Properties of Determinants

Q.

The value of is equal to

1. 0

2. 1

3. xyz

4. log xyz

Q. The determinant $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right|$ is not equal to

1. $\left|\begin{array}{ccc}2& 1& 1\\ 2& 2& 3\\ 2& 3& 6\end{array}\right|$
2. $\left|\begin{array}{ccc}2& 1& 1\\ 3& 2& 3\\ 4& 3& 6\end{array}\right|$
3. $\left|\begin{array}{ccc}1& 2& 1\\ 1& 5& 3\\ 1& 9& 6\end{array}\right|$
4. $\left|\begin{array}{ccc}3& 1& 1\\ 6& 2& 3\\ 10& 3& 6\end{array}\right|$

Q. Let $f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|.$ The value of $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}$ is equal to

1. $1$
2. $-1$
3. $0$
4. $2$

Q. $\left|\begin{array}{ccc}a+b& a+2b& a+3b\\ a+2b& a+3b& a+4b\\ a+4b& a+5b& a+6b\end{array}\right|=$

1. $a^2+b^2+c^2-3abc$
2. 3ab
3. 3a+5b
4. \N

Q. Let a , b be arbitrary real numbers find the smallest natural number b for which the equation x^2 +2(a+b)x + (a-b+8)=0 has an equal real roots for all a belong to R

Q. If a and b are rational and α and β be the roots of x^2+2ax+b=0, then the equation with rational coefficients one of whose roots is α+β+\sqrt{α^2+β^2}

Q. If ω is an imaginary number cube root of unity then (1-ω^4)(1-ω^8)(1-ω^{22})(1-ω^{44})equals to (1)ω^{2 } (2)ω (3)9 (4)0

Q. 13. Suppose that g(x)=1+\sqrt{}x and f(g(x)=3+2\sqrt{}x+x .then f(x) is

Q.

If $x\sin \theta =y\sin (\theta +2\pi /3)=z\sin (\theta +4\pi /3)$ then $xy+yz+zx$ is equal to:

1. $1$

2. $\frac{1}{2}$

3. $0$

4. None of these

Q. Find the value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$, where a, b and c are respectively the pth, qth and rth terms of a harmonic progression.

1. 1
2. pqr
3. \N
4. $\frac{1}{abc}$

Q. Prove that though covariance is independent of the choice of origin, it depends upon the scale. If u=ax+b, v=cy+d, show that cov(u, v)=a.c. cov(x, y)

Q. Given, $x=cy+bz,y=az+cx,z=bx+ay$, where x, y, z are not all zero, and a, b, c are real numbers. The value of $a^2+b^2+c^2+2abc$ is

1. \N
2. 4
3. 2
4. 1

Q. 134. Prove that (1+w)-(1-w) =0 , where w is cube root of unity

Q.

Is the rank of a matrix equal to the rank of its transpose?

Q.

The value of $\log \left(\tan \left(\frac{\mathrm{\pi }}{4}+\frac{ix}{2}\right)\right)$ is

1. $i{\tan }^{-1}\left(\sin hx\right)$

2. $i{\tan }^{-1}\left(\cos hx\right)$

3. None of these

Q. find the real values of the parameter a such that (2a +1)x^2 - a(x-1)=2 has one root greater than 1 and other less than 1

Q. If a+b+c=-46 and the roots x, y, z of x^3+ax^2+bx+c are integers and greater than 2 then x-y+z is equal to

Q.

Read the following information carefully and answer the following request. Let $p$ be an odd prime number and $T_p$ be the following set of $2\times 2$ matrices. The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both and $det\left(A\right)$ is divisible by $p$ is

1. ${(p-1)}^2$

2. $2(p-1)$

3. ${(p-1)}^2+1$

4. $2p-1$

Q. $A=\left[\begin{array}{ccc}3& a& -1\\ 2& 5& c\\ b& 8& 2\end{array}\right]$ is symmetric and $B=\left[\begin{array}{ccc}d& 3& a\\ b-a& e& -2b-c\\ -2& 6& -f\end{array}\right]$ is skew-symmetric, then $AB$ is

1. $\left[\begin{array}{ccc}4& -3& 6\\ 31& -54& 26\\ 28& -9& 50\end{array}\right]$
2. $\left[\begin{array}{ccc}-4& -31& -28\\ 3& 54& 9\\ -6& -26& -50\end{array}\right]$
3. $\left[\begin{array}{ccc}-4& 3& -6\\ -31& 54& -26\\ -28& 9& -50\end{array}\right]$
4. $\left[\begin{array}{ccc}4& 31& 28\\ -3& -54& -9\\ 6& 26& 50\end{array}\right]$

Q. If $\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$ ; then for all

$\theta \in (\frac{3\pi }{4},\frac{5\pi }{4}),det\left(A\right)$ lies in the interval :

1. $(0,\frac{3}{2}]$
2. $(1,\frac{5}{2}]$
3. $(\frac{3}{2},3]$
4. $[\frac{5}{2},4)$

Q. 19 If omega is a non real imaginary cube root of unity then find arg(iw) + arg(iw).

Q.

How to find the inverse of the matrix?

Q. Suppose a, b, c are three real numbers, such that the quadratic equation x²-(a+b+c)x + (ab+bc+ac) = 0 has roots of the form d±ie where d>0 and e≠0 are real numbers [ here i = √-1 ]. Show that (I) the numbers a, b and c are all positive. (II) the numbers √a , √b, √c, form the sides of triangle.

Q. The least positive integer n such that ${\left(\frac{2i}{1+i}\right)}^n$ is a positive integer, is
(a) 16
(b) 8
(c) 4
(d) 2

Q.

What is the elementary transformation of matrices ?

Q. Let $A$ and $B$ be two invertible matrices of order $3\times 3$. If $\text{det}\left(AB{A}^T\right)=8$ and $\text{det}\left(A{B}^{-1}\right)=8$, then $\text{det}\left(B{A}^{-1}{B}^T\right)$ is equal to :

1. $1$
2. $\frac{1}{4}$
3. $16$
4. $\frac{1}{16}$

Q. $\left|\begin{array}{ccc}{b}^2-ab& b-c& bc-ac\\ ab-a^2& a-b& b^2-ab\\ bc-ac& c-a& ab-a^2\end{array}\right|=$

1. abc(a+b+c)
2. $3a^2{b}^2{c}^2$
3. \N
4. None of these

Q. If a, b , c are rational and a +b+ c=0, prove that the roots of the
Quadratic equation ax^2+bx+ c =0 are rational.

Q.

Q. Let $f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1$ and $g$ is either odd or even function. If $fog$ is defined, then

1. $f\left(x\right)$ is even function
2. $fog$ is odd function
3. $fog$ is even function
4. $f\left(x\right)$ is odd function