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Determinants Applied Maths - Determinants

CBSE - STD 12 Science - Applied Maths (CBSE - English Medium). 75 questions in this chapter.

Question types

Determinants question types

75 questions across 5 question groups — pick any mix to generate a Applied Maths paper with step-by-step answer keys.

75
Questions
5
Question groups
5
Question types
Sample Questions

Determinants questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ1 Mark
If $A$ and $B$ are two square matrices, then $|A . B|$ is same as which of the following?
  • ✓
    $| A | \cdot| B |$
  • B
    $| B | \cdot| A |$
  • C
    $|B-A|$
  • D
    $| A - B |$

Answer: A.

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Q 2MCQ1 Mark
For a determinant containing 4 elements $a_1, b_1, a_2$ and $b_2;$ what will the elements of leading diagonals?
  • A
    $a_1, b_1$
  • B
    $a_1, b_2$
  • C
    $a_2, b_1$
  • D
    $a_1, b_2$
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Q 3MCQ1 Mark
$\left|\begin{array}{ccc}2 & 3 & 4 \\5 & 6 & 8 \\6 x & 9 x & 12 x\end{array}\right|=\ldots\ldots$
  • ✓
    $0$
  • B
    $x$
  • C
    $3 x$
  • D
    None of these

Answer: A.

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Q 4MCQ1 Mark
If $\left|\begin{array}{cc}x-2 & -3 \\ 3 x & 2 x\end{array}\right|=3$, then $x=\ldots . .(x \in I)$
  • A
    3
  • ✓
    -3
  • C
    2
  • D
    -2

Answer: B.

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Q 5MCQ1 Mark
Cramer's rule is not suitable for which type of problems?
  • A
    Small systems with 4 unknowns
  • B
    Systems with 2 unknowns
  • ✓
    Large systems
  • D
    Systems with 3 unknowns

Answer: C.

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Q 62 Marks Questions2 Marks
A factory produces three products every day. Their production on a particular day is 45 tons. It is found that production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product.
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Q.1. It $x, y$ and $z$ respectively denotes the quantity (in tons) of first, second and third product produced, then construct the system of equations and write it in matrix form.
Q.2. If $\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & -1 & 1\end{array}\right]^{-1}=\frac{1}{6}\left[\begin{array}{ccc}2 & 2 & 2 \\ 3 & 0 & -3 \\ 1 & -2 & 1\end{array}\right]$, then find the inverse of $\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1\end{array}\right]$.
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Q 92 Marks Questions2 Marks
If $A_{i j}$ is the cofactor of the element $a_{i j}$ of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$, then write the value of $a_{32} A_{32}$.
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Q 102 Marks Questions2 Marks
Without expanding at any stage, find the value of the determinant:
$\Delta=\left|\begin{array}{lll}2 & x & y+z \\2 & y & z+x \\2 & z & x+y\end{array}\right|$
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Q 113 Marks Question3 Marks
On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However if there were 16 children more, everyone would have got ₹ 10 less. Using matrix method, find the number of children and the amount distributed by Seema.
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Q 123 Marks Question3 Marks
Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain the same. But if length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by $5300 \text{m}^2.$ Using matrices, find the dimensions of the plot.
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Q 133 Marks Question3 Marks
Let $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{cc}4 & -6 \\ -2 & 4\end{array}\right]$. Then compute $A B$. Hence, solve the following system of equations: $2 x+y=4,3 x+2 y=1.$
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Q 143 Marks Question3 Marks
If $A=\left[\begin{array}{cc}1 & -2 \\ 2 & 1\end{array}\right]$ then using $A^{-1}$, solve the following system of equations : $x-2 y=-1,2 x+y=2$.
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Q 153 Marks Question3 Marks
Using properties of determinants, prove the following:
$\left|\begin{array}{ccc}1+a^2-b^2 & 2 a b & -2 b \\2 a b & 1-a^2+b^2 & 2 a \\2 b & -2 a & 1-a^2-b^2\end{array}\right|=\left(1+a^2+b^2\right)^3 .$
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Q 165 Marks Questions5 Marks
Using determinants, show that the following system of linear equation is inconsistent:
$\begin{aligned}x-3 y+5 z & =4 \\2 x-6 y+10 z & =11 \\3 x-9 y+15 z & =12\end{aligned}$
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Q 175 Marks Questions5 Marks
Solve the following system of equations by Cramer's rule :
$\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4, \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 \text { and } \frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$
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Q 185 Marks Questions5 Marks
For what values of $a$ and $b$, the system of equations
$\begin{array}{r}2 x+a y+6 z=8 \\x+2 y+b z=5 \\x+y+3 z=4,\end{array}$
has (i) unique solution (ii) no solution?
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Q 195 Marks Questions5 Marks
In $A=\left(\begin{array}{ccc}2 & 3 & 10 \\ 4 & -6 & 5 \\ 6 & 9 & -20\end{array}\right)$ find $A^{-1}$. Using $A^{-1}$ solve the system of equations $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=2 ; \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=5$; $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=-4$
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Q 205 Marks Questions5 Marks
Use product $\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]\left[\begin{array}{ccc}-2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2\end{array}\right]$ to solve the system of equations $x+3 z=9,-x+2 y-2 z=4$, $2 x-3 y+4 z=-3$
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Q 21Case study (4 Marks)4 Marks
The management committee of a residential colony decided to award some of its members (say $x$ ) for honesty, some (say $y$ ) for helping others and some others (say $z$ ) for supervising the workers to kept the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. The sum of the number of awardees for honesty and supervision is twice the number of awardees for helping.
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Q.1. $x+y+z= $ __________.
(A) 3 $\quad$ (B) 5
(C) 7 $\quad$ (D) 12
Q.2. $x-2 y=$ __________.
(A) $z$ $\quad$ (B) $-z$
(C) $2 z$$\quad$(D) $-2 z$
Q.3. The value of $z$ is __________.
(A) 3 $\quad$ (B) 4
(C) 5 $\quad$ (D) 6
Q.4. The value of $x+2 y=$ __________.
(A) 9 $\quad$ (B) 10
(C) 11$\quad$(D) 12
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Q 22Case study (4 Marks)4 Marks
A trust invested some money in two types of bonds. The first bond pays $10\%$ interest and second bond pays $12 \%$ interest the trust received ₹ $2800$ as interest. However if trust had interchanged money in bonds, they would have got ₹ $100$ less as interest.
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Q.1. For the given information, the system of equations can be written as
(A)
$\begin{array}{l}5 x+6 y=135000 \\6 x+5 y=140000\end{array}$
(B)
$5 x+6 y=140000$
$6 x+5 y=135000$
(C)
$\begin{array}{l}5 x+5 y=140000 \\6 x+6 y=135000\end{array}$
(D)
$\begin{aligned}x+y & =140000 \\6 x+5 y & =135000\end{aligned}$
Q.2. If the given system of equations can be written as $A X=B$, then find $|A|$.
(A) -11
(B) 11
(C) -5
(D) -10
Q.3. The amount invested in first type of bond is
(A) ₹ 9000
(B) ₹ 12000
(C) ₹ 15000
(D) ₹ 10000
Q.4. The amount invested in first type of bond is
(A) ₹ 9000
(B) ₹ 12000
(C) ₹ 15000
(D) ₹ 10000
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Q 23Case study (4 Marks)4 Marks
A shopkeeper has 3 varieties of pen 'A', 'B' and 'C'. Meenu purchase 1 pen of each variety for a total of ₹ 21 . Jeevan purchase 4 pens of 'A' variety, 3 pens say 'B' variety and 2 pens of 'C' variety for ₹ 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for ₹ 70.
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Q.1. For the given information, the system of equations can be written as:
(A)
$\begin{array}{r}x+y+z=21 \\4 x+3 y+2 z=60 \\6 x+2 y+3 z=70\end{array}$
(B)
$\begin{aligned}x+y+z & =60 \\4 x+3 y+2 z & =21 \\6 x+2 y+3 z & =70\end{aligned}$
(C)
$\begin{aligned} 4 x+3 y+2 z & =21 \\ x+y+z & =70 \\ 6 x+2 y+3 z & =60\end{aligned}$
(D)
$\begin{aligned} x+y+z & =70 \\ 4 x+3 y+2 z & =60 \\ 6 x+2 y+3 z & =21\end{aligned}$
Q.2. If the given system of equations can be written as $A X=B$, then find $|A|$.
(A) 10 $\quad$ (B) 5
(C) -5 $\quad$ (D) -10
Q.3. The cost of '$A$' variety pen is
(A) ₹ 4 $\quad$ (B) ₹ 5
(C) ₹ 8 $\quad$ (D) ₹ 2
Q.4. The cost of '$B$' variety pen is
(A) ₹ 4 $\quad$ (B) ₹ 5
(C) ₹ 8 $\quad$(D) ₹ 2
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Q 24Case study (4 Marks)4 Marks
Two schools P and Q want to award their selected students on the values of the Discipline, Politeness and Punctuality. The school P wants to award ₹ $x$ each, ₹ $y$ each and ₹ $z$ each for the three respective values to its 3, 2 and 1 students with the total award money of ₹ $1,000.$ School Q wants spend ₹ $1,500$ to award its 4, 1 and 3 students on their respective values (by giving the same award money for the three values as before). The total amount of awards for one prize on each value is ₹ $600.$
Image
Q.1. For the given information, the system of equations can be written as
(A)
$\begin{aligned}3 x+2 y+z & =1000 \\4 x+y+3 z & =1500 \\x+y+z & =600\end{aligned}$
(B)
$\begin{aligned}3 x+2 y+z & =1500 \\4 x+y+3 z & =1000 \\x+y+z & =600\end{aligned}$
(C)
$3 x+2 y+z=1000$
$\begin{aligned} x+y+z & =15000 \\ 4 x+y+3 z & =600\end{aligned}$
(D)
$\begin{aligned} x+2 y+z & =1000 \\ 4 x+y+3 z & =1500 \\ 3 x+2 y+z & =600\end{aligned}$
Q.2. If the given system of equations can be written as $A X=B$, then find $|A|$ :
(A) 10 $\quad$ (B) 5
(C) -5 $\quad$ (D) -10
Q.3. The award money for Discipline is:
(A) ₹ 100 $\quad$ (B) ₹ 200
(C) ₹ 300 $\quad$ (D) ₹ 400
Q.4. The award money for Politeness is:
(A) ₹ 100 $\quad$ (B) ₹ 200
(C) ₹ 300 $\quad$ (D) ₹ 400
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