Question

The number of integral values of $K$, for which the equation $7\cos x + 5\sin x = 2K + 1$ has a solution, is
A.4
B.8
C.10
D.12

Answer 1

Here, we will use the general form of the given equation and will find the range in which $K$ lies. Using this range, we will be able to find the integral values which $K$ can take. Hence, counting them, will help us to know the required number of integral values.
Formula Used: $- r \le a\cos \theta + b\sin \theta \le r$ where, $r = \sqrt {{a^2} + {b^2}}$

Complete step-by-step answer:
The given equation is: $7\cos x + 5\sin x = 2K + 1$
Now, we know that, if an equation is in the form of $a\cos \theta + b\sin \theta$, then,$- r \le a\cos \theta + b\sin \theta \le r$ where, $r = \sqrt {{a^2} + {b^2}}$.
Comparing this with LHS of the given equation, i.e. $7\cos x + 5\sin x$
Here, $a = 7$ and $b = 5$
Thus, $r = \sqrt {{7^2} + {5^2}}$
Applying the exponent on the terms, we get
$\Rightarrow r = \sqrt {49 + 25}$
Adding the terms, we get
$\Rightarrow r = \sqrt {74}$
Therefore the range becomes,
$- \sqrt {74} \le 7\cos x + 5\sin x \le \sqrt {74}$
Now, we know that $7\cos x + 5\sin x = 2K + 1$, hence, writing the RHS instead of the LHS, we get,
$\Rightarrow - \sqrt {74} \le 2K + 1 \le \sqrt {74}$
We know that $\sqrt {74} = 8.6$, therefore,
$\Rightarrow - 8.6 \le 2K + 1 \le 8.6$
Now, subtracting 1 from each side of the inequality,
$\Rightarrow - 9.6 \le 2K \le 7.6$
Dividing each side of the inequality by 2,
$\Rightarrow - 4.8 \le 2K \le 3.8$
Hence, the integral values of $K$ are:
$- 4, - 3, - 2, - 1,0,1,2,3$
On counting, we will find that there are 8 possible integral values of $K$.
Hence, option B is the correct answer.

Note: In mathematics, integral is either a numerical value equal to the area under the graph of a function for some definite integral or it is a new function whose derivative is the original function. By looking at the question, we can clearly observe that we are required to find the integral i.e. the numerical values which $K$ can take (which can’t be in the form of a fraction or decimal). Hence, in order to find the required number of possible integral values, we had found the range in which $K$ lies. This means that the largest and the smallest possible values in which the value of $K$ lies. Hence, counting all the integrals lying in that range helps us to find the required answer.