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1.

In a rectangle, the length is 3.5 times its width. Its perimeter is 90 cm. Then the area of the rectangle is1. 350 cm22. 35 cm23. 250 cm24. 45 cm2

Answer» Correct Answer - Option 1 : 350 cm2

Concept:

Area of rectangle = Length × Breadth

Perimeter of rectangle = 2 × (Length + Breadth)

Calculation:

Given:

The length and breadth of a rectangle are in the ratio of 3.5 : 1, The perimeter of the rectangle is 90 cm.

Let the length of the rectangle be 3.5x breadth of the recatangle is x

The perimeter of the rectangle is 2 × (L +  B)

Perimeter = 2(3.5x + x)

Perimeter = 9x

∴ 9x = 90

∴ x = 10.

Length = 35 cm and Breadth = 10 cm

Area of rectangle = Length × Breadth

Area = 35 × 10

∴ Area = 350 cm2.
Discussion
2.

If A (3, 2), C (- 3, 2) and D(k, - 1) are the vertices of the quadrilateral ABCD where |AD| = |CD| and the area of Δ ABC is 9 sq. unit. Then the area of quadrilateral ABCD will be:1. 15 sq. unit2. 12 sq. unit3. 36 sq. unit4. 18 sq. unit

Answer» Correct Answer - Option 4 : 18 sq. unit

Concept:

1. Let A (x1, y1), B (x2, y2) and C (x3, y3) be the vertices of a Δ ABC, then area of Δ ABC = \(\frac{1}{2} \cdot \left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ {{x_3}}&{{y_3}}&1 \end{array}} \right|\)

2. Distance formula:  A(x, y) and B(a, b) be any two points, by distance formula,

 \(\rm AB = \sqrt {(x - a)^2 + (y - b)^2}\)

3. (a + b)2 = a2 + 2ab + b2

4. (a - b)2 = a2 - 2ab + b2

Calculation:

Given that, coordinates of point A, C and D are (3, 2), (- 3, 2) and (k, -1) respectively. 

Area of Δ ABC = 9 sq. unit

According to the question, |AD| = |CD| 

Using distance formula,

\(\sqrt {(3 - k)^2 + (2 + 1)^2}\ =\ \sqrt {(-3 \ -\ k)^2 + (2\ +\ 1)^2}\)

Taking square of both side

⇒ (3 - k)2 + 9 = (3 + k)2 + 9

⇒ 3 + k2 - 6k = 9 + k2 + 6k

⇒ 12k = 0 ⇒ k = 0

Therefore, A (3, 2), C (- 3, 2) and D (0, -1)
⇒ Area of Δ ACD = \(\)\(\frac{1}{2} \cdot \left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ {{x_3}}&{{y_3}}&1 \end{array}} \right|\)
⇒ Area of Δ ACD = 9 sq unit
⇒ Area of quadilateral ABCD = Area of ΔABC + Area of Δ ACD
⇒  (9 + 9) sq. units = 18 sq. units
Hence, option 4 is the correct answer.
Discussion
3.

The area enclosed within the curve |x| + |y| = 1 is:1. 2 sq. unit2. 3 sq. unit3. 1 sq. unit4. 5 sq. unit

Answer» Correct Answer - Option 1 : 2 sq. unit

Concept:

1. The intercept form of the line is  \({\rm{\;}}\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\) 

Where a is the x-intercept and b is the y-intercept.

2. Consider a square of side 'a'.

Area of square = a2

Diagonal of square = √2 a

Pythagoras theorem: It states that In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

Distance formula: Let  A = (x1, y1) and  B = (x2, y2) be any two points.

Then Distance between A And B is given by the distance formula.

AB = \(\rm \sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}\) 

Calculation:

Given curve is 

|x| + |y| = 1

⇒ ± x ± y = 1 

On comparing the above equation with the intercept form of line 

\({\rm{\;}}\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\)

We can see that, x & y-intercept are A(1, 0), B(0, 1), C(-1, 0) & D(0, -1)

Clearly ABCD represent square.

Using Pythagoras theorem,

The length of each side = Distance b/w AB or BC or CD or DA

⇒ \(a\ =\ \sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}\)

\(a \ =\ \sqrt{1^1\ +\ 1^2}\ =\ √{2}\)

Therefore, the area of square ABCD

A = (√2)2 = 2 sq. unit.

Hence, option 1 is correct.
Discussion