1. If P=$\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right],A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and Q=${\mathrm{PAP}}^T$, then $P^T{Q}^{2005}P=$

2. A=$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right],I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $A^{-1}=\frac{1}{6}(A^2+\mathrm{cA}+\mathrm{dI},$ then c and d are

3. If A=$\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and det $A^3$=125, then $\alpha =$

4. If A=$\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and B=$\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$, then $A^2=B$ for

5. The number of values of k for which the system of equations $(k+1)x+8y=4k,\mathrm{kx}+(k+3)y=3k-1$has infinity of solutions is

6. The value of $\lambda$ such that the system $x-2y+z=-4,2x-y+2z=2,x+y+\lambda z=4$ has no solution is

7. If the system of equations x+ ay=0, az +y=0, ax +z=o has infinite number of solutions, then a=

8. The number of distinct real roots of the equation $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in $[-\frac{\pi }{4},\frac{\pi }{4}]$ is

9. If the system of equations $x-\mathrm{ky}-z=0,\mathrm{kx}-y-z=0,x+y-z=0$ has a non-zero solution, then k=

10. If $f(x)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& (x+1)x\\ 3x(x-1)& x(x-1)(x-2)& (x+1)x(x-1)\end{array}\right|$ then $f(100)=$

11. The parameter, on which the value of $\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos \mathrm{px}& \cos (p+d)x\\ \sin (p-d)x& \sin \mathrm{px}& \sin (p+d)x\end{array}\right|$ does not depend upon, is

12. $\left|\begin{array}{ccc}\mathrm{xp}+y& x& y\\ \mathrm{yp}+z& y& z\\ 0& \mathrm{xp}+y& \mathrm{yp}+z\end{array}\right|=0\mathrm{if}$

13. If x, y, z be positive, then $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|=$

14. $\left|\begin{array}{ccc}1& a& a^2-\mathrm{bc}\\ 1& b& b^2-\mathrm{ca}\\ 1& b& c^2-\mathrm{ab}\end{array}\right|=$

15. If $\left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & {\cos }^2\theta & 1+4\sin 4\theta \end{array}\right|=0$ then $\theta =$

16. The determinant $\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0\mathrm{if}$

17. If $\left|\begin{array}{ccc}\left(\begin{array}{c}x\\ r\end{array}\right)& \left(\begin{array}{c}x+1\\ r+1\end{array}\right)& \left(\begin{array}{c}x+2\\ r+2\end{array}\right)\\ \left(\begin{array}{c}y\\ r\end{array}\right)& \left(\begin{array}{c}y+1\\ r+1\end{array}\right)& \left(\begin{array}{c}y+2\\ r+2\end{array}\right)\\ \left(\begin{array}{c}z\\ r\end{array}\right)& \left(\begin{array}{c}z+1\\ r+1\end{array}\right)& \left(\begin{array}{c}z+2\\ r+2\end{array}\right)\end{array}\right|=\left|\begin{array}{ccc}\left(\begin{array}{c}x\\ r\end{array}\right)& \left(\begin{array}{c}x\\ r+1\end{array}\right)& \left(\begin{array}{c}x\\ r+2\end{array}\right)\\ \left(\begin{array}{c}y\\ r\end{array}\right)& \left(\begin{array}{c}y\\ r+1\end{array}\right)& \left(\begin{array}{c}y\\ r+2\end{array}\right)\\ \left(\begin{array}{c}z\\ r\end{array}\right)& \left(\begin{array}{c}z\\ r+1\end{array}\right)& \left(\begin{array}{c}z\\ r+2\end{array}\right)\end{array}\right|$ then $\lambda$

18. The number of real values of $\lambda$ for which the system of equations $\lambda x+y+z=0,x-\lambda y-z=0,x+y-\lambda z=0$ will have nontrivial solution is

19. If $\left|\begin{array}{ccc}1& a& \mathrm{bc}\\ 1& b& \mathrm{ca}\\ 1& c& \mathrm{ab}\end{array}\right|=\lambda \left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 1& 1& 1\end{array}\right|,\mathrm{then}\lambda =$

20. Given x=-9 is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0,$ the roots are

21. 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 A consisting of all determinants with value -1. Then

22. The solution set of the equation $\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|=0$ is

23. Let $p{x}^4+q{x}^3+r{x}^2+\mathrm{sx}+t=$ $\left|\begin{array}{ccc}{x}^2+3x& x-1& x+3\\ x+1& -2x& x-4\\ x-3& x+4& 3x\end{array}\right|$ be an identity, where p, q, r,s, t are constants. Then t=0

24. If a, b, c are positive and not all equal, then the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

25. If $\left|\begin{array}{ccc}{b}^2+c^2& a^2& a^2\\ b^2& c^2+a^2& b^2\\ c^2& c^2& {a^2+b}^2\end{array}\right|=\lambda a^2{b}^2{c}^2,\mathrm{then}\lambda =$

26. If $x>0,y>0,2x-5y=20,3x+\mathrm{my}=m,\mathrm{then}m\in$

27. If the system of equations $x+\mathrm{ky}+3z=0$, $3x+\mathrm{ky}-2z=0$, $2x+3y-4z=0$ has nontrivial solution, then $\frac{\mathrm{xy}}{z^2}=$

28. If x= cy+bz, y=az+cx, z=bx+ay, where x, y, z are not all zero, then $a^2+b^2+c^2=$

29. If $a+b+c\ne 0$ and
$(b+c)(y+z)-\mathrm{ax}=b-c$
$(c+a)(z+x)-\mathrm{by}=c-a$
$(a+b)(x+y)-\mathrm{cz}=a-b,$
then $x:y:z=$

30. A column matrix has only

31. If A, B are two matrices of the same type, the (A + B)' is equal to

32. If A is 3 x 4 matrix and B is a matrix such that A'B and B'A are both defined. Then B is of the type

33. If A and B are two matrices, then

34. If A is a square matrix, then A + A' is

35. If A and B are symmetric matrices of order n (A ≠ B), then

36. If A is a matrix of order 3 x 4, then each row of A has

37. If A and B are square matrices of same order, then ${(A+B)}^2$ = $A^2$ + 2AB + $B^2$ if

38. If A and B are square matrices of order 3 such that │A│= -1│B│= 3, then the determinant of 3 AB is equal to

39. If A, B, C be three square matrices such that A = B + C, then det A is equal to

40. If A is a square matrix such that $A^2$= 1, then $A^{-1}$is equal to

41. If A is any square matrix, then A (Adj A) is equal to

42. Matrices A and B will be inverse of each other, only if

43. If A and B be two invertible matrices of order 3, then ${(\mathrm{AB})}^{-1}$ is equal to

44. The system of equations AX = B of n equation in n unknown has infinitely many solutions if

45. The rotation through ${180}^o$ is identical to

46. Matrix theory was introduced by

47. The system of linear equations ax + by = 0, cx + dy = 0 has a non trivial solution if

48. A matrix is a