PYQ Boards Imp formulas | 3D Geometry Class 12 Chapter 11 Maths | CBSE NCERT with Solutions PDF

 

Important Concepts Covered in This Revision Lecture

(Chapter 11 – Three-Dimensional Geometry | Class 12 NCERT CBSE | Extremely Useful for 2026 Boards)

• Cartesian equation of a line$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$
• Vector form of a line$\vec{r} = \vec{a} + \lambda \vec{b}$
• Parametric form / Any point on the line Using parameter λ (or t, μ): $x = x_1 + a\lambda,\ y = y_1 + b\lambda,\ z = z_1 + c\lambda$
• Direction ratios (DRs) and Direction cosines (DCs) of a line
• Condition for two lines to be parallel$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Condition for two lines to be perpendicular$a_1a_2 + b_1b_2 + c_1c_2 = 0$
• Direction ratios of line joining two points$(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$$x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1$
• Mid-point formula in 3D$\left( \frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2},\ \frac{z_1+z_2}{2} \right)$
• Foot of the perpendicular from a point to a line (most frequent 4–6 mark question)
• Image (reflection) of a point in a given line Key properties used:
  1. Foot of perpendicular Q is the mid-point of A and A'
  2. Line AA' is perpendicular to the given line → Solve using parametric point + perpendicular condition
• Equation of the line joining a point and its image (A and A')
• Distance of a point from a line = 7 units type questions → Use distance formula between point and general point on line → solve quadratic in λ → two possible points P1, P2
• Line perpendicular to two given lines (common normal) Direction ratios = $\vec{b_1} \times \vec{b_2}$ (vector cross product)
• Shortest distance between two skew lines (non-parallel, non-intersecting) $\mid \frac{(\overrightarrow{a_2}-\overrightarrow{a_1})\cdot (\overrightarrow{b_1}\times \overrightarrow{b_2})}{\mathrm{\mid }\overrightarrow{b_1}\times \overrightarrow{b_2}\mathrm{\mid }}$
• Equation of a line parallel to a given line and passing through a given point Same DRs, new point
• Conversion between Cartesian and vector forms of a line
• Case-study/application-based questions (metro lines, solar panels, pedestrian pathway) that combine parallel, perpendicular, shortest distance, and equation of line concepts

CBSE Class 12 Maths | 3D Geometry Chapter 11 Revision + 2025 Board Questions Full Solution | 2026 Boards Complete last-minute revision of Three Dimensional Geometry Class 12 CBSE Chapter 11 with all 2025 board paper questions solved step-by-step. Perfect for CBSE 2026 students targeting 95%+ in Maths! q1 Find the image A' of the point A(1, 6, 3) in the line x/1=y-1/2=z-2/3  Also, find the equation of the line joining A and A'. q2. Find the image A' of the point A(2, 1, 2) in the line l : 7 = 41 + 2j + 2k + 2 (i - j -k). Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l. q3 Find a point P on the line x+5/1=y+3/4=z-6/-9 from point Q(2, 4,-1) is 7 units. Also, find the equation of line joining P and Q. q4. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by Class 12 3d chapter cbse pdf Class 12 3D Geometry NCERT PDF 3D Geometry Class 12 NCERT Solutions PDF Class 12 3d chapter cbse solutions 3D Geometry Class 12 Notes PDF download 3D Geometry Class 12 PYQ boards PDF Class 12 3d chapter cbse notes Class 12 Maths Chapter 11 PDF 00:00 imp formulas 06:00 q1 13:38 q2 25:45 q3 34:30 q4 3d geometry class 12 three dimensional geometry class 12 class 12 maths chapter 11 3d geometry cbse 2026 3d geometry 2025 board paper image of point in a line foot of perpendicular 3d geometry shortest distance between skew lines cbse class 12 maths 2026 class 12 maths revision 3d geometry important questions direction ratios class 12 equation of line in 3d cbse 2025 maths solutions 3d geometry case study metro case study 3d geometry class 12 maths chapter 11 ncert 3d geometry one shot 3d geometry revision 2026 class 12 maths board exam 2026 cbse class 12 maths important questions 3d geometry pyq previous year questions 3d geometry class 12 maths chapter wise revision cbse maths 2026 strategy 3d geometry ncert solutions vector form of line class 12 cartesian form of line perpendicular lines in 3d parallel lines in 3d

Amazon Basics Rechargeable LED Table Lamp Review: Features, Battery Life, Drawbacks & Real-User Verdict

 In this article, I share a detailed hands-on review, including the lamp’s best features, battery performance, everyday usability, and the one major drawback that buyers often discover too late.

👉 Watch the full video review here: https://youtu.be/URIiMSa2Yo4

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⭐ 1. Design & Flexibility: A Lamp That Moves the Way You Need

One of the first things you notice is the lamp’s highly flexible neck. Unlike rigid metal stems, this one bends effortlessly, allowing you to adjust it to any position—perfect for studying in bed, reading a book on a couch, or directing focused light onto your laptop keyboard.

Key design benefits:

• Lightweight and portable

• Fully flexible stem for angle adjustment

• Minimalist design suitable for bedrooms, study tables, and home offices

This feature alone makes it one of the most user-friendly study lamps in its price range.

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⭐ 2. Three Light Modes: White, Yellow & Mixed

Lighting preferences change depending on the task—white for reading, yellow for relaxing, and mixed for balanced usage.
The Amazon Basics lamp offers three touch-sensitive modes:

1. Cool White Light – Bright and sharp; ideal for reading textbooks or writing notes

2. Warm Yellow Light – Comfortable for evening use and reduces eye strain

3. Mixed Light Mode – A blend of both, suitable for long study sessions

Switching between modes is intuitive:
✔ Tap once → White
✔ Tap twice → Yellow
✔ Tap thrice → Mixed
✔ Tap again → OFF

This simplicity makes it ideal for students and working professionals alike.

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⭐ 3. Adjustable Brightness: Press-and-Hold Dimming

The lamp also offers smooth brightness control.
All you need to do is press and hold the touch button:

• The brightness gradually decreases

• Hold again to increase the light intensity

This dimmable feature is extremely useful during late-night study sessions when harsh lighting can cause fatigue.

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⭐ 4. Battery Life: 4–6 Hours on a Full Charge

With regular use, the lamp delivers around 4 to 5 hours of continuous light, depending on the mode and brightness level.

A fully charged battery easily lasts an entire evening of studying or binge-reading. Once the brightness begins to fade, it's a sign that the lamp needs recharging.

This makes the lamp perfect for:

• Hostel rooms

• Power cuts

• Late-night reading

• Portable table lighting without messy cables

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⭐ 5. The Major Drawback: Charging Socket Loosens Over Time

Here’s where things get tricky—and this is the main reason many users hesitate to repurchase the product.

After a few months of constant plugging and unplugging, the DT-type charging socket tends to loosen.
Once loose, the lamp stops charging properly, and eventually becomes unusable.

In my experience:

• This issue appeared on multiple units

• It is a recurring manufacturing flaw

• Once the port loses grip, repairing it is difficult

For a lamp designed to be recharged daily, this becomes a critical long-term problem.

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⭐ 6. Packaging & What You Get Inside

Amazon Basics usually maintains good packaging standards.

Inside the box, you will find:

• The Rechargeable LED Table Lamp

• A DT-type charging cable

• A user instruction manual

Note:
An adapter is not included, but you can use any standard mobile charging adapter from home.

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⭐ 7. Should You Buy This Lamp in 2025? – Verdict

The Amazon Basics Rechargeable LED Table Lamp is ideal if you want:

• A budget-friendly study lamp

• Multiple light modes

• Flexible design

• Adjustable brightness

• Portability

However, the recurring charging socket issue makes it a risky long-term investment.

✔ Buy it if:

You need a short-term or occasional-use lamp.

✘ Avoid it if:

You want a lamp that lasts for years without charging problems.

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⭐ Conclusion: A Good Lamp Held Back by One Serious Flaw

Based on features alone, the lamp performs exceptionally well for its price.
But durability is where it struggles.

If Amazon improves the charging port in future versions, this could easily become one of the best budget study lamps in India.

For a complete demonstration—flexibility, modes, dimming, charging, and live usage—
👉 Watch the full video review here: https://youtu.be/URIiMSa2Yo4

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50 ideas in maths you should know | 50 ideas in physics you should know

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Osho Rajneesh Jyun Tha Tyun Thaharaya pdf

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Beginner’s Guide to Essential Electronic Components (With Images & Easy Examples)

A complete introduction for school students, engineering beginners, and DIY hobbyists

Electronics can look intimidating when you’re just getting started—so many components, so many pins, so many small markings! But once you understand the basics, building circuits becomes one of the most enjoyable ways to learn science.

This guide walks you through the 10 most important components you’ll use in almost every project. Whether you’re making a school model, preparing for engineering, or exploring Arduino, these components form the foundation of real-world electronics.

At the end, you’ll also understand how to connect everything on a breadboard using a simple power source like an Arduino Uno.

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🔌 1. Resistors: The Guardians of Safe Current

A resistor limits the flow of current so that other components are not damaged.
Resistors come in different values—220Ω, 330Ω, 1kΩ, 4.7kΩ, 10kΩ and more.

How to identify them:
Color bands painted on the resistor follow a universal code. Once you decode the colors, you can read any resistor’s value instantly.

Where you’ll use it:
LED protection, voltage dividers, sensors, logic circuits.

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➡️ 2. Diodes: One-Way Gates for Current

A diode allows current to flow only in one direction.
They always have polarity, meaning one side is positive and the other is negative.

How to identify polarity:
A silver (or white) band on one end marks the negative (cathode) side.

Where you’ll use it:
Rectifiers, reverse-current protection, switching circuits.

---

💡 3. LEDs: The Most Fun Component for Beginners

LEDs glow when current passes through them.
They too have polarity:

• Long leg = Positive (Anode)

• Short leg = Negative (Cathode)

Different colours have different voltage ratings—red LEDs glow at ~2V, while blue/green/white need ~3–3.3V.

Where you’ll use it:
Indicators, Arduino projects, displays, mini-lamps.

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⚡ 4. Electrolytic Capacitors: Mini Charge Storage Devices

Capacitors store small amounts of electrical charge.
Electrolytic capacitors have polarity and are identified by:

• Short leg = Negative

• Long leg = Positive

Markings like “100µF 25V” tell you the capacitance and voltage rating.

Where you’ll use it:
Filtering, smoothing power supplies, timing circuits.

---

🔲 5. MOSFET: The Efficient Electronic Switch

A MOSFET behaves like a high-speed switch or amplifier.
It has three terminals: Source → Gate → Drain.

How to identify the terminals:
Face the flat side of the MOSFET towards you.
From left to right, the terminals are:

1. Source

2. Gate

3. Drain

Where you’ll use it:
Motor control, high-power switching, battery-powered circuits.

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🔘 6. Push Button: The Simplest Control Element

A push button has no polarity.
When it’s pressed, it closes the circuit and allows current to flow.
When released, the circuit opens.

Where you’ll use it:
Doorbells, reset buttons, project inputs, counters.

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🎚️ 7. Potentiometer: A Variable Resistor

A pot has three terminals and allows you to vary resistance manually.

• The two outer pins give a fixed resistance (e.g., 10kΩ).

• The middle pin gives a variable resistance based on the rotation.

Where you’ll use it:
Volume control, brightness adjustment, sensitivity control.

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🌀 8. 5V DC Fan: A Perfect Motor for Starters

A 5V DC fan can run safely on USB power or Arduino.

• Red wire = +5V

• Black wire = GND

The connector at the end makes it easy to plug into jumper wires.

Where you’ll use it:
Cooling, mini robots, airflow sensors.

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🔗 9. Jumper Wires: The Nervous System of the Breadboard

Jumper wires connect everything together.

Types you’ll commonly use:

• Male-to-Male

• Male-to-Female

• Female-to-Female

Each one has a different purpose depending on the connection required.

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🧱 10. Breadboard: Your First Real Workspace

A breadboard lets you build circuits without soldering.
You simply push the component legs into the board and connect them using jumper wires.

But remember:
A breadboard does not generate its own power.

So you need a power source like…

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💻 Powering Your Breadboard Using an Arduino Uno

The Arduino Uno receives power from your computer via USB.
You can then use its 5V and GND pins to power the breadboard:

• Connect 5V → Positive rail

• Connect GND → Negative rail

Once powered, all your components become active, and you can start experimenting with real circuits.

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🧪 What You Can Build After Learning These Components

Once you know these basics, you can make:

• LED blink circuits

• Automatic night lamp

• Touch sensor

• Temperature-based fan

• Simple motor driver

• RGB light controller

• Basic alarm systems

These are perfect for school projects, engineering mini-projects, and learning Arduino.

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📥 Download the PDF with High-Res Images

A separate PDF with images, labels, and identification markers is available in the video description.
Perfect for printing or revision.

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🎯 Final Thoughts

Electronics doesn’t have to be complicated.
Start small. Understand each component. Combine them gradually.
Within a few projects, you’ll find yourself confidently building impressive circuits.

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