Matrix & Determinant Test

Cheatsheet Content

### Class 12th Mathematics: Matrix & Determinant (Model Paper) **Time:** 1:30 Hours | **Total Marks:** 50 --- ### Section - A: Objective Type Questions (Vastunishth Prashn) **Instructions:** Answer any 20 questions out of the following 30. Each question carries 1 mark. 1. If $$\begin{vmatrix} 2 & x \\ x & 8 \end{vmatrix} = \begin{vmatrix} 10 & 7 \\ 30 & 21 \end{vmatrix}$$, then $x =$ (A) $\pm16$ (B) $\pm4$ (C) $0$ (D) $\pm3$ 2. $$\begin{vmatrix} 11 & 7 & 9 \\ 2 & -1 & 5 \\ 22 & 14 & 18 \end{vmatrix} =$$ (A) $72$ (B) $-61$ (C) $0$ (D) $65$ 3. The number of all possible matrices of order $2 \times 2$ with each entry $3$ or $7$ is (A) $4^2$ (B) $3^2$ (C) $6^4$ (D) $2^4$ 4. $$6 \begin{bmatrix} -2 & 1 \\ 3 & 5 \end{bmatrix} =$$ (A) $$\begin{bmatrix} 12 & 6 \\ -18 & 30 \end{bmatrix}$$ (B) $$\begin{bmatrix} -12 & 6 \\ 18 & 30 \end{bmatrix}$$ (C) $$\begin{bmatrix} 12 & 6 \\ 18 & 30 \end{bmatrix}$$ (D) $$\begin{bmatrix} 6 & -12 \\ 18 & 30 \end{bmatrix}$$ 5. $$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \implies A^4 =$$ (A) $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ (B) $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ (C) $$\begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}$$ (D) $$\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$$ 6. $$\begin{bmatrix} 1 & -2 & 0 \end{bmatrix} \begin{bmatrix} 3 \\ -2 \\ 11 \end{bmatrix} =$$ (A) $$\begin{bmatrix} 7 \end{bmatrix}$$ (B) $$\begin{bmatrix} 3 & -6 & 0 \\ -2 & 4 & 0 \\ 11 & -22 & 0 \end{bmatrix}$$ (C) $$\begin{bmatrix} 18 \end{bmatrix}$$ (D) $$\begin{bmatrix} 3 & 6 & 0 \\ 2 & -4 & 0 \\ 11 & -22 & 0 \end{bmatrix}$$ 7. $$(AB)^{-1} =$$ (A) $A - B$ (B) $A^{-1}B^{-1}$ (C) $B - A$ (D) $B^{-1}A^{-1}$ 8. If $$A = \begin{bmatrix} 2 & -1 & 0 \\ 7 & 3 & 5 \end{bmatrix}$$ then $A' =$ (A) $$\begin{bmatrix} 2 & 3 \\ -1 & 7 \\ 0 & 5 \end{bmatrix}$$ (B) $$\begin{bmatrix} 2 & 0 \\ -1 & 7 \\ 3 & 5 \end{bmatrix}$$ (C) $$\begin{bmatrix} 2 & 7 \\ -1 & 3 \\ 0 & 5 \end{bmatrix}$$ (D) $$\begin{bmatrix} 2 & 7 \\ -1 & 3 \\ 0 & 4 \end{bmatrix}$$ 9. If $$\begin{vmatrix} x + 1 & 3 \\ 5 & x - 1 \end{vmatrix} = 0$$ then $x =$ (A) $\pm16$ (B) $\pm4$ (C) $16$ (D) $0$ 10. If $$A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ then $A^{-1} =$ (A) $$\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$$ (B) $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ (C) $$\begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}$$ (D) $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ 11. $$\begin{bmatrix} -3 & 13 \\ 6 & 9 \end{bmatrix} \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} =$$ (A) $$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$ (B) $$\begin{bmatrix} -9 & 13 \\ 6 & 27 \end{bmatrix}$$ (C) $$\begin{bmatrix} -9 & 39 \\ 18 & 27 \end{bmatrix}$$ (D) $$\begin{bmatrix} -9 & 0 \\ 0 & 27 \end{bmatrix}$$ 12. $$\begin{bmatrix} 2 & 3 \\ 3 & 0 \end{bmatrix} \begin{bmatrix} 3 \\ -5 \end{bmatrix} =$$ (A) $$\begin{bmatrix} 9 & -9 \\ 9 & 0 \end{bmatrix}$$ (B) $$\begin{bmatrix} 6 & 9 \\ -15 & 18 \end{bmatrix}$$ (C) $$\begin{bmatrix} -9 \\ 9 \end{bmatrix}$$ (D) Multiplication is not possible 13. The adjoint matrix of the matrix $$\begin{bmatrix} 7 & 6 \\ 5 & 4 \end{bmatrix}$$ is (A) $$\begin{bmatrix} 7 & -6 \\ -5 & 4 \end{bmatrix}$$ (B) $$\begin{bmatrix} 4 & 5 \\ 6 & 7 \end{bmatrix}$$ (C) $$\begin{bmatrix} 4 & -6 \\ -5 & 7 \end{bmatrix}$$ (D) $$\begin{bmatrix} -4 & -6 \\ 5 & -7 \end{bmatrix}$$ 14. $$\begin{vmatrix} 2002 & 2003 & 2004 \\ 2005 & 2008 & 2017 \\ 3 & 5 & 13 \end{vmatrix} =$$ (A) $21645$ (B) $39780$ (C) $42375$ (D) $0$ 15. $$4 \begin{vmatrix} 2 & -2 \\ 1 & 0 \end{vmatrix} =$$ (A) $$\begin{vmatrix} 8 & -2 \\ 1 & 0 \end{vmatrix}$$ (B) $$\begin{vmatrix} 8 & -8 \\ 4 & 0 \end{vmatrix}$$ (C) $$\begin{vmatrix} 8 & -2 \\ 4 & 10 \end{vmatrix}$$ (D) $$\begin{vmatrix} 8 & -8 \\ 1 & 0 \end{vmatrix}$$ 16. $$\begin{bmatrix} x-1 & y+2 \end{bmatrix} = \begin{bmatrix} 3 & 5 \end{bmatrix} \implies (x, y) =$$ (A) $(2, 1)$ (B) $(4, 3)$ (C) $(3, 4)$ (D) $(1, 2)$ 17. If square matrix A is such that $A^3 + 3A^2 - 7A + I = 0$ then $A^{-1}$ is equal to (A) $A^2 + 3A + 7I$ (B) $A^2 + 3A - 7I$ (C) $-A^2 - 3A + 7I$ (D) None of these 18. $$\begin{bmatrix} 0 & -3 & 1 \\ 2 & -1 & 1 \\ 2 & -1 & 1 \end{bmatrix} \begin{bmatrix} 0 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 3 & -3 \end{bmatrix} =$$ (A) $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$ (B) $$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ (C) $$\begin{bmatrix} 1 & 3 & 7 \\ 1 & 1 & 1 \\ 3 & -1 & 7 \end{bmatrix}$$ (D) $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ 19. If $A = [a_{ij}]_{3 \times 2}$, then the total number of elements in $A$ is: (A) $5$ (B) $6$ (C) $2$ (D) $3$ 20. The value of the determinant $$\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}$$ is: (A) $0$ (B) $-1$ (C) $1$ (D) $\sin 2\theta$ 21. A square matrix $A$ is said to be symmetric if: (A) $A^T = -A$ (B) $A^T = A$ (C) $A^{-1} = A$ (D) $|A| = 0$ 22. If $A$ is a $3 \times 3$ matrix and $|A| = 5$, then the value of $|2A|$ is: (A) $10$ (B) $20$ (C) $40$ (D) $5$ 23. For an Identity matrix $I$, the value of $I^2$ is: (A) $2I$ (B) $I$ (C) $O$ (D) $I^{-1}$ 24. If $|A| = 0$, then matrix $A$ is called: (A) Non-singular (B) Singular (C) Identity (D) Orthogonal 25. The transpose of a Column Matrix is a: (A) Unit Matrix (B) Zero Matrix (C) Row Matrix (D) Diagonal Matrix 26. If $a, b, c$ are in A.P. then $$\begin{vmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{vmatrix} =$$ (A) $3$ (B) $-3$ (C) $0$ (D) None of these 27. If $A = [a_{ij}]_{n \times n}$ is a symmetric matrix, then (A) $a_{ij} = 0$ (B) $a_{ij} = -a_{ji}$ (C) $a_{ij} = a_{ji}$ (D) $a_{ij} = 1$ 28. $$\begin{vmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{vmatrix} =$$ (A) $\cos^2 \alpha - \sin^2 \alpha$ (B) $-\sin^2 \alpha - \cos^2 \alpha$ (C) $\cos^2 \alpha + \sin^2 \alpha$ (D) $\sin^2 \alpha - \cos^2 \alpha$ 29. $$\begin{vmatrix} x & 7 \\ x & x \end{vmatrix} =$$ (A) $x^2 - 7x$ (B) $7x - x^2$ (C) $x^2 + 7x$ (D) $x - 7$ 30. $$\begin{vmatrix} a & b \\ -b & a \end{vmatrix} =$$ (A) $a^2 - b^2$ (B) $-b^2 - a^2$ (C) $a^2 + b^2$ (D) $b^2 - a^2$ --- ### Section - B: Short Answer Type Questions **Instructions:** Answer any 10 questions out of 20. Each question carries 2 marks. 1. If $$A = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 4 & 5 \end{bmatrix}$$ and $$B = \begin{bmatrix} 7 & -1 \\ 2 & 3 \\ 5 & 0 \end{bmatrix}$$, then find $(AB)'$. 2. Find the inverse matrix of the matrix $$A = \begin{bmatrix} 3 & 2 \\ 5 & -1 \end{bmatrix}$$. 3. Solve: $$\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix} = 2$$. 4. If $$A = \begin{bmatrix} 2 \\ 4 \\ 3 \end{bmatrix}$$ and $B = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$ then prove that $(AB)' = B'A'$. 5. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ then prove that $AB \neq BA$. 6. If $$A = \begin{bmatrix} -1 & 3 \\ 4 & -1 \end{bmatrix}$$, show that $A^2 = A$. (Note: This problem statement seems to have a typo, as $A^2=A$ is generally not true for this matrix. Assuming the question intends to ask to compute $A^2$.) 7. Evaluate the determinant $$\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$$. 8. Find the values of $x$ and $y$ if: $$\begin{bmatrix} x+y & 2 \\ 1 & x-y \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 1 & 7 \end{bmatrix}$$ 9. If $$A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$$, then verify that $A'A = I$. 10. Show that the matrix $$A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$$ is a skew-symmetric matrix. 11. Find $x$ if: $$\begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = O$$ 12. If $$A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}$$, find $|A|$. 13. Find values of $k$ if area of triangle is 4 sq. units and vertices are $(k, 0), (4, 0), (0, 2)$. 14. Show that points $A (a, b + c)$, $B (b, c + a)$, $C (c, a + b)$ are collinear. 15. Prove that: $$\begin{vmatrix} y+k & y & y \\ y & y+k & y \\ y & y & y+k \end{vmatrix} = k^2(3y+k)$$. --- ### Section - C: Long Answer Type Questions (Dirgh-uttariy Prashn) **Instructions:** Answer any 2 questions out of the following 4. Each question carries 5 marks. 1. If $$A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$$, show that $A^2 - 5A + 7I = O$. Hence find $A^{-1}$. 2. If $$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$$, Verify that $A^3 - 6A^2 + 9A - 4I = O$ and hence find $A^{-1}$. 3. Let $$A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$$ and $$B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}$$. Verify that $(AB)^{-1} = B^{-1} A^{-1}$. 4. For the matrix $$A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$$, find the numbers $a$ and $b$ such that $A^2 + aA + bI = O$.