Chapter 1,2,4

Kalaimagal Academy

Periodic Test (Mathematics)

Time: 1 Hr 30 Mins	Max. Marks: 50
Class: 10th Standard	Portion: Chapters 1, 2 & 4
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:
   (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:
   (a) 8

   (b) 20

   (c) 12

   (d) 16
3. 3. If \( f(x) = 2x^2 \) and \( g(x) = \frac{1}{3x} \), then \( f \circ g \) is:
   (a) \(\frac{3}{2x^2}\)

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

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

   (d) \(\frac{1}{6x^2}\)
4. 4. 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) \( 0 < r \le b \)
5. 5. 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
6. 6. The sum of the exponents of the prime factors in the prime factorization of 1729 is:
   (a) 1

   (b) 2

   (c) 3

   (d) 4
7. 7. If in \(\Delta ABC\), \(DE \parallel BC\), \(AD = 3\text{ cm}\), \(DB = 4\text{ cm}\), and \(AE = 6\text{ cm}\), then \(EC\) is:
   (a) \( 7\text{ cm} \)

   (b) \( 8\text{ cm} \)

   (c) \( 9\text{ cm} \)

   (d) \( 10\text{ cm} \)
8. 8. If \(\Delta ABC \sim \Delta PQR\) such that \(\text{Area}(\Delta ABC) = 36\text{ cm}^2\) and \(\text{Area}(\Delta PQR) = 64\text{ cm}^2\), if \(PQ = 16\text{ cm}\), then \(AB\) is:
   (a) \( 12\text{ cm} \)

   (b) \( 9\text{ cm} \)

   (c) \( 10\text{ cm} \)

   (d) \( 8\text{ cm} \)
9. 9. A line which intersects a circle at two distinct points is called a:
   (a) Point of contact

   (b) Tangent

   (c) Secant

   (d) Chord
10. 10. How many tangents can be drawn to a circle from an external point?
    (a) 1

    (b) 2

    (c) Infinite

    (d) 0

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 \( B \times A \).
12. 12. 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.
13. 13. If \( f(x) = 2x + 1 \) and \( g(x) = x^2 - 2 \), find \( f \circ g \) and \( g \circ f \).
14. 14. Find the HCF of 252 and 105 using Euclid's Division Algorithm.
15. 15. Solve \( 8x \equiv 1 \pmod{11} \).
16. 16. If \( a \) and \( b \) are two positive integers such that \( a^b \cdot b^a = 800 \), find \( a \) and \( b \).
17. 17. In \(\Delta ABC\), \(D\) and \(E\) are points on the sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = 8x - 7\), \(DB = 5x - 3\), \(AE = 4x - 3\), and \(EC = 3x - 1\), find the value of \(x\).
18. 18. Check whether \(\Delta ABC\) is a right-angled triangle with side lengths \(a = 7\text{ cm}\), \(b = 24\text{ cm}\), and \(c = 25\text{ cm}\).
19. 19. In \(\Delta ABC\), \(AD\) is the bisector of \(\angle A\). If \(AB = 10\text{ cm}\), \(AC = 14\text{ cm}\), and \(BC = 6\text{ cm}\), find \(BD\) and \(DC\).
20. 20. The length of the tangent to a circle from a point \(P\), which is at a distance of \(25\text{ cm}\) from the centre of the circle, is \(24\text{ cm}\). Find the radius of the circle.

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

21. 21. A function \( f: [-5, 9] \rightarrow \mathbb{R} \) is defined as follows: \[ f(x) = \begin{cases} 6x + 1 & \text{if } -5 \le x < 2 \\ 5x^2 - 1 & \text{if } 2 \le x < 6 \\ 3x - 4 & \text{if } 6 \le x \le 9 \end{cases} \] 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 \( (f \circ g) \circ h = f \circ (g \circ h) \).
23. 23. Find the sum to $n$ terms of the series \( 5 + 55 + 555 + \dots \)
24. 24. State and prove Basic Proportionality Theorem (Thales Theorem).

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