answer question and explain 99 103 104 105
Answers
99)
If a number is divisible by 11, then the difference between the sum of alternate digits of that number will be divisible by 11.
E. g.: Check whether 54763786 is a multiple of 11.
First take the sum of alternate digits of the number.
54763786:
5 + 7 + 3 + 8 = 23
4 + 6 + 7 + 6 = 23
Then take the difference of the sums obtained. The difference can be the value of all integers, including 0 and negatives.
23 - 23 = 0
Here, 0 is got. ∴ 54763786 is a multiple of 11.
Okay, let's have a look at 1372x413.
Sum of alternate digits is to be taken first.
1372x413:
1 + 7 + x + 1 = x + 9
3 + 2 + 4 + 3 = 12
Here, 12 - (x + 9) is a multiple of 11.
12 - (x + 9)
= 12 - x - 9
= 3 - x
∴ 3 - x is a multiple of 11.
If x = 3, 3 - x = 3 - 3 = 0 is a multiple of 11.
If x = 14, 3 - x = 3 - 14 = -11 is a multiple of 11. But x can't be 14 as x is a digit of 1372x413 and 14 is a two-digit number.
∴ The answer is 3, i. e., option (D).
103)
The diagonals of any square makes the angles on either sides of the diagonal 45°. That's a feature of a square.
Here, BD is a diagonal.
∴∠CDB = ∠DBA = ∠ADB = ∠DBC = 45°
Let the point where x° is located be P. The point is not marked in the question. So I suggested a name for the point.
Here, ∠DOC = ∠POB = 85° (Alternate angles)
Found that ∠DBA = 45°
By considering ΔPOB,
∠POB = 85°
∠DBA (Or ∠OBP for better) = 45°
∴ ∠BPO = 180° - (85° + 45°) = 180° - 130° = 50°
∠APC and ∠BPC (i. e., ∠BPO) are linear pairs.
∴ ∠APC = x° = 180° - ∠BPC = 180° - 50° = 130°
∴ 130° is the answer.
104)
∠EOS = ∠BOT = 50° (Alternate angles)
In any rectangle, the diagonals have equal length and bisect each other.
∴ OB = OE = OS = OT
In ΔBOT,
∠BOT = 50°
OB = OT
∴ ∠OTB = ∠OBT
∠OTB + ∠OBT = 180° - ∠BOT = 180° - 50° = 130°
∠OTB = ∠OBT = 130° ÷ 2 = 65°
∴ 65° is the answer.
105)
Let the sides be 4x and 5x.
Perimeter = 2(l + b) = 90 cm
= 2(4x + 5x) = 90 cm
= 2 × 9x = 90 cm
= 18x = 90
x = 90 ÷ 18 = 5
4x = 4 × 5 = 20 cm
5x = 5 × 5 = 25 cm
∴ Sides are 20, 25.
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