Class 12 maths samacheer kalvi || Chapter 1

Kalaimagal Academy

12th Standard Mathematics Unit Test

Time: 1 Hr 30 Mins	Max. Marks: 50
Class: 12th Standard	Portion: Chapters 1, 2, and 3
Student Name: _______________________	Roll No: ____________
Date: ____________

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

1. 1. If \( A = \begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix} \), then \( \text{adj}(A) \) is:
   (a) \( \begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix} \)

   (b) \( \begin{pmatrix} -2 & 5 \\ 1 & -3 \end{pmatrix} \)

   (c) \( \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} \)

   (d) \( \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \)
2. 2. If \( A \) is a non-singular matrix of order 3, then \( |\text{adj}(A)| \) is equal to:
   (a) \( |A| \)

   (b) \( |A|^2 \)

   (c) \( |A|^3 \)

   (d) \( 1/|A| \)
3. 3. If \( i^2 = -1 \), then \( i + i^2 + i^3 + i^4 \) is equal to: [Ex 2.1 - 3]
   (a) 0

   (b) 1

   (c) -1

   (d) \( i \)
4. 4. The value of \( |z - 1| \) represents the distance between:
   (a) \( z \) and origin

   (b) \( z \) and \( (1, 0) \)

   (c) \( z \) and \( (0, 1) \)

   (d) \( z \) and \( (-1, 0) \)
5. 5. If \( \omega \neq 1 \) is a cubic root of unity, then \( 1 + \omega + \omega^2 \) is: [Ex 2.8]
   (a) 0

   (b) 1

   (c) -1

   (d) 3
6. 6. A polynomial equation of degree \( n \) has exactly:
   (a) \( n \) real roots

   (b) \( n \) roots

   (c) at least \( n \) roots

   (d) maximum 2 roots
7. 7. If \(\alpha, \beta, \gamma\) are the roots of \(x^3 + px^2 + qx + r = 0\), then \(\sum \alpha\) is: [Ex 3.1]
   (a) \( -p \)

   (b) \( p \)

   (c) \( q \)

   (d) \( -r \)
8. 8. If \( 2 + i \) is a root of a polynomial with real coefficients, another root must be: [Ex 3.2 - 3]
   (a) \( -2 + i \)

   (b) \( 2 - i \)

   (c) \( -2 - i \)

   (d) \( 1 + 2i \)
9. 9. The number of positive real roots of \( x^9 - 5x^5 + 4x^3 + 2x^2 + 1 = 0 \) is at most: [Ex 3.6 - 3]
   (a) 2

   (b) 3

   (c) 4

   (d) 5
10. 10. If the rank of a matrix \( A \) of order \( 3 \times 3 \) is 3, then \( A \) is:
    (a) Singular

    (b) Non-singular

    (c) Symmetric

    (d) Zero matrix

PART - II: SHORT ANSWER QUESTIONS (Answer Any 5) 5 × 2 = 10 Marks

11. 11. If \( A \) is a non-singular matrix of odd order, prove that \( |\text{adj}(A)| \) is positive. [Example 1.4]
12. 12. Find the rank of the matrix \( \begin{pmatrix} 2 & -4 \\ -1 & 2 \end{pmatrix} \) by row reduction. [Ex 1.2 - 1(iv)]
13. 13. Simplify: \( \sum_{n=1}^{12} i^n \). [Ex 2.1 - 3]
14. 14. Show that \( (2 + i\sqrt{3})^{10} + (2 - i\sqrt{3})^{10} \) is purely real. [Example 2.8 / Ex 2.4 - 7(2)]
15. 15. If \( z_1 = 2 - i \) and \( z_2 = -4 + 3i \), find the inverse of \( z_1 z_2 \). [Ex 2.3 - 2(1)]
16. 16. Construct a cubic equation with roots 1, 2, and 3. [Ex 3.1 - 1]
17. 17. Find a polynomial equation of minimum degree with rational coefficients having \( 2 + \sqrt{3}i \) as a root. [Ex 3.2 - 3]

PART - III: BRIEF ANSWER QUESTIONS (Answer Any 5) 5 × 3 = 15 Marks

18. 18. Verify \( (AB)^{-1} = B^{-1}A^{-1} \) with \( A = \begin{pmatrix} 0 & -3 \\ 1 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} -2 & -3 \\ 0 & -1 \end{pmatrix} \). [Example 1.9]
19. 19. Solve the system of linear equations by Matrix Inversion Method: \( 2x - y = 8 \), \( 3x + 2y = -2 \). [Ex 1.3 - 1(1)]
20. 20. If \( |z| = 1 \), show that \( 2 \le |z^2 - 3| \le 4 \). [Example 2.7 / Ex 2.5 - 5]
21. 21. Show that the equation \( z^3 + 2\bar{z} = 0 \) has five solutions. [Example 2.24]
22. 22. If \( 1 + 2i \) and \( 3 \) are roots of \( x^3 - 5x^2 + 11x - 15 = 0 \), solve the equation completely. [Example 3.11]
23. 23. Solve the equation: \( x^3 - 5x^2 - 4x + 20 = 0 \). [Example 3.25]
24. 24. Discuss the maximum possible number of positive and negative roots of the polynomial \( x^9 - 5x^8 - 14x^7 = 0 \). [Ex 3.6 - 5]

PART - IV: LONG ANSWER QUESTIONS (Answer All) 3 × 5/7 = 15 Marks

25. 25. (a) Solve the system of equations by Cramer's rule: \( x_1 - x_2 = 3 \), \( 2x_1 + 3x_2 + 4x_3 = 17 \), \( x_2 + 2x_3 = 7 \). [Example 1.25]
    OR
    (b) Test the consistency of the system of linear equations and if possible solve: \( x - y + z = -9 \), \( 2x - 2y + 2z = -18 \), \( 3x - 3y + 3z = -27 \). [Ex 1.6 - 1(4)]
26. 26. (a) Solve the equation \( z^3 + 8i = 0 \), where \( z \in \mathbb{C} \). [Example 2.34]
    OR
    (b) If \( 2\cos\alpha = x + \frac{1}{x} \) and \( 2\cos\beta = y + \frac{1}{y} \), show that \( \frac{x^m}{y^n} - \frac{y^n}{x^m} = 2i\sin(m\alpha - n\beta) \). [Ex 2.8 - 4]
27. 27. (a) Solve the equation: \( 6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0 \). [Example 3.27 / Ex 3.5 - 7]
    OR
    (b) Solve the equation \( 2x^3 - 9x^2 + 10x - 3 = 0 \). [Example 3.19]

*** ALL THE BEST ***