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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 8301. |
If A is a skew symmetric matrix and n is a positive integer, then \[{{A}^{n}}\]is |
| A. | A symmetric matrix |
| B. | Skew-symmetric matrix |
| C. | Diagonal matrix |
| D. | None of these |
| Answer» E. | |
| 8302. |
If A is a square matrix, then which of the following matrices is not symmetric |
| A. | \[A+{A}'\] |
| B. | \[A{A}'\] |
| C. | \[{A}'A\] |
| D. | \[A-{A}'\] |
| Answer» E. | |
| 8303. |
If \[A=\left[ \begin{matrix} 5 & 2 \\ 3 & 1 \\ \end{matrix} \right],\]then \[{{A}^{-1}}\]= [EAMCET 1988] |
| A. | \[\left[ \begin{matrix} 1 & -2 \\ -3 & 5 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} -1 & 2 \\ 3 & -5 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} -1 & -2 \\ -3 & -5 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 1 & 2 \\ 3 & 5 \\ \end{matrix} \right]\] |
| Answer» C. \[\left[ \begin{matrix} -1 & -2 \\ -3 & -5 \\ \end{matrix} \right]\] | |
| 8304. |
If A and B be symmetric matrices of the same order, then \[AB-BA\] will be a |
| A. | Symmetric matrix |
| B. | Skew symmetric matrix |
| C. | Null matrix |
| D. | None of these |
| Answer» C. Null matrix | |
| 8305. |
Inverse of the matrix \[\left[ \begin{matrix} 3 & -2 & -1 \\ -4 & 1 & -1 \\ 2 & 0 & 1 \\ \end{matrix} \right]\] is [MP PET 1990] |
| A. | \[\left[ \begin{matrix} 1 & 2 & 3 \\ 3 & 3 & 7 \\ -2 & -4 & -5 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 1 & -3 & 5 \\ 7 & 4 & 6 \\ 4 & 2 & 7 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 1 & -3 & 5 \\ 7 & 4 & 6 \\ 4 & 2 & -7 \\ \end{matrix} \right]\] |
| Answer» D. \[\left[ \begin{matrix} 1 & -3 & 5 \\ 7 & 4 & 6 \\ 4 & 2 & -7 \\ \end{matrix} \right]\] | |
| 8306. |
If \[A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\] and I is the unit matrix of order 3, then \[{{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}\] is equal to |
| A. | \[7{{A}^{8}}\] |
| B. | \[7{{A}^{7}}\] |
| C. | 8I |
| D. | 6I |
| Answer» B. \[7{{A}^{7}}\] | |
| 8307. |
If \[{{A}_{1}},{{A}_{3}},.......,{{A}_{2n-1}}\] are n skew-symmetric matrices of same order, then \[B=\sum\limits_{r=1}^{n}{(2r-1){{({{A}_{2r-1}})}^{2r-1}}}\] will be |
| A. | Symmetric |
| B. | Skew-symmetric |
| C. | Neither symmetric nor skew-symmetric |
| D. | Data is adequate |
| Answer» C. Neither symmetric nor skew-symmetric | |
| 8308. |
If A and B are two matrices such that AB = A and BA=B, then which one of the following is correct? |
| A. | \[{{({{A}^{T}})}^{2}}={{A}^{T}}\] |
| B. | \[{{({{A}^{T}})}^{2}}={{B}^{T}}\] |
| C. | \[{{({{A}^{T}})}^{2}}={{({{A}^{-1}})}^{-1}}\] |
| D. | None of the above |
| Answer» B. \[{{({{A}^{T}})}^{2}}={{B}^{T}}\] | |
| 8309. |
If \[\left[ \begin{matrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\]then find the value of x |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{5}\] |
| C. | No unique value of 'x' |
| D. | None of these |
| Answer» C. No unique value of 'x' | |
| 8310. |
Let \[A=\left[ \begin{matrix} 0 & \alpha \\ 0 & 0 \\ \end{matrix} \right]\] and \[{{(A+I)}^{50}}-50A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right],\]find\[abc+abd+bcd+acd\] |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | None of these |
| Answer» B. -1 | |
| 8311. |
If \[X=\left[ \begin{matrix} 1 & -2 \\ 0 & 3 \\ \end{matrix} \right]\], and I is a \[2\times 2\] identity matrix, then \[{{X}^{2}}-2X+3I\] equals to which one of the following? |
| A. | #NAME? |
| B. | -2X |
| C. | 2X |
| D. | 4X |
| Answer» D. 4X | |
| 8312. |
Let A, B, C, D be (not necessarily square) real matrices such that \[{{A}^{T}}=BCD;\text{ }{{B}^{T}}=CDA;\] \[{{C}^{T}}=DAB\] and \[{{D}^{T}}=ABC\] for the matrix \[S=ABCD,{{S}^{3}}=\] |
| A. | I |
| B. | \[{{S}^{2}}\] |
| C. | S |
| D. | O |
| Answer» D. O | |
| 8313. |
If \[A=\left[ \begin{matrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right]\] then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}{{A}^{n}}\] is |
| A. | A null matrix |
| B. | An identity matrix |
| C. | \[\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix} \right]\] |
| D. | None of these |
| Answer» B. An identity matrix | |
| 8314. |
If A is symmetric as well as skew-symmetric matrix, then A is |
| A. | Diagonal |
| B. | Null |
| C. | Triangular |
| D. | None of these |
| Answer» C. Triangular | |
| 8315. |
Let \[A+2B=\left[ \begin{matrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \\ \end{matrix} \right]\] and\[2A-B=\left[ \begin{matrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \\ \end{matrix} \right]\], then \[\operatorname{tr}(A) tr(B)\] is |
| A. | 1 |
| B. | 3 |
| C. | 2 |
| D. | 0 |
| Answer» D. 0 | |
| 8316. |
If \[B=\left[ \begin{matrix} 3 & 4 \\ 2 & 3 \\ \end{matrix} \right]\] and \[C=\left[ \begin{matrix} 3 & -4 \\ -2 & 3 \\ \end{matrix} \right]\] and \[X=BC\],find \[{{X}^{n}}\] |
| A. | 0 |
| B. | I |
| C. | 2I |
| D. | None of these |
| Answer» C. 2I | |
| 8317. |
The values of a, b, c if \[\left[ \begin{matrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \\ \end{matrix} \right]\] is orthogonal are |
| A. | \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{6}};c=\pm \frac{1}{\sqrt{3}}\] |
| B. | \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{3}};c=\pm \frac{1}{\sqrt{6}}\] |
| C. | \[a=\pm \frac{1}{\sqrt{6}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{3}}\] |
| D. | \[a=\pm \frac{1}{\sqrt{3}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{6}}\] |
| Answer» B. \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{3}};c=\pm \frac{1}{\sqrt{6}}\] | |
| 8318. |
If \[A=\left[ \begin{matrix} 1 & 2 \\ 0 & 3 \\ \end{matrix} \right]\] is a \[2\times 2\] matrix and \[f(x)={{x}^{2}}-x+2\] is a polynomial, then what is f(A)? |
| A. | \[\left[ \begin{matrix} 1 & 7 \\ 1 & 7 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 2 & 6 \\ 0 & 8 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} 2 & 6 \\ 0 & 6 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 2 & 6 \\ 0 & 7 \\ \end{matrix} \right]\] |
| Answer» C. \[\left[ \begin{matrix} 2 & 6 \\ 0 & 6 \\ \end{matrix} \right]\] | |
| 8319. |
If A is a square matrix, then \[A{{A}^{T}}\] is a |
| A. | Skew-symmetric matrix |
| B. | Symmetric matrix |
| C. | Diagonal matrix |
| D. | None of these |
| Answer» C. Diagonal matrix | |
| 8320. |
Let A and B be \[3\times 3\] matrices of real numbers, where A is symmetric, B is skew symmetric, and \[(A+B)(A-B)=(A-B)(A+B).\] If \[{{(AB)}^{t}}={{(-1)}^{k}}AB\]where \[{{(AB)}^{t}}\] is the transpose of the matrix AB, then k is |
| A. | Any integer |
| B. | Odd integer |
| C. | Even integer |
| D. | Cannot say anything |
| Answer» C. Even integer | |
| 8321. |
If \[A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\] and \[{{A}^{2}}=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right]\], then |
| A. | \[\alpha =2ab,\beta ={{a}^{2}}+{{b}^{2}}\] |
| B. | \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =ab\] |
| C. | \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =2ab\] |
| D. | \[\alpha ={{a}^{2}}+{{b}^{2}},\beta ={{a}^{2}}-{{b}^{2}}\] |
| Answer» D. \[\alpha ={{a}^{2}}+{{b}^{2}},\beta ={{a}^{2}}-{{b}^{2}}\] | |
| 8322. |
If \[A=\left[ \begin{matrix} 1 & 0 \\ -1 & 7 \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\], then the value of k so that \[{{A}^{2}}=8A+kI\] is |
| A. | \[k=7\] |
| B. | \[k=-7\] |
| C. | \[k=0\] |
| D. | None of these |
| Answer» C. \[k=0\] | |
| 8323. |
Let \[A=\left[ \begin{align} & \begin{matrix} 5 & 6 & 1 \\ \end{matrix} \\ & \begin{matrix} 2 & -1 & 5 \\ \end{matrix} \\ \end{align} \right]\]. Let there exist a matrix B such that \[AB=\left[ \begin{matrix} 35 & 49 \\ 29 & 13 \\ \end{matrix} \right]\]. What is B equal to? |
| A. | \[\left[ \begin{align} & \begin{matrix} 5 & 1 & 4 \\ \end{matrix} \\ & \begin{matrix} 2 & 6 & 3 \\ \end{matrix} \\ \end{align} \right]\] |
| B. | \[\left[ \begin{align} & \begin{matrix} 2 & 6 & 3 \\ \end{matrix} \\ & \begin{matrix} 5 & 1 & 4 \\ \end{matrix} \\ \end{align} \right]\] |
| C. | \[\left[ \begin{align} & \begin{matrix} 5 & 2 \\ \end{matrix} \\ & \begin{matrix} 1 & 6 \\ \end{matrix} \\ & \begin{matrix} 4 & 3 \\ \end{matrix} \\ \end{align} \right]\] |
| D. | \[\left[ \begin{align} & \begin{matrix} 2 & 5 \\ \end{matrix} \\ & \begin{matrix} 6 & 1 \\ \end{matrix} \\ & \begin{matrix} 3 & 4 \\ \end{matrix} \\ \end{align} \right]\] |
| Answer» D. \[\left[ \begin{align} & \begin{matrix} 2 & 5 \\ \end{matrix} \\ & \begin{matrix} 6 & 1 \\ \end{matrix} \\ & \begin{matrix} 3 & 4 \\ \end{matrix} \\ \end{align} \right]\] | |
| 8324. |
If C is skew-symmetric matrix of order n and X is \[n\times 1\] column matrix, then X'CX is a |
| A. | Scalar matrix |
| B. | Unit matrix |
| C. | Null matrix |
| D. | None of these |
| Answer» D. None of these | |
| 8325. |
If \[{{A}^{k}}=0\] (A is nilpotent with index k),\[{{(I-A)}^{p}}=I+A+{{A}^{2}}+....+{{A}^{k-1}},\] thus p is, |
| A. | -1 |
| B. | -2 |
| C. | ½ |
| D. | None of these |
| Answer» B. -2 | |
| 8326. |
The matrix \[A=\left[ \begin{matrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \\ \end{matrix} \right]\] is |
| A. | Idempotent matrix |
| B. | Involutory matrix |
| C. | Nilpotent matrix |
| D. | None of these |
| Answer» C. Nilpotent matrix | |
| 8327. |
Let \[A=\left[ \begin{matrix} x+y & y \\ 2x & x-y \\ \end{matrix} \right],B=\left[ \begin{matrix} 2 \\ -1 \\ \end{matrix} \right]\] and \[C=\left[ \begin{matrix} 3 \\ 2 \\ \end{matrix} \right]\] If \[AB=C,\] then what is \[{{A}^{2}}\] equal to? |
| A. | \[\left[ \begin{matrix} 6 & -10 \\ 4 & 26 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} -10 & 5 \\ 4 & 24 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} -5 & -6 \\ -4 & -20 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} -5 & -7 \\ -5 & 20 \\ \end{matrix} \right]\] |
| Answer» B. \[\left[ \begin{matrix} -10 & 5 \\ 4 & 24 \\ \end{matrix} \right]\] | |
| 8328. |
If \[A=\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \\ \end{matrix} \right]\], then \[{{A}^{16}}\] is equal to: |
| A. | \[\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} -1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\] |
| Answer» E. | |
| 8329. |
If B, C are square matrices of order n and if \[A=B+C,\text{ }BC=CB,\text{ }{{C}^{2}}=0\], then for any positive integer \[N,{{A}^{N+1}}={{B}^{K}}[B+(N+1)C],\] then K/N is |
| A. | 1 |
| B. | ½ |
| C. | 2 |
| D. | None of these |
| Answer» B. ½ | |
| 8330. |
If the matrix B is the adjoint of the square matrix A and \[\alpha \] is the value of the determinant of A, then what is AB equal to? |
| A. | \[\alpha \] |
| B. | \[\left( \frac{1}{\alpha } \right)I\] |
| C. | \[I\] |
| D. | \[\alpha I\] |
| Answer» E. | |
| 8331. |
If A, B, and C are the angles of a triangle and \[\left| \begin{matrix} 1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+{{\sin }^{2}}A & \sin B+{{\sin }^{2}}B & \sin C+{{\sin }^{2}}C \\ \end{matrix} \right|=0,\]then the triangle must be |
| A. | Isosceles |
| B. | Equilateral |
| C. | Right-angled |
| D. | None of these |
| Answer» B. Equilateral | |
| 8332. |
If in a triangle ABC, \[\left| \begin{matrix} 1 & \sin A & {{\sin }^{2}}A \\ 1 & \sin B & {{\sin }^{2}}B \\ 1 & \sin C & {{\sin }^{2}}C \\ \end{matrix} \right|=0\] then the triangle is |
| A. | Equilateral or isosceles |
| B. | Equilateral or right-angled |
| C. | Right angled or isosceles |
| D. | None of these |
| Answer» B. Equilateral or right-angled | |
| 8333. |
If \[\omega \] is a complex cube root of unity, then value of\[\Delta =\left| \begin{matrix} {{a}_{1}}+{{b}_{1}}\omega & {{a}_{1}}{{\omega }^{2}}+{{b}_{1}} & {{c}_{1}}+{{b}_{1}}\bar{\omega } \\ {{a}_{2}}+{{b}_{2}}\omega & {{a}_{2}}{{\omega }^{2}}+{{b}_{2}} & {{c}_{2}}+{{b}_{2}}\bar{\omega } \\ {{a}_{3}}+{{b}_{3}}\omega & {{a}_{3}}{{\omega }^{2}}+{{b}_{3}} & {{c}_{3}}+{{b}_{3}}\bar{\omega } \\ \end{matrix} \right|\] is |
| A. | \[0\] |
| B. | \[-1\] |
| C. | \[2\] |
| D. | None of these |
| Answer» B. \[-1\] | |
| 8334. |
The equations \[2x+3y+4=0;\] \[3x+4y+6=0\] and \[4x+5y+8=0\]are |
| A. | Consistent with unique solution |
| B. | Inconsistent |
| C. | Consistent with infinitely many solutions |
| D. | None of the above |
| Answer» B. Inconsistent | |
| 8335. |
If \[\text{l}_{r}^{2}+m_{r}^{2}+n_{r}^{2}=\text{1};\] \[r=1,2,3\] and \[{{\text{l}}_{r}}{{\text{l}}_{s}}+{{m}_{r}}{{m}_{s}}+{{n}_{r}}{{n}_{s}}=0;\]\[r\ne s,\]\[r=1,2,3;\] \[s=1,2,3,\]then the value of \[\left| \begin{matrix} {{\text{l}}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{\text{l}}_{2}} & {{m}_{2}} & {{n}_{2}} \\ {{\text{l}}_{3}} & {{m}_{3}} & {{n}_{3}} \\ \end{matrix} \right|\] is |
| A. | \[0\] |
| B. | \[\pm 1\] |
| C. | \[2\] |
| D. | None of these |
| Answer» C. \[2\] | |
| 8336. |
The value of \[\left| \begin{matrix} ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0,\] when m is equal to |
| A. | \[6\] |
| B. | \[5\] |
| C. | \[4\] |
| D. | \[1\] |
| Answer» C. \[4\] | |
| 8337. |
If \[A\left[ \begin{matrix} 1 & 2 \\ 3 & 5 \\ \end{matrix} \right],\] then the value of the determinant \[|{{A}^{2009}}-5{{A}^{2008}}|\] is |
| A. | \[-6\] |
| B. | \[-5\] |
| C. | \[-4\] |
| D. | \[4\] |
| Answer» B. \[-5\] | |
| 8338. |
If \[a,b,c,d>0,x\text{ }\in \text{R}\] and \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{x}^{2}}-2(ab+bc+cd)x+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\le 0.\]Then, \[\left| \begin{matrix} 33 & 14 & \log a \\ 65 & 27 & \log b \\ 97 & 40 & \log c \\ \end{matrix} \right|\] is equal to |
| A. | \[1\] |
| B. | \[-1\] |
| C. | \[2\] |
| D. | \[0\] |
| Answer» E. | |
| 8339. |
If \[A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{matrix} \right],\] then the value of \[|adj\,\,A|\] is |
| A. | \[{{a}^{27}}\] |
| B. | \[{{a}^{9}}\] |
| C. | \[{{a}^{6}}\] |
| D. | \[{{a}^{2}}\] |
| Answer» D. \[{{a}^{2}}\] | |
| 8340. |
If \[a\ne p,\] \[b\ne q,\] \[c\ne r\]and \[\left| \begin{matrix} p & b & c \\ a & q & c \\ a & b & r \\ \end{matrix} \right|=0\] then the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is equal to |
| A. | \[-1\] |
| B. | \[1\] |
| C. | \[-2\] |
| D. | \[2\] |
| Answer» E. | |
| 8341. |
Let \[A={{[{{a}_{ij}}]}_{m\times m}}\] be a matrix and \[C={{[{{c}_{ij}}]}_{m\times m}}\] be another matrix where \[{{c}_{ij}}\] is the cofactor of \[{{a}_{ij}}\]Then, what is the value of \[|AC|\]? |
| A. | \[|A{{|}^{m-1}}\] |
| B. | \[|A{{|}^{m}}\] |
| C. | \[|A{{|}^{m+1}}\] |
| D. | Zero |
| Answer» D. Zero | |
| 8342. |
If the value of the determinant \[\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{matrix} \right|\] is positive, where \[a\ne b\ne c,\] then the value of abc |
| A. | Cannot be less than 1 |
| B. | Is greater than \[-8\] |
| C. | Is less than \[-8\] |
| D. | Must be greater than 8 |
| Answer» C. Is less than \[-8\] | |
| 8343. |
If \[{{e}^{i\theta }}=\cos \theta +i\sin \theta ,\] then the value of \[\left| \begin{matrix} 1 & {{e}^{i\pi /3}} & {{e}^{i\pi /4}} \\ {{e}^{-i\pi /3}} & 1 & {{e}^{i2\pi /3}} \\ {{e}^{-i\pi /4}} & {{e}^{-i2\pi /3}} & 1 \\ \end{matrix} \right|\]is |
| A. | \[-2+\sqrt{2}\] |
| B. | \[2-\sqrt{2}\] |
| C. | \[-2-\sqrt{2}\] |
| D. | 1 |
| Answer» D. 1 | |
| 8344. |
If \[C=2cos\theta ,\] then the value of the determinant\[\Delta =\left[ \begin{matrix} C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C \\ \end{matrix} \right]\] is |
| A. | \[\frac{2{{\sin }^{2}}2\theta }{\sin \theta }\] |
| B. | \[8{{\cos }^{3}}\theta -4\cos \theta +6\] |
| C. | \[\frac{2\sin 2\theta }{\sin \theta }\] |
| D. | \[8{{\cos }^{3}}\theta +4\cos \theta +6\] |
| Answer» C. \[\frac{2\sin 2\theta }{\sin \theta }\] | |
| 8345. |
For all values of A, B, C and P, Q, R the value of the determinant\[{{(x+a)}^{3}}\left| \begin{matrix} \cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R) \\ \end{matrix} \right|\] is |
| A. | \[1\] |
| B. | \[0\] |
| C. | \[2\] |
| D. | None of these |
| Answer» C. \[2\] | |
| 8346. |
If the system of linear equations \[x+2ay+az=0;\] \[x+3by+bz=0;\] \[x+4cy+cz=0\] has a non - zero solution, then a, b, c. |
| A. | Satisfy \[a+2b+3c=0\] |
| B. | Are in A.P |
| C. | Are in G.P |
| D. | Are in H.P. |
| Answer» E. | |
| 8347. |
The number of values of k for which the system of equations \[(k+1)x+8y=4k;\] \[kx+(k+3)y=3k-1\] has infinitely many solutions is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | infinite |
| Answer» C. 2 | |
| 8348. |
If and if A is invertible, then which of the following is not true? |
| A. | \[\left| A \right|=\left| B \right|\] |
| B. | \[\left| A \right|=-\left| B \right|\] |
| C. | \[\left| adj\,A \right|=\left| adj\,B \right|\] |
| D. | A is invertible if and only if B is invertible |
| Answer» B. \[\left| A \right|=-\left| B \right|\] | |
| 8349. |
\[{{(-A)}^{-1}}\] is always equal to (where A is nth-order square matrix) |
| A. | \[{{(-1)}^{n}}{{A}^{-1}}\] |
| B. | \[-{{A}^{-1}}\] |
| C. | \[{{(-1)}^{n-1}}{{A}^{-1}}\] |
| D. | none of these |
| Answer» C. \[{{(-1)}^{n-1}}{{A}^{-1}}\] | |
| 8350. |
The value of the determinant\[\left| \begin{matrix} kb & {{k}^{^{2}}}+{{a}^{2}} & 1 \\ kb & {{k}^{2}}+{{b}^{2}} & 1 \\ kc & {{k}^{2}}+{{c}^{2}} & 1 \\ \end{matrix} \right|\]is |
| A. | \[k(a+b)(b+c)(c+a)\] |
| B. | \[k\,abc({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] |
| C. | \[k(a-b)(b-c)(c-a)\] |
| D. | \[k(a+b-c)(b+c-a)(c+a-b)\] |
| Answer» D. \[k(a+b-c)(b+c-a)(c+a-b)\] | |