Binomial Theorem quiz

The number of terms in the expansion of (x + y)^7 is:

The sum of the coefficients in the expansion of (1 + x)^5 is:

The coefficient of x^2 in (1 + x)^5 is:

The value of 5C2 is:

In Pascal's triangle, the third row (for n = 2) is:

The last term in the expansion of (2 - x)^4 is:

The coefficient of x^3 in (x + 2)^5 is:

The ancient Indian name for Pascal's triangle is:

(a + b)^3 - (a - b)^3 equals:

The middle entry in the row of Pascal's triangle for n = 4 is:

Expand (x + 2)^5. (2 marks)

Expand (1 - 2x)^3. (2 marks)

Write the expansion of (x/3 + 1/x)^4. (2 marks)

Using Pascal's triangle, write the coefficients of (a + b)^5. (2 marks)

Find the value of 6C0 + 6C1 + 6C2 + ... + 6C6. (2 marks)

Evaluate (101)^4 using the binomial theorem. (2 marks)

Expand (sqrt x - 1)^4 for x > 0. (2 marks)

Show that the coefficients equidistant from the beginning and end of a binomial expansion are equal. (2 marks)

Find the value of (1.02)^6 correct to four decimal places. (2 marks)

Expand (3x^2 - 2)^3. (2 marks)

Evaluate 99^5 using the binomial theorem. (3 marks)

Find (1.1)^5 using the binomial theorem. (3 marks)

Evaluate (sqrt2 + 1)^4 + (sqrt2 - 1)^4. (3 marks)

Find the sum of all coefficients in the expansion of (3x - 1)^5 by putting x = 1. (3 marks)

Show that 6^n - 5n always leaves the remainder 1 when divided by 25, for n a positive integer. (3 marks)

Read the passage and answer the questions. A teacher wrote Pascal's triangle on the board up to n = 5. Students noticed that each row begins and ends with 1, that each other number is the sum of the two numbers just above it, and that the numbers in each row are symmetric. (i) Write the row for n = 5. (ii) Write the expansion of (x + y)^5 using this row. (iii) What is the sum of the numbers in the row for n = 5? (5 marks)

Read the passage and answer the questions. A bank pays 2% compound interest per year. The amount after 5 years on a deposit of Rs 10,000 is 10,000 (1.02)^5. (i) Write (1.02)^5 using the binomial theorem. (ii) Find its value correct to four decimal places. (iii) Find the amount. (5 marks)

Read the passage and answer the questions. A student claimed that (1.01)^100 is less than 2. Her friend used the binomial theorem to check: (1 + 0.01)^100 = 1 + 100(0.01) + 100C2 (0.01)^2 + ... (i) What are the first two terms? (ii) Evaluate the third term. (iii) Is the claim correct? Explain. (5 marks)

Read the passage and answer the questions. In the expansion of (a + b)^n, the general pattern is that the power of a decreases from n to 0 and the power of b increases from 0 to n, while the coefficients are nC0, nC1, ..., nCn. (i) Write the expansion of (a + b)^3. (ii) Write the expansion of (a - b)^3. (iii) Find (a + b)^3 + (a - b)^3. (5 marks)

Read the passage and answer the questions. The number of subsets of a set with n elements equals the sum of nC0 + nC1 + ... + nCn. A club has 6 members and wants to know how many different groups (including the empty group) can be formed. (i) Express this as a binomial sum. (ii) Find the number of groups. (iii) How many groups have exactly 2 members? (5 marks)

State and prove the binomial theorem for positive integral index. (6 marks)

Expand (x^2 + 2/x)^5 and simplify each term. (6 marks)

Using the binomial theorem, find (0.98)^5 and 1.05^4 correct to four decimal places. (6 marks)

Prove that the sum of the binomial coefficients nC0 + nC1 + ... + nCn = 2^n, and that nC0 - nC1 + nC2 - ... + (-1)^n nCn = 0. (6 marks)

Find (x + 1)^6 + (x - 1)^6. Hence or otherwise evaluate (sqrt2 + 1)^6 + (sqrt2 - 1)^6. (6 marks)