የምዕራፍ ማስታወሻዎችና የመልመጃ ጥያቄዎች
በዚህ ገጽ ያለው ሁሉ ለ12ኛ ክፍል የሒሳብ መፅሐፍ ምዕራፍ 4 ነው፦ ምዕራፉ የሚሸፍነው፣ አጭር ማስታወሻዎቹ፣ የመልመጃ ጥያቄዎች ከመልሶቻቸው ጋር፣ እና ቪዲዮ ባለበት የትምህርቱ ቪዲዮ።
በ12ኛ ክፍል የሒሳብ ገጽ ላይ በዚህ ምዕራፍ ሥር የተመዘገቡት ካርዶች። አንድ በአንድ ይክፈቱ።
| የኢ-እኩልነት አይነት | ምልክቶች | የወሰን መስመር | በመስመሩ ላይ ያሉ ነጥቦች ይካተታሉ |
|---|---|---|---|
| ጥብቅ | የተቆራረጠ | አይካተቱም | |
| ላላ | ሙሉ መስመር | ይካተታሉ |
መቼ እንደሚጠቀሙበት
ትክክለኛውን የመስመር አይነት ለመምረጥ እና የወሰን መስመሩ በመፍትሄ ክልል ውስጥ መካተቱን ለመወሰን ይህንን ሰንጠረዥ ይጠቀሙ።
ይጠንቀቁ
ለጥብቅ ኢ-እኩልነት ድፍን መስመር መሳል ወይም ለላላ ኢ-እኩልነት የተቆራረጠ መስመር መሳል በግራፍ ስራ ላይ ነጥብ ያስቀንሳል።
ከGrade 12 Mathematics፣ ገጽ 237-302 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል
መቼ እንደሚጠቀሙበት
የኢ-እኩልነትን ሁለቱንም ጎኖች በኔጌቲቭ ቁጥር ሲያባዙ ወይም ሲያካፍሉ ይህንን ህግ ይጠቀሙ።
ይጠንቀቁ
በኔጌቲቭ ቁጥር ማባዛት ወይም ማካፈል የኢ-እኩልነት ምልክትን ይቀለብሳል፣ ነገር ግን ማንኛውንም ቁጥር መደመር ወይም መቀነስ ምልክቱን አይለውጥም።
ከGrade 12 Mathematics፣ ገጽ 237-302 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል
መቼ እንደሚጠቀሙበት
ባለሁለት ተለዋዋጭ መስመራዊ ኢ-እኩልነትን በግራፍ ሲስሉ የትኛውን የወሰን መስመር ጎን ማጥቆር እንዳለቦት ለመለየት ይህንን ዘዴ ይጠቀሙ።
ይጠንቀቁ
በወሰን መስመሩ ላይ የሚገኝ መፈተሻ ነጥብ ፈጽሞ አይምረጡ። መስመሩ በመነሻ ነጥብ በኩል የሚያልፍ ከሆነ ከ(0, 0) ይልቅ እንደ (0, 1) ወይም (1, 0) ያለ ነጥብ ይፈትሹ።
ከGrade 12 Mathematics፣ ገጽ 237-302 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል
መቼ እንደሚጠቀሙበት
የቃላት ጥያቄዎችን እና የገሃዱ ዓለም ገደቦችን ወደ መስመራዊ ኢ-እኩልነት ሲቀይሩ ይጠቀሙ።
ይጠንቀቁ
ቢበዛ የሚለው ቃል ከ ወይም እኩል ያነሰ ወደሚለው ምልክት ይተረጎማል እንጂ ከሚበልጥ ጋር አይደለም። ቢያንስ የሚለው ደግሞ ከ ወይም እኩል የሚበልጥ ወደሚለው ይተረጎማል።
ከGrade 12 Mathematics፣ ገጽ 237-302 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል
በዚህ ምዕራፍ ላይ የተጻፉ 10 የምርጫ ጥያቄዎች፣ እያንዳንዳቸው መልሱና ምክንያቱ አላቸው። መልሱን ከመክፈትዎ በፊት ይሞክሩ። ጥያቄዎቹ በእንግሊዝኛ ናቸው።
ጥያቄ 1
Solve the inequality 7x + 3 < 5x + 9.
መልስ፦ B x < 3
ምክንያቱ
Subtract 5x from both sides to get 2x + 3 < 9. Then subtract 3 from both sides to get 2x < 6. Divide both sides by the positive number 2 to get x < 3. The sign stays the same because you did not divide by a negative number.
ጥያቄ 2
Alem has Birr 500 to buy wheat flour. One packet costs Birr 130. If x is the number of packets, the inequality is 130x < 500. Which value of x is not a solution?
መልስ፦ D 4
ምክንያቱ
When x = 4, 130 × 4 = 520. But 520 is not less than 500, so the statement is false. The values 0, 1, 2 and 3 all make the inequality true.
ጥያቄ 3
In the inequality y < 5x + 7, the boundary line is drawn as a dashed (broken) line. Why?
መልስ፦ C The points on the line are not included, because the sign is strict (<).
ምክንያቱ
When the sign is < or >, the points on the line are not part of the solution. So you draw a dashed line. When the sign is ≤ or ≥, the points on the line are included, and you draw a solid line.
ጥያቄ 4
A taxi in Addis Ababa charges the total cost T = 0.75x + 1.75, where x is the distance in km and T is in Birr. Selam has at most Birr 10 to spend. What is the maximum distance she can travel?
መልስ፦ C 11 km
ምክንያቱ
Write 0.75x + 1.75 < 10. Subtract 1.75 to get 0.75x < 8.25. Divide by 0.75 to get x < 11. So the maximum distance is 11 km.
ጥያቄ 5
A furniture dealer wants to maximize profit. The objective function is Z = 300x + 100y, where x is the number of tables and y is the number of chairs. The corner points of the feasible region are (0, 0), (20, 0), (10, 50) and (0, 60). What is the maximum profit?
መልስ፦ C Birr 8,000
ምክንያቱ
Substitute each corner point into Z. At (0, 0), Z = 0. At (20, 0), Z = 6000. At (10, 50), Z = 3000 + 5000 = 8000. At (0, 60), Z = 6000. The largest value is 8000, at (10, 50), which means 10 tables and 50 chairs.
ጥያቄ 6
The dealer has Birr 50,000 to invest. A table costs Birr 2500 and a chair costs Birr 500. Which simplified inequality shows this investment limit? (x = number of tables, y = number of chairs)
መልስ፦ B 5x + y ≤ 100
ምክንያቱ
The total spent is 2500x + 500y, and it must be at most 50000. Divide every term by 500 to get 5x + y ≤ 100.
ጥያቄ 7
A linear programming problem has the objective function Z = x + y. The feasible region has corner points (0, 0), (1, 0), (0, 2) and (2, 1). At which point is Z largest?
መልስ፦ D (2, 1)
ምክንያቱ
Calculate Z at each corner. At (0, 0), Z = 0. At (1, 0), Z = 1. At (0, 2), Z = 2. At (2, 1), Z = 3. The largest value is 3, at (2, 1).
ጥያቄ 8
Ficho's budget constraints are: the number of answer sheets (x) is at least 40 more than the number of pencils (y), and the cost is x + 2y ≤ 400 (sheets cost Birr 1 each, pencils cost Birr 2 each). Can Ficho buy 150 answer sheets and 100 pencils?
መልስ፦ C Yes, because both conditions are met.
ምክንያቱ
Check the first condition: 150 is at least 100 + 40 = 140, so it holds. Check the cost: 150 + 2 × 100 = 350, which is at most 400, so it holds. Both conditions are met, so the purchase is possible.
ጥያቄ 9
A world feed programme needs at least 800 kg of special feed each day. The minimum cost objective is Z = 30x + 90y, where x is kg of corn and y is kg of soybean meal. The two corner points that are candidates for the optimum are (400, 400) and (480, 320). What is the minimum daily cost?
መልስ፦ A Birr 43,200
ምክንያቱ
At (400, 400), Z = 30 × 400 + 90 × 400 = 48,000. At (480, 320), Z = 30 × 480 + 90 × 320 = 14,400 + 28,800 = 43,200. The smaller value is the minimum cost, which is Birr 43,200 per day.
ጥያቄ 10
A problem asks to minimize Z = 2x + 4y. The corner points are (0, 10), (2, 4) and (10, 0), giving Z values of 40, 20 and 20. Which statement is correct?
መልስ፦ B The minimum value is 20, and every point on the segment between (2, 4) and (10, 0) gives this value, so the solution is not unique.
ምክንያቱ
Both (2, 4) and (10, 0) give Z = 20, which is the smallest value at the corners. Every point on the line segment between them also gives Z = 20. So the optimal solution is not unique. The value 40 at (0, 10) is larger, not smaller.
መተግበሪያው በዚህ ምዕራፍ ላይ 12 ጥያቄዎች አሉት፤ እየሠሩ ያርምዎታል። እዚህ ያሉት የመጀመሪያዎቹ 10 ናቸው።
ምዕራፍ 4 በየትምህርት ሚኒስቴር እትም ከገጽ 237 እስከ ገጽ 302 ይደርሳል።
የትምህርት ሚኒስቴር እትም, እንግሊዝኛ, 426 ገጾች
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