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Q54 jee_main_2025_07_april_evening Range of Rational Functions

If the range of the function

f(x) = (5 - x)/(x² - 3x + 2)

,

x \neq 1, 2

, is

(-\infty , α ] [ β , \infty)

, then

α² +β²

is equal to :

A. $190$ $190$
B. $192$ $192$
C. $188$ $188$
D. $194$ $194$

Solution

Related Formula

For a quadratic equation $Ax² + Bx + C = 0$ $Ax^2 + Bx + C = 0$ to yield real roots, its discriminant must satisfy:

D = B² - 4AC \geq 0

Core Logic

Set $y = (5 - x)/(x² - 3x + 2)$ $y = \frac{5 - x}{x^2 - 3x + 2}$:

y(x² - 3x + 2) = 5 - x

yx² - 3xy + 2y + x - 5 = 0

Rearranging into a standard quadratic equation in terms of $x$ $x$:

yx² + (1 - 3y)x + (2y - 5) = 0

Step 1: Discriminant Method

Case I: If $y = 0$ $y = 0$, the equation simplifies to $x - 5 = 0 x = 5$ $x - 5 = 0 \implies x = 5$, which is a valid part of the domain. Thus, $0$ $0$ belongs to the range.

Case II: If $y \neq 0$ $y \neq 0$, for $x$ $x$ to be real, $D \geq 0$ $D \ge 0$:

(1 - 3y)² - 4(y)(2y - 5) \geq 0

9y² + 1 - 6y - 8y² + 20y \geq 0

y² + 14y + 1 \geq 0

Step 2: Solving the Inequality

Completing the square for $y² + 14y + 1 \geq 0$ $y^2 + 14y + 1 \ge 0$:

(y + 7)² - 48 \geq 0 (y + 7)² \geq (4\sqrt(3))²

This gives:

y \leq -7 - 4\sqrt(3) or y \geq -7 + 4\sqrt(3)

Comparing with the interval $(-\infty , α ] [ β , \infty)$ :

α = -7 - 4\sqrt(3)

β = -7 + 4\sqrt(3)

Step 3: Finding alpha^2 + beta^2

Using algebraic identities:

α² + β² = (-7 - 4\sqrt(3))² + (-7 + 4\sqrt(3))²

= 2(7² + (4\sqrt(3))²) = 2(49 + 48) = 2(97) = 194

Pattern Recognition

For rational expressions of the form $LinearQuadratic$ , converting to a quadratic in $x$ $x$ and forcing $D \geq 0$ $D \ge 0$ establishes the range boundaries elegantly.

Chapter Mix

Class 11 Mathematics: Sets, Relations and Functions

Q52 jee_main_2025_24_jan_evening One-One and Onto Functions

The function

f:(-\infty,\infty)arrow(-\infty,1)

, defined by

f(x)= 2x-2-x2x+2-x

is:

A. $One-one but not onto$ $\text{One-one but not onto}$
B. $Onto but not one-one$ $\text{Onto but not one-one}$
C. $Both one-one and onto$ $\text{Both one-one and onto}$
D. $Neither one-one nor onto$ $\text{Neither one-one nor onto}$

Solution

Related Formula

A function is one-one if its derivative is strictly monotonic (always positive or always negative) across its domain. It is onto if its range equals its co-domain.

Core Logic

Rewrite the function by multiplying the numerator and denominator by $2^x$ $2^x$:

f(x) = 22x - 122x + 1 = 1 - 222x + 1

Step 1: Check One-One property

Differentiating $f(x)$ $f(x)$ with respect to $x$ $x$ :

f'(x) = 2(22x + 1)² \cdot 2 \cdot 22x \cdot ln 2 = 4 \cdot 22x \cdot ln 2(22x + 1)²

Since $22x > 0$ $2^{2x} > 0$ and $ln 2 > 0$ $\ln 2 > 0$, $f'(x) > 0$ $f'(x) > 0$ always. Thus, $f(x)$ $f(x)$ is strictly increasing, confirming it is a one-one function.

Step 2: Check Onto property

Analyze the limits at boundaries:

x \to -\infty f(x) = 1 - (2)/(0 + 1) = -1

x \to \infty f(x) = 1 - 0 = 1

Thus, the range of the function is $(-1, 1)$ $(-1, 1)$. Since the given co-domain is $(-\infty, 1)$ $(-\infty, 1)$ and $Range \neq Co-domain$ $\text{Range} \neq \text{Co-domain}$ , the function is not onto.

Pattern Recognition

The expression given is a shifted form of the hyperbolic tangent function $(x ln 2)$ $\tanh(x \ln 2)$. Hyperbolic tangent always maps to $(-1, 1)$ $(-1, 1)$, making its restriction against $(-\infty, 1)$ $(-\infty, 1)$ non-surjective (not onto).

Chapter Mix

Class 12 Mathematics: Relations and Functions

Q58 jee_main_2025_24_jan_evening Linear Programming and Inequalities in Two Variables

Let the points

((11)/(2),α)

lie on or inside the

with sides

x+y=11

,

x+2y=16

and

2x+3y=29

Then the product of the smallest and the largest values of a is equal to:

A. $22$ $22$
B. $44$ $44$
C. $33$ $33$
D. $55$ $55$

Solution

Related Formula

For a vertical line $x = x₀$ $x = x_0$ crossing a bounded region, the valid coordinates of $y$ $y$ sit between the boundary lines intersecting that specific line vertical plane.

Core Logic

The point given is fixed at $x = (11)/(2) = 5.5$ $x = \frac{11}{2} = 5.5$. We evaluate the values of $y$ $y$ along this vertical line segment across each boundary edge.

Linear Programming region graph for Q58 - JEE Main 2025 Evening

Step 1: Evaluate Intersections

Substitute $x = (11)/(2)$ $x = \frac{11}{2}$ into the three linear constraints:

From $x + y = 11$ $x + y = 11$:

(11)/(2) + y = 11 \Rightarrow y = 11 - 5.5 = 5.5

From $x + 2y = 16$ $x + 2y = 16$:

(11)/(2) + 2y = 16 \Rightarrow 2y = 16 - 5.5 = 10.5 \Rightarrow y = 5.25

From $2x + 3y = 29$ $2x + 3y = 29$:

2((11)/(2)) + 3y = 29 \Rightarrow 11 + 3y = 29 \Rightarrow 3y = 18 \Rightarrow y = 6

Step 2: Define Extrema and Multiply

Checking the internal region of the $\triangle$ bounded by these lines, the valid range for $α$ $\alpha$ along the section line is delimited by $y = 5.5$ $y = 5.5$ and $y = 6$ $y = 6$:

α = (11)/(2) = 5.5

α = 6

The product of the limits is :

α \cdot α = (11)/(2) \times 6 = 33

Pattern Recognition

Instead of drawing full coordinate diagrams or computing all three vertex points, evaluating values directly at the fixed coordinate constraint $x = 5.5$ $x = 5.5$ saves time during multi-line area problems.

Chapter Mix

Class 11 Mathematics: Linear Inequalities Class 11 Mathematics: Straight Lines

Q71 jee_main_2025_24_jan_evening Number of Functions under Constraints

Number of functions

f:1,2, ,100arrow0,1

, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to \_\_\_\_.

Numerical Answer. Answer: 392

Solution

Related Formula

Fundamental Counting Principle: If an operation can be performed in $n₁$ $n_1$ ways, followed by a second operation in $n₂$ $n_2$ ways, the total configurations equal $n₁ \times n₂$ $n_1 \times n_2$.

Core Logic

The domain set contains integers from 1 to 100. We divide the mapping requirements across distinct subsets of this domain.

Function mapping grid for Q71 - JEE Main 2025 Evening

Step 1: Choose the single element from $1, 2, , 98$ $\{1, 2, \dots, 98\}$

We must assign the image value 1 to exactly one positive integer from the subset $1, 2, , 98$ $\{1, 2, \dots, 98\}$. The number of ways to pick this single element is :

981 = 98 ways

Step 2: Mapping remaining elements

The remaining 97 elements in the $1, 2, , 98$ $\{1, 2, \dots, 98\}$ subset cannot map to 1, so they must map to 0. This leaves exactly 1 choice per remaining element.

For the final two elements in the domain, 99 and 100, there are no structural constraints:

Element 99 can map to either 0 or 1 (2 options) .
Element 100 can map to either 0 or 1 (2 options).

Step 3: Total functions combination

Multiply the independent choices together :

Total functions = 98 \times 2 \times 2 = 392

Pattern Recognition

Separate domains tightly into restricted blocks vs completely free components. Realizing that elements 99 and 100 behave independently with full co-domain targets leaves a clear product formulation.

Chapter Mix

Class 12 Mathematics: Relations and Functions Class 11 Mathematics: Permutations and Combinations

Q jee_main_2025_24_jan_morning Symmetric Property of Functions

Let

f(x) = 2x + 2 + 1622x + 1 + 2x + 4 + 32

. Then the value of

8(f((1)/(15)) + f((2)/(15)) + \dots + f((59)/(15)))

is equal to:

A. $118$ $118$
B. $92$ $92$
C. $102$ $102$
D. $108$ $108$

Solution

Related Formula

Many finite fractional sum questions involving functional terms rely on identifying an underlying symmetric summation invariant, typically of the form $f(x) + f(k-x) = constant$ $f(x) + f(k-x) = \text{constant}$.

Core Logic

First simplify the expression for $f(x)$ $f(x)$ algebraically:

f(x) = (4 \cdot 2^x + 16)/(2 \cdot (2^x)² + 16 \cdot 2^x + 32)

Factor out $4$ $4$ from the numerator and $2$ $2$ from the denominator:

f(x) = (4(2^x + 4))/(2[(2^x)² + 8 \cdot 2^x + 16]) = (2(2^x + 4))/((2^x + 4)²) = (2)/(2^x + 4)

Step 1: Establish Symmetry Pairings

Let's check the value of $f(x) + f(4-x)$ $f(x) + f(4-x)$:

f(4-x) = 224-x + 4 = (2)/((16)/(2^x) + 4) = (2 \cdot 2^x)/(16 + 4 \cdot 2^x) = (2 \cdot 2^x)/(4(2^x + 4)) = (2^x)/(2(2^x + 4))

Now compute the sum directly:

f(x) + f(4-x) = (2)/(2^x + 4) + (2^x)/(2(2^x + 4)) = (4 + 2^x)/(2(2^x + 4)) = (1)/(2)

Hence, whenever two input arguments sum up to $4$ $4$, the sum of their functional values is exactly $(1)/(2)$ $\frac{1}{2}$.

Step 2: Group the Finite Series Terms

Consider the terms inside the requested sequence:

(1)/(15) + (59)/(15) = (60)/(15) = 4 f((1)/(15)) + f((59)/(15)) = (1)/(2)

(2)/(15) + (58)/(15) = (60)/(15) = 4 f((2)/(15)) + f((58)/(15)) = (1)/(2)

This complementary pairing continues up to:

f((29)/(15)) + f((31)/(15)) = (1)/(2)

This yields exactly $29$ $29$ distinct pairs. The single middle term left unpaired corresponds to:

Middle Term = f((30)/(15)) = f(2) = (2)/(2² + 4) = (2)/(8) = (1)/(4)

Step 3: Evaluate Final Expression

Compute the total value by multiplying the grouped sum by 8:

Total = 8 \cdot [ 29 \cdot ((1)/(2)) + (1)/(4) ]

Total = 8 \cdot (29)/(2) + 8 \cdot (1)/(4) = 116 + 2 = 118

Pattern Recognition

Whenever a symmetric set of arguments is presented inside a summation matching $(k)/(n) + (N-k)/(n) = constant$ , look for an algebraic reduction of $f(x)$ $f(x)$ that yields a uniform constant sum for symmetric pairs.

Chapter Mix

Class 11 Mathematics: Functions

Solution & Explanation

Related Formula

Core Logic

Step 1: Testing x = 11

Step 2: Testing x = 10

Pattern Recognition

Chapter Mix

Reference Study Guides

More Sets Previous-Year Questions — Page 9

Solution

Related Formula

Core Logic

Step 1: Discriminant Method

Step 2: Solving the Inequality

Step 3: Finding alpha^2 + beta^2

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Check One-One property

Step 2: Check Onto property

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Evaluate Intersections

Step 2: Define Extrema and Multiply

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Choose the single element from 1, 2, , 98$\{1, 2, \dots, 98\}$

Step 2: Mapping remaining elements

Step 3: Total functions combination

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Establish Symmetry Pairings

Step 2: Group the Finite Series Terms

Step 3: Evaluate Final Expression

Pattern Recognition

Chapter Mix

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Step 1: Choose the single element from $1, 2, , 98$ $\{1, 2, \dots, 98\}$