = 7

log_2(x^2 + 4x + 5) < 1

x^2 + 4x + 5 < 2^1 = 2

x^2 + 4x + 3 < 0 implies (x+1)(x+3) < 0

Hence,

x in (-3, -1)

, which gives

alpha = -3

and

beta = -1

. ### Step 1: Finding the domain of g(x) For the function

g(x) = sin^-1left(frac7x+10x-2right)

, the argument must lie within

[-1, 1]

-1 le frac7x+10x-2 le 1

Let's break this into two separate inequalities: Inequality A:

frac7x+10x-2 ge -1 implies frac7x+10+x-2x-2 ge 0 implies frac8x+8x-2 ge 0 implies x in (-infty, -1] cup (2, infty)

Inequality B:

frac7x+10x-2 le 1 implies frac7x+10-x+2x-2 le 0 implies frac6x+12x-2 le 0 implies x in [-2, 2)

Taking the intersection of both intervals:

x in [-2, -1]

Thus,

gamma = -2

and

delta = -1

. ### Step 2: Computing the final sum of squares Now we calculate

alpha^2 + beta^2 + gamma^2 + delta^2

alpha^2 + beta^2 + gamma^2 + delta^2 = (-3)^2 + (-1)^2 + (-2)^2 + (-1)^2

= 9 + 1 + 4 + 1 = 15

### Pattern Recognition For nested logs, start from the outermost log condition and work your way inward step-by-step. Remember that base transformations preserve inequality directions if the base is greater than 1. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Relations and Functions

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Reference Study Guides

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