The number of integral values of k for which the equation $7 \cos x + 5 \sin x = 2k + 1$ has solution is
- 4
- 8
- 10
- 12
The Correct Option is B
Solution and Explanation
We know that $-r \le a \cos\theta +b \sin\theta \le r$ where $r=\sqrt{a^{2} +b^{2}}$
$ \therefore \, \, - \sqrt{49 +25} \le 7\cos x +5 \sin x \le\sqrt{49+25}$
$ \Rightarrow - \sqrt{74} \le 2k +1 \le \sqrt{74}\Rightarrow -8.6 \le2k + 1 \le 8.6$
$ \Rightarrow -9.6 \le2k \le 7.6 \Rightarrow -4.8 \le k \le 3.8 $
$ \Rightarrow \, \, k $ can take only 8 integral values.
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Concepts Used:
General and Middle Terms
In the expansion of a binomial term (a + b) raised to the power of n, on the basis of value ‘n’, we can write the general and middle terms. Prior to getting into the general and middle terms in binomial expansion,
Here,
First-term = nC0 anb0
Second-term = nC1 an–1b1
Third-term = nC2 an–2b2
Read More: Binomial Expansion Formula
Middle Term of the Binomial Expansion
As it is known, the expansion of (a + b)n contains an (n + 1) number of terms. On the basis of value ‘n’, we can write the middle term or terms of (a + b)n.
That means, that if ‘n’ is even, there will be only one middle term and if ‘n’ is odd, there will be two middle terms.
When ‘n’ is Even
In case ‘n’ is even, then the number of terms in the expansion will be n + 1. Since n is even so n + 1 is odd. Therefore, the ((n+1+1)/2)th