15 extra multiple-choice questions for Numbers and Sequences (10th Standard Mathematics, Samacheer Kalvi), beyond the ones printed in the textbook — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
If the HCF of 120 and 45 is expressible in the form \(120x - 225\), then the value of \(x\) is
- A. 1
- B. 2Correct
- C. 3
- D. 4
Explanation. The HCF of 120 and 45 is 15. Equating \(120x - 225 = 15\) gives \(120x = 240\), which simplifies to \(x = 2\).
Q2
If a positive integer \(n\) is divided by \(7\), the remainder is \(5\). What is the remainder when \(n^2\) is divided by \(7\)?
- A. 1
- B. 2
- C. 3
- D. 4Correct
Explanation. Since \(n \equiv 5 \pmod{7}\), we have \(n^2 \equiv 25 \pmod{7}\). Dividing \(25\) by \(7\) leaves a remainder of \(4\).
Q3
If \(2^a \times 3^b \times 5^c = 3600\), where \(a\), \(b\), and \(c\) are natural numbers, then the value of \(a + b + c\) is
- A. 6
- B. 7
- C. 8Correct
- D. 9
Explanation. The prime factorization of \(3600\) is \(2^4 \times 3^2 \times 5^2\). Thus, \(a=4\), \(b=2\), and \(c=2\), so \(a+b+c = 8\).
Q4
The least positive integer value of \(x\) satisfying the congruence \(5x \equiv 4 \pmod{6}\) is
- A. 2Correct
- B. 4
- C. 5
- D. 8
Explanation. The congruence \(5x \equiv 4 \pmod{6}\) means \(5x - 4\) is a multiple of \(6\). For \(x=2\), \(5(2)-4 = 6\), which is divisible by \(6\).
Q5
If the general term of a sequence is \(a_n = (-1)^n (n^2 + 1)\), then the value of \(a_4 - a_3\) is
- A. 7
- B. 17
- C. 27Correct
- D. 37
Explanation. We calculate \(a_4 = (-1)^4(4^2 + 1) = 17\) and \(a_3 = (-1)^3(3^2 + 1) = -10\). Their difference \(a_4 - a_3 = 17 - (-10) = 27\).
Q6
If the \(7^{\text{th}}\) term of an Arithmetic Progression is \(40\) and the \(13^{\text{th}}\) term is \(76\), then its common difference is
- A. 4
- B. 5
- C. 6Correct
- D. 8
Explanation. Subtracting \(a + 6d = 40\) from \(a + 12d = 76\) gives \(6d = 36\), which yields the common difference \(d = 6\).
Q7
The sum of the first \(20\) terms of an Arithmetic Progression with first term \(5\) and common difference \(4\) is
- A. 800
- B. 820
- C. 860Correct
- D. 900
Explanation. Using the formula \(S_n = \frac{n}{2}[2a + (n-1)d]\), we get \(S_{20} = 10[10 + 19(4)] = 10(86) = 860\).
Q8
If the \(3^{\text{rd}}\) term of a Geometric Progression is \(24\) and the \(6^{\text{th}}\) term is \(192\), then its first term is
- A. 3
- B. 4
- C. 6Correct
- D. 12
Explanation. Dividing \(ar^5 = 192\) by \(ar^2 = 24\) gives \(r^3 = 8\), so \(r = 2\). Substituting \(r=2\) into \(ar^2 = 24\) yields \(a = 6\).
Q9
The sum to infinity of the Geometric Progression \(12, 4, \frac{4}{3}, \dots\) is
- A. 16
- B. 18Correct
- C. 20
- D. 24
Explanation. Here, \(a = 12\) and \(r = \frac{1}{3}\). The sum to infinity is \(S_\infty = \frac{a}{1-r} = \frac{12}{1 - 1/3} = \frac{12}{2/3} = 18\).
Q10
The sum of the squares of the first \(10\) natural numbers is
- A. 285
- B. 330
- C. 385Correct
- D. 505
Explanation. Using the formula \(\sum n^2 = \frac{n(n+1)(2n+1)}{6}\) for \(n=10\), we get \(\frac{10 \times 11 \times 21}{6} = 385\).
Q11
If the sum of the first \(n\) natural numbers is \(15\), then the sum of their cubes is
- A. 125
- B. 225Correct
- C. 625
- D. 3375
Explanation. The sum of cubes of first \(n\) natural numbers is the square of the sum of the first \(n\) natural numbers, which is \(15^2 = 225\).
Q12
Using Euclid's division lemma, when the integer \(-25\) is divided by \(4\), the quotient and remainder are respectively
- A. \(-6, -1\)
- B. \(-6, 1\)
- C. \(-7, 3\)Correct
- D. \(-7, 1\)
Explanation. According to \(a = bq + r\) with \(0 \le r < |b|\), we write \(-25 = 4(-7) + 3\). Thus, quotient \(q = -7\) and remainder \(r = 3\).
Q13
The number of distinct prime factors in the prime factorization of \(13824\) is
- A. 2Correct
- B. 3
- C. 5
- D. 6
Explanation. The prime factorization of \(13824\) is \(2^9 \times 3^3\). The distinct prime factors are only \(2\) and \(3\), so there are \(2\) distinct prime factors.
Q14
If today is Sunday, then the day of the week \(100\) days from now (excluding today) will be
- A. Monday
- B. TuesdayCorrect
- C. Wednesday
- D. Thursday
Explanation. We compute \(100 \equiv 2 \pmod{7}\). Adding \(2\) days to Sunday gives Tuesday.
Q15
If the three consecutive terms \(x + 2\), \(2x + 3\), and \(5x - 1\) form an Arithmetic Progression, then the value of \(x\) is
- A. 2
- B. 3
- C. 2.5Correct
- D. 3.5
Explanation. Since they are in A.P., we have \(2(2x + 3) = (x + 2) + (5x - 1)\), which simplifies to \(4x + 6 = 6x + 1\). Thus, \(2x = 5\), so \(x = 2.5\).
