Comprehensive Mathematics Examination Questions (Linear Programming to Trigonometry)
Published:
20. Find the maximum value of the objective function $z = 2x + 3y$ under the constraints $x + y \le 6$, $x - y \ge 2$ and $y \ge 0$.
21. If the function $f(x) = 4x - 1$ is defined,
(a) Find the value of $f(2.999)$ and $f(3.001)$
(b) Is function $f(x)$ continuous at the point $x = 3$? Give reason.
22. Solve by matrix method:
$3x + y = 9$ and $2x - y = 1$
23. The angle between a pair of straight lines represented by equation $2x^2 + kxy + 3y^2 = 0$ is $45^\circ$, find the value of $k$.
24. Prove that:
$\frac{1 + \tan\theta}{1 - \tan\theta} = \frac{1 + \sin2\theta}{\cos2\theta}$
25. If $A + B + C = \pi^C$, prove that:
$\cos2A + \cos2B - \cos2C = 1 - 2\sin A \cdot \sin B \cdot \cos C$
26. From a point 45 m and 20 m away from a tower, the angles of elevation of the top of the tower are complementary, find the height of the tower.
Animated Video Solution
The first half plays free, the full solution is in the app.
Step by Step Written Solution
In this problem, we need to find the value of k if the angle between the pair of straight lines represented by the equation two x squared plus k x y plus three y squared equals zero is forty-five degrees.
Angle Between Pair of Lines
First, let's compare our given equation with the general form of a pair of lines passing through the origin, which is a x squared plus two h x y plus b y squared equals zero.
By comparing coefficients, we find that a equals two, two h equals k, which means h equals k over two, and b equals three.
Now, we use the formula for the angle theta between such lines, which is tangent theta equals plus or minus two times the square root of h squared minus a b, all divided by a plus b.
Let's substitute our values into the formula. Tangent of forty-five degrees equals plus or minus two times the square root of k over two squared minus two times three, all over two plus three.
Substitution
We know that tangent of forty-five degrees is one. On the right side, the denominator becomes five.
Multiplying both sides by five, we get five equals plus or minus two times the square root of k squared over four minus six.
The rest of this solution is on Solvi
6 more steps are locked. Watch the full animated, narrated solution for free.
Snap a photo, solve any question like this.
Watch the Rest for Free
Free to download · First solutions are on us
