Explanation

The given inequality is 3 < x < 7, which can be rewritten as |x-5|<2.

Explanation

A cubic equation is a polynomial equation of degree 3, and it can have up to 3 real roots.

Explanation

• A cycle of length 2 is called a transposition.
• A transposition is a permutation that swaps two elements, leaving the others unchanged.
• For example, the permutation (1 2) is a transposition that swaps the elements 1 and 2.

Explanation

• The graph of a linear function is always a straight line.
• Linear functions have the form: f(x) = mx + b
• Where m is the slope and b is the y-intercept. The graph of such a function is a straight line with a constant slope.

Explanation

Ext(A) represents the exterior of set A, which is the set of all points in the universal set X that are not in the closure of A (denoted by A°).

The interior of the closure of A (Int(A°)) consists of all points in A° that are not boundary points.

Since the exterior of A consists of all points in X that are not in A°, Ext(A) is equal to Int(A°).

Explanation

Eccentricity of ellipse is 0 < e < 1

If e = 0, ellipse becomes a perfect circle

Explanation

Let the unknown number = x
(1/5 + x)/2 = 1/4 → Multiply both sides by 2:
1/5 + x = 1/2 → x = 1/2 - 1/5 = (5−2)/10 = 3/10 = ⅖

Explanation

Ahmad stands at point D, 2m in front of a spotlight at point A. He is 1.6m tall and is facing the wall of a building which is 10.5m away from him. How tall (BC) is his shadow on the wall of the building (5 marks)

To find the height $BC$ of Ahmad's shadow on the wall of the building, we can use the properties of similar triangles.

Given:

1. Ahmad's height $DE = 1.6$ m.
2. Distance from the spotlight at $A$ to Ahmad at $D$is $2$ m.
3. Distance from Ahmad at $D$ to the wall at $B$ is $10.5$ m.
4. Total distance $AB = 2 + 10.5 = 12.5$m.

Using similar triangles:

$$$$BCAB=DEAD​

Substitute the values:

$$frac{BC}{12.5} = frac{1.6}{2}$$ $$BC = 12.5 times frac{1.6}{2} = 12.5 times 0.8 = 10 text{ m}$$

Height $BC = 10$ m.

Explanation

• 1. The ratio of length to breadth is 3:2. Let's assume the length is 3x and the breadth is 2x.
• 2. The perimeter of the rectangular park is 2(3x + 2x) = 10x.
• 3. The man completes one round in 8 minutes at a speed of 12 km/hr. Convert the speed to meters per minute:

12 km/hr = 12,000 m/60 min = 200 m/min

• 1. The distance covered in 8 minutes is:

200 m/min × 8 min = 1600 m
Since the man completes one round, the perimeter of the park is 1600 m:
10x = 1600
x = 160
Now, find the length and breadth:
Length = 3x = 3 × 160 = 480 m
Breadth = 2x = 2 × 160 = 320 m
Finally, calculate the area:
Area = Length × Breadth = 480 × 320 = 153,600 sq. m

Explanation

This is the formula for the tangent of the sum of two angles: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ)