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Graph of a Function

Graph of a Function

by Sooraj Prakash -
Number of replies: 1

I had a doubt in the following question of COORDINATE GEOMETRY.

The graph of the function

f(x) = cos x .cos (x+2) – cos2(x+1) is

(a)         a straight line passing through (0,0)

(b)        a straight line passing through (0,-sin21) with slope 2.

(c)         a parabola with vertex (1,- sin21)

(d)        a straight line passing through the point

       (pie/2, - sin21) and parallel to the x-axis.

Please ans. it.

The ans. to this question is option (d) but explanation is required.

 

 

 

 

In reply to Sooraj Prakash

Re: Graph of a Function

by Ankul Garg -
Dear Sooraj,
Just simplify the fn. as follows:
f(x) = cos x .cos (x+2) – cos2(x+1)
=> f(x) = 1/2 [2cosx . cos(x+2)] - cos2(x+1)
           = 1/2[cos2(x+1) + cos2] - cos2(x+1)                     [using 2cosA.cosB =
                                                                                      cos(A+B)+cos(A-B)]
           = 1/2[2cos2(x+1) - 1 + cos2] - cos2(x+1)        [ use cos2A = 2cos2A - 1]
           = cos2(x+1) - 1/2 + 1/2cos2 - cos2(x+1)
           = -1/2(1-cos2)
           = - sin21                         [ use cos2A = 1- 2sin2A]
Now, the fn. reduces to y = - sin21
where (- sin21) is a constant term
i.e. eqn. of curve is of the form y = constant i.e. the curve will pass through a general point (x , constant) and will be parallel to X-axis.
Here only the (d) option suffises for these conditions.
In option (d) the pt. is (pie/2, - sin21) i.e. x = pie/2 (as abscissa can have any value) and y = constant = - sin21.