Sarrus Rule

Q. Find value of Determinant using SarrUs Rule $\left|\begin{array}{ccc}1& 2& 6\\ 3& 1& 5\\ 7& 2& 1\end{array}\right|$

1. 40
2. 49
3. 54
4. 60

Q. $\text{Let}x\in R$ and let $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right],$ $Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}.$ Then which of the following options is/are correct?

1. There exists a real number $x$ such that $PQ=QP$
2. $\text{det}R=\text{det}\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 5\end{array}\right]+8,\text{for all}x\in R$
3. $\text{For}x=0,\text{if}R\left[\begin{array}{c}1\\ a\\ b\end{array}\right]=6\left[\begin{array}{c}1\\ a\\ b\end{array}\right],$ then $a+b=5$
4. $\text{For}x=1$, there exists a unit vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\text{for which}R\left[\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Q. If p + q + r = 0 and $\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|=k\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|,thenK=$

1. 0
2. abc
3. pqr
4. a + b +c

Q. The equation $\left|\begin{array}{ccc}x& a& a\\ a& x& a\\ a& x& a\end{array}\right|+\left|\begin{array}{ccc}b& b& x\\ b& x& b\\ x& b& b\end{array}\right|$ = 0 is satisfied when x equals

1. 0
2. a-b
3. a+b
4. none of these

Q. For $x\in R$, let $\left[x\right]$ denotes the greatest integer $\le x$, then the value of $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{2}{100}]+...+[-\frac{1}{3}-\frac{99}{100}]$ is

1. $-100$
2. $-123$
3. $-135$
4. $-153$

Q. Sarrus Rule says that determinant of the three colums on the left is the sum of products along dashed diagonals minus the sum of products along solid diagonals.

1. True
2. False

Q. Given $3\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ z+w& 3\end{array}\right],$ find the values of $x,y,z$ and $w$

Q. Using properties of determinants, prove that $\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$

Q.

Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is$___

Q. Find value of Determinant using SarrUs Rule $\left|\begin{array}{ccc}1& 2& 6\\ 3& 1& 5\\ 7& 2& 1\end{array}\right|$

1. 40
2. 49
3. 54
4. 60

Q. Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is$

1. 5
2. 6
3. 7
4. 8

Q. Sarrus Rule says that determinant of the three colums on the left is the sum of products along dashed diagonals minus the sum of products along solid diagonals.

1. True
2. False

Q. If $P=\left|\begin{array}{cc}x& 1\\ 1& x\end{array}\right|$ and $Q=\left|\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right|,$ then $\frac{dQ}{dx}=$

1. $3P+1$
2. $1-3P$
3. $-3P$
4. $3P$

Q. $\left|\begin{array}{ccc}x& 1& y+z\\ y& 1& z+x\\ z& 1& x+y\end{array}\right|$ =

1. $x+y+z$
2. $0$
3. $1$
4. $1+x+y+z$

Q. If three points $(k,2k),(2k,3k),(3,1)$ are collinear, then $k$ is equal to:

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

Q. If $a,b$ and $c$ are real numbers and $\Delta =\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|=0$,
show that either $a+b+c=0$ or $a=b=c$.

Q. $f\left(x\right)=4+5x^2+6x^4$ has

1. Neither maximum nor minimum
2. Only one minimum
3. Only one maximum
4. No minimum

Q.

Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is$___

Q. $\left|\begin{array}{ccc}a-b& b-c& c-a\\ x-y& y-z& z-x\\ p-q& q-r& r-p\end{array}\right|$

1. None of these
2. 0
3. abc+xyz+pqr
4. a(x+y+z)+b(p+q+r)+c

Q.

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Q.

Sum of series $\left(1+2\right)+\left(3+5\right)+\left(6+7\right)+\left(9+10\right)+...+\left(93+94\right)+\left(95+97\right)+\left(98+99\right)$ will be:

1. $5050$

2. $3750$

3. $1350$

4. $4250$

Q. If $\left|\begin{array}{ccc}a& -b& -c\\ -a& b& -c\\ -a& -b& -c\end{array}\right|+\lambda abc=0$, then $\lambda$ is equal to

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

Q.

Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is..............$

___

Q. Let $D_1=\left|\begin{array}{ccc}a& b& a+b\\ c& d& c+d\\ a& b& a-b\end{array}\right|,D_2=\left|\begin{array}{ccc}a& c& a+c\\ b& d& b+d\\ a& c& a+b+c\end{array}\right|$, then the value of $\left|\frac{D_1}{D_2}\right|$, where $b\ne 0$ and $ad\ne bc$, is ............

Q. $\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|=(1-x^3{)}^2$

Q. The number of different numbers each of six digits that can be formed by using all the digits 1, 2, 1, 0, 2, 2 is:

1. $600$
2. $100$
3. $50$
4. $120$

Q. $\left|\begin{array}{ccc}a-b& b-c& c-a\\ x-y& y-z& z-x\\ p-q& q-r& r-p\end{array}\right|$

1. a(x+y+z)+b(p+q+r)+c
2. 0
3. abc+xyz+pqr
4. None of these

Q. The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$ is ?

1. -1
2. -2
3. 1
4. 2

Q. Let $k$ be an integer such that the sum of the digits of $k$ is $4$ and ${10}^5<k<{10}^6$. How many values can $k$ have?

1. $36$
2. $46$
3. $45$
4. $56$

Q.

Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is$___