Chapter 1,2,and geometry and graph

Kalaimagal Academy

Model Test Paper (Mathematics)

Time: 1 Hr 30 Mins	Max. Marks: 50
Class: 10th Standard	Portion: Ch 1, Ch 2, Geometry & Graph
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. Let \( f(x) = \sqrt{1 + x^2} \), then:
   (a) \( f(xy) = f(x) \cdot f(y) \)

   (b) \( f(xy) \ge f(x) \cdot f(y) \)

   (c) \( f(xy) \le f(x) \cdot f(y) \)

   (d) None of these
3. 3. 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) \( 0 < r \le b \)
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. If the sequence \( t_1, t_2, t_3, \dots \) is in A.P., then the sequence \( t_6, t_{12}, t_{18}, \dots \) is:
   (a) an A.P.

   (b) a G.P.

   (c) neither A.P. nor G.P.

   (d) a constant sequence
6. 6. If in triangles \( \Delta ABC \) and \( \Delta DEF \), \( \frac{AB}{DE} = \frac{BC}{FD} \), then they will be similar, when:
   (a) \( \angle B = \angle E \)

   (b) \( \angle A = \angle D \)

   (c) \( \angle B = \angle D \)

   (d) \( \angle A = \angle F \)
7. 7. In a \( \Delta ABC \), \( AD \) is the bisector of \( \angle BAC \). If \( AB = 8 \text{ cm} \), \( BD = 6 \text{ cm} \), and \( DC = 3 \text{ cm} \), then the length of the side \( AC \) is:
   (a) \( 6 \text{ cm} \)

   (b) \( 4 \text{ cm} \)

   (c) \( 3 \text{ cm} \)

   (d) \( 8 \text{ cm} \)
8. 8. How many tangents can be drawn to a circle from an exterior point?
   (a) One

   (b) Two

   (c) Infinite

   (d) Zero
9. 9. The graph of a quadratic equation \( y = ax^2 + bx + c \) is a:
   (a) Straight line

   (b) Circle

   (c) Parabola

   (d) Hyperbola
10. 10. If the graph of a polynomial does not intersect the x-axis at all, then the number of real roots is:
    (a) 0

    (b) 1

    (c) 2

    (d) Infinite

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 \( A \times B \).
12. 12. If \( f(x) = 3x - 2 \) and \( g(x) = 2x + k \), find the value of \( k \) if \( f \circ g = g \circ f \).
13. 13. Find the HCF of 252 and 105 using Euclid's Division Algorithm.
14. 14. If \( a \) and \( b \) are two positive integers such that \( a^b \cdot b^a = 800 \), find \( a \) and \( b \).
15. 15. Find the 19th term of an A.P. \( -11, -15, -19, \dots \)
16. 16. State Thales Theorem (Basic Proportionality Theorem).
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. State Pythagoras Theorem.
19. 19. Check whether the nature of roots for the quadratic equation \( x^2 - x - 12 = 0 \) using its discriminant before drawing its graph.
20. 20. If a linear relation is given by \( y = 3x \), state whether it represents a direct or indirect variation.

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

21. 21. Let \( A = \{x \in \mathbb{W} \mid x < 2\} \), \( B = \{x \in \mathbb{N} \mid 1 < x \le 4\} \), and \( C = \{3, 5\} \). Verify that:
    

    \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)
22. 22. Find the sum of all natural numbers between 300 and 600 which are divisible by 7.
23. 23. State and prove Angle Bisector Theorem.
24. 24. Graph the quadratic equation \( y = x^2 - 4x + 3 \) and state the nature of its roots.

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