10 maths chapter 1 and 2 unit test

Kalaimagal Academy

Unit Test II (Mathematics)

Time: 1 Hr 30 Mins	Max. Marks: 50
Class: 10th Standard	Portion: Chapters 1 & 2
Student Name: _______________________	Roll No: ____________
Date: ____________

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

1. 1. If \( n(A \times B) = 6 \) and \( A = \{1, 3\} \), then \( n(B) \) is: [Ex 1.6 (2)]
   (a) 1

   (b) 2

   (c) 3

   (d) 6
2. 2. If \( A = \{a, b, p\} \), \( B = \{2, 3\} \), and \( C = \{p, q, r, s\} \), then \( n[(A \cup C) \times B] \) is: [Ex 1.6 (3)]
   (a) 8

   (b) 20

   (c) 12

   (d) 16
3. 3. If there are 1024 relations from a set \( A = \{1, 2, 3, 4, 5\} \) to a set \( B \), then the number of elements in \( B \) is: [Ex 1.6 (5)]
   (a) 3

   (b) 2

   (c) 4

   (d) 8
4. 4. The range of the relation \( R = \{(x, x^2) \mid x \text{ is a prime number less than } 13\} \) is: [Ex 1.6 (6)]
   (a) {2, 3, 5, 7}

   (b) {2, 3, 5, 7, 11}

   (c) {4, 9, 25, 49, 121}

   (d) {1, 4, 9, 25, 49, 121}
5. 5. If \( f(x) = 2x^2 \) and \( g(x) = \frac{1}{3x} \), then \( f \circ g \) is: [Ex 1.6 (9)]
   (a) \(\frac{3}{2x^2}\)

   (b) \(\frac{2}{3x^2}\)

   (c) \(\frac{2}{9x^2}\)

   (d) \(\frac{1}{6x^2}\)
6. 6. 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) \( 0 < r \le b \)
7. 7. 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
8. 8. 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
9. 9. The sum of the exponents of the prime factors in the prime factorization of 1729 is:
   (a) 1

   (b) 2

   (c) 3

   (d) 4
10. 10. 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) \(\frac{1}{32}\)

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

11. 11. Let \( A = \{1, 2, 3\} \) and \( B = \{x \mid x \text{ is a prime number less than } 10\} \). Find \( A \times B \) and \( B \times A \). [Example 1.3]
12. 12. Let \( A = \{3, 4, 7, 8\} \) and \( B = \{1, 7, 10\} \). Which of the following sets are relations from \( A \) to \( B \)? [Example 1.4]
    (i) \( R_1 = \{(3,7), (4,7), (7,10), (8,1)\} \)
    (ii) \( R_2 = \{(3,1), (4,12)\} \)
13. 13. Let \( A = \{1, 2, 3, 4\} \) and \( B = \{-1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12\} \). Let \( f = \{(1, 2), (2, 5), (3, 8), (4, 11)\} \) be a function. Show that \( f \) is a one-to-one function. [Example 1.16]
14. 14. If \( f(x) = 2x + 1 \) and \( g(x) = x^2 - 2 \), find \( f \circ g \) and \( g \circ f \). [Example 1.25]
15. 15. Let \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) and \( R \) be the relation defined on \( A \) by "$x$ is a square of $y$". Write $R$ as a set of ordered pairs and find its domain and range. [Ex 1.2 (7)]
16. 16. We have 3404 bold nuts and 3010 bolts. How many maximum boxes are required to pack them so that each box contains an equal number of nuts and bolts? [Example 2.1]
17. 17. Find the HCF of 252 and 105 using Euclid's Division Algorithm. [Example 2.3]
18. 18. Solve \( 8x \equiv 1 \pmod{11} \). [Example 2.12]
19. 19. If \( a \) and \( b \) are two positive integers such that \( a^b \cdot b^a = 800 \), find \( a \) and \( b \). [Ex 2.2 (4)]
20. 20. Find the general term of the Arithmetic Progression (A.P.): \( 3, 15, 27, 39, \dots \) [Ex 2.5 (2)]

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

21. 21. A function \( f: [-5, 9] \rightarrow \mathbb{R} \) is defined as follows: [Example 1.19] \[ f(x) = \begin{cases} 6x + 1 & \text{if } -5 \le x < 2 \\ 5x^2 - 1 & \text{if } 2 \le x < 6 \\ 3x - 4 & \text{if } 6 \le x \le 9 \end{cases} \] Find the values of:

    1. \( f(-3) + f(2) \)
    2. \( f(7) - f(1) \)
    3. \( 2f(4) + f(8) \)
    4. \( \frac{2f(-2) - f(6)}{f(4) + f(-2)} \)
22. 22. If \( f(x) = x - 1 \), \( g(x) = 3x + 1 \) and \( h(x) = x^2 \), show that \( (f \circ g) \circ h = f \circ (g \circ h) \). [Example 1.26]
23. 23. Find the sum to $n$ terms of the series \( 5 + 55 + 555 + \dots \) [Example 2.31]
24. 24. Find the sum of all natural numbers between 300 and 600 which are divisible by 7. [Ex 2.6 (2)]

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