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Mathematics · Class 12 Science

Ch 2Matrices — Class 12 Mathematics, concept-first.

A matrix of order is a square arrangement of elements, and every such square matrix has a corresponding determinant -- the same elements, expanded out into a single numerical value.

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Elementary Transformations of a Matrix

An elementary transformation is one of six basic, reversible operations performed on the rows or columns of a matrix.

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Chapter contents

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Introduction

A matrix of order is a square arrangement of elements, and every such square matrix has a corresponding determinant -- the same elements, expanded out into a single numerical value.

2.1

Elementary Transformation

The six elementary transformations. An elementary transformation of a matrix is one of six operations -- three acting on rows, and three (their mirror images) acting on columns:

+Exercise 2.1i9 questions
  1. Q1Apply the given elementary transformation on the following matrix. $A = \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix}$, $R_1 \leftrightarrow…Free
  2. Q2Apply the given elementary transformation on the following matrix. $B = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 5 & 4 \end{bmatrix}$, $R_1 \righta…Free
  3. Q3Apply the given elementary transformation on each of the following matrices. $A = \begin{bmatrix} 5 & 4 \\ 1 & 3 \end{bmatrix}$, $C_1 \leftr…Free
  4. Q4Apply the given elementary transformation on each of the following matrices. $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & 1 & 3 \end{bmatrix}$, $2…Preview
  5. Q5$A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}$, apply $3R_3$ and then $C_3 + 2C_2$.Preview
  6. Q6$A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}$, apply $C_3 + 2C_2$ and then $3R_3$. What do you conclude from ex.…Preview
  7. Q7Use suitable transformation on $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ to convert it into an upper triangular matrix.Preview
  8. Q8Convert $\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$ into an identity matrix by suitable row transformations.Preview
  9. Q9Transform $\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4 \end{bmatrix}$ into an upper triangular matrix by suitable column transforma…Preview
2.2

Inverse of a Matrix

Definition. If is a square matrix of order and there exists another square matrix of the same order such that , where is the identity matrix of order , then is called the inverse of , written .

+Exercise 2.2i15 questions
  1. Q10Find the co-factors of the elements of the following matrix. $\begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix}$Free
  2. Q11Find the co-factors of the elements of the following matrix. $\begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{bmatrix}$Free
  3. Q12Find the matrix of co-factors for the following matrix. $\begin{bmatrix} 1 & 3 \\ 4 & -1 \end{bmatrix}$Free
  4. Q13Find the matrix of co-factors for the following matrix. $\begin{bmatrix} 1 & 0 & 2 \\ -2 & 1 & 3 \\ 0 & 3 & -5 \end{bmatrix}$Preview
  5. Q14Find the adjoint of the following matrix. $\begin{bmatrix} 2 & -3 \\ 3 & 5 \end{bmatrix}$Preview
  6. Q15Find the adjoint of the following matrix. $\begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{bmatrix}$Preview
  7. Q16If $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}$, verify that $A(\text{adj }A) = (\text{adj }A)A = |A|\,I$Preview
  8. Q17Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$Preview
  9. Q18Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$Preview
  10. Q19Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$Preview
  11. Q20Find the inverse of the following matrix by the adjoint method. $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$Preview
  12. Q21Find the inverse of the following matrix. $\begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$Preview
  13. Q22Find the inverse of the following matrix. $\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$Preview
  14. Q23Find the inverse of the following matrix. $\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$Preview
  15. Q24Find the inverse of the following matrix. $\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$Preview
2.2.1

Inverse of a Nonsingular Matrix by Elementary Transformation

By the definition of inverse, if exists then . Consider the equation : here is the given matrix of order , is the identity matrix of order , and the only unknown is .

2.2.2

Inverse of a Square Matrix by Adjoint Method

The elementary-transformation method of section 2.2.1 works but is elaborate, needing a series of transformations tracked carefully. This section develops a second, direct route: the adjoint method.

2.3

Application of Matrices

Having covered the inverse of a matrix, this section turns to a major application: solving a system of linear equations using matrices.

+Exercise 2.3i10 questions
  1. Q25Solve the following equations by inversion method. $x + 2y = 2,\ 2x + 3y = 3$Free
  2. Q26Solve the following equations by inversion method. $x + y = 4,\ 2x - y = 5$Free
  3. Q27Solve the following equations by inversion method. $2x + 6y = 8,\ x + 3y = 5$Free
  4. Q28Solve the following equations by reduction method. $2x + y = 5,\ 3x + 5y = -3$Preview
  5. Q29Solve the following equations by reduction method. $x + 3y = 2,\ 3x + 5y = 4$Preview
  6. Q30Solve the following equations by reduction method. $3x - y = 1,\ 4x + y = 6$Preview
  7. Q31Solve the following equations by reduction method. $5x + 2y = 4,\ 7x + 3y = 5$Preview
  8. Q32The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90, whereas the cost of 6 pencils,…Preview
  9. Q33If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third number we get 8 and i…Preview
  10. Q34The total cost of 3 T.V. sets and 2 V.C.R.s is Rs. 35000. The shop-keeper wants profit of Rs. 1000 per television and Rs. 500 per V.C.R. He…Preview
2.3.1

Method of Inversion

As the name suggests, this method uses the inverse of the coefficient matrix directly. Consider three linear equations As explained in section 2.3, these can be expressed as where are of orders , , re…

2.3.2

Method of Reduction

As the name suggests, in this method the given equations are reduced -- through row transformations -- to a form from which the solution is read off directly, without ever computing a matrix inverse.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 19 questions19 questions
  1. Q1Find $(AB)^{-1}$ if $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -2 & -3 \end{bmatrix}$, $B = \begin{bmatrix} 1 & -1 \\ 1 & 2 \\ 1 & -2 \end{bmatri…Preview
  2. Q2The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹90 w…Preview
  3. Q3The inverse of the matrix $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$ is (a) $\dfrac{1}{13}\begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatri…Preview
  4. Q4Solve the following equations by method of reduction: $x - y + z = 4$, $2x + y - 3z = 0$, $x + y + z = 2$Preview
  5. Q5If $A = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix}$, then adjoint of matrix A is ______. (a) $\begin{bmatrix} 1 & 3 \\ -4 & 2 \end{bmatri…Preview
  6. Q6Find the inverse of the matrix, $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$ using elementary row transformati…Preview
  7. Q7If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 an…Preview
  8. Q8Find the matrix of co-factors for the matrix $\begin{bmatrix}1 & 3\\4 & -1\end{bmatrix}$Preview
  9. Q9Solve the following equations by the method of reduction: $x+3y+3z=12$; $x+4y+4z=15$; $x+3y+4z=13$Preview
  10. Q10Find the cofactors of the elements of the matrix $\begin{bmatrix}-1 & 2\\-3 & 4\end{bmatrix}$Preview
  11. Q11Solve the following system of equations by the method of inversion: $x - y + z = 4$, $2x + y - 3z = 0$, $x + y + z = 2$Preview
  12. Q12If $A = \begin{bmatrix}x & 0 & 0\\0 & y & 0\\0 & 0 & z\end{bmatrix}$ is a non singular matrix, then find $A^{-1}$ by elementary row transfor…Preview
  13. Q13If $A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ verify that $A(\text{adj}A) = (\text{adj}A)A = |A|I$Preview
  14. Q14Check whether the matrix $\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}$ is invertible or not.Preview
  15. Q15Solve the following system of equations by the method of reduction: $x+y+z=6$, $y+3z=11$, $x+z=2y$.Preview
  16. Q16Find the adjoint of the matrix $\begin{bmatrix}2 & -2\\ 4 & 3\end{bmatrix}$.Preview
  17. Q17Find the inverse of $\begin{bmatrix}\cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1\end{bmatrix}$ by elementary row t…Preview
  18. Q18If $A=\begin{bmatrix}2 & -4\\ 3 & 1\end{bmatrix}$, then the adjoint of matrix $A$ is ____. (a) $\begin{bmatrix}1 & 3\\ 4 & -2\end{bmatrix}$…Preview
  19. Q19If $A=\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}$, prove that $A\cdot(\text{adj }A)=(\text{adj }A)\cdot A=|A|\cdot I$Preview

More questions

68 Q
+Show 39 questions39 questions
  1. Q35If $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}$ then reduce it to $I_3$ by using column transformations.Free
  2. Q36If $A = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ then reduce it to $I_3$ by using row transformations.Free
  3. Q37Check whether the following matrix is invertible or not. $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$Free
  4. Q38Check whether the following matrix is invertible or not. $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$Preview
  5. Q39Check whether the following matrix is invertible or not. $\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}$Preview
  6. Q40Check whether the following matrix is invertible or not. $\begin{bmatrix} 2 & 3 \\ 10 & 15 \end{bmatrix}$Preview
  7. Q41Check whether the following matrix is invertible or not. $\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$Preview
  8. Q42Check whether the following matrix is invertible or not. $\begin{bmatrix} \sec\theta & \tan\theta \\ \tan\theta & \sec\theta \end{bmatrix}$Preview
  9. Q43Check whether the following matrix is invertible or not. $\begin{bmatrix} 3 & 4 & 3 \\ 1 & 1 & 0 \\ 1 & 4 & 5 \end{bmatrix}$Preview
  10. Q44Check whether the following matrix is invertible or not. $\begin{bmatrix} 1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3 \end{bmatrix}$Preview
  11. Q45Check whether the following matrix is invertible or not. $\begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{bmatrix}$Preview
  12. Q46Find $AB$, if $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -2 & -3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 1 & 2 \\ 1 & -2 \end{bmatrix}$…Preview
  13. Q47If $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is a nonsingular matrix then find $A^{-1}$ by elementary row tran…Preview
  14. Q48If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $X$ is a $2\times2$ matrix such that $AX = I$, then find $X$.Preview
  15. Q49Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$Preview
  16. Q50Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}$Preview
  17. Q51Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}$Preview
  18. Q52Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 2 & -3 \\ 5 & 7 \end{bmatrix}$Preview
  19. Q53Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}$Preview
  20. Q54Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 3 & -10 \\ 2 & -7 \end{bmatrix}$Preview
  21. Q55Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix}$Preview
  22. Q56Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}$Preview
  23. Q57Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$Preview
  24. Q58Find the inverse of each of the following matrices (if they exist). $\begin{bmatrix} 1 & 2 & -2 \\ 0 & -2 & 1 \\ -1 & 3 & 0 \end{bmatrix}$Preview
  25. Q59Find the inverse of $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$ by elementa…Preview
  26. Q60Find the inverse of $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$ by elementa…Preview
  27. Q61If $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$, find $AB$ and $(AB)^{-1}$. Verify…Preview
  28. Q62If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$, then show that $A^{-1} = \dfrac{1}{6}(A - 5I)$Preview
  29. Q63Find matrix $X$ such that $AX = B$, where $A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 2 & 4 \end{b…Preview
  30. Q64Find $X$, if $AX = B$ where $A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 \\ 2 \\ 3 \e…Preview
  31. Q65If $A = \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix}$ and $C = \begin{bmatrix} 24 & 7 \\…Preview
  32. Q66Find the inverse of $\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$ by adjoint method.Preview
  33. Q67Find the inverse of $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ by adjoint method.Preview
  34. Q68Find $A^{-1}$ by adjoint method and by elementary transformations if $A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}…Preview
  35. Q69Find the inverse of $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ by elementary column transformations.Preview
  36. Q70Find the inverse of $\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$ by elementary row transformations.Preview
  37. Q71Show with usual notations that for any matrix $A = [a_{ij}]_{3\times3}$: $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0$Preview
  38. Q72Show with usual notations that for any matrix $A = [a_{ij}]_{3\times3}$: $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = |A|$Preview
  39. Q73If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end…Preview
+Show 12 questions12 questions
  1. Q74Choose the correct alternative. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $\text{adj }A = \begin{bmatrix} 4 & a \\ -3 & b \end{…Free
  2. Q75The inverse of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ is (A) $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ (B) $\begin{bmatrix} 0 &…Free
  3. Q76If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $A(\text{adj }A) = kI$ then the value of $k$ is ..... (A) $1$ (B) $-1$ (C) $0$ (D)…Free
  4. Q77If $A = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}$ then the adjoint of matrix $A$ is (A) $\begin{bmatrix} -1 & 3 \\ 4 & 1 \end{bmatrix}$…Preview
  5. Q78If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $A(\text{adj }A) = kI$ then the value of $k$ is (A) $2$ (B) $-2$ (C) $10$ (D) $-10…Preview
  6. Q79If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$ then $A^{-1}$ does not exist if $\lambda = $ (A) $0$ (B) $\pm 1$ (C) $2$…Preview
  7. Q80If $A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$ then $A^{-1} = $ (A) $\begin{bmatrix} 1/\cos\alph…Preview
  8. Q81If $F(\alpha) = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$ where $\alpha \in \m…Preview
  9. Q82The inverse of $A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is (A) $I$ (B) $A$ (C) $A'$ (D) $-I$Preview
  10. Q83The inverse of a symmetric matrix is - (A) Symmetric (B) Non-symmetric (C) Null matrix (D) Diagonal matrixPreview
  11. Q84For a $2\times2$ matrix $A$, if $A(\text{adj }A) = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$ then determinant $A$ equals (A) $20$ (B)…Preview
  12. Q85If $A^{-1} = -\dfrac{1}{2}\begin{bmatrix} 1 & -4 \\ -1 & 2 \end{bmatrix}$ then $A =$ (a) $\begin{bmatrix} 2 & 4 \\ -1 & 1 \end{bmatrix}$ (b)…Preview
+Show 17 questions17 questions
  1. Q86Solve the following equations by the methods of inversion. $2x-y=-2,\ x+5y=5$Free
  2. Q87Solve the following equations by the methods of inversion. $x+y+z=1,\ 2x+3y+2z=2$ and $ax+ay+2az=4$Free
  3. Q88Solve the following equations by the methods of inversion. $5x-y+4z=5,\ 2x+3y+5z=2$ and $5x-2y+6z=-1$Free
  4. Q89Solve the following equations by the methods of inversion. $2x+3y=-5,\ 3x+y=3$Preview
  5. Q90Solve the following equations by the methods of inversion. $x+y+z=-1,\ y+z=2$ and $x+y-z=3$Preview
  6. Q91Express the following equation in matrix form and solve them by the method of reduction. $x-y+z=1,\ 2x-y=1,\ 3x+3y-4z=2$Preview
  7. Q92Express the following equation in matrix form and solve them by the method of reduction. $x+y=1,\ y+z=\dfrac{5}{3},\ z+x=\dfrac{4}{3}$Preview
  8. Q93Express the following equation in matrix form and solve them by the method of reduction. $2x - [y] + z = 1$ (the source textbook prints this…Preview
  9. Q94Express the following equation in matrix form and solve them by the method of reduction. $x+y+z=6,\ 3x-y+3z=10$ and $5x+5y-4z=3$Preview
  10. Q95Express the following equation in matrix form and solve them by the method of reduction. $x+2y+z=8,\ 2x+3y-z=1$ and $3x-y-2z=5$Preview
  11. Q96Express the following equation in matrix form and solve them by the method of reduction. $x+3y+2z=6,\ 3x-2y+5z=5$ and $2x-3y+6z=7$Preview
  12. Q97The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding first and the third num…Preview
  13. Q98The cost of 4 pencils, 3 pens and 2 books is Rs. 150. The cost of 1 pencil, 2 pens and 3 books is Rs. 125. The cost of 6 pencils, 2 pens and…Preview
  14. Q99The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding three times first number to the sum…Preview
  15. Q100The sum of three numbers is 2. If twice the second number is added to the sum of first and third number, we get o adding five times the firs…Preview
  16. Q101An amount of Rs. 5000 is invested in three types of investments, at interest rates 6.7, 7.7, 8% per annum respectively. The total annual inc…Preview
  17. Q102The sum of the costs of one book each of Mathematics, Physics and Chemistry is Rs. 210. Total cost of a mathematics book, 2 physics books, a…Preview