Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:a(xy) + 2b(yx) +c(xy)2 Solution: $a(x-y)+2 b(y-x)+c(x-y)^{2}$ $=a(x-y)-2 b(x-y)+c(x-y)^{2} \quad[\because(y-x)=-(x-y)]$ $=[a-2 b+c(x-y)](x-y)$ $=(a-2 b+c x-c y)(x-y)$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:6(a+ 2b) 4(a+ 2b)2 Solution: $6(a+2 b)-4(a+2 b)^{2}$ $=[6-4(a+2 b)](a+2 b) \quad[$ Taking $(a+2 b)$ as the common factor $]$ $=2[3-2(a+2 b)](a+2 b) \quad\{$ Taking 2 as the common factor of $[6-4(a+2 b)]\}$ $=2(3-2 a-4 b)(a+2 b)$...
Read More →The diameters of the front and rear wheels of a tractor are 80 cm and 2 m respectively.
Question: The diameters of the front and rear wheels of a tractor are 80 cm and 2 m respectively. Find the number of revolutions that a rear wheel makes to cover the distance which the front wheel covers is 800 revolutions. Solution: Radius of the front wheel $=40 \mathrm{~cm}=\frac{2}{5} \mathrm{~m}$ Circumference of the front wheel $=\left(2 \pi \times \frac{2}{5}\right) \mathrm{m}=\frac{4 \pi}{5} \mathrm{~m}$ Distance covered by the front wheel in 800 revolutions $=\left(\frac{4 \pi}{5} \time...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:(xy)2+ (xy) Solution: $(x-y)^{2}+(x-y)$ $=(x-y)(x-y)+(x-y) \quad[$ Taking $(x-y)$ as the common factor $]$ $=(x-y+1)(x-y)$...
Read More →Find the value of the determinant
Question: Find the value of the determinant $\left[\begin{array}{lll}101 102 103 \\ 104 105 106 \\ 107 108 109\end{array}\right]$ Solution: Let $\Delta=\mid \begin{array}{lll}101 102 103\end{array}$ $\begin{array}{lll}104 105 106\end{array}$ $\begin{array}{lll}107 108 109\end{array}$ $\Delta=\mid \begin{array}{lll}101 1 2\end{array}$ $\begin{array}{lll}104 1 2\end{array}$ $\begin{array}{lll}107 1 2\end{array}$ $\left[\right.$ Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $\left.C_{3} \rightarrow ...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:a2(x+y) +b2(x+y) +c2(x+y) Solution: $\mathrm{a}^{2}(\mathrm{x}+\mathrm{y})+\mathrm{b}^{2}(\mathrm{x}+\mathrm{y})+\mathrm{c}^{2}(\mathrm{x}+\mathrm{y})$ $=\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right)(\mathrm{x}+\mathrm{y})$$\quad[$ Taking $(\mathrm{x}+\mathrm{y})$ as the common factor $]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:3a(x 2y) b(x 2y) Solution: $3 a(x-2 y)-b(x-2 y)$ $3 \mathrm{a}(\mathrm{x}-2 \mathrm{y})-\mathrm{b}(\mathrm{x}-2 \mathrm{y})$ $=(3 \mathrm{a}-\mathrm{b})(\mathrm{x}-2 \mathrm{y})$$\quad[$ Taking $(\mathrm{x}-2 \mathrm{y})$ as the common factor $]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:16(2l 3m)212(3m 2l) Solution: $16(2 l-3 m)^{2}-12(3 m-2 l)$ $=16(2 l-3 m)^{2}+12(2 l-3 m) \quad[\because(3 m-2 l)=-(2 l-3 m)]$ $=[16(2 l-3 m)+12](2 l-3 m) \quad[$ Taking $(2 l-3 m)$ as the common factor $]$ $=4[4(2 l-3 m)+3](2 l-3 m) \quad$ Taking 4 as the common factor of $[16(2 l-3 m)+12]\}$ $=4(8 l-12 m+3)(2 l-3 m)$...
Read More →A diagonal of a parallelogram
Question: A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus. Solution: Given Let $A B C D$ is a parallelogram and diagonal $A C$ bisects the angle $A$. $\therefore \quad \angle C A B=\angle C A D \quad \ldots$ (i) To show $A B C D$ is a rhombus. Proof Since, $A B C D$ is a parallelogram, therefore $A B \| C D$ and $A C$ is a transversal. $\begin{array}{lll}\therefore \angle C A B=\angle A C D \text { [alternate interior angles] }\end{array}$ Again, $A D \| B C$ a...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:5(x 2y)2+ 3(x 2y) Solution: $5(x-2 y)^{2}+3(x-2 y)$ $=[5(x-2 y)+3](x-2 y) \quad[$ Taking $(x-2 y)$ as the common factor $]$ $=(5 x-10 y+3)(x-2 y)$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:9a(6a 5b) 12a2(6a 5b) Solution: $9 a(6 a-5 b)-12 a^{2}(6 a-5 b)$ $=\left(9 a-12 a^{2}\right)(6 a-5 b) \quad[$ Taking $(6 a-5 b)$ as the common factor $]$ $=3 a(3-4 a)(6 a-5 b) \quad\left[\right.$ Taking $3 a$ as the common factor of the quadratic $\left.\left(9 a-12 a^{2}\right)\right]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:7a(2x 3) + 3b(2x 3) Solution: $7 a(2 x-3)+3 b(2 x-3)$ $=(7 a+3 b)(2 x-3) \quad$ [Taking $(2 x-3)$ as the common factor $]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:2r(yx) +s(xy) Solution: $2 r(y-x)+s(x-y)$ $=2 r(y-x)-s(y-x)$ $[\because(x-y)=-(y-x)]$ $=(2 r-s)(y-x)$ $[$ Taking $(y-x)$ as the common factor $]$...
Read More →Factorize each of the following algebraic expressions:
Question: Factorize each of the following algebraic expressions:6x(2xy) + 7y(2xy) Solution: $6 x(2 x-y)+7 y(2 x-y)$ $=(6 x+7 y)(2 x-y)$$\quad[$ Taking $(2 x-y)$ as the common factor $]$...
Read More →The wheel of a motorcycle is of radius 35 cm.
Question: The wheel of a motorcycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/hr? Solution: The radius of wheel of a motorcycle = 35 cm = 0.35 m. So, the distance covered by this wheel in 1 revolution will be equal to perimeter of wheel i.e. $2 \pi r=2 \times \frac{22}{7} \times 0.35=2.2 \mathrm{~m}$. Since speed is given to be $66 \mathrm{~km} / \mathrm{hr}=66 \times \frac{1000 \mathrm{~m}}{60 \mathrm{~min}}=1100 \mathrm{~m} / \mathrm...
Read More →Find the value of the determinant
Question: Find the value of the determinant $\left[\begin{array}{ll}4200 4201 \\ 4205 4203\end{array}\right]$. Solution: Let $\Delta=\mid \begin{array}{ll}4200 4201\end{array}$ $\begin{array}{ll}4202 4203 \mid\end{array}$ $\Delta=\mid 42001$ 42021 [Applying $C_{2} \rightarrow C_{2}-C_{1}$ ] $=4200-4202$ $=-2$...
Read More →P, Q, R and S are respectively the mid-points of sides AB,
Question: P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC BD. Prove that PQRS is a square. Solution: Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively. Also, AC = BD and AC BD. To prove PQRS is a square. Proof Now, in ΔADC, S and R are the mid-points of the sides AD and DC respectively, then by mid-point theorem, $S R \| A C$ and $S R=\frac{1}{2} A C$$\ldots($ i) In $\t...
Read More →A boy is cycling in such a way that the wheels of his bicycle are making 140 revolutions per minute.
Question: A boy is cycling in such a way that the wheels of his bicycle are making 140 revolutions per minute. If the diameter of a wheel is 60 cm, calculate the speed (in km/h) at which the boy is cycling. Solution: Diameter of the wheel = 60 cm Radius of the wheel = 30 cm Circumference of the wheel $=2 \pi \mathrm{r}$ $=2 \times \frac{22}{7} \times 30$ $=\frac{1320}{7} \mathrm{~cm}$ Distance covered by the wheel in 1 revolution $=\frac{1320}{7} \mathrm{~cm}$ $\therefore$ Distance covered by t...
Read More →The wheels of a car make 2500 revolutions in covering a distance of 4.95 km.
Question: The wheels of a car make 2500 revolutions in covering a distance of 4.95 km. Find the diameter of a wheel. Solution: Distance $=4.95 \mathrm{~km}=4.95 \times 1000 \times 100 \mathrm{~cm}$ $\therefore$ Distance covered by the wheel in 1 revolution $=\frac{\text { Total distance covered }}{\text { Number of revolutions }}$ $=\frac{4.95 \times 1000 \times 100}{2500}$ $=198 \mathrm{~cm}$ Now,Circumference of the wheel = 198 cm $\Rightarrow 2 \pi r=198$ $\Rightarrow 2 \times \frac{22}{7} \t...
Read More →State whether the matrix
Question: State whether the matrix $\left[\begin{array}{ll}2 3 \\ 6 4\end{array}\right]$ is singular or non-singular. Solution: Let $\Delta=\mid \begin{array}{ll}2 3\end{array}$ $6 \quad 4 \mid=\{(2 \times 4)-(6 \times 3)\}=8-18=-10$ A matrix is said to be singular if its determinant is equal to zero. Since $\Delta=-10 \neq 0$, the given matrix is non $-$ singular....
Read More →Factorize the following:
Question: Factorize the following:ax2y+bxy2+cxyz Solution: The greatest common factor of the terms ax $^{2} y, b x y^{2}$ and cxyz of the expression ax $^{2} y+b x y^{2}+c x y z$ is $x y$. Also, we can write $a x^{2} y=x y \times a x, b x y^{2}=x y \times b y$ and $c x y z=x y \times c z$. $\therefore a x^{2} y+b x y^{2}+c x y z=x y \times a x+x y \times b y+x y \times c z$ $=x y(a x+b y+c z)$...
Read More →Write the value of the determinant
Question: Write the value of the determinant $\left[\begin{array}{ccc}2 3 4 \\ 2 x 3 x 4 x \\ 5 6 8\end{array}\right]$. Solution: Let $\Delta=\mid \begin{array}{lll}2 3 4\end{array}$ $\begin{array}{ccc}2 x 3 x 4 x \\ 5 6 8\end{array}$ $=x \mid 2 \quad 3 \quad 4$ $\begin{array}{lll}2 3 4 \\ 5 6 8\end{array}$ $\left[\right.$ Taking out $x$ common from $\left.R_{2}\right]$ $=0$...
Read More →Write the value of the determinant
Question: Write the value of the determinant $\left[\begin{array}{ccc}2 3 4 \\ 2 x 3 x 4 x \\ 5 6 8\end{array}\right]$. Solution: Let $\Delta=\mid \begin{array}{lll}2 3 4\end{array}$ $\begin{array}{ccc}2 x 3 x 4 x \\ 5 6 8\end{array}$ $=x \mid 2 \quad 3 \quad 4$ $\begin{array}{lll}2 3 4 \\ 5 6 8\end{array}$ $\left[\right.$ Taking out $x$ common from $\left.R_{2}\right]$ $=0$...
Read More →Factorize the following:
Question: Factorize the following:x2yz+xy2z+xyz2 Solution: The greatest common factor of the terms $x^{2} y z, x y^{2} z$ and $x y z^{2}$ of the expression $x^{2} y z+x y^{2} z+x y z^{2}$ is $x y z$. Also, we can write $x^{2} y z=x y z \times x, x y^{2} z=x y z \times y$ and $x y z^{2}=x y z \times z$. $\therefore x^{2} y z+x y^{2} z+x y z^{2}=x y z \times x+x y z \times y+x y z \times z$ $=x y z(x+y+z)$...
Read More →P, Q, R and S are respectively the mid-points
Question: P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC BD. Prove that PQRS is a rectangle. Solution: Given In quadrilateral ABCD, P, O, S and S are the mid-points of the sides AB, BC, CD and DA, respectively. Also, AC BD To prove PQRS is a rectangle. Proof Since, AC BD . COD = AOD= AOB= COB = 90 In $\triangle A D C, S$ and $R$ are the mid-points of $A D$ and $D C$ respectively, then by mid-point theorem $S R \| A C$ and $S R=\f...
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