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🔥 Class 12 Maths Chapter 5 PYQs 🔥

Want to score high in Continuity & Differentiability? 📈
These Previous Year Questions reveal the most important concepts, exam patterns, and frequently asked problems.

✅ Continuity
✅ Differentiability
✅ Chain Rule
✅ Product Rule
✅ Quotient Rule
✅ Implicit Differentiation

Practice smart, not just hard! 💯

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🔗 Product Rule = Teamwork in Calculus!

When two functions are multiplied, their derivative isn’t just the product of their derivatives.

(uv)’ = u’v + uv’

👉 Differentiate the first, keep the second.
👉 Then keep the first, differentiate the second.
👉 Add the results.

Remember:
“First derivative × Second + First × Second derivative”

Master this rule and many differentiation problems become much easier!

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🔗 Chain Rule — The Rule That Never Breaks!

When one function is hidden inside another, differentiation follows the chain.
Differentiate the outer function, then the inner function.

📌 Differentiate layer by layer, link by link.

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📊 Same continuity, different story for differentiability!

🔹 Corner → Different slopes from left and right.
🔹 Cusp → Slopes become infinite and no unique tangent exists.

Both graphs are continuous, yet neither is differentiable at that point.

Mathematics isn’t just about formulas—it’s about understanding the shape of a graph! ✨

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🔺 Same shape, different height!

Vertical shifting of the modulus function is simple:
📈 |x| + k → Graph moves up
📉 |x| − k → Graph moves down

The V-shape never changes—only its position on the y-axis does.

Visualize it, and graph transformations become effortless! ✨

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🔄 Shift the graph, not the shape!

For modulus functions:
➡️ |x − h| → Shift Right
⬅️ |x + h| → Shift Left

The V-shape stays the same, only the vertex moves.

Remember: The sign inside the modulus works in the opposite direction! 🎯

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🔍 Modulus removes the sign, not the value!

Whether the number is positive or negative, modulus tells us its distance from zero.

|-8| = 8 and |8| = 8

📈 Simple concept, powerful applications!

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📈 Vertical Shifting of a Parabola Made Easy!

A parabola can move up or down without changing its shape.

🔹 y = f(x) + k → Shift Up by k units
🔹 y = f(x) - k → Shift Down by k units

Remember:
➕ Outside the function = Move Up
➖ Outside the function = Move Down

Visualize the movement, don’t just memorize the rule! 🎯

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📉➡️📈 Horizontal shifting changes the position of a parabola without changing its shape!

🔹 x-h → Shift Right
🔹 x+h → Shift Left

Remember: The sign inside the bracket works in the opposite direction. 🎯

Visualize it, don’t memorize it! 🚀

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😄 Positive x² → Parabola Smiles Upward

😞 Negative x² → Parabola Frowns Downward

One sign tells the whole story! 📈

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