Angle Between A Line And A Plane

We are asked to find the area enclosed by the curves

y%20%3D%20%5Csin%20x%20%2B%20%5Ccos%20x

and

y%20%3D%20%7C%20%5Ccos%20x%20-%20%5Csin%20x%20%7C

over the interval

%5Cleft%5B%200%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright%5D

. Step 1: Analyze the functions - The curve

y%20%3D%20%5Csin%20x%20%2B%20%5Ccos%20x

is a smooth continuous curve. - The curve

y%20%3D%20%7C%20%5Ccos%20x%20-%20%5Csin%20x%20%7C

involves the absolute value function, so we need to split it into two cases based on the value of

%5Ccos%20x%20-%20%5Csin%20x

. For

x%20%5Cin%20%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5D

, we know: -

%5Ccos%20x%20-%20%5Csin%20x%20%5Cgeq%200

for

x%20%5Cin%20%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5D

%5Ccos%20x%20-%20%5Csin%20x%20%5Cleq%200

for

x%20%5Cin%20%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5D

Thus, the function

y%20%3D%20%7C%20%5Ccos%20x%20-%20%5Csin%20x%20%7C

becomes: -

y%20%3D%20%5Ccos%20x%20-%20%5Csin%20x

for

x%20%5Cin%20%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5D

y%20%3D%20%5Csin%20x%20-%20%5Ccos%20x

for

x%20%5Cin%20%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5D

Step 2: Set up the integrals The area enclosed by the curves is the sum of two areas: 1. The area between

y%20%3D%20%5Csin%20x%20%2B%20%5Ccos%20x

and

y%20%3D%20%5Ccos%20x%20-%20%5Csin%20x

over the interval

%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5D

. 2. The area between

y%20%3D%20%5Csin%20x%20%2B%20%5Ccos%20x

and

y%20%3D%20%5Csin%20x%20-%20%5Ccos%20x

over the interval

%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5D

. Area for

x%20%5Cin%20%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5D

A_1%20%3D%20%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%20%5Cleft%5B%20(%5Csin%20x%20%2B%20%5Ccos%20x)%20-%20(%5Ccos%20x%20-%20%5Csin%20x)%20%5Cright%5D%20dx%20%3D%20%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%202%5Csin%20x%20%5C%2C%20dx

Integrating:

A_1%20%3D%202%20%5Cleft%5B%20-%5Ccos%20x%20%5Cright%5D_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%20%3D%202%20%5Cleft(%20-%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%2B%20%5Ccos%200%20%5Cright)%20%3D%202%20%5Cleft(%20-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%2B%201%20%5Cright)

A_1%20%3D%202%20%5Cleft(%201%20-%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%5Cright)%20%3D%202%5Cleft(%20%5Csqrt%7B2%7D%20-%201%20%5Cright)

Area for

x%20%5Cin%20%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5D

A_2%20%3D%20%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%20%5Cleft%5B%20(%5Csin%20x%20%2B%20%5Ccos%20x)%20-%20(%5Csin%20x%20-%20%5Ccos%20x)%20%5Cright%5D%20dx%20%3D%20%5Cint_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%202%5Ccos%20x%20%5C%2C%20dx

Integrating:

A_2%20%3D%202%20%5Cleft%5B%20%5Csin%20x%20%5Cright%5D_%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%20%3D%202%20%5Cleft(%20%5Csin%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20-%20%5Csin%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cright)%20%3D%202%20%5Cleft(%201%20-%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%5Cright)

A_2%20%3D%202%20%5Cleft(%20%5Csqrt%7B2%7D%20-%201%20%5Cright)

Step 3: Total area The total area enclosed by the curves is:

A%20%3D%20A_1%20%2B%20A_2%20%3D%202%5Cleft(%20%5Csqrt%7B2%7D%20-%201%20%5Cright)%20%2B%202%5Cleft(%20%5Csqrt%7B2%7D%20-%201%20%5Cright)%20%3D%202%5Csqrt%7B2%7D(%5Csqrt%7B2%7D%20-%201)

Thus, the correct answer is Option B.

A harmonic progression (HP) is the sequence of numbers whose reciprocals form an arithmetic progression (AP). Let $a_1%2C%20a_2%2C%20a_3%2C%20%5Cdots$ be a harmonic progression.
Then, the reciprocals $%5Cfrac%7B1%7D%7Ba_1%7D%2C%20%5Cfrac%7B1%7D%7Ba_2%7D%2C%20%5Cfrac%7B1%7D%7Ba_3%7D%2C%20%5Cdots$ form an arithmetic progression. Let the terms of this AP be denoted by $b_1%2C%20b_2%2C%20b_3%2C%20%5Cdots$ , where: $b_n%20%3D%20%5Cfrac%7B1%7D%7Ba_n%7D$ Given that $a_1%20%3D%205$ and $a_%7B20%7D%20%3D%2025$ , we know the values of $b_1%20%3D%20%5Cfrac%7B1%7D%7B5%7D$ and $b_%7B20%7D%20%3D%20%5Cfrac%7B1%7D%7B25%7D$ . The general term of an arithmetic progression is given by: $b_n%20%3D%20b_1%20%2B%20(n%20-%201)%20%5Ccdot%20d$ where $d$ is the common difference. Substituting the known values for $b_1$ and $b_%7B20%7D$ , we have: $%5Cfrac%7B1%7D%7B25%7D%20%3D%20%5Cfrac%7B1%7D%7B5%7D%20%2B%20(20%20-%201)%20%5Ccdot%20d$
Simplifying: $%5Cfrac%7B1%7D%7B25%7D%20%3D%20%5Cfrac%7B1%7D%7B5%7D%20%2B%2019d$ $%5Cfrac%7B1%7D%7B25%7D%20-%20%5Cfrac%7B1%7D%7B5%7D%20%3D%2019d$ $%5Cfrac%7B-4%7D%7B25%7D%20%3D%2019d$ $d%20%3D%20%5Cfrac%7B-4%7D%7B25%20%5Ctimes%2019%7D%20%3D%20%5Cfrac%7B-4%7D%7B475%7D$ Thus, the common difference is $d%20%3D%20%5Cfrac%7B-4%7D%7B475%7D$ .
Now, the general term of the harmonic progression is: $a_n%20%3D%20%5Cfrac%7B1%7D%7Bb_n%7D%20%3D%20%5Cfrac%7B1%7D%7Bb_1%20%2B%20(n%20-%201)%20%5Ccdot%20d%7D$ Substituting $b_1%20%3D%20%5Cfrac%7B1%7D%7B5%7D$ and $d%20%3D%20%5Cfrac%7B-4%7D%7B475%7D$ , we get: $a_n%20%3D%20%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B5%7D%20%2B%20(n%20-%201)%20%5Ccdot%20%5Cfrac%7B-4%7D%7B475%7D%7D$
Simplifying the expression: $a_n%20%3D%20%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B5%7D%20-%20%5Cfrac%7B4(n%20-%201)%7D%7B475%7D%7D$ Now, to find the least $n$ such that $a_n%3C0$ , we need to solve the inequality: $%5Cfrac%7B1%7D%7B5%7D%20-%20%5Cfrac%7B4(n%20-%201)%7D%7B475%7D%3C0$ Multiplying through by 475: $95%20-%204(n%20-%201)%3C0$ $95%20-%204n%20%2B%204%3C0$ $99%20-%204n%3C0$ $4n%3E99$ $n%3E%5Cfrac%7B99%7D%7B4%7D%20%3D%2024.75$ Thus, the least integer $n$ is $n%20%3D%2025$ . Therefore, the correct answer is Option D.

About Angle Between A Line And A Plane - BITSAT

Angle Between A Line And A Plane is a vital chapter for BITSAT aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Angle Between A Line And A Plane PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Angle Between A Line And A Plane carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
5 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Angle Between A Line And A Plane - BITSAT

Frequently Asked Questions

Why focus on Angle Between A Line And A Plane PYQs?

How to best use this analysis?

Angle Between A Line And A Plane

High-Yield Trend

Chapter Questions 5 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Angle Between A Line And A Plane - BITSAT

Frequently Asked Questions

Why focus on Angle Between A Line And A Plane PYQs?

How to best use this analysis?

Chapter Questions
5 MCQs