chapter 2 and 3.17,3.18,3.19 || English Medium ||

Kalaimagal Academy

10th Standard Mathematics - Special Test Paper

Time: 1 Hr 30 Mins	Max. Marks: 50
Class: 10th Standard	Portion: Chapter 2 & Matrices (Ex 3.17 - 3.19)
Student Name: _______________________	Roll No: ____________
Date: ____________

PART - I: CHOOSE THE CORRECT ANSWER 10 × 1 = 10 Marks

1. 1. Euclid's division lemma states that for positive integers \( a \) and \( b \), there exist unique integers \( q \) and \( r \) such that \( a = bq + r \), where \( r \) must satisfy:
   (a) \( 1 < r < b \)

   (b) \( 0 < r < b \)

   (c) \( 0 \le r < b \)

   (d) \( 0 < r \le b \)
2. 2. Using Euclid's division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are:
   (a) 0, 1, 8

   (b) 1, 4, 8

   (c) 0, 1, 3

   (d) 1, 3, 5
3. 3. If the HCF of 65 and 117 is expressible in the form \( 65m - 117 \), then the value of \( m \) is:
   (a) 4

   (b) 2

   (c) 1

   (d) 3
4. 4. The sum of the exponents of the prime factors in the prime factorization of 1729 is:
   (a) 1

   (b) 2

   (c) 3

   (d) 4
5. 5. The next term of the sequence \( \frac{3}{16}, \frac{1}{8}, \frac{1}{12}, \frac{1}{18}, \dots \) is:
   (a) \(\frac{1}{24}\)

   (b) \(\frac{1}{27}\)

   (c) \(\frac{2}{27}\)

   (d) \(\frac{1}{32}\)
6. 6. If matrix \( A = [a_{ij}]_{2 \times 3} \), then the total number of elements in \( A \) is:
   (a) 5

   (b) 6

   (c) 2

   (d) 3
7. 7. If \( A \) is a matrix of order \( 3 \times 4 \) and \( B \) is a matrix of order \( 4 \times 2 \), then the order of \( AB \) is:
   (a) \( 3 \times 2 \)

   (b) \( 4 \times 4 \)

   (c) \( 3 \times 4 \)

   (d) Product not possible
8. 8. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), then \( A + A^T \) is a:
   (a) Diagonal Matrix

   (b) Symmetric Matrix

   (c) Skew-Symmetric Matrix

   (d) Zero Matrix
9. 9. Transpose of a row matrix is a:
   (a) Unit matrix

   (b) Diagonal matrix

   (c) Column matrix

   (d) Square matrix
10. 10. If \( \begin{pmatrix} y & 0 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 2x & 0 \\ 1 & 3 \end{pmatrix} \), then \( y \) in terms of \( x \) is:
    (a) \( x \)

    (b) \( 2x \)

    (c) \( x/2 \)

    (d) \( x^2 \)

PART - II: SHORT ANSWER QUESTIONS 10 × 2 = 20 Marks

11. 11. Find the HCF of 252 and 105 using Euclid's Division Algorithm.
12. 12. Solve: \( 8x \equiv 1 \pmod{11} \).
13. 13. If \( a \) and \( b \) are two positive integers such that \( a^b \cdot b^a = 800 \), find \( a \) and \( b \).
14. 14. Find the 19th term of an A.P. \( -11, -15, -19, \dots \)
15. 15. Find the sum of the first 15 terms of the A.P. \( 8, 3, -2, \dots \)
16. 16. Construct a \( 3 \times 3 \) matrix whose elements are given by \( a_{ij} = |i - 2j| \). [Ex 3.17]
17. 17. If \( A = \begin{pmatrix} 7 & 8 & 6 \\ 1 & 3 & 9 \\ -4 & 3 & -1 \end{pmatrix} \), find the transpose of \( A \) (i.e., \( A^T \)). [Ex 3.17]
18. 18. If \( A = \begin{pmatrix} 0 & 4 & 9 \\ 8 & 3 & 7 \end{pmatrix} \) and \( B = \begin{pmatrix} 7 & 3 & 8 \\ 1 & 4 & 9 \end{pmatrix} \), find \( 3A - 9B \). [Ex 3.18]
19. 19. If \( A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \), find \( AB \). [Ex 3.19]
20. 20. Show that the matrices \( A = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & -2 \\ -3 & 1 \end{pmatrix} \) commute under multiplication or check whether \( AB = BA \). [Ex 3.19]

PART - III: DETAILED ANSWER QUESTIONS 4 × 5 = 20 Marks

21. 21. Find the sum of all natural numbers between 300 and 600 which are divisible by 7.
22. 22. Find the sum to \( n \) terms of the series \( 5 + 55 + 555 + \dots \)
23. 23. If \( A = \begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix} \), show that \( A^2 - 4A + 4I_2 = O \). [Ex 3.19]
24. 24. Solve for \( X \) and \( Y \) if \( X + Y = \begin{pmatrix} 7 & 0 \\ 3 & 5 \end{pmatrix} \) and \( X - Y = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \). [Ex 3.18]

*** ALL THE BEST ***