Consistency of Linear System of Equations

Q.

What is the formula of $a^2-b^2$?

Q.

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide R 30000 amoung the two types of bonds, if the trust fund must obtain an annual total interest of
(a) Rs 2000

Q. Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students respectively with a total award money of Rs 1, 600. School B wants to spend Rs 2, 300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Q. The sum of three numbers is $6$. If we multiply the third number by $3$ and add it to the second number to it, we get $11$. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.

Q. The number of values of $\theta \in (0,\pi )$ for which the system of linear equations $x+3y+7z=0$
$-x+4y+7z=0$
$\left(\sin 3\theta \right)x+\left(\cos 2\theta \right)y+2z=0$ has a non-trivial solution, is :

1. four
2. three
3. two
4. one

Q. If $3A+4B^{\prime }$ =$\left[\begin{array}{ccc}7& -10& 17\\ 0& 6& 31\end{array}\right]$ and $2B-3A^{\prime }$=$\left[\begin{array}{cc}-1& 18\\ 4& 0\\ -5& -7\end{array}\right]$ then $B$=

1. $\left[\begin{array}{cc}-1& -18\\ 4& -16\\ -5& -7\end{array}\right]$
2. $\left[\begin{array}{cc}1& 3\\ -1& 1\\ 2& 4\end{array}\right]$
3. $\left[\begin{array}{cc}1& 3\\ -1& 1\\ 2& -4\end{array}\right]$
4. $\left[\begin{array}{cc}1& -3\\ -1& 1\\ 2& 4\end{array}\right]$

Q.

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of 1 kg food is given below.

$\begin{array}{cccc}Food& VitaminA& VitaminB& VitaminC\\ X& 1& 2& 3\\ Y& 2& 2& 1\end{array}$

1 kg of food X costs of Rs. 16 and 1 kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet ?

Q.

The most general solution of $\tan$ $\theta =1$ and $\cos \theta$ $=\frac{-1}{\sqrt{2}}$ is

Q. Let $a,\lambda ,\mu \in R.$ Consider the system of linear equations
$ax+2y=\lambda 3x-2y=\mu$
Which of the following statement(s) is(are) correct?

1. If $a=-3,$ then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
2. If $a\ne -3,$ then the system has a unique solution for all values of $\lambda$ and $\mu$
3. If $\lambda +\mu =0,$ then the system has infinitely many solutions for $a=-3$
4. If $\lambda +\mu \ne 0,$ then the system has no solution for $a=-3$

Q.

If the lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ are concurrent, then:

1. $a^3+b^3+c^3+3abc=0$

2. $a^3+b^3+c^3-abc=0$

3. $a^3+b^3+c^3-3abc=0$

4. None of these

Q. If the system of linear equations
$2x+2y+3z=a$
$3x-y+5z=b$
$x-3y+2z=c$
where $a,b,c$ are non-zero real numbers, has more than one solution, then:

1. $b+c-a=0$
2. $b-c+a=0$
3. $b-c-a=0$
4. $a+b+c=0$

Q. If $A=\left[\begin{array}{ccc}0& 6& 7\\ -6& 0& 8\\ 7& -8& 0\end{array}\right],B=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 2\\ 1& 2& 0\end{array}\right],C=\left[\begin{array}{c}2\\ -2\\ 3\end{array}\right]$, Calculate $AC,BC$ and $(A+B)C.$ Also, verify that $(A+B)C=AC+BC$.

Q. A pair of straight lines drawn through the origin form an isosceles right angled triangle with the line 2x + 3y = 6, then the lines and the area of the triangle thus formed is

1. x-5y=0
   5x+y=0
   
2. 3x-y=0
   x+3y=0
   
3. 5x-y=0
   x+5y=0
   
4. None of these

Q.

Write four solutions for the following equations: $\mathrm{\pi }x+y=9$

Q. If the system of equations $ax+y+z=0;x+by+z=0$ and $x+y+cz=0$ has a non-trivial solution, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

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

Q.

If x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zeros, then the value of $a^2+b^2+c^2+2abc$ is

1. 0

2. 1

3. -1

4. None of these

Q.

Find the roots of the equation $x^2-2xcos\theta$ + 1 = 0.

1. ,

2. ,

3. ,

4. ,

Q. If $abc=p$ and $A=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$ such that $A{A}^{\prime }=I$ and $A^{\prime }$ is transpose of $A,$ then $a,b,c$ are the roots of the equation

1. $x^3-p=0$
2. $x^3+x^2-p=0$
3. $x^3+3x^2-p=0$
4. $x^3+2x^2-p=0$

Q.

Solve the following system of linear equations, using matrix method

$x-y+z=4,2x+y-2z=0,x+y+z=2$

Q.

Using matrix method, solve the system of equations 3x + 2y - 2z = 3, x + 2y + 3z = 6 and 2x - y + z = 2.

Q.

If x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zeros, then the value of $a^2+b^2+c^2+2abc$ is

1. 0

2. 1

3. -1

4. None of these

Q. The number of values of $p$ for which the lines $x+y-1=0,$ $px+2y+1=0$ and $4x+2py+7=0$ are concurrent, is

Q.

Solve the following system of linear equations, using matrix method

$2x+3y+3z=5,x-2y+z=-4,3x-y-2z=3$

Q. Let $\lambda$ and $\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations
$\lambda x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0$
$x+\left(cos\alpha \right)y+\left(sin\alpha \right)z=0$
$-x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0$
has a non-trivial solution. For $\lambda =1$, find all values of $\alpha$ which are possible

1. $\pi /8$
2. $7\pi /8$
3. $15\pi /8$
4. $9\pi /8$

Q. For the system of linear equations :
$x-2y=1,x-y+kz=-2,ky+4z=6,k\in R,$
consider the following statements :
$\text{(}A)$ The system has unique solution if $k\ne 2,k\ne -2.$
$\left(B\right)$ The system has unique solution if $k=-2.$
$\left(C\right)$ The system has unique solution if $k=2.$
$\left(D\right)$ The system has no-solution if $k=2.$
$\left(E\right)$ The system has infinite number of solutions if $k\ne -2.$

Which of the following statements are correct ?

1. $\left(B\right)$ and $\left(E\right)$ only
2. $\left(C\right)$ and $\left(D\right)$ only
3. $\left(A\right)$ and $\left(D\right)$ only
4. $\left(A\right)$ and $\left(E\right)$ only

Q.

Solve the following system of linear equations, using matrix method

$2x+y+z=1,x-2y-z=\frac{3}{2},3y-5x=9$

Q. Consider the system of equations
$kx+(c-1)y+z=2$
$cx+(k+1)y+kz=4$
$x+cy+z=1$

Then correct statement is/are

1. If $c=1$, then the system will be inconsistent.
2. If $c=1$, then the may have infinite solutions.
3. If $k=2\cos \frac{\pi }{5}$ then the system will be inconsistent.
4. The system can never possess infinite solutions.

Q.

Examine the consistency of the system of equations

$x+3y=5,2x+6y=8$

Q. The value of sum of the given determinants,
$|A|=\left|\begin{array}{ccc}103& 115& 114\\ 111& 108& 106\\ 104& 113& 116\end{array}\right|,|B|=\left|\begin{array}{ccc}113& 116& 104\\ 108& 106& 111\\ 115& 114& 103\end{array}\right|$ is

Q. If the system of linear equations
$x+ay+z=3$
$x+2y+2z=6$
$x+5y+3z=b$
has no solution, then:

1. $a=-1,b=9$
2. $a=-1,b\ne 9$
3. $a\ne -1,b=9$
4. $a=1,b\ne 9$