Summation of Determinant

Q. If a, b and care non zero numbers, then $\Delta =\left|\begin{array}{ccc}{b}^2{c}^2& bc& b+c\\ c^2{a}^2& ca& c+a\\ a^2{b}^2& ab& a+b\end{array}\right|$ is equal to

1. abc
2. $a^2{b}^2{c}^2$
3. ab+bc+ca
4. None of these

Q.

1. (2008)²

2. 2007

3. (2007)²

4. 2008

Q. For non zero, a, b, c if $\Delta =\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=0$, then the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=$

1. $Noneofthese$
2. $abc$
3. $\frac{1}{abc}$
4. $-a(a+b+c)$

Q. The approximate value of $(1.0002{)}^{3000}$ is equal to

1. $1.06$
2. $1.008$
3. $1.6$
4. $1.08$

Q. Simplify 9^1/2(log₅3)_{+ 27}^log₆³⁶₊₃^6/log₇9

Q. prove y= [4sinx/(2+cosx)] - x is increasing on [0, /2]

Q. Find the zeros of, x2+x-p(p-1).

Q.

The value of $0÷7$ is:

1. $1$

2. $0$

3. $70$

4. $50$

Q. If $p+q+r=0=a+b+c,$ then the determinant $△=\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ is equal to

1. $0$
2. $1$
3. $pa+qb+rc$
4. None of these

Q. $\mathrm{If}{U}_n=\left(\begin{array}{ccc}n& 1& 5\\ n^2& 2N+1& 2N+1\\ n^3& 3N^2& 3N\end{array}\right|,\mathrm{then}\underset{n=1}{\overset{N}{\sum U_n=}}$
.

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

Q. $If{x}^a{y}^b=e^m,x^c{y}^d=e^n,{\Delta }_1=\left|\begin{array}{cc}m& b\\ n& d\end{array}\right|{\Delta }_2=\left|\begin{array}{cc}a& m\\ c& n\end{array}\right|$ and ${\Delta }_3=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|,$then the values of x and y are respectively

1. $\frac{{\Delta }_2}{{\Delta }_1}$ and $\frac{{\Delta }_3}{{\Delta }_1}$
2. $\frac{{\Delta }_1}{{\Delta }_3}$ and $\frac{{\Delta }_2}{{\Delta }_3}$
3. $e^{\frac{{\Delta }_1}{{\Delta }_3}}$ and $e^{\frac{{\Delta }_2}{{\Delta }_3}}$
4. $log\left(\frac{{\Delta }_1}{{\Delta }_3}\right)$ and $log\left(\frac{{\Delta }_2}{{\Delta }_3}\right)$

Q.

Which of the following is or are correct statements for conditions of col-linearity of three points

$A(x_1,y_1),B(x_2,y_2)andC(x_3,y_3)?$
1.slope of line AB = slope of line BC = slope of line CA
2.Area of $△$ABC = 0
3. AC = AB + BC or AB ˜BC

1. only 1

2. only 1 and 2

3. all 1, 2 and 3

4. only 2 and 3

Q. Evaluate the determinants
(i)$\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& -0\end{array}\right|$ (ii)$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

Q. If $\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right|=16$, then the value of $\left|\begin{array}{ccc}p+x& a+x& a+p\\ q+y& b+y& b+q\\ r+z& c+z& c+r\end{array}\right|$ is

(a) 4
(b) 8
(c) 16
(d) 32

Q. Prove that

Q.

1. 1

2. -1

3. 0

4. 2

Q. Let $\Delta =\left|\begin{array}{ccc}Ax& x^2& 1\\ By& y^2& 1\\ Cz& z^2& 1\end{array}\right|$ and ${\Delta }_1=\left|\begin{array}{ccc}A& B& C\\ x& y& z\\ zy& zx& xy\end{array}\right|$ then

1. ${\Delta }_1=-\Delta$
2. ${\Delta }_1=\Delta$
3. ${\Delta }_1\ne \Delta$
4. ${\Delta }_1=2\Delta$

Q. Write the value of $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|.$

Q.

Verify whether the following are true or false

$0÷(-18)=-18$

1. True
2. False

Q. The value of the determinant $\left|\begin{array}{ccc}{b}^2-ab& b-c& bc-ac\\ ab-a^2& a-b& b^2-ab\\ bc-ac& c-a& ab-a^2\end{array}\right|$

1. $abc$
2. $a+b+c$
3. $0$
4. $ab+bc+ca$

Q. Prove that $\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right|=1+a^2+b^2+c^2$

Q. Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]$ and $U_1,U_2,U_3$ be column
matrices satisfying $A{U}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],A{U}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right],A{U}_3=\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]$. If U is
$3\times 3$ matrix whose columns are $U_1,U_2,U_3$, then $\left|U\right|=$

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

Q. ntSolven nt3x(x+y-2) = 2yn nty(x+y-1) = 9xn

Q. Let $a,b,c\in R$ be such that $a+b+c>0$ and $abc=2$. Let $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ If $A^2=I$, then value of $a^3+b^3+c^3$ is

1. 0
2. 2
3. -1
4. 7

Q. If A satisfies the equation $x^3-5x^2+4x+\lambda =0$ then A⁻¹ exists if
(a) $\lambda \ne 1$
(b) $\lambda \ne 2$
(c) $\lambda \ne -1$
(d) $\lambda \ne 0$

Q. Evaluate the following:

$\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|$

Q. Convert $\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ into an identity matrix by suitable row transformations.

Q. If $D_k=\left|\begin{array}{ccc}1& n& n\\ 2k& n^2+n+1& n^2+n\\ 2k-1& n^2& n^2+n+1\end{array}\right|$ and ${\sum }_{k=1}^n{D}_k=56$, then n equals

1. 4
2. 6
3. 8
4. None of these

Q. If $\omega \ne 1$ is a cube root of unity, then

1. $A$ is singular
2. $|A|=0$
3. $A$ is symmetric
4. none of these

Q. ntSolve:n nt(x-x) dy/dx + (2x-1)y = axn