MCQOPTIONS
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MCQOPTIONS
This section includes 72 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct? |
| A. | A must be a unit matrix. |
| B. | B must be a unit matrix. |
| C. | Both A and B must be unit matrices. |
| D. | A and B need not be unit matrices. |
| Answer» D. A and B need not be unit matrices. | |
| 52. |
If a ≠ b ≠ c, then one value of x which satisfies the equation \(\left| {\begin{array}{*{20}{c}} 0&{{\rm{x}} - {\rm{a}}}&{{\rm{x}} - {\rm{b}}}\\ {{\rm{x}} + {\rm{a}}}&0&{{\rm{x}} - {\rm{c}}}\\ {{\rm{x}} + {\rm{b}}}&{{\rm{x}} + {\rm{c}}}&0 \end{array}} \right| = 0\) is given by. |
| A. | a |
| B. | b |
| C. | c |
| D. | 0 |
| Answer» E. | |
| 53. |
If ω is a cube root of unity, then find the value of the determinant \(\rm \begin{vmatrix} 1 + \omega & \omega^2 & -\omega \\\ 1 + \omega^2 & \omega & -\omega^2 \\\ \omega^2 + \omega & \omega & -\omega^2 \end{vmatrix}\) is |
| A. | 3ω |
| B. | -3ω |
| C. | 3ω2 |
| D. | -3ω2 |
| Answer» E. | |
| 54. |
If \(\left| {\;\begin{array}{*{20}{c}} {x + 2}&2&2\\ 2&{x + 2}&2\\ 2&2&{x + 2} \end{array}} \right|\) = 0, then values of x satisfying this equation are |
| A. | 0, -2, -6 |
| B. | 0, -1, -2 |
| C. | 0, 0, -2 |
| D. | 0, 0, -6 |
| Answer» E. | |
| 55. |
If Δ is the value of the determinant\(\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|\)then what is the value of the following determinant?\(\left| {\begin{array}{*{20}{c}} {{pa_1}}&{{b_1}}&{{qc_1}}\\ {{pa_2}}&{{b_2}}&{{qc_2}}\\ {{pa_3}}&{{b_3}}&{{qc_3}} \end{array}} \right|\)(p ≠ 0 or 1, q ≠ 0 or 1) |
| A. | pΔ |
| B. | qΔ |
| C. | (p + q)Δ |
| D. | pqΔ |
| Answer» E. | |
| 56. |
If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations then |
| A. | it can be reduced to a single equation and so a solution does not exist |
| B. | it has only the trivial solution x = y = z = 0 |
| C. | it is consistent |
| D. | determinant of the matrix of coefficient is zero |
| Answer» C. it is consistent | |
| 57. |
If the system of equation 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 be consistent, then the values of k are - |
| A. | \(6, \ \dfrac{-17}{5}\) |
| B. | \(-1, \ \dfrac{1}{5}\) |
| C. | \(-6, \ \dfrac{17}{5}\) |
| D. | \(6, \ \dfrac{-12}{5}\) |
| Answer» B. \(-1, \ \dfrac{1}{5}\) | |
| 58. |
Let \[\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}\]. Then the value of the determinant \[\left| \,\begin{matrix} 1 & 1 & 1\\ 1 & -1-{{\omega }^{2}} & {{\omega }^{2}}\\ 1 & {{\omega }^{2}} & {{\omega }^{4}}\\ \end{matrix}\, \right|\]is [IIT Screening 2002] |
| A. | \[3\omega \] |
| B. | \[3\omega (\omega -1)\] |
| C. | \[3{{\omega }^{2}}\] |
| D. | \[3\omega (1-\omega )\] |
| Answer» C. \[3{{\omega }^{2}}\] | |
| 59. |
If \[A=\left| \,\begin{matrix} -1 & 2 & 4\\ 3 & 1 & 0\\ -2 & 4 & 2\\ \end{matrix}\, \right|\]and \[B=\left| \,\begin{matrix} -2 & 4 & 2\\ 6 & 2 & 0\\ -2 & 4 & 8\\ \end{matrix}\, \right|\], then B is given by [Tamilnadu (Engg.) 2002] |
| A. | \[B=4A\] |
| B. | \[B=-4A\] |
| C. | \[B=-A\] |
| D. | \[B=6A\] |
| Answer» C. \[B=-A\] | |
| 60. |
If \[\left| \,\begin{matrix} a & b & a+b\\ b & c & b+c\\ a+b & b+c & 0\\ \end{matrix}\, \right|=0\]; then \[a,b,c\] are in [AMU 2000] |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | None of these |
| Answer» C. H. P. | |
| 61. |
If \[a,b,c\] are in A.P., then the value of \[\left| \,\begin{matrix} x+2 & x+3 & x+a\\ x+4 & x+5 & x+b\\ x+6 & x+7 & x+c\\ \end{matrix}\, \right|\] is [RPET 1999] |
| A. | \[x-(a+b+c)\] |
| B. | \[9{{x}^{2}}+a+b+c\] |
| C. | \[a+b+c\] |
| D. | 0 |
| Answer» E. | |
| 62. |
If\[a,b,c\] are different and \[\left| \,\begin{matrix} a & {{a}^{2}} & {{a}^{3}}-1\\ b & {{b}^{2}} & {{b}^{3}}-1\\ c & {{c}^{2}} & {{c}^{3}}-1\\ \end{matrix}\, \right|=0\], then [EAMCET 1989] |
| A. | \[a+b+c=0\] |
| B. | \[abc=1\] |
| C. | \[a+b+c=1\] |
| D. | \[ab+bc+ca=0\] |
| Answer» C. \[a+b+c=1\] | |
| 63. |
If \[a\ne b\ne c,\] the value of x which satisfies the equation \[\left| \,\begin{matrix} 0 & x-a & x-b\\ x+a & 0 & x-c\\ x+b & x+c & 0\\ \end{matrix}\, \right|=0\], is [EAMCET 1988; Karnataka CET 1991; MNR 1980;MP PET 1988, 99, 2001; DCE 2001] |
| A. | \[x=0\] |
| B. | \[x=a\] |
| C. | \[x=b\] |
| D. | \[x=c\] |
| Answer» B. \[x=a\] | |
| 64. |
If \[a,b,c\] are positive integers, then the determinant \[\Delta =\left| \,\begin{matrix} {{a}^{2}}+x & ab & ac\\ ab & {{b}^{2}}+x & bc\\ ac & bc & {{c}^{2}}+x\\ \end{matrix}\, \right|\] is divisible by |
| A. | \[{{x}^{3}}\] |
| B. | \[{{x}^{2}}\] |
| C. | \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] |
| D. | None of these |
| Answer» C. \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] | |
| 65. |
If \[\Delta =\left| \,\begin{matrix} a & b & c\\ x & y & z\\ p & q & r\\ \end{matrix}\, \right|\], then \[\left| \,\begin{matrix} ka & kb & kc\\ kx & ky & kz\\ kp & kq & kr\\ \end{matrix}\, \right|\]= [RPET 1986] |
| A. | \[\Delta \] |
| B. | \[k\Delta \] |
| C. | \[3k\Delta \] |
| D. | \[{{k}^{3}}\Delta \] |
| Answer» E. | |
| 66. |
The value of the determinant \[\left| \,\begin{matrix} 4 & -6 & 1\\ -1 & -1 & 1\\ -4 & 11 & -1\,\\ \end{matrix} \right|\]is [RPET 1992] |
| A. | -75 |
| B. | 25 |
| C. | 0 |
| D. | -25 |
| Answer» E. | |
| 67. |
If \[A=\left| \,\begin{matrix} 1 & 1 & 1\\ a & b & c\\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}}\\ \end{matrix}\, \right|,B=\left| \,\begin{matrix} 1 & 1 & 1\\ {{a}^{2}} & {{b}^{2}} & {{c}^{2}}\\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}}\\ \end{matrix}\, \right|,C=\left| \,\begin{matrix} a & b & c\\ {{a}^{2}} & {{b}^{2}} & {{c}^{2}}\\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}}\\ \end{matrix}\, \right|,\] then which relation is correct |
| A. | \[A=B\] |
| B. | \[A=C\] |
| C. | \[B=C\] |
| D. | None of these |
| Answer» E. | |
| 68. |
\[\left| \,\begin{matrix} a & b & c\\ b & c & a\\ c & a & b\\ \end{matrix}\, \right|=\] [MP PET 1991] |
| A. | \[3abc+{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] |
| B. | \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] |
| C. | \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] |
| D. | \[abc+{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] |
| Answer» C. \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] | |
| 69. |
\[\left| \,\begin{matrix} 1 & a & b\\ -a & 1 & c\\ -b & -c & 1\\ \end{matrix}\, \right|=\] [MP PET 1991] |
| A. | \[1+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| B. | \[1-{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| C. | \[1+{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\] |
| D. | \[1+{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\] |
| Answer» B. \[1-{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] | |
| 70. |
If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=-2\]and \[f(x)=\left| \begin{matrix} 1+{{a}^{2}}x & (1+{{b}^{2}})x & (1+{{c}^{2}})x\\ (1+{{a}^{2}})x & 1+{{b}^{2}}x & (1+{{c}^{2}})x\\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & 1+{{c}^{2}}x\\ \end{matrix} \right|\] then f(x) is a polynomial of degree [AIEEE 2005] |
| A. | 3 |
| B. | 2 |
| C. | 1 |
| D. | 0 |
| Answer» C. 1 | |
| 71. |
If \[\left| \,\begin{matrix} x-1 & 3 & 0\\ 2 & x-3 & 4\\ 3 & 5 & 6\\ \end{matrix}\, \right|=0\], then x = [RPET 2003] |
| A. | 0 |
| B. | 2 |
| C. | 3 |
| D. | 1 |
| Answer» E. | |
| 72. |
\[\left| \,\begin{matrix} 1/a & 1 & bc\\ 1/b & 1 & ca\\ 1/c & 1 & ab\\ \end{matrix}\, \right|=\] [RPET 2002] |
| A. | 0 |
| B. | abc |
| C. | 1/abc |
| D. | None of these |
| Answer» B. abc | |