Exercise 3.19 - Chapter 3 Algebra 10th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $3.19$
Multiple choice questions.
Question $1 .$
A system of three linear equations in three variables is inconsistent if their planes
(1) intersect only at a point
(2) intersect in a line
(3) coincides with each other
(4) do not intersect.
Solution:
(4) do not intersect
 

Question 2.
The solution of the system $x+y-3 z=-6,-7 y+7 z=7,3 z=9$ is
(1) $x=1, y=2, z=3$
(2) $x=-1, y=2, z=3$
(3) $x=-1, y=-2, z=3$
(4) $x=1, y=2, z=3$
Answer:
(1) $x=1, y=2, z=3$
Hint.
$\begin{aligned}
&x+y-3 x=-6 \ldots(1) \\
&-7 y+7 z=7 \ldots(2) \\
&3 z=9
\end{aligned}$
From (3) we get
$\mathrm{z}=\frac{9}{3}=3$
Substitute the value of $z$ in (2)
$\begin{aligned}
&-7 y+7(3)=7 \\
&-7 y=-14
\end{aligned}$
Substitute the value of $y=2$ and $z=3$ in (1)
$\begin{aligned}
&x+2-3(3)=-6 \\
&x+2-9=-6
\end{aligned}$

$\begin{aligned}
&x=-6+7 \\
&x=1
\end{aligned}$
The value of $x=1, y=2$ and $z=3$
 

Question $3 .$
If $(x-6)$ is the HCF of $x^{2}-2 x-24$ and $x^{2}-k x-6$ then the value of $k$ is
(1) 3
(2) 5
(3) 6
(4) 8
Solution:
(2) 5

Question $4 .$

$\frac{3 y-3}{y} \div \frac{7 y-7}{3 y^{2}}$ is
(1) $\frac{9 y}{7}$
(2) $\frac{9 y^{3}}{(21 y-21)}$
(3) $\frac{21 y^{2}-42 y+21}{3 y^{3}}$
(4) $\frac{7\left(y^{2}-2 y+1\right)}{y^{2}}$
Solution:
(1) $\frac{9 y}{7}$

Question 5 .
$\mathbf{y}^{2}+\frac{1}{y^{2}}$ is not equal to
(1) $\frac{y^{4}+1}{y^{2}}$
(2) $\left(y+\frac{1}{y}\right)^{2}$
(3) $\left(y-\frac{1}{y}\right)^{2}+2$
(4) $\left(y+\frac{1}{y}\right)^{2}-2$
Solution:
(2) $\left(y+\frac{1}{y}\right)^{2}$
Hint:
$y^{2}+\frac{1}{y^{2}} \neq\left[y+\frac{1}{y}\right]^{2}$

Question 6.

$\frac{x}{x^{2}-25}-\frac{8}{x^{2}+6 x+5}$ gives
(1) $\frac{x^{2}-7 x+40}{(x+5)(x-5)}$
(2) $\frac{x^{2}+7 x+40}{(x+5)(x-5)(x+1)}$

(3) $\frac{x^{2}-7 x+40}{\left(x^{2}-25\right)(x+1)}$
(4) $\frac{x^{2}+10}{\left(x^{2}-25\right)(x+1)}$
Solution:
(3) $\frac{x^{2}-7 x+40}{(x+5)(x-5)(x+1)}$

Question $7 .$
The square root of $\frac{256 x^{8} y^{4} z^{10}}{25 x^{6} y^{6} z^{6}}$ is equal to
(1) $\frac{16}{5}\left|\frac{x^{2} z^{4}}{y^{2}}\right|$
(2) $16\left|\frac{y^{2}}{x^{2} z^{4}}\right|$
(3) $\frac{16}{5}\left|\frac{y}{x z^{2}}\right|$
(4) $\frac{16}{5}\left|\frac{x z^{2}}{y}\right|$
Solution:
(4) $\frac{16}{5}\left|\frac{x z^{2}}{y}\right|$
Hint:
$\frac{16 x^{4} y^{2} z^{5}}{5 x^{3} y^{3} z^{3}}=\frac{16}{5} \frac{x z^{2}}{y}=\left|\frac{16 x z^{2}}{5 y}\right|$
 

Question 8.
Which of the following should be added to make $x^{4}+64$ a perfect square
(1) $4 x^{2}$
(2) $16 x^{2}$
(3) $8 x^{2}$
(4) $-8 x^{2}$
Answer:
(2) $16 x^{2}$

Question $9 .$
The solution of $(2 x-1)^{2}=9$ is equal to
(1) -1
(2) 2
(3) $-1,2$
(4) None of these
Solution:
(3) $-1,2$
Hint:
$\begin{aligned}
&(2 \mathrm{x}-1)^{2}=(\pm 3)^{2} \\
&\Rightarrow 2 \mathrm{x}-1=+3 \\
&2 \mathrm{x}-1=3,2 \mathrm{x}-1=-3 \\
&2 \mathrm{x}=4,2 \mathrm{x}=-2 \\
&\mathrm{x}=2,-1
\end{aligned}$

Question $10 .$
The values of $a$ and $b$ if $4 x^{4}-24 x^{3}+76 x^{2}+a x+b$ is a perfect square are
(1) 100,120
(2) 10,12
(3) $-120,100$
(4) 12,10
Solution:
(3) $-120,100$

Question $11 .$
If the roots of the equation $\mathrm{q}^{2} \mathrm{x}^{2}+\mathrm{p}^{2} \mathrm{x}+\mathrm{r}^{2}=0$ are the squares of the roots of the equation $\mathrm{qx}{ }^{2}+\mathrm{px}$ $+r=0$, then $q, p, r$ are in
(1) A.P
(2) G.P
(3) Both A.P and G.P
(4) none of these
Solution:
(2) G.P
Hint: $\mathrm{q}^{2} \mathrm{x}^{2}+\mathrm{p}^{2} \mathrm{x}+\mathrm{r}^{2}=0$
(2) G.P.

Question $12 .$
Graph of a linear polynomial is a
(1) straight line
(2) circle
(3) parabola
(4) hyperbola
Answer:
(1) straight line

Question 13.
The number of points of intersection of the T quadratic polynomial $\mathrm{x}^{2}+4 \mathrm{x}+4$ with the $\mathrm{X}$ axis.
(1) 0
(2) 1
(3) 0 or 1
(4) 2
Solution:
(2) 1
$(x+2)^{2}=(x+2)(x+2)$ $=x=-2,-2=1$
 

Question $14 .$
For the given matrix $A=\left[\begin{array}{cccc}1 & 3 & 5 & 7 \\ 2 & 4 & 6 & 8 \\ 9 & 11 & 13 & 15\end{array}\right]$ the order of the matrix $A^{T}$ is
(1) $2 \times 3$
(2) $3 \times 2$
(3) $3 \times 4$
(4) $4 \times 3$
Solution:
(3) $3 \times 4$

Question $15 .$
If $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 4$ matrix, how many columns does $A B$ have
(1) 3
(2) 4
(3) 2
(4) 5
Solution:
(2) 4

Question $16 .$
If a number of columns and rows are not equal in a matrix then it is said to be a
(1) diagonal matrix
(2) rectangular matrix
(3) square matrix
(4) identity matrix
Answer:
(2) rectangular matrix

Question $17 .$
Transpose of a column matrix is
(1) unit matrix
(2) diagonal matrix
(3) column matrix
(4) row matrix
Solution:
(4) row matrix

Question $18 .$
Find the matrix $X$ if $2 X+\left[\begin{array}{ll}1 & 3 \\ 5 & 7\end{array}\right]=\left[\begin{array}{ll}5 & 7 \\ 9 & 5\end{array}\right]$
(1) $\left(\begin{array}{rr}-2 & -2 \\ 2 & -1\end{array}\right)$
(2) $\left(\begin{array}{rr}2 & 2 \\ 2 & -1\end{array}\right)$
(3) $\left(\begin{array}{ll}1 & 2 \\ 2 & 2\end{array}\right)$
(4) $\left(\begin{array}{ll}2 & 1 \\ 2 & 2\end{array}\right)$
Solution:
(2) $\left(\begin{array}{cc}2 & 2 \\ 2 & -1\end{array}\right)$

Question $19 .$
Which of the following can be calculated from the given matrices $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4 \\ 5 & 6\end{array}\right], B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$
(i) $\mathrm{A}^{2}$
(ii) $\mathrm{B}^{2}$
(iii) $\mathrm{AB}$
(iv) $\mathrm{BA}$
(1) (i) and (ii) only
(2) (ii) and (iiii) only
(3) (ii) and (iv) only
(4) all of these
Solution:
(3) (ii) and (iv) only

Question $20 .$
If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 2 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & 0 \\ 2 & -1 \\ 0 & 2\end{array}\right]$ and $C=\left[\begin{array}{rr}0 & 1 \\ -2 & 5\end{array}\right]$ Which of the following statements are correct? (i) $\mathbf{A B}+C=\left[\begin{array}{ll}5 & 5 \\ 5 & 5\end{array}\right]$
(ii) $\mathrm{BC}=\left[\begin{array}{rr}0 & 1 \\ 2 & -3 \\ -4 & 10\end{array}\right]$ (iii) $\mathrm{BA}+\mathrm{C}=\left[\begin{array}{ll}2 & 5 \\ 3 & 0\end{array}\right]$
(iv) $(A B) C=\left[\begin{array}{cc}-8 & 20 \\ -8 & 13\end{array}\right]$
(1) (i) and (ii) only
(2) (ii) and (iii) only
(3) (ii) and (iv) only
(4) all of these
Solution:
(1) (i) and (ii) only