Mathematics · Textbook solutions

Determinants

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 92 questions

4.2 Determinant

18 q

Solved Examples

Worked · 5
  1. 4.1 Eg.1
    Evaluate ∣24−12∣\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}.
  2. 4.1 Eg.2
    Evaluate ∣xx+1x−1x∣\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}.
  3. 4.1 Eg.3
    Evaluate the determinant Δ=∣124−130410∣\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}.
  4. 4.1 Eg.4
    Evaluate Δ=∣0sin⁡α−cos⁡α−sin⁡α0sin⁡βcos⁡α−sin⁡β0∣\Delta = \begin{vmatrix} 0 & \sin\alpha & -\cos\alpha \\ -\sin\alpha & 0 & \sin\beta \\ \cos\alpha & -\sin\beta & 0 \end{vmatrix}.
  5. 4.1 Eg.5
    Find values of xx for which ∣3xx1∣=∣3241∣\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}.

Exercise 4.1

Practice · 13
  1. Ex 4.1 Q1
    Evaluate the determinant ∣24−5−1∣\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}.
  2. Evaluate the determinant.
    Ex 4.1 Q2 (i)
    ∣cos⁡θ−sin⁡θsin⁡θcos⁡θ∣\begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix}
  3. Ex 4.1 Q2 (ii)
    ∣x2−x+1x−1x+1x+1∣\begin{vmatrix} x^{2} - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}
  4. Ex 4.1 Q3
    If A=[1242]A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}, then show that ∣2A∣=4∣A∣|2A| = 4|A|.
  5. Ex 4.1 Q4
    If A=[101012004]A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}, then show that ∣3A∣=27∣A∣|3A| = 27|A|.
  6. Evaluate the determinants.
    Ex 4.1 Q5 (i)
    ∣3−1−200−13−50∣\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}
  7. Ex 4.1 Q5 (ii)
    ∣3−4511−2231∣\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}
  8. Ex 4.1 Q5 (iii)
    ∣012−10−3−230∣\begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}
  9. Ex 4.1 Q5 (iv)
    ∣2−1−202−13−50∣\begin{vmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{vmatrix}
  10. Ex 4.1 Q6
    If A=[11−221−354−9]A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}, find ∣A∣|A|.
  11. Find values of xx, if the following holds.
    Ex 4.1 Q7 (i)
    ∣2451∣=∣2x46x∣\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}
  12. Ex 4.1 Q7 (ii)
    ∣2345∣=∣x32x5∣\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}
  13. Ex 4.1 Q8
    If ∣x218x∣=∣62186∣\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}, then xx is equal to
    1. A.
      66
    2. B.
      ±6\pm 6
    3. C.
      −6-6
    4. D.
      00

4.3 Area of a Triangle

11 q

Solved Examples

Worked · 2
  1. 4.2 Eg.6
    Find the area of the triangle whose vertices are (3,8)(3, 8), (−4,2)(-4, 2) and (5,1)(5, 1).
  2. 4.2 Eg.7
    Find the equation of the line joining A(1,3)A(1, 3) and B(0,0)B(0, 0) using determinants and find kk if D(k,0)D(k, 0) is a point such that area of triangle ABD is 3 sq units.

Exercise 4.2

Practice · 9
  1. Find area of the triangle with vertices at the point given in each of the following:
    Ex 4.2 Q1 (i)
    (1,0)(1, 0), (6,0)(6, 0), (4,3)(4, 3)
  2. Ex 4.2 Q1 (ii)
    (2,7)(2, 7), (1,1)(1, 1), (10,8)(10, 8)
  3. Ex 4.2 Q1 (iii)
    (−2,−3)(-2, -3), (3,2)(3, 2), (−1,−8)(-1, -8)
  4. Ex 4.2 Q2
    Show that points A(a,b+c)A(a, b + c), B(b,c+a)B(b, c + a), C(c,a+b)C(c, a + b) are collinear.
  5. Find values of kk if area of triangle is 4 sq. units and vertices are:
    Ex 4.2 Q3 (i)
    (k,0)(k, 0), (4,0)(4, 0), (0,2)(0, 2)
  6. Ex 4.2 Q3 (ii)
    (−2,0)(-2, 0), (0,4)(0, 4), (0,k)(0, k)
  7. Ex 4.2 Q4 (i)
    Find equation of line joining (1,2)(1, 2) and (3,6)(3, 6) using determinants.
  8. Ex 4.2 Q4 (ii)
    Find equation of line joining (3,1)(3, 1) and (9,3)(9, 3) using determinants.
  9. Ex 4.2 Q5
    If area of triangle is 35 sq units with vertices (2,−6)(2, -6), (5,4)(5, 4) and (k,4)(k, 4). Then kk is
    1. A.
      1212
    2. B.
      −2-2
    3. C.
      −12,−2-12, -2
    4. D.
      12,−212, -2

4.4 Minors and Cofactors

11 q

Solved Examples

Worked · 4
  1. 4.3 Eg.8
    Find the minor of element 6 in the determinant Δ=∣123456789∣\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}
  2. 4.3 Eg.9
    Find minors and cofactors of all the elements of the determinant ∣1−243∣\begin{vmatrix} 1 & -2 \\ 4 & 3 \end{vmatrix}
  3. 4.3 Eg.10
    Find minors and cofactors of the elements a11a_{11}, a21a_{21} in the determinant Δ=∣a11a12a13a21a22a23a31a32a33∣\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}
  4. 4.3 Eg.11
    Find minors and cofactors of the elements of the determinant ∣2−3560415−7∣\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix} and verify that a11A31+a12A32+a13A33=0a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33} = 0

Exercise 4.3

Practice · 7
  1. Write Minors and Cofactors of the elements of following determinants:
    Ex 4.3 Q1 (i)
    ∣2−403∣\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}
  2. Ex 4.3 Q1 (ii)
    ∣acbd∣\begin{vmatrix} a & c \\ b & d \end{vmatrix}
  3. Write Minors and Cofactors of the elements of following determinants:
    Ex 4.3 Q2 (i)
    ∣100010001∣\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}
  4. Ex 4.3 Q2 (ii)
    ∣10435−1012∣\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}
  5. Ex 4.3 Q3
    Using Cofactors of elements of second row, evaluate Δ=∣538201123∣\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}.
  6. Ex 4.3 Q4
    Using Cofactors of elements of third column, evaluate Δ=∣1xyz1yzx1zxy∣\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}.
  7. Ex 4.3 Q5
    If Δ=∣a11a12a13a21a22a23a31a32a33∣\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} and AijA_{ij} is Cofactors of aija_{ij}, then value of Δ\Delta is given by
    1. A.
      a11A31+a12A32+a13A33a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33}
    2. B.
      a11A11+a12A21+a13A31a_{11} A_{11} + a_{12} A_{21} + a_{13} A_{31}
    3. C.
      a21A11+a22A12+a23A13a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13}
    4. D.
      a11A11+a21A21+a31A31a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}

4.5 Adjoint and Inverse of a Matrix

22 q

Solved Examples

Worked · 4
  1. 4.4 Eg.12
    Find adj⁡A\operatorname{adj} A for A=[2314]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}.
  2. 4.4 Eg.13
    If A=[133143134]A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, then verify that Aadj⁡A=∣A∣IA \operatorname{adj} A = |A| I. Also find A−1A^{-1}.
  3. 4.4 Eg.14
    If A=[231−4]A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix} and B=[1−2−13]B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}, then verify that (AB)−1=B−1A−1(AB)^{-1} = B^{-1} A^{-1}.
  4. 4.4 Eg.15
    Show that the matrix A=[2312]A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} satisfies the equation A2−4A+I=OA^2 - 4A + I = O, where II is the 2×22 \times 2 identity matrix and OO is the 2×22 \times 2 zero matrix. Using this equation, find A−1A^{-1}.

Exercise 4.4

Practice · 18
  1. Find adjoint of each of the matrices in Exercises 1 and 2.
    Ex 4.4 Q1
    Find the adjoint of the matrix [1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.
  2. Find adjoint of each of the matrices in Exercises 1 and 2.
    Ex 4.4 Q2
    Find the adjoint of the matrix [1−12235−201]\begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix}.
  3. Verify A(adj⁡A)=(adj⁡A)A=∣A∣IA(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I in Exercises 3 and 4.
    Ex 4.4 Q3
    Verify A(adj⁡A)=(adj⁡A)A=∣A∣IA(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I for A=[23−4−6]A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix}.
  4. Verify A(adj⁡A)=(adj⁡A)A=∣A∣IA(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I in Exercises 3 and 4.
    Ex 4.4 Q4
    Verify A(adj⁡A)=(adj⁡A)A=∣A∣IA(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I for A=[1−1230−2103]A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}.
  5. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q5
    Find the inverse of the matrix (if it exists) [2−243]\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}.
  6. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q6
    Find the inverse of the matrix (if it exists) [−15−32]\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}.
  7. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q7
    Find the inverse of the matrix (if it exists) [123024005]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}.
  8. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q8
    Find the inverse of the matrix (if it exists) [10033052−1]\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}.
  9. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q9
    Find the inverse of the matrix (if it exists) [2134−10−721]\begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix}.
  10. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q10
    Find the inverse of the matrix (if it exists) [1−1202−33−24]\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}.
  11. Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
    Ex 4.4 Q11
    Find the inverse of the matrix (if it exists) [1000cos⁡αsin⁡α0sin⁡α−cos⁡α]\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}.
  12. Ex 4.4 Q12
    Let A=[3725]A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix} and B=[6879]B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}. Verify that (AB)−1=B−1A−1(AB)^{-1} = B^{-1} A^{-1}.
  13. Ex 4.4 Q13
    If A=[31−12]A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, show that A2−5A+7I=OA^2 - 5A + 7I = O. Hence find A−1A^{-1}.
  14. Ex 4.4 Q14
    For the matrix A=[3211]A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}, find the numbers aa and bb such that A2+aA+bI=OA^2 + aA + bI = O.
  15. Ex 4.4 Q15
    For the matrix A=[11112−32−13]A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}, show that A3−6A2+5A+11I=OA^3 - 6A^2 + 5A + 11 I = O. Hence, find A−1A^{-1}.
  16. Ex 4.4 Q16
    If A=[2−11−12−11−12]A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}, verify that A3−6A2+9A−4I=OA^3 - 6A^2 + 9A - 4I = O and hence find A−1A^{-1}.
  17. Ex 4.4 Q17
    Let AA be a nonsingular square matrix of order 3×33 \times 3. Then ∣adj⁡A∣|\operatorname{adj} A| is equal to
    1. A.
      ∣A∣|A|
    2. B.
      ∣A∣2|A|^2
    3. C.
      ∣A∣3|A|^3
    4. D.
      3∣A∣3|A|
  18. Ex 4.4 Q18
    If AA is an invertible matrix of order 2, then det⁡(A−1)\det(A^{-1}) is equal to
    1. A.
      det⁡(A)\det(A)
    2. B.
      1det⁡(A)\dfrac{1}{\det(A)}
    3. C.
      11
    4. D.
      00

4.6 Applications of Determinants and Matrices

20 q

Solved Examples

Worked · 4
  1. 4.5 Eg.16
    Solve the system of equations 2x+5y=12x + 5y = 1 3x+2y=73x + 2y = 7
  2. 4.5 Eg.17
    Solve the following system of equations by matrix method. 3x−2y+3z=83x - 2y + 3z = 8 2x+y−z=12x + y - z = 1 4x−3y+2z=44x - 3y + 2z = 4
  3. 4.5 Eg.18
    The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.
  4. 4.5 Eg.19
    Use product [1−1202−33−24][−20192−361−2]\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix} to solve the system of equations x−y+2z=1x - y + 2z = 1 2y−3z=12y - 3z = 1 3x−2y+4z=23x - 2y + 4z = 2

Exercise 4.5

Practice · 16
  1. Examine the consistency of the system of equations.
    Ex 4.5 Q1
    x+2y=2x + 2y = 2 2x+3y=32x + 3y = 3
  2. Examine the consistency of the system of equations.
    Ex 4.5 Q2
    2x−y=52x - y = 5 x+y=4x + y = 4
  3. Examine the consistency of the system of equations.
    Ex 4.5 Q3
    x+3y=5x + 3y = 5 2x+6y=82x + 6y = 8
  4. Examine the consistency of the system of equations.
    Ex 4.5 Q4
    x+y+z=1x + y + z = 1 2x+3y+2z=22x + 3y + 2z = 2 ax+ay+2az=4ax + ay + 2az = 4
  5. Examine the consistency of the system of equations.
    Ex 4.5 Q5
    3x−y−2z=23x - y - 2z = 2 2y−z=−12y - z = -1 3x−5y=33x - 5y = 3
  6. Examine the consistency of the system of equations.
    Ex 4.5 Q6
    5x−y+4z=55x - y + 4z = 5 2x+3y+5z=22x + 3y + 5z = 2 5x−2y+6z=−15x - 2y + 6z = -1
  7. Solve system of linear equations, using matrix method.
    Ex 4.5 Q7
    5x+2y=45x + 2y = 4 7x+3y=57x + 3y = 5
  8. Solve system of linear equations, using matrix method.
    Ex 4.5 Q8
    2x−y=−22x - y = -2 3x+4y=33x + 4y = 3
  9. Solve system of linear equations, using matrix method.
    Ex 4.5 Q9
    4x−3y=34x - 3y = 3 3x−5y=73x - 5y = 7
  10. Solve system of linear equations, using matrix method.
    Ex 4.5 Q10
    5x+2y=35x + 2y = 3 3x+2y=53x + 2y = 5
  11. Solve system of linear equations, using matrix method.
    Ex 4.5 Q11
    2x+y+z=12x + y + z = 1 x−2y−z=32x - 2y - z = \frac{3}{2} 3y−5z=93y - 5z = 9
  12. Solve system of linear equations, using matrix method.
    Ex 4.5 Q12
    x−y+z=4x - y + z = 4 2x+y−3z=02x + y - 3z = 0 x+y+z=2x + y + z = 2
  13. Solve system of linear equations, using matrix method.
    Ex 4.5 Q13
    2x+3y+3z=52x + 3y + 3z = 5 x−2y+z=−4x - 2y + z = -4 3x−y−2z=33x - y - 2z = 3
  14. Solve system of linear equations, using matrix method.
    Ex 4.5 Q14
    x−y+2z=7x - y + 2z = 7 3x+4y−5z=−53x + 4y - 5z = -5 2x−y+3z=122x - y + 3z = 12
  15. Ex 4.5 Q15
    If A=[2−3532−411−2]A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}, find A−1A^{-1}. Using A−1A^{-1} solve the system of equations 2x−3y+5z=112x - 3y + 5z = 11 3x+2y−4z=−53x + 2y - 4z = -5 x+y−2z=−3x + y - 2z = -3
  16. Ex 4.5 Q16
    The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Miscellaneous Exercise on Chapter 4

10 q

Miscellaneous Exercise

Practice · 10
  1. Misc Q1
    Prove that the determinant ∣xsin⁡θcos⁡θ−sin⁡θ−x1cos⁡θ1x∣\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix} is independent of θ\theta.
  2. Misc Q2
    Evaluate ∣cos⁡αcos⁡βcos⁡αsin⁡β−sin⁡α−sin⁡βcos⁡β0sin⁡αcos⁡βsin⁡αsin⁡βcos⁡α∣\begin{vmatrix} \cos\alpha \cos\beta & \cos\alpha \sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha \cos\beta & \sin\alpha \sin\beta & \cos\alpha \end{vmatrix}.
  3. Misc Q3
    If A−1=[3−11−156−55−22]A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix} and B=[12−2−1300−21]B = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}, find (AB)−1(AB)^{-1}.
  4. Let A=[121231115]A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}. Verify that
    Misc Q4 (i)
    [adj A]−1=adj (A−1)[adj\, A]^{-1} = adj\,(A^{-1})
  5. Misc Q4 (ii)
    (A−1)−1=A(A^{-1})^{-1} = A
  6. Misc Q5
    Evaluate ∣xyx+yyx+yxx+yxy∣\begin{vmatrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{vmatrix}.
  7. Misc Q6
    Evaluate ∣1xy1x+yy1xx+y∣\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}.
  8. Misc Q7
    Solve the system of equations 2x+3y+10z=4\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4, 4x−6y+5z=1\dfrac{4}{x} - \dfrac{6}{y} + \dfrac{5}{z} = 1, 6x+9y−20z=2\dfrac{6}{x} + \dfrac{9}{y} - \dfrac{20}{z} = 2.
  9. Misc Q8
    If x,y,zx, y, z are nonzero real numbers, then the inverse of matrix A=[x000y000z]A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} is
    1. A.
      [x−1000y−1000z−1]\begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}
    2. B.
      xyz[x−1000y−1000z−1]xyz \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}
    3. C.
      1xyz[x000y000z]\dfrac{1}{xyz} \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}
    4. D.
      1xyz[100010001]\dfrac{1}{xyz} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
  10. Misc Q9
    Let A=[1sin⁡θ1−sin⁡θ1sin⁡θ−1−sin⁡θ1]A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}, where 0≤θ≤2π0 \le \theta \le 2\pi. Then
    1. A.
      Det (A)=0\mathrm{Det}\,(A) = 0
    2. B.
      Det (A)∈(2,∞)\mathrm{Det}\,(A) \in (2, \infty)
    3. C.
      Det (A)∈(2,4)\mathrm{Det}\,(A) \in (2, 4)
    4. D.
      Det (A)∈[2,4]\mathrm{Det}\,(A) \in [2, 4]