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The rectangular components of a vector lying in the xy plane are 1 and p+1. If coordinate system turned by 30, they are p and 4 respectively the value of p is:
A) 2
B) 4
C) 3.5
D) 7

Answer
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Hint:
First, we will assume that the components after rotation be x and y respectively, such that we have x=xcosθ+ysinθ and y=ycosθxsinθ. Then we will find these values from the problem and then substitute the values in the assumed expression to find the value of p.

Complete step by step solution:
We are given that the rectangular components of a vector lying in xy plane are 1 and p+1.
Let us assume that the components after rotation be x and y respectively, such that we have
x=xcosθ+ysinθ
y=ycosθxsinθ
Since we are given that when θ=30, the coordinates are p and 4.
Finding the value of x, y, x and y, we get
x=1
y=p+1
x=p
y=4
Substituting these above values x, y and x in the equation for x, we get
p=1cos30+(p+1)sin30p=cos30+(p+1)sin30
Using the value of cos30=32 and sin30=12 in the above equation, we get
p=32+(p+1)2
Substituting these above values x, y and y in the equation for y, we get
4=(p+1)cos301sin304=(p+1)cos30sin30
Using the value of cos30=32 and sin30=12 in the above equation, we get
4=(p+1)3212
Multiplying the above equation by 2 on both sides, we get
42=2((p+1)3212)8=(p+1)318=p3+31
Adding the above equation with 1 on both sides, we get
8+1=p3+31+19=p3+3
Taking 3 common from the right hand side of the above equation, we get
9=(p+1)3
Dividing the above equation by 3 on both sides, we get
93=(p+1)3393=p+1
Rationalizing the left hand side of the above equation by multiplying 3 with numerator and denominator, we get
9×33×3=p+19×33=p+133=p+1
Subtracting the above equation by 1 on both sides, we get
331=p+11331=pp=331p=4

Hence, option B is correct.

Note:
We need to know that rectangular components are from a vector, one for the x–axis and the second one for the y–axis. Students should use the values of trigonometric functions really carefully. Some angles can also be resolved along with these vectors. If A is a vector then its x component is Ax and its y component is Ay.