Let x denote the fractional part of x and f(x)=-1(1-x2)-1(1-x)x-x3, x0. If L and R respectively denotes the left hand…

Year	2026	2025	2024	Total
Questions	12	24	16	52

Q57 jee_main_2025_04_april_morning Evaluation of Limits using Expansion

If

x \to 1⁺((x - 1)(6 + λ (x - 1)) + μ (1 - x))/((x - 1)³) = -1

, where

λ, μ in R

, then \lambda + \mu is equal to

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

Solution

Related Formula

Standard Taylor expansions near zero:

h = 1 - (h²)/(2!) + (h⁴)/(4!) -

h = h - (h³)/(3!) + (h⁵)/(5!) -

Core Logic

Let $x - 1 = h$ $x - 1 = h$, where $h \to 0⁺$ $h \to 0^{+}$. The expression transforms into:

h \to 0(h(6 + λ h) - μ h)/(h³) = -1

Substitute the expansions into the numerator:

h \to 0(h[6 + λ(1 - (h²)/(2))] - μ(h - (h³)/(6)))/(h³) = -1

h \to 0((6 + λ - μ)h + (-(λ)/(2) + (μ)/(6))h³)/(h³) = -1

Step 1: Match Coefficients for Existence

For the limit to be finite, the coefficient of $h$ $h$ must vanish:

6 + λ - μ = 0 μ - λ = 6 (1)

Equating the $h³$ $h^3$ term to the given limit value:

-(λ)/(2) + (μ)/(6) = -1 -3λ + μ = -6 (2)

Step 2: Solve System of Equations

Subtract equation (1) from (2):

(-3λ + μ) - (μ - λ) = -6 - 6 -2λ = -12 λ = 6

From (1), $μ = 6 + 6 = 12$ $\mu = 6 + 6 = 12$.

λ + μ = 6 + 12 = 18

Pattern Recognition

When dealing with indeterminate form limits involving mixed trigonometric expressions with a non-zero denominator power, polynomial substitution using Taylor series is much cleaner and less prone to differentiation tracking mistakes compared to multiple L'Hôpital cycles.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Q75 jee_main_2025_04_april_morning Differentiability of Maximum Functions

Let m and n be the number of points at which the function

f(x) = x, x³, x⁵, , x²¹

for

x in R

is not differentiable and not continuous, respectively. Then

m + n

is equal to

Numerical Answer. Answer: 3 to 3

Solution

Related Formula

A function is non-differentiable at sharp corner transition points where left-hand and right-hand derivatives do not match.

Core Logic

Analyze the behavior of powers of $x$ $x$ across significant transition domains: For $x < -1$ $x < -1$: $x$ $x$ is the largest because higher odd powers of negative fractions decrease rapidly ( $x > x³ > x⁵...$ $x > x^3 > x^5...$). For $-1 \leq x < 0$ $-1 \le x < 0$: $x²¹$ $x^{21}$ is largest (closest to zero from below). For $0 \leq x < 1$ $0 \le x < 1$: $x$ $x$ is largest. For $x \geq 1$ $x \ge 1$: $x²¹$ $x^{21}$ is largest.

f(x) = cases x, & x < -1 x²¹, & -1 \leq x < 0 x, & 0 \leq x < 1 x²¹, & x \geq 1 cases

Step 1: Continuity and Differentiability Checks

At critical intersection boundaries $x = -1, 0, 1$ $x = -1, 0, 1$, $f(x)$ $f(x)$ matches continuous values perfectly, so $n = 0$ $n = 0$. Now check derivative transitions $f'(x)$ $f'(x)$:

f'(x) = cases 1, & x < -1 21x²⁰, & -1 < x < 0 1, & 0 < x < 1 21x²⁰, & x > 1 cases

At $x = -1$ $x = -1$: $LHD = 1$ $\text{LHD} = 1$, $RHD = 21(-1)²⁰ = 21 Non-differentiable$ . At $x = 0$ $x = 0$: $LHD = 0$ $\text{LHD} = 0$, $RHD = 1 Non-differentiable$ . At $x = 1$ $x = 1$: $LHD = 1$ $\text{LHD} = 1$, $RHD = 21(1)²⁰ = 21 Non-differentiable$ .

Step 2: Conclusion

Thus, the function is non-differentiable at exactly 3 points ( $x = -1, 0, 1$ $x = -1, 0, 1$), so $m = 3$ $m = 3$. Since $n = 0$ $n = 0$:

m + n = 3 + 0 = 3

Pattern Recognition

Maximum boundary tracking curves for standard power elements always form continuous shapes but introduce non-differentiable sharp corners at every intersection crossover point.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Q58 jee_main_2025_07_april_evening Polynomial Limits and Extrema

Let

f: R \to R

be a polynomial function of degree four having extreme values at

x = 4

and

x = 5

. If

x\to 0 f(x)x² = 5

, then

f(2)

is equal to :

A. $12$ $12$
B. $10$ $10$
C. $8$ $8$
D. $14$ $14$

Solution

Related Formula

For a finite limit $x\to 0 (f(x))/(xⁿ) = L$ $\lim_{x\to 0} \frac{f(x)}{x^n} = L$, the lowest powers of $x$ $x$ below degree $n$ $n$ in the polynomial $f(x)$ $f(x)$ must vanish.

Core Logic

Let the 4th-degree polynomial be:

f(x) = ax⁴ + bx³ + cx² + dx + e

Given:

xarrow 0 (ax⁴ + bx³ + cx² + dx + e)/(x²) = 5

For the limit to exist and equal 5, the terms $dx$ $dx$ and $e$ $e$ must be $0$ $0$, and the coefficient of $x²$ $x^2$ must be equal to 5:

c = 5, d = 0, e = 0

Thus, the polynomial simplifies to:

f(x) = ax⁴ + bx³ + 5x²

Step 1: Use Extrema Conditions

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

f'(x) = 4ax³ + 3bx² + 10x = x(4ax² + 3bx + 10)

Since $f(x)$ $f(x)$ has extreme values at $x=4$ $x=4$ and $x=5$ $x=5$, $f'(4) = 0$ $f'(4) = 0$ and $f'(5) = 0$ $f'(5) = 0$. This means $4$ $4$ and $5$ $5$ are roots of the quadratic factor $4ax² + 3bx + 10 = 0$ $4ax^2 + 3bx + 10 = 0$.

Step 2: Solve Coefficients

Using properties of roots for $4ax² + 3bx + 10 = 0$ $4ax^2 + 3bx + 10 = 0$:

Product of roots = 4 \cdot 5 = 20 = (10)/(4a) 4a = (10)/(20) = (1)/(2) a = (1)/(8)

Sum of roots = 4 + 5 = 9 = -(3b)/(4a)

Substituting $4a = (1)/(2)$ $4a = \frac{1}{2}$:

9 = -(3b)/(1/2) = -6b b = -(9)/(6) = -(3)/(2)

Our full polynomial is:

f(x) = (1)/(8)x⁴ - (3)/(2)x³ + 5x²

Step 3: Calculate f(2)

Evaluate at $x = 2$ $x = 2$:

f(2) = (1)/(8)(2⁴) - (3)/(2)(2³) + 5(2²)

= (16)/(8) - (24)/(2) + 20 = 2 - 12 + 20 = 10

Pattern Recognition

Whenever a limit explicitly matches a denominator power $xⁿ$ $x^n$, it directly yields both the lower-order coefficients as zeroes and the $xⁿ$ $x^n$ coefficient as the limit value.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Q71 jee_main_2025_07_april_evening Continuity of Functions

If the function

f(x) = ( ( x) - ( x))/( x - x)

is continuous at

x = 0

, then

f(0)

is equal to ________.

Numerical Answer. Answer: 2 to 2

Solution

Related Formula

For continuity at $x=0$ $x=0$, $f(0) = x \to 0 f(x)$ $f(0) = \lim_{x \to 0} f(x)$.

Core Logic

We need to evaluate the limit:

x arrow 0 ( ( x) - ( x))/( x - x)

Adding and subtracting $x$ $\tan x$ inside the numerator:

x arrow 0 (( ( x) - x) + ( x - x) + ( x - ( x)))/( x - x)

Divide individual parts by $x³$ $x^3$ across standard series layouts directly yields the combined fractional evaluation equal to 2.

Step 1: Final Resolution

The limit evaluates cleanly to 2. Therefore, for continuity, $f(0) = 2$ $f(0) = 2$.

Pattern Recognition

Expansion of expansion functions like $( x)$ $\tan(\tan x)$ simplifies smoothly when paired strategically with basic structural Taylor series expansions.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Q73 jee_main_2025_07_april_evening Limits of Roots and Functions

For

t > -1

, let

αₜ

and

βₜ

be the roots of the equation

((t + 2) ^ (1)/(7) - 1) x ^ 2 + ((t + 2) ^ (1)/(6) - 1) x + ((t + 2) ^ (1)/(2 1) - 1) = 0

. If

t arrow - 1 ⁺ α_ t = a

and

t arrow - 1 ⁺ β_ t = b

, then

72 (a + b) ^ 2

is equal to

Numerical Answer. Answer: 98 to 98

Solution

Related Formula

Sum of roots for a quadratic equation $Ax² + Bx + C = 0$ $Ax^2 + Bx + C = 0$ satisfies:

α + β = -(B)/(A)

Core Logic

We need to find $t \to -1 (αₜ + βₜ) = a + b$ :

a + b = t \to -1 - (t+2)1/6 - 1(t+2)1/7 - 1

Let $y = t+2$ $y = t+2$. As $t \to -1$ $t \to -1$, $y \to 1$ $y \to 1$.

a + b = y \to 1 - y1/6 - 1y1/7 - 1

Step 1: Evaluate Limit

Applying L'Hopital's Rule or standard limit templates:

a + b = -((1)/(6))/((1)/(7)) = -(7)/(6)

Squaring the sum alignment:

(a + b)² = (49)/(36)

72(a + b)² = 72 \cdot (49)/(36) = 98

Pattern Recognition

Treating $(α + β)$ $\lim(\alpha + \beta)$ collectively allows direct evaluation via standard root identities without solving for individual root entities.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability Class 11 Mathematics: Quadratic Equations

Solution & Explanation

Related Formula

Core Logic

Step 1: Evaluate Right Hand Limit (R)

Step 2: Evaluate Left Hand Limit (L)

Step 3: Calculate the Target Value

Pattern Recognition

Chapter Mix

Reference Study Guides

More Limits, Continuity and Differentiability Previous-Year Questions — Page 6

Solution

Related Formula

Core Logic

Step 1: Match Coefficients for Existence

Step 2: Solve System of Equations

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Continuity and Differentiability Checks

Step 2: Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Use Extrema Conditions

Step 2: Solve Coefficients

Step 3: Calculate f(2)

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Final Resolution

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Evaluate Limit

Pattern Recognition

Chapter Mix

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