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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
If `A = [[1,2],[3,-5]]` and `B=[[1,0],[0,2]]` and X is a matrix such that `A=BX`, then X=A. `(1)/(2)[{:(,2,4),(,4,5):}]`B. `(1)/(2)[{:(,-2,4),(,3,5):}]`C. `[{:(,2,4),(,3,-5):}]`D. none of these |
| Answer» Correct Answer - A | |
| 2. |
If `|(x-4,2x,2x),(2x,x-4,2x),(2x,2x,x-4)|=(A+Bx)(x-A)^2` then the ordered pair (A,B) is equal toA. `(-4,5)`B. `(4,5)`C. `(-4,-5)`D. `(-4,3)` |
| Answer» Correct Answer - A | |
| 3. |
Let `A=|{:(,-2,7,sqrt3),(,0,0,-2),(,0,2,0):}| and A^(4)=lambda,I,"then"lambda "is"`A. `-16`B. 16C. 8D. `-8` |
| Answer» Correct Answer - B | |
| 4. |
Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value `-1`. ThenA. Statement-1is true, Statement-2 is true and Statement-2 is correct explantion for Statement-1.B. Statement-1 is true, Statement-2 is true and Statement-2 is not correct explantion for Statement-1.C. Statement-1 is true, Statement-2 is false.D. Statement-1 is false, Statement-2 is false. |
| Answer» Correct Answer - B | |
| 5. |
Obtain the inverse of the matrix `A=[{:(2,3),(1,1):}]` using elementary operations. |
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Answer» Write `A=IA. i.e. [{:(,0,1,2),(,1,2,3),(,3,1,1):}]=[{:(,1,0,0),(,0,1,0),(,0,1,1):}]A` or `[{:(,1,2,3),(,0,1,2),(,3,1,1):}]=[{:(,0,1,0),(,1,0,0),(,0,0,1):}]A ("applying "R_(1) Leftrightarrow R_(2)` `or [{:(,1,2,3),(,0,1,2),(,0,-5,-8):}]=[{:(,0,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(3) rightarrow R_(3)-3R_(1))` or `or [{:(,1,0,-1),(,0,1,2),(,0,-5,-8):}]=[{:(,-2,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(1) rightarrow R_(1)-2R_(2))` `or [{:(,1,0,-1),(,0,1,2),(,0,0,2):}]=[{:(,-2,1,0),(,1,0,0),(,5,-3,1):}]A ("applying "R_(3) rightarrow R_(3)+5R_(2))` `or [{:(,1,0,-1),(,0,1,2),(,0,0,1):}]=A[{:(,-2,1,0),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(3) rightarrow (1)/(2)R_(3))` `or [{:(,1,0,0),(,0,1,2),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(1) rightarrow R_(1)+R_(3))` `or [{:(,1,0,0),(,0,1,0),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(2) rightarrow R_(2)-2R_(3))` `"Hence"=A^(-1)[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]` |
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| 6. |
How many 3 x 3 skew symmetric matrices can be formed using numbers -2,-1,1,2,3,4. 0 (any num can be used any number of times but O can be used at most 3 times)A. 8B. 27C. 64D. 54 |
| Answer» Correct Answer - C | |
| 7. |
Which of the following is true for matrox `A=[{:(,1,-1),(,2,3):}]`A. `A+4I` is a symmetric matrixB. `A^(2)-4A+5I_(2)=0`C. A-B is a diagonal matrix for any value of `alpha "if" =[{:(,alpha,-1),(,2,5):}]`D. A-4I is a skew symmetric matrix. |
| Answer» Correct Answer - B::C | |
| 8. |
If `a , b , c`are non-zero, then the system of equations`(alpha+a)x+alphay+alphaz=0,alphax+(alpha+b)y+alphaz=0,alphax+alphay+(alpha+c)z=0`has a non-trivial solution if`alpha^(-1)=-(a^(-1)+b^(-1)+c^(-1))`b. `alpha^(-1)=a+b+c`c.`alpha+a+b+c=1`d. none of theseA. `alpha^(-1)=-(a^(-1)+b^(-1)+c^(-1))`B. `alpha^(-1)=a+b+c`C. `alpha+a+b+c=1`D. none of these |
| Answer» Correct Answer - A | |
| 9. |
Let AX=B where A is `3 xx 3` and X and B are `3 xx 1` matrices then which of the following is correct?A. If |A|=0 then AX=B has infinite solutionsB. If AX=B has infinite solutions then |A|=0C. If (adj (A))B=0 `|A|ne0` then AX=B has unique solutionsD. If (adj (A))Bne0 `|A|ne0` then AX=B has no solution |
| Answer» Correct Answer - B::C::D | |
| 10. |
Let zt be the set of all `3 xx 3` summetric matrices whose entries are `1,1,1,0,0,0,-1,-1,`.B is one of the matrix in set zt and `X=[{:(,x),(,y),(,z):}] U=[{:(,0),(,0),(,0):}] V=[{:(,1),(,0),(,0):}]` ltBRgt Number of matrices B in set is `lambda, "then" lamda` lies in the intervalA. (30,40)B. (38,40)C. (34,38)D. (25,35) |
| Answer» Correct Answer - A::C | |
| 11. |
Let zt be the set of all `3 xx 3` summetric matrices whose entries are `1,1,1,0,0,0,-1,-1,`.B is one of the matrix in set zt and `X=[{:(,x),(,y),(,z):}] U=[{:(,0),(,0),(,0):}] V=[{:(,1),(,0),(,0):}]` The equations BX=VA. is inconsistent for atleast 3 matrices, B.B. is inconsistent for all matrices, B.C. is inconsistent for all most 12 matrices, B.D. has infinite number of solutions for at least 3 matrices, B |
| Answer» Correct Answer - A::C | |
| 12. |
Let zt be the set of all `3 xx 3` summetric matrices whose entries are `1,1,1,0,0,0,-1,-1,`.B is one of the matrix in set zt and `X=[{:(,x),(,y),(,z):}] U=[{:(,0),(,0),(,0):}] V=[{:(,1),(,0),(,0):}]` Number of matrices B such that equations BX=U has infinite solutions.A. is at least 6B. is not more than 10C. lie between 8 to 16D. is zero |
| Answer» Correct Answer - A::C | |
| 13. |
Construct a `3 xx 2` matrix whose elements ar given by `a_(ij)=(1)/(2)(i-3j)` |
| Answer» In general a `3 xx 2` is given by `A=[{:(,a_(11),a_(12)),(,a_(21),22),(,a_(31),a_(32)):}]` | |
| 14. |
Let `f(x)=|{:(,1//x,logx,x^(n)),(,1,-1//n,(-1)^(n)),(,1,a,a^(2)):}|` where `("where"f^(n)(x) "donotes "n^(th)` derivative of f(x).A. `f^(n)(1)"is indepent of a"`B. `f^(n)(1)"is indepent of n"`C. `f^(n)(1)"is indepent a of n"`D. `y=a(x-f^(n)(1))` represent a straight line through the origin |
| Answer» Correct Answer - A::B::D | |
| 15. |
If a non-singular matrix and `A^(T)` denotes the tranpose of A, thenA. `|A|ne |A^(T)|`B. `|A.A^(T)|=|A|^(2)`C. `|A^(T).A|=|A^(T)|^(2)`D. `|A|=|A|^(T)ne0` |
| Answer» Correct Answer - B::C::D | |
| 16. |
If A=diag(2,-1,3), B=diag(-1,3,2)then `A^(2)B`A. diag(5,4,11)B. diag`(-4,3,18)`C. diag(3,1,8)D. diag(3,1,19) |
| Answer» Correct Answer - B | |
| 17. |
A matrix `A=[a_(ij)]` is an upper triangular matrix, ifA. `I lt i`B. `i=j`C. `I gt j`D. `I lt j` |
| Answer» Correct Answer - C | |
| 18. |
If I=`[{:(,1,0),(,0,1):}], J=[{:(,0,1),(,-1,0):}]and B=[{:(,cos theta,sin theta),(,-sin theta,cos theta)]:}"then B"=`A. `Icos theta+J sin theta`B. `Icostheta-Jsintheta`C. `I sin theta+Jcos theta`D. `-Icos theta+Jsin theta` |
| Answer» Correct Answer - A | |
| 19. |
If `A=[{:(,1),(,2),(,3):}]and B =[{:(,-5,4,0),(,0,2,-1),(,1,-3,2):}]"then"`A. `AB=[{:(,-5,8,0),(,0,4,-2),(,3,-9,6):}]`B. `AB=[-2,-1,4)`C. `AB=[{:(,-1),(,1),(,1)]`D. AB does not exist |
| Answer» Correct Answer - D | |
| 20. |
`[{:(,x^(2)+x,-1),(,3,2):}]+[{:(,0,-1),(,-x+1,x):}]=[{:(,0,-2),(,5,1):}]` then x is equalto-A. `-1`B. 2C. `4`D. No value of x |
| Answer» Correct Answer - A | |
| 21. |
If A and B are two matrices such that AB=B and BA=A , then `A^2+B^2=`A. 2ABB. 2BAC. A+BD. AB |
| Answer» Correct Answer - C | |
| 22. |
If , then value of `X^(n)` is (where n is a natural number)A. `{:[(,3n,-4),(,n,-n)]:}`B. `{:[(,2n+n,5-n),(,n,-n)]:}`C. `{:[(,3^(n),(-4)^(n)),(,1^(n),(-1)^(n))]:}`D. `{:[(,2n+1,-4),(,n,-(2n-1))]:}` |
| Answer» Correct Answer - D | |
| 23. |
Let A=`[{:p q q p:}]` such that det(A)=r where p,q,r all prime number, then trace of A is equal toA. 6B. 5C. 2D. 3 |
| Answer» Correct Answer - A | |
| 24. |
Let `A=[(0,1),(2,0)] and (A^(8)+A^(5)+A^(2)+I)V=[(32),(62)]` where is the `(2xx2` identity matrix). Then the product of all elements of matrix V isA. 2B. 1C. 3D. `-2` |
| Answer» Correct Answer - A | |
| 25. |
`|[-1,2,1] , [3+2sqrt2,2+2sqrt2,1] , [3-2sqrt2,2-2sqrt2,1]|=`A. `16sqrt2`B. `8sqrt2`C. 8D. none of these |
| Answer» Correct Answer - A | |
| 26. |
Let `A=[(cos^(- 1)x,cos^(- 1)y,cos^(- 1)z),(cos^(- 1)y,cos^(- 1)z,cos^(- 1)x),(cos^(- 1)z,cos^(- 1)x,cos^(- 1)y)]`such that `|A| = 0`, then maximum value of `x + y + z` isA. 3B. 0C. 1D. 2 |
| Answer» Correct Answer - A | |
| 27. |
If A and B are squar matrices of order 3 such that |A|=-1, |B|=3 then |3AB| is equal toA. `-9`B. `-81`C. `-27`D. 81 |
| Answer» Correct Answer - B | |
| 28. |
Let A=`[{:(,3x^(2)),(,1),(,6x):}], B=[a,b,c] and C=[{:(,(x+2)^(2),5x^(2),2x),(,5x^(2),2x,(x+1)^(2)),(,2x,(x+2)^(2),5x^(2))]:}` where, a,b c and `x in R`, Given that tr (AB)=tr(C ). Then the value of (a+b+c)A. 7B. 2C. 1D. 4 |
| Answer» Correct Answer - A | |
| 29. |
If `A=[a b c d]`(where `b c!=0`) satisfies the equations `x^2+k=0,t h e n``a+d=0`b. `K=-|A|`c. `k=|A|`d. none of theseA. `a+d=0&k=|A|`B. `a-d=0 & k=|A|`C. `a+d=0 & k=-|A|`D. `a+d ne 0,& k=|A|` |
| Answer» Correct Answer - A | |
| 30. |
If A=`[{:(,1,0,2),(,0,2,1),(,2,0,3):}]` is a root of polynomial `x^(3)-6x^(2)+7x+k=0` then the value of k isA. 2B. 4C. `-2`D. 1 |
| Answer» Correct Answer - A | |
| 31. |
How many `3xx3`matrices `M`with entries from `{0,1,2}`are there, for which the sum of the diagonal entries of `M^T Mi s5?`126 (b)198 (c) 162(d) 135A. 198B. 162C. 126D. 135 |
| Answer» Correct Answer - A | |
| 32. |
The number of values of k, for which the system of eauations: `(k+1)x+8y=4k` `kx+(k+3)y=3k-1` has no solution is,A. infiniteB. 1C. 2D. 3 |
| Answer» Correct Answer - B | |
| 33. |
If D=`|{:(,a^(2)+1,ab,ac),(,ba,b^(2)+1,bc),(,ca,cb,c^(2)+1):}|` then D=A. `1+a^(2)+b^(2)+c^(2)`B. `a^(2)+b^(2)+c^(2)`C. `(a+b+c)^(2)`D. none of these |
| Answer» Correct Answer - A | |
| 34. |
If the trivial solution is the only solution of the system of equations `x-ky + z = 0, kx + 3y-kz=0, 3x + y-z = 0` Then the set of all values of `k` is:A. `R-(2-3)`B. `R-(2)`C. `R-(-3)`D. `(2,-3)` |
| Answer» Correct Answer - A | |
| 35. |
If `a,b,c` are non-zero real numbers then `D=|[b^2 c^2, bc, b+c] , [c^2a^2, ca, c+a] , [a^2b^2, ab, a+b]|=` (A) abc (B) `a^2 b^2 c^2` (C) bc+ca+ab (D) 0A. abcB. `a^(2)b^(2)c^(2)`C. bc+ca+abD. zero |
| Answer» Correct Answer - D | |
| 36. |
If `x,y,z in R & Delta =|(x,x+y,x+y+z),(2x,5x+2y,7x+5y+2z),(3x,7x+3y,9x+7y+3z)|=-16` then the value of x isA. `-2`B. `-3`C. 2D. 3 |
| Answer» Correct Answer - C | |
| 37. |
`Delta =|(1+a^2+a^4,1+ab+a^2b^2, 1+ac+a^2c^2), (1+ab+a^2b^2, 1+b^2+b^4, 1+bc+b^2c^2),(1+ac+a^2c^2, 1+bc+b^2c^2, 1+c^2c^4)|` is equal toA. `(a-b)^(2)(b-c)^(2)(c-a)^(2)`B. `2(a-b)(b-c)(c-a)`C. `4(a-b)(b-c)(c-a)`D. `(a+b+c)^(3)` |
| Answer» Correct Answer - A | |
| 38. |
Let `theta=(pi)/(5),X=[{:(,cos theta,-sin theta),(,sin theta,cos theta):}]`, O is null matrix and I is an identity of order `2 xx 2`, and if `I+X+X^(2)+....+X^(n)=0` then n can beA. 9B. 19C. 4D. |
| Answer» Correct Answer - A::B::D | |
| 39. |
Flind the product of two matrices `A =[[cos^(2) theta , cos theta sin theta],[cos theta sin theta ,sin^(2)theta]] B= [[cos^(2) phi,cos phi sin phi],[cos phisin phi,sin^(2)phi]]` Show that, AB is the zero matrix if `theta and phi` differ by an odd multipl of `pi/2`.A. 5(6!)B. 3(6!)C. 12(6!)D. 8(6!) |
| Answer» Correct Answer - D | |
| 40. |
Let `Delta =|(sin theta cos phi, sin theta sin phi, cos theta), (cos theta cos phi, cos theta sin phi, -sin theta), (-sin theta sin phi, sin theta cos phi, 0)|` thenA. `Delta` is independent of `theta`B. `Delta` is independt of `phi`C. `Delta` is a constantD. none of these |
| Answer» Correct Answer - B | |
| 41. |
If `Delta_1=|[2a,b,e],[2d,e,f],[4x,2y,2z]|,Delta_2=|[f,2d,e],[2z,4x,2y],[e,2a,b]|`, then the value of `Delta_1-Delta_2` isA. `x+(y)/(z)+z`B. 2C. 0D. 3 |
| Answer» Correct Answer - C | |
| 42. |
If `[{:(,x+3,z+4,2y-7),(,-6,a-1,0),(,b-3,-21,0):}]=[{:(,0,6,3y-2),(,-6,-3,2c+2),(,2b+4,-21,0),(,2b+4,-21,0):}]` then find the values of a,b,c,x,y and z. |
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Answer» As the given matrices are equal. Therefore, their, corresponding elements must be equal. Comparing the corrsponding elements, we get `{:(,x+3=0,z+4=6,2y-7=3y-2),(,a-1=-3,0=2c+2,b-3=2b+4):}` `Rightarrow a=-2,b=-7,c=-1,x=-3,y=-5,z=2` |
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| 43. |
If `A=|{:(,5a,-b),(,3,2):}|` and A adj `A=A A^(T)`, then 5a+b is equal toA. 5B. 4C. 13D. `-1` |
| Answer» Correct Answer - A | |
| 44. |
If `alpha,beta!=0`, and `f(n)""=alpha^n+beta^n`and `|3 1+f(1)1+f(2)1+f(1)1+f(2)1+f(3)1+f(2)1+f(3)1+f(4)|=K(1-alpha)^2(1-beta)^2(alpha-beta)^2`, then K isequal to(1) `alphabeta`(2) `1/(alphabeta)`(3) 1(4) `-1`A. 1B. `-1`C. `alphabeta`D. `(1)/(alphabeta)` |
| Answer» Correct Answer - A | |
| 45. |
If `A=|{:(,2,-3),(,-4,1):}|` then adj `(3A^(2)+12A)` is equal toA. `[{:(,72,-84),(,-63,51):}]`B. `[{:(,51,63),(,84,72):}]`C. `[{:(,51,84),(,63,72):}]`D. `[{:(,72,-63),(,-84,51):}]` |
| Answer» Correct Answer - B | |
| 46. |
The set of the all values of `lamda` for which the system of linear equations `2x_(1) - 2x_(2) + x_(3) = lamdax_(1)` `2x_(1) - 3x_(2) + 2x_(3) = lamda x_(2)` `-x_(1) + 2x_(2) = lamda x_(3)` has a non-trivial solution,A. is an empty setB. is a singletonC. contains two elementsD. contains more than two elements |
| Answer» Correct Answer - C | |
| 47. |
Let `d in R, and A[{:(,-2,4+d,(sin theta-2)),(,1,(sin theta)+2,d),(,5,(2sin theta)d,(-sin theta)+2+2d):}]=theta in [0,2pi]` If the minimum value of det(A) is B. Then the value of d is:A. `-5`B. `2(sqrt2+2)`C. `2(sqrt2+1)`D. `-7` |
| Answer» Correct Answer - A | |
| 48. |
Let `A=|{:(,0,21,r),(,p,q,-r),(,p,-q,r):}|=A A^(T)=I_(3)` then |p| isA. `(1)/(sqrt5)`B. `(1)/(sqrt3)`C. `(1)/(sqrt6)`D. `(1)/(sqrt2)` |
| Answer» Correct Answer - D | |
| 49. |
Let p be an odd prime number and `T_p`, be the following set of `2 xx 2` matrices `T_p={A=[(a,b),(c,a)]:a,b,c in {0,1,2,.........p-1}}` The number of A in `T_p`, such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p isA. `(p-1)^(2)`B. 2(p-1)C. `(p-1)^(2)+1`D. 2p-1 |
| Answer» Correct Answer - D | |
| 50. |
Let a,b, and c be three real numbers satistying `[a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0]` Let `omega` be a solution of `x^3-1=0` with `Im(omega)gt0. I fa=2` with b nd c satisfying (E) then the vlaue of `3/omega^a+1/omega^b+3/omega^c` is equa to (A) -2 (B) 2 (C) 3 (D) -3A. `-2`B. 2C. 3D. `-3` |
| Answer» Correct Answer - A | |