If A=[aij] is a 3 × 3 diagonal matrix
Question: If $A=\left[a_{i j}\right]$ is a $3 \times 3$ diagonal matrix such that $a_{11}=1, a_{22}=2 a_{33}=3$, then find $|A| .$ Solution: If $A=\left[\mathrm{a}_{\mathrm{ij}}\right]$ is a diagonal matrix of order $\mathrm{n}$, then $|A|=\mathrm{a}_{11} \times \mathrm{a}_{22} \times \mathrm{a}_{33} \times \ldots \times \mathrm{a}_{\mathrm{nn}}$. Given : $a_{11}=1, \mathrm{a}_{22}=2$ and $\mathrm{a}_{33}=3$ $\Rightarrow|A|=1 \times 2 \times 3=6 \quad$ [Applying the above property]...
Read More →In the given figure, ABCD is a trapezium of area 24.5 cm2 ,
Question: In the given figure, ABCD is a trapezium of area 24.5 cm2, If AD || BC, DAB = 90, AD = 10 cm, BC = 4 cm and ABE is quadrant of a circle, then find the area of the shaded region. Solution: Area of trapezium $=\frac{1}{2}(\mathrm{AD}+\mathrm{BC}) \times \mathrm{AB}$ $\Rightarrow 24.5=\frac{1}{2}(10+4) \times \mathrm{AB}$ $\Rightarrow \mathrm{AB}=3.5 \mathrm{~cm}$ Area of shaded region = Area of trapezium ABCD Area of quadrant ABE $=24.5-\frac{1}{4} \pi(\mathrm{AB})^{2}$ $=24.5-\frac{1}{4...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{cc}1 2 \\ 3 -1\end{array}\right]$ and $B=\left[\begin{array}{cc}1 -4 \\ 3 -2\end{array}\right]$, find $|A B|$. Solution: $A=\left[\begin{array}{rr}1 2 \\ 3 -1\end{array}\right]$ $\Rightarrow|A|=-1-6=-7$ $B=\left[\begin{array}{cc}1 -4 \\ 3 -2\end{array}\right]$ $\Rightarrow|B|=-2+12=10$ If $A$ and $B$ are square matrices of the same order, then $|A B|=|A||B|$. $\Rightarrow|A B|=|A||B|=-7 \times 10=-70$...
Read More →If three circles of radius a each, are drawn such that each touches the other
Question: If three circles of radius a each, are drawn such that each touches the other two, prove that the area included between them is equal to $\frac{4}{25} a^{2}$. Solution: When three circles touch each other, their centres form an equilateral triangle, with each side being 2a. Area of the triangle $=\frac{\sqrt{3}}{4} \times 2 a \times 2 a=\sqrt{3} a^{2}$ Total area of the three sectors of circles $=3 \times \frac{60}{360} \times \frac{22}{7} \times a^{2}=\frac{1}{2} \times \frac{22}{7} \...
Read More →If three circles of radius a each, are drawn such that each touches the other
Question: If three circles of radius a each, are drawn such that each touches the other two, prove that the area included between them is equal to $\frac{4}{25} a^{2}$. Solution: When three circles touch each other, their centres form an equilateral triangle, with each side being 2a. Area of the triangle $=\frac{\sqrt{3}}{4} \times 2 a \times 2 a=\sqrt{3} a^{2}$ Total area of the three sectors of circles $=3 \times \frac{60}{360} \times \frac{22}{7} \times a^{2}=\frac{1}{2} \times \frac{22}{7} \...
Read More →Show that the quadrilateral formed
Question: Show that the quadrilateral formed by joining the consecutive sides of a square is also a square. Solution: Given In a square $A B C D, P, Q, R$ and $S$ are the mid-points of $A B$, $B C, C D$ and $D A$, respectively. To show $P Q R S$ is a square. Construction Join $A C$ and $B D$. Proof Since, $A B C D$ is a square. $\therefore \quad A B=B C=C D=A D$ Also, $P, Q, R$ and $S$ are the mid-points of $A B, B C, C D$ and $D A$, respectively. Then, in $\triangle A D C$, $S R \| A C$ and$S R...
Read More →If w is an imaginary cube root of unity,
Question: If $w$ is an imaginary cube root of unity, find the value of $\left|\begin{array}{ccc}1 w w^{2} \\ w w^{2} 1 \\ w^{2} 1 w\end{array}\right|$. Solution: $\mid \begin{array}{lll}1 w w^{2}\end{array}$ $w \quad w^{2} \quad 1$ $w^{2} \quad 1 \quad w \mid$ $=\mid 1+w+w^{2} \quad w \quad w^{2}$ $w+w^{2}+1 \quad w^{2} \quad 1$ $w^{2}+1+w \quad 1 \quad w \mid$ $\left[\right.$ Applying $\left.\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\right]$ $=\mid \begin{array}{lll...
Read More →E is the mid-point of a median AD of ΔABC
Question: E is the mid-point of a median AD of ΔABC and BE is produced to meet AC at F. Show that AF = 1/3 AC. Solution: Given In a $\triangle A B C, A D$ is a median and $E$ is the mid-point of $A D$. Construction Draw DP $\| E F$. Proof In $\triangle A D P, E$ is the mid-point of $A D$ and $E F \| D P$. So, $F$ is mid-point of $A P$. [by converse of mid-point theorem] In $\Delta F B C, D$ is mid-point of $B C$ and $D P \| B F$. So, $P$ is mid-point of $F C$. Thus, $A F=F P=P C$ $\therefore$ $A...
Read More →Solve the following equations
Question: Evaluate $\left|\begin{array}{ll}4785 4787 \\ 4789 4791\end{array}\right|$ Solution: Let $\Delta=\mid \begin{array}{ll}4785 4787\end{array}$ $\begin{array}{ll}4789 4791 \mid\end{array}$ $\Rightarrow \Delta=\mid 4785 \quad 2$ $\begin{array}{lll}4789 2 {\left[\text { Applying } C_{2} \rightarrow C_{2}-C_{1}\right]}\end{array}$ $=2 \times \mid 4785 \quad 1$ $\begin{array}{ll}4789 1\end{array}$ $=2 \times(4785-4789)=2 \times(-4)=-8$ $\Rightarrow \mid 4785 \quad 4787$ $4789 \quad 4791 \mid=...
Read More →Three equal circles, each of radius 6 cm,
Question: Three equal circles, each of radius 6 cm, touch one another as shown in the figure. Find the area of enclosed between them. Solution: Join ABC. All sides are equal, so it is an equilateral triangle.Now, Area of the equilateral triangle $=\frac{\sqrt{3}}{4} \times \operatorname{Side}^{2}$ $=\frac{1.73}{4} \times 12 \times 12$ $=62.28 \mathrm{~cm}^{2}$ Area of the arc of the circle $=\frac{60}{360} \pi \mathrm{r}^{2}$ $=\frac{1}{6} \pi r^{2}$ $=\frac{1}{6} \times \frac{22}{7} \times 6 \t...
Read More →In figure, AB || DE, AB = DE, AC|| DF
Question: In figure, $A B\|D E, A B=D E, A C\| D F$ and $A C=O F$. Prove that $B C \| E F$ and $B C=E F$. Solution: Given In figure AB || DE and AC || DF, also AB = DE and AC = DF To prove BC ||EF and BC = EF Proof In quadrilateral ABED, AB||DE and AB = DE So, ABED is a parallelogram. AD || BE and AD = BE Now, in quadrilateral ACFD, AC || FD and AC = FD ..(i) Thus, ACFD is a parallelogram. AD || CF and AD = CF (ii) From Eqs.(i)and(ii), AD = BE = CF and CF || BE (iii) Now, in quadrilateral BCFE, ...
Read More →Solve this
Question: If $A=\left[\begin{array}{cc}1 2 \\ 3 -1\end{array}\right]$ and $B=\left[\begin{array}{cc}1 0 \\ -1 0\end{array}\right]$, find $|A B|$. Solution: $A=\left[\begin{array}{ll}1 2\end{array}\right.$ $3-1]$ $B=\left[\begin{array}{ll}1 0\end{array}\right.$ $\left.\begin{array}{ll}-1 0\end{array}\right]$ $A B=\left[\begin{array}{ll}1 2\end{array}\right.$ $3-1]\left[\begin{array}{ll}1 0\end{array}\right.$ $-1 \quad 0]=\left[\begin{array}{ll}1-2 0+0 \\ 3+1 0+0\end{array}\right]=\left[\begin{arr...
Read More →ABCD is a quadrilateral in which AB || DC
Question: $\mathrm{ABCD}$ is a quadrilateral in which $\mathrm{AB} \| \mathrm{DC}$ and $\mathrm{AD}=\mathrm{BC}$. Prove that $\angle \mathrm{A}=\angle \mathrm{B}$ and $\angle \mathrm{C}=\angle \mathrm{D}$. Solution: Given $A B C D$ is a quadrilateral such that $A B \| D C$ and $A D=B C$ Construction Extend $A B$ to $E$ and draw a line $C E$ parallel to $A D$. Proof Since, $A D \| C E$ and transversal $A E$ cuts them at $A$ and $E$, respectively. $\therefore \quad \angle A+\angle E=180^{\circ}$ [...
Read More →Four equal circles, each of radius a units, touch each other.
Question: Four equal circles, each of radius a units, touch each other. Show that the area between them is $\left(\frac{6}{7} a^{2}\right)$ sq units. Solution: When four circles touch each other, their centres form the vertices of a square. The sides of the square are 2aunits. Area of the square $=(2 a)^{2}=4 a^{2}$ sq. units Area occupied by the four sectors $=4 \times \frac{90}{260} \times \pi \times a^{2}$ $=\pi a^{2}$ sq. units Area between the circles = Area of the squareArea of the four se...
Read More →Find the value
Question: If $A=\left[\begin{array}{ll}0 i \\ i 1\end{array}\right]$ and $B=\left[\begin{array}{ll}0 1 \\ 1 0\end{array}\right]$, find the value of $|A|+|B|$. Solution: $A=\left[\begin{array}{ll}0 i\end{array}\right.$ $\left.\begin{array}{ll}i 1\end{array}\right]$ $\Rightarrow|A|=0-i^{2}$ $=-(-1)=1$ Also, $B=\left[\begin{array}{ll}0 1\end{array}\right.$ $\left.\begin{array}{ll}1 0\end{array}\right]$ $\Rightarrow|B|=0-1=-1$ So, $|A|+|B|=1-1=0$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:4(x + y) (3a b) +6(x + y) (2b3a) Solution: $4(x+y)(3 a-b)+6(x+y)(2 b-3 a)$ $=2(x+y)[2(3 a-b)+3(2 b-3 a)] \quad\{$ Taking $[2(x+y)]$ as the common factor $\}$ $=2(x+y)(6 a-2 b+6 b-9 a)$ $=2(x+y)(4 b-3 a)$...
Read More →Four equal circles, each of radius 5 cm, touch each other, as shown in the figure.
Question: Four equal circles, each of radius 5 cm, touch each other, as shown in the figure. Find the area included between them. Solution: Radius = 5 cmAB = BC = CD = AD = 10 cmAll sides are equal, so it is a square. Area of a square $=\mathrm{Side}^{2}$ Area of the square $=10^{2}=100 \mathrm{~cm}^{2}$ Area of the quadrant of one circle $=\frac{1}{4} \pi r^{2}$ $=\frac{1}{4} \times \frac{22}{7} \times 5 \times 5$ $=19.64 \mathrm{~cm}^{2}$ Area of the quadrants of four circles $=19.64 \times 4=...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:(2x 3y)(a+b) + (3x 2y)(a+b) Solution: $(2 x-3 y)(a+b)+(3 x-2 y)(a+b)$ $=(2 x-3 y+3 x-2 y)(a+b) \quad[$ Taking $(a+b)$ as the common factor $]$ $=(5 x-5 y)(a+b)$ $=5(x-y)(a+b)$$\quad[$ Taking 5 as the common factor of $(5 x-5 y)]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:x3(a 2b) +x2(a 2b) Solution: $x^{3}(a-2 b)+x^{2}(a-2 b)$ $=\left(x^{3}+x^{2}\right)(a-2 b) \quad[$ Taking $(a-2 b)$ as the common factor $]$ $=x^{2}(x+1)(a-2 b) \quad\left[\right.$ Taking $x^{2}$ as the common factor of $\left.\left(x^{3}+x^{2}\right)\right]$...
Read More →P and Q are the mid-points of the opposite
Question: P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram. Solution: Given In a parallelogram ABCD, P and Q are the mid-points of AS and CD, respectively. To show PRQS is a parallelogram. Proof Since, ABCD is a parallelogram. AB||CD = AP || QC Also, $A B=D C$ $\frac{1}{2} A B=\frac{1}{2} D C$ [dividing both sides by 2$]$ $\Rightarrow \quad A P=Q C$ [since, $P$ and $Q$ are the mi...
Read More →Solve this
Question: If $A=\left[\begin{array}{ll}0 i \\ i 1\end{array}\right]$ and $B=\left[\begin{array}{ll}0 1 \\ 1 0\end{array}\right]$, find the value of $|A|+|B|$. Solution: $A=\left[\begin{array}{ll}0 i\end{array}\right.$...
Read More →Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure.
Question: Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm. Solution: Side of the square = 14 cm Radius of the circle $=\frac{14}{2}=7 \mathrm{~cm}$ Area of the quadrant of one circle $=\frac{1}{4} \pi r^{2}$ $=\frac{1}{4} \times \frac{22}{7} \times 7 \times 7$ $=38.5 \mathrm{~cm}^{2}$ Area of the quadrants of four circles $=38.5 \times ...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:4(x 2y)2+ 8(x2y) Solution: $-4(x-2 y)^{2}+8(x-2 y)$ $=[-4(x-2 y)+8](x-2 y) \quad[$ Taking $(x-2 y)$ as the common factor $]$ $=4[-(x-2 y)+2](x-2 y) \quad\{$ Taking 4 as the common factor of $[-4(x-2 y)+8]\}$ $=4(2 y-x+2)(x-2 y)$...
Read More →Write the value of the determinant
Question: Write the value of the determinant $\left|\begin{array}{lll}a 1 b+c \\ b 1 c+a \\ c 1 a+b\end{array}\right|$. Solution: Let $\Delta=\mid \begin{array}{lll}a 1 b+c\end{array}$ $\begin{array}{lll}b 1 c+a\end{array}$ $\begin{array}{lll}c 1 a+b \mid\end{array}$ $=\mid a+b+c \quad 1 \quad b+c$ $\begin{array}{llll}a+b+c 1 c+a \\ a+b+c 1 a+b \mid\end{array}$ $\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}+C_{3}\right]$ $=a+b+c \mid 1 \quad 1 \quad b+c$ $\begin{array}{lll}1 1 c+a \\ 1 ...
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