Question 1 – Solution

a) The maximum number of books that can be arranged on each shelf without mixing genres is the highest common factor (HCF) of 36, 54, and 90. HCF = 18.

b) Drama books = 36/18 = 2 shelves, History books = 54/18 = 3 shelves, Science books = 90/18 = 5 shelves. Total = 2 + 3 + 5 = 10 shelves.

c) Total books = 36 + 54 + 90 = 180. If a student can borrow 5 books per day, it would take 180/5 = 36 days to borrow all books.

d) The least common multiple (LCM) of 36, 54, and 90 is 540.

Question 2 – Solution

a) Prime factorization of 225 = 3 x 3 x 5 x 5 or \(3^2 \times 5^2\).

b) The maximum number of flowers each bouquet can contain is the HCF of 270, 225, and 405. HCF = 45.

c) Number of bouquets = 270/45 + 225/45 + 405/45 = 6 + 5 + 9 = 20 bouquets.

d) 270, 225, and 405 are not coprime as they have common factors other than 1.

Question 3 – Solution

a) Total spent on snacks = ₹1550 – (2 x ₹400) = ₹750. Possible orders: Pizza + Chips + Water + Ice Cream = ₹200 + ₹130 + ₹30 + ₹100 = ₹460.

b) Combo cost = Pizza + Ice Cream + Soda = ₹200 + ₹100 + ₹150 = ₹450. Additional amount needed = ₹450 – amount not used from ₹460 = ₹10.

Question 4 – Solution

i) EG = FH because C is the midpoint of FH, which implies FC = CH and GH = EF. So, EG = EF + FG = EF + FC = FH.

ii) Since C is the midpoint of FH, CH = 1/2 FH. So, EF = FH – GH = FH – CH = 1/2 FH. EH = EF + FG + GH = EF + FC + CH = EF + 1/2 FH = 1/3 EH.

Question 5 – Solution

a) A rough figure would show two lines intersecting at X.

b) True. Two chords in a circle will always intersect, either inside the circle or, if extended, outside.

c) The sketch would show a four-sided shape with lines drawn between opposite corners. The intersection of the diagonals would be inside the quadrilateral.

Question 6 – Solution

a) Sum on the number line = -6 – 2 = -8.

b) Computation result = -250 + 324 – 237 + 630 = 467.

c) Using the given values, pq = -|12-18| + 5 = -|-6| + 5 = -6 + 5 = -1.

Question 7 – Solution

a) The HCF of two consecutive even numbers is 2.

b) For a number to be divisible by 9, the sum of its digits must be divisible by 9. Summing up the digits we have: 7 + 9 + 2 + 8 + 4 + 5 = 35. To make it divisible by 9, the sum must be the next multiple of 9 after 35, which is 36. Thus, the smallest value of * is 1.

c) The LCM of 15, 25, and 40 is 600. Adding 4 to the LCM, we get 604. However, this isn’t a 4-digit number. To get the smallest 4-digit number that fulfills the condition, we can add 600 (the LCM) to 604. So, 1204 is the number.

Question 8 – Solution

First, compute the sums: -256 + 152 = -104 and 452 – 782 = -330. Subtracting these results, -330 – (-104) = -226.

Question 9 – Solution

A trapezium (or trapezoid in some countries) is a four-sided figure with one pair of parallel sides. A trapezium is called an isosceles trapezium when its non-parallel sides are equal in length. An isosceles trapezium will also have angles that are equal where the sides meet the longer base.

Question 10 – Solution

a) Using the distributive property: 12345 * (28 + 12 + 60) = 12345 * 100 = 1,234,500.

b) Using multiplication: 1098 * 97 = 106,506.

Question 11 – Solution

Let the unknown number be x. From the equation x + (-9) = 13, x = 13 + 9 = 22.