### CBSE Assignments class 09 Mathematics

Mathematics Assignments & Worksheets  For  Class IX Chapter-wise mathematics assignment for class 09. Important and useful extra questions strictly according to the CBSE syllabus and pattern with answer key CBSE Mathematics is a very good platform for the students and is contain the assignments for the students from 9 th  to 12 th  standard.  Here students can find very useful content which is very helpful to handle final examinations effectively.  For better understanding of the topic students should revise NCERT book with all examples and then start solving the chapter-wise assignments.  These assignments cover all the topics and are strictly according to the CBSE syllabus.  With the help of these assignments students can easily achieve the examination level and  can reach at the maximum height. Class 09 Mathematics    Assignment Case Study Based Questions Class IX

## Math Assignment  Class XII  Ch - 04DETERMINANTS

Extra questions of chapter 04 Determinants class XII  with answers and  hints to the difficult questions, strictly according to the CBSE syllabus. Important and useful math. assignment for the students of class XII

## Strictly according to the CBSE Board

Question 1

Find the value of k for which matrix A is a singular matrix

$\inline \bg{black}{A=\left [ \begin{matrix}k & 8 \\4 & 2k \\\end{matrix} \right ]}$

Ans: 土 4

Question 2

If A is a square matrix of order 3 and |A| = -4, then find the value of |adj A|

Ans: 16

Question 3

$\inline \bg{black}If\: A=\left [ \begin{matrix}\alpha & 2 \\2 & \alpha \\\end{matrix} \right ]$ ,

and |A3| = 27, then find the value of α

Ans: √7

Question 4

$\inline \bg{black}If\: \: \left|\begin{matrix}5 & 3 & -1 \\-7 & x & -3 \\9 & 6 & -2 \\\end{matrix} \right|=0$ , then find the value of x

Ans : x = 9

Question 5

$\inline \bg{black}If\: A=\left [ \begin{matrix}2 & -3 \\5 & -4 \\\end{matrix} \right ],$ $\inline \bg{black}\: B=\left [ \begin{matrix}1 & -2 \\3 & 5 \\\end{matrix} \right ]$

then verify that |AB| = |A||B|

Solution Hint: |AB| = 77 and |A||B| = 7 x 11 = 77

Question 6

$\inline \bg{black}Evaluate\: \left [ \begin{matrix}a+ib & c+id \\-c+id & a-ib \\\end{matrix} \right ]$

Ans: a2 + b2 + c2 + d2

Question 7

$\inline \bg{black}Evaluate\: \left [ \begin{matrix}cos15^{o} & sin15^{o} \\sin75^{o} & cos75^{o} \\\end{matrix} \right ]$

Ans: 0

Question 8

For what value of x, the matrix A is singular

$\inline \bg{black}A= \left [ \begin{matrix}5-x & x+1 \\2 & 4 \\\end{matrix} \right ]$

Question 9

Find the area of triangle with vertices A(5, 4), B(-2, 4), C(2, -6)

Ans: 35 sq unit

Question 10

Using determinants show that the points (2, 3), (-1, -2) and (5, 8) are collinear

Solution Hint:

Find area of triangle by taking above given points as vertices.

If area of triangle = 0 then points are collinear.

Question 11

Using determinants find the value of k so that the points (k, 2 - 2k), (- k + 1, 2k), and (- 4 - k, 6 - 2k) may be collinear

Ans: k = -1, 1/2

Question 12
Using determinants, find the equation of line joining the points (3,1), and (9,3)

Ans: x - 3y = 0

Question 13

Find the value of k, if area of triangle is 4 square units whose vertices are (-2,0), (0,4), and (0, k)

Ans: K = 0, 8

Question 14

Find the value of |AB| if matrices A and B are given below

$\bg{black}A=\left [ \begin{matrix}1 & 2 \\3 & -1 \\\end{matrix} \right ]$ and $\bg{black}B=\left [ \begin{matrix}1 & 0 \\-1 & 0 \\\end{matrix} \right ]$

Ans: 0

Question 15

Find the value of x if matrix A is a singular matrix
$A=\left [ \begin{matrix}5x & 2 \\-10 & 1 \\\end{matrix} \right ]$

Ans: x = -4

Question 16

Find the product : $\left [ \begin{matrix}1 & -1 & 0 \\2 & 3 & 4 \\0 & 1 & 2 \\\end{matrix} \right ]\left [ \begin{matrix}2 & 2 & -4 \\-4 & 2 & -4 \\2 & -1 & 5 \\\end{matrix} \right ]$

Hence solve the following system of equations
x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Ans: x = 2, y = -1, z = 4

Solution Hint
Let given matrices are A and C

$AC=\left [ \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\\end{matrix} \right ]=6I$
Now find the product AC we get

$A\times \frac{1}{6}C=I\: \: \Rightarrow \: \: A^{-1}=\frac{1}{6}C$
Now given system of equations can be written as

$A=\left [ \begin{matrix}1 & -1 & 0 \\2 & 3 & 4 \\0 & 1 & 2 \\\end{matrix} \right ], X=\left [ \begin{matrix}x \\y \\z\end{matrix} \right ], B=\left [ \begin{matrix} 3\\17 \\7\end{matrix} \right ]$

AX = B ⇒ X =
A-1B
$\Rightarrow X=\frac{1}{6}CB$
⇒ x = 2, y = -1, z = 4 is the required solution

Question 17

$If \: \: A =\left [ \begin{matrix}3 & -4 & 2 \\2 & 3 & 5 \\1 & 0 & 1 \\\end{matrix} \right ],$
Find A-1  and hence solve following system of equations

3x - 4y + 2z = -1, 2x + 3y + 5z = 7, x + z = 2

Ans: x = 3, y = 2, z = -1

Solution Hint:

Find |A| we find |A| = -9 ⇒ A is invertible

Find cofactors of A and then find Adj. A we get
$Adj. A =\left [ \begin{matrix}3 & 4 & -26 \\3 & 1 & -11 \\-3 & -4 & 17 \\\end{matrix} \right ],$
Find A-1  we get

$A^{-1} =\frac{1}{9}\left [ \begin{matrix}-3 & -4 & 26 \\-3 & -1 & 11 \\3 & 4 & -17 \\\end{matrix} \right ],$

Given system of equations can be written as AX = B
⇒ X= A-1B

$\left [ \begin{matrix}x \\y \\z\end{matrix} \right ] =\frac{1}{9}\left [ \begin{matrix}-3 & -4 & 26 \\-3 & -1 & 11 \\3 & 4 & -17 \\\end{matrix} \right ]\left [ \begin{matrix} -1\\7 \\2\end{matrix} \right ]=\left [ \begin{matrix} 3\\2 \\-1\end{matrix} \right ]$

⇒ x = 3, y = 2, z = -1

Question 18

$\inline \bg{black} \mathbf{\; If\; \; A=\left [ \begin{matrix}2 & -3 & 5 \\3 & 2 & -4 \\1 & 1 & -2 \\\end{matrix} \right ]}$

Find A-1. Use A-1 to solve the system of equations.

2x – 3y + 5z = 11, 3x + 2y – 4z = - 5, x + y - 2z = - 3

Ans: x = 1, y = 2, z = 3

Solution Hint

Find the |A| we get |A| = -1

$\inline \bg{black}Agjoint \: A=\left| \begin{matrix}0 & -1 & 2 \\2 & -9 & 23 \\1 & -5 & 13 \\\end{matrix}\right|$

Solve: X = A-1B we get x = 1, y = 2, z = 3

Question 19

Show that the matrix A satisfies the equation A - 4A - 5I = O and hence find A-1
$\bg{black}A=\left [ \begin{matrix}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1 \\\end{matrix} \right ]$

Ans:
$\bg{black}A=\frac{1}{5}\left [ \begin{matrix}-3 & 2 & 2 \\2 & -3 & 2 \\2 & 2 & -3 \\\end{matrix} \right ]$

Question 20

Find the matrix X for which

$\bg{black}\left [ \begin{matrix}1 & -4 \\3 & -2 \\\end{matrix} \right ]X=\left [ \begin{matrix}-16 & -6 \\ 7&2 \\\end{matrix} \right ]$

Ans
$\bg{black}X=\left [ \begin{matrix}6 & 2 \\11/2 & 2 \\\end{matrix} \right ]$

Question 21

$\inline If\: \: A^{-1}=\left [ \begin{matrix}3 & -1 & 1 \\-15 & 6 & -5 \\5 & -2 & 2 \\\end{matrix} \right ]$  $\inline and\: \: B=\left [ \begin{matrix}1 & 2 & -2 \\-1 & 3 & 0 \\0 & -2 & 2 \\\end{matrix} \right ]$

Find  (AB)-1

Ans:
$\inline Ans \: \: \: (AB)^{-1}=\left [ \begin{matrix}9 & -3 & 5 \\-2 & 1 & 0 \\1 & 0 & 2 \\\end{matrix} \right ]$
Question 22
Solve the following system of equations
x + y + z = 3, 2x - y + z = -1, 2x + y - 3z = -9
Ans:  x = -8/7,  y=10/7,  z=19/7

Question 23
Solve the following system of equations

$\inline \frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4,$
$\inline \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1,$
$\inline \frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$  where x, y, z ≠ 0
Ans: x = 2, y = 3, z = 5