CBSE Class 12 Maths 2014 Previous Year Question Paper
This is a CBSE Class 12 Maths previous year question paper from 2014, focusing on the topic of Tangents and Normals. The paper includes questions carrying 1 mark and 4 marks, testing students' ability to find the slope of tangents to curves at given points and to determine the equations of tangents and normals to parametric curves. For instance, one 1-mark question asks for the slope of the tangent to the curve y = 3x² - 6 at x = 2. A 4-mark question requires finding the equations of the tangent and normal to the curves x = a sin³θ and y = a cos³θ at θ = π/4. Solving this board question paper helps students understand the exam pattern, identify important concepts within Tangents & Normals, and enhance their problem-solving skills for the upcoming CBSE board examinations.
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Quick info
| Board | CBSE |
|---|---|
| Class | 12 |
| Subject | Maths |
| Session | 2014 |
| Language | English |
| Type | Previous Year Question Paper |
| Exam type | Board Exam |
Paper pattern
The paper includes 1-mark and 4-mark questions related to tangents and normals.
Topics covered
Paper topics
- Tangents
- Normals
- Slope of Tangent
- Parametric Curves
Important topics
- Slope of tangent to curves
- Equations of tangent and normal to parametric curves
PDF preview
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Question paper text
Tangentsand Normals
1 Marks Questions
- Find the slope of tangent to the curve <math>y = 3x^2 - 6</math> at the point on it whose x-coordinate is 2.
All India 2009C
Firstly, differentiate the given function with respect to x and then determine the value of <math>\frac{dy}{dx}</math> at <math>x=2</math>.
Given, <math>y = 3x^2 - 6</math>
On differentiating both sides w.r.t. x, we get
<math display="block">\frac{dy}{dx} = 6x</math>
At <math>x = 2</math>, slope of tangent
<math display="block">\Rightarrow \left[ \frac{dy}{dx} \right]_{x=2} = 6 (2) = 12</math> F , 7
∴ Required slope = 12
- Find the slope of tangent to the curve <math>y = 3x^2 - 4x</math> at point whose x-coordinate is 2. Delhi 2009C
Given, <math>y = 3x^2 - 4x</math>
On differentiating both sides w.r.t. x, we get
<math>\frac{dy}{dx} = 6x - 4</math>
At <math>x = 2</math>, slope of tangent
<math display="block">\Rightarrow \left[\frac{dy}{dx}\right]_{x=2} = 6(2) - 4 = 12 - 4 = 8</math>
∴ Required slope = 8
(1)
3. Find the slope of the tangent to the curve <math>y = 3x^4 - 4x</math> at <math>x = 1</math>.
All India 2008C
Do same as Que. 2.
[Ans. 8]
- For the curve <math>y = 3x^2 + 4x</math>, find the slope of tangent to the curve at point, where x-coordinate is -2. Delhi 2008C
Do same as Que. 2.
<math>[Ans. - 8]</math>
4 Marks Questions
- Find the equations of the tangent and normal to the curves <math>x = a \sin^3 \theta</math> and <math>y = a \cos^3 \theta</math> at <math>\theta = \frac{\pi}{4}</math>.
Delhi 2014
Given, <math>x = a \sin^3 \theta</math>
On differentiating w.r.t. <math>\theta</math>, we get
<math display="block">\frac{dx}{d\theta} = a(3\sin^2\theta\cos\theta)</math>
<math display="block">\frac{dx}{d\theta} = 3a \sin^2 \theta \cos \theta</math>
<math>y = a \cos^3 \theta</math> and (1)
On differentiating w.r.t. <math>\theta</math>, we get
<math display="block">\frac{dy}{d\theta} = -3a\cos^2\theta\sin\theta</math>
Then,
<math display="block">\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{dx}} = \frac{-3a \cos^2 \theta \sin \theta}{3a \sin^2 \theta \cos \theta}</math>
<math>d\theta</math>
- cot θ dx (1)
Frequently asked questions
What is this document?
This is a CBSE Class 12 Maths previous year board question paper from 2014, focusing on the topic of Tangents and Normals.
What is the main topic covered?
The main topic covered in this paper is Tangents and Normals, including finding slopes and equations of tangents and normals.
What types of questions are included?
The paper includes 1-mark questions for finding the slope of a tangent and 4-mark questions for finding equations of tangents and normals to curves.
How does solving this paper help?
Solving this previous year question paper helps students understand the exam pattern, practice important concepts, and improve their scores in the CBSE Class 12 Maths board exam.
What is the year and class for this paper?
This is a CBSE Class 12 Maths previous year question paper from the year 2014.
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