CBSE Class 6 Mathematics Patterns in Mathematics MCQs Set 11

Answer: (b) number theory

Explanation:
1. Patterns in the whole numbers 0, 1, 2, 3, 4, … are studied in number theory.
2. Geometry deals with patterns in shapes, so it does not fit here.

Teacher's Note:
1. Number theory is for numbers; geometry is for shapes.
2. Learn the two names as a pair.

Answer: (d) 21

Explanation:
1. Triangular numbers are made by adding 1, 2, 3, 4, …: 1, 3, 6, 10, 15, 21, 28, …
2. 21 appears in this list; 12, 20 and 24 do not.
3. So the answer is 21.

Teacher's Note:
1. Write the list by adding the next counting number each time.
2. Stop once you pass the largest option.

Answer: (a) Virahānka numbers

Explanation:
1. Each term is the sum of the two terms just before it: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13.
2. This rule gives the Virahānka numbers.

Teacher's Note:
1. Look for 'add the two previous terms'.
2. Virahānka was an ancient Indian scholar of poetry and rhythm.

Answer: (c) 10

Explanation:
1. The pictures are triangles of dots with 1, 2 and 3 rows, holding 1, 3 and 6 dots.
2. The 4th picture adds a new bottom row of 4 dots.
3. So it has 6 + 4 = 10 dots.

Teacher's Note:
1. Each new picture gets one longer row at the bottom.
2. These are the triangular numbers.

Answer: (a) geometry

Explanation:
1. Patterns in shapes, in 1D, 2D, 3D or more dimensions, are studied in geometry.
2. Number theory studies patterns in whole numbers instead.

Teacher's Note:
1. Geometry means shapes.
2. Do not mix it up with number theory.

Answer: (a) Only Statement I is true.

Explanation:
1. Powers of 3: 1, 3, 9, 27, 81, … and 27 × 3 = 81, so Statement I is true.
2. Powers of 2: 1, 2, 4, 8, 16, 32, … and 16 × 2 = 32, not 24, so Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Powers of 2 double; they do not add 8.
2. Check each statement separately.

Answer: (c) nonagon

Explanation:
1. Heptagon has 7 sides, octagon 8, nonagon 9 and decagon 10.
2. So a 9-sided regular polygon is a nonagon.

Teacher's Note:
1. Hepta = 7, octa = 8, nona = 9, deca = 10.
2. Make a small chart of polygon names.

Answer: (d) 16

Explanation:
1. 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16.
2. Adding odd numbers from 1 gives a square: four odd numbers give 4 × 4 = 16.

Teacher's Note:
1. Sum of the first n odd numbers is n × n.
2. Here n = 4.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 1 + 2 + 3 + … + 8 = 36, so 36 is the 8th triangular number; and 6 × 6 = 36, so it is a square number.
2. Both A and R are true.
3. Being triangular is exactly why 36 dots form a triangle, and being square is why they form a square, so R explains A.

Teacher's Note:
1. A number can have more than one shape.
2. Check whether R gives the reason for A.

Answer: (b) (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)

Explanation:
1. 1, 4, 9, 16, 25, … are squares (s); 1, 8, 27, 64, 125, … are cubes (p).
2. 2, 4, 6, 8, 10, … are even numbers (q); 1, 3, 6, 10, 15, … are triangular numbers (r).
3. So the match is (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r).

Teacher's Note:
1. Spot the easiest pair first, then check the options.
2. Squares: n × n; cubes: n × n × n.

Answer: (d) 61

Explanation:
1. The gaps are 6, 12, 18, so the next gap is 24: 37 + 24 = 61.
2. Check: 6 × 10 + 1 = 61, using the triangular number 10.
3. The next hexagonal number is 61.

Teacher's Note:
1. Gaps between hexagonal numbers go up by 6.
2. Six times a triangular number plus 1 is a good check.

Answer: (c) Only Statement II is true.

Explanation:
1. Mathematics searches for patterns and for the reasons behind them, so Statement I is false.
2. Mathematicians do see mathematics as both an art and a science, so Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. The word 'never' makes Statement I wrong.
2. Finding the 'why' is half of mathematics.

Answer: (a) 100

Explanation:
1. The sum of the first n odd numbers is n × n.
2. Here n = 10, so the total is 10 × 10 = 100.

Teacher's Note:
1. Count how many odd numbers there are: 1 to 19 gives 10.
2. 55 is the sum of 1 to 10, a common trap.

Answer: (d) a square number

Explanation:
1. 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25.
2. All these are square numbers.
3. Two neighbouring dot triangles fit together to make a square of dots.

Teacher's Note:
1. Test two or three examples first.
2. Picture one triangle turned upside down on the other.

Answer: (b) Adding up odd numbers starting from 1 gives square numbers.

Explanation:
1. Starting from the corner, the L-shaped layers hold 1, 3, 5, 7 and 9 dots.
2. Together they fill the 5 × 5 square of 25 dots: 1 + 3 + 5 + 7 + 9 = 25.
3. So the picture shows that adding odd numbers from 1 gives square numbers.

Teacher's Note:
1. Count the dots in each L layer.
2. Each new L layer turns a square into the next bigger square.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Row by row from the top, a stacked triangle holds 1, 3, 5, 7, … little triangles, so R is true.
2. Adding these odd numbers gives 1, 4, 9, 16, 25, …, so A is true.
3. R gives exactly the reason for A.

Teacher's Note:
1. Count a stacked triangle row by row.
2. R here really is the 'why' of A.

Answer: (c) Both statements are true.

Explanation:
1. 1, 1 + 1, 1 + 1 + 1, … gives 1, 2, 3, …, the counting numbers. Statement I is true.
2. 1, 1 + 2, 1 + 2 + 3, … gives 1, 3, 6, 10, …, the triangular numbers. Statement II is true.
3. So both statements are true.

Teacher's Note:
1. 'Adding up' means taking running totals.
2. Write the first four totals to check.

Answer: (d) 3 times the powers of 4

Explanation:
1. Every segment is replaced by 4 smaller segments, so the count is multiplied by 4 each time.
2. 3 = 3 × 1, 12 = 3 × 4, 48 = 3 × 16.
3. So these are 3 times the powers of 4.

Teacher's Note:
1. Divide each term by the one before it.
2. A constant multiplier means powers.

Answer: (b) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. Sides of regular polygons: 3, 4, 5, 6, 7, … (r). Lines in complete graphs: 1, 3, 6, 10, 15, … (p).
2. Little squares in stacked squares: 1, 4, 9, 16, 25, … (s). Koch snowflake segments: 3, 12, 48, 192, … (q).
3. So the match is (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q).

Teacher's Note:
1. Polygons start at 3 sides.
2. Complete graphs give triangular numbers.

Answer: (a) 900

Explanation:
1. Adding counting numbers up to a top number and back down gives the square of the top number.
2. The top number is 30, so the sum is 30 × 30 = 900.

Teacher's Note:
1. Up-and-down sums give squares.
2. 465 is only the 'up' half, a common trap.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. A is true: in any closed figure, each side ends where the next begins, so sides and corners match one to one.
2. R is true: it is the meaning of 'regular'.
3. But equal sides and corners do not explain why the counts are equal; that is because the figure is closed. So R is not the explanation.

Teacher's Note:
1. Even an irregular polygon has as many corners as sides.
2. Ask: would A still hold without R?

Answer: (c) one less than a power of 2

Explanation:
1. The totals are 1, 3, 7, 15, 31, …
2. Adding 1 gives 2, 4, 8, 16, 32, …, which are powers of 2.
3. So each total is one less than a power of 2.

Teacher's Note:
1. Add 1 to each total and look again.
2. 7 = 8 − 1, 15 = 16 − 1.

Answer: (b) 10

Explanation:
1. The figure is the complete graph K₅ on 5 dots.
2. Each dot is joined to 4 others, but each line is counted twice: \( \frac{5 \times 4}{2} \) = 10.
3. So there are 10 lines (5 sides of the pentagon + 5 diagonals).

Teacher's Note:
1. Do not count each line from both ends.
2. Lines in complete graphs are triangular numbers: 1, 3, 6, 10, …

Answer: (d) Only Statement I is true.

Explanation:
1. 1 + 7 + 19 + 37 + 61 = 125, so Statement I is true.
2. 125 = 5 × 5 × 5 is a cube; 11 × 11 = 121 and 12 × 12 = 144, so 125 is not a square. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Sums of hexagonal numbers give cubes, not squares.
2. Check squares near 125 to be sure.

Answer: (d) 10000

Explanation:
1. The sum of the first n odd numbers is n × n.
2. For n = 100: 100 × 100 = 10000.

Teacher's Note:
1. 5050 is the sum of 1 to 100, not of odd numbers.
2. Use the n × n rule.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. 1 + 7 + 19 + 37 = 64, so A is true.
2. 6 × 1 + 1 = 7, 6 × 3 + 1 = 19, 6 × 6 + 1 = 37, so R is true.
3. But the total is 64 because sums of hexagonal numbers give cubes (4 × 4 × 4); R is a different fact, so it does not explain A.

Teacher's Note:
1. Both facts are true but unrelated here.
2. The reason for A is the cube pattern.

Answer: (c) Only Statement II is true.

Explanation:
1. 3, 6, 10 and 15 are triangular but not square, so Statement I is false.
2. 36 is the 8th triangular number and 6 × 6, so Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. One counterexample is enough to reject 'every'.
2. 1 and 36 are both triangular and square.

Answer: (a) (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)

Explanation:
1. Powers of 2 add to 1, 3, 7, 15, …, one less than powers of 2 (s). Odd numbers add to 1, 4, 9, 16, …, squares (p).
2. All 1’s add to 1, 2, 3, 4, …, counting numbers (q). Hexagonal numbers add to 1, 8, 27, 64, …, cubes (r).
3. So the match is (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r).

Teacher's Note:
1. Work out the first three totals of each row.
2. Then name the sequence you see.

Answer: (d) A is false, but R is true.

Explanation:
1. The counts are 1, 4, 9, 16, …, the square numbers, so R is true.
2. The 7th shape has 7 × 7 = 49 little triangles, not 28, so A is false.
3. 28 is the 7th triangular number, not the 7th square number.

Teacher's Note:
1. Do not confuse triangular numbers with stacked triangles.
2. Stacked triangles give squares.

Answer: (c) A is true, but R is false.

Explanation:
1. The counts are 3, 12, 48, 192, …, so the 4th shape has 48 × 4 = 192 segments. A is true.
2. Each segment is replaced by a 'speed bump' of 4 segments, not 3, so R is false.

Teacher's Note:
1. The speed bump has 4 pieces.
2. Multiply by 4 at each step.