Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2+ 12x 45 Solution: To factorise $\mathrm{x}^{2}+12 \mathrm{x}-45$, we will find two numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=12$ and $\mathrm{pq}=-45$. Now $15+(-3)=12$ and $15 \times(-3)=-45$ Splitting the middle term $12 \mathrm{x}$ in the given quadratic as $-3 \mathrm{x}+15 \mathrm{x}$, we get: $x^{2}+12 x-45=x^{2}-3 x+15 x-45$ $=\left(\mathrm{x}^{2}-3 \mathrm{x}\right)+(15 \mathrm{x}-45)$ $=\mathr...
Read More →Two lines I and m intersect at the
Question: Two lines $I$ and $m$ intersect at the point 0 and $P$ is a point on a line $n$ passing through the point 0 such that $P$ is equidistant from $I$ and $m$. Prove that $n$ is the bisector of the angle formed by $I$ and $m$. Solution: Given Two lines $l$ and $m$ intersect at the point $O$ and $P$ is a point on a line $n$ passing through O such that $P$ is equidistant from $l$ and $m$. i.e., $P Q=P R$. To prove $n$ is the bisector of the angle formed by $l$ and $m$ i.e., $n$ is the bisecto...
Read More →A parallelogram and a rhombus are equal in area.
Question: A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m add 44 m. If one of the sides of the ∥gm is 66 m long, find its corresponding altitude. Solution: Area of the rhombus $=\frac{1}{2}$ (Product of diagonals) $=\frac{1}{2}(120 \times 44)=2640 \mathrm{~m}^{2}$ Area of the parallelogram $=$ Base $\times$ Height $=66 \times$ Height Given:The area of the rhombus is equal to the area of the parallelogram. Thus, we have: $66 \times$ Height $=2640$ $\Rig...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2y2 4xz+ 4z2 Solution: $x^{2}-y^{2}-4 x z+4 z^{2}$ $=\left(\mathrm{x}^{2}-4 \mathrm{xz}+4 \mathrm{z}^{2}\right)-\mathrm{y}^{2}$ $=\left[\mathrm{x}^{2}-2 \times \mathrm{x} \times 2 \mathrm{z}+(2 \mathrm{z})^{2}\right]-\mathrm{y}^{2}$ $=(\mathrm{x}-2 \mathrm{z})^{2}-\mathrm{y}^{2}$ $=[(\mathrm{x}-2 \mathrm{z})-\mathrm{y}][(\mathrm{x}-2 \mathrm{z})+\mathrm{y}]$ $=(\mathrm{x}-2 \mathrm{z}-\mathrm{y})(\mathrm{x}-2 \mathrm{z}+\mathrm{y})$...
Read More →In a right triangle, prove that the line-segment
Question: In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse. Solution: Given In $\triangle A B C, \angle B=90^{\circ}$ and $D$ is the mid-point of $A C$. Construction Produce $B D$ to $E$ such that $B D=D E$ and join $E C$. To prove $B D=\frac{1}{2} A C$ Proof in $\triangle A D B$ and $\triangle C D E$, $A D=D C$ $[\because D$ is mid-point of $A C]$ $B D=D E$ [by construction] and $\angle A D B=\angle C D E$ [ve...
Read More →If the matrix
Question: If the matrix $A=\left[\begin{array}{ccc}1 3 x+2 \\ 2 4 8 \\ 3 5 10\end{array}\right]$ is singular, then $x=$_______ Solution: Given: The matrix $A=\left[\begin{array}{ccc}1 3 x+2 \\ 2 4 8 \\ 3 5 10\end{array}\right]$ is singular $A$ is singular $\Rightarrow|A|=0$ Thus, $\left|\begin{array}{ccc}1 3 \mathrm{x}+2 \\ 2 4 8 \\ 3 5 10\end{array}\right|=0$ $\Rightarrow 1(40-40)-3(20-24)+(\mathrm{x}+2)(10-12)=0$ $\Rightarrow 1(0)-3(-4)+(\mathrm{x}+2)(-2)=0$ $\Rightarrow 12-2 \mathrm{x}-4=0$ $...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2+ 4b2 4ab 4c2 Solution: $a^{2}+4 b^{2}-4 a b-4 c^{2}$ $=\left(a^{2}-4 a b+4 b^{2}\right)-4 c^{2}$ $=\left[a^{2}-2 \times a \times 2 b+(2 b)^{2}\right]-4 c^{2}$ $=(a-2 b)^{2}-(2 c)^{2}$ $=[(a-2 b)-2 c][(a-2 b)+2 c]$ $=(a-2 b-2 c)(a-2 b+2 c)$...
Read More →Find the area of quad. ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diag.
Question: Find the area of quad.ABCDin whichAB= 42 cm,BC= 21 cm,CD= 29 cm,DA= 34 cm and diag.BD= 20 cm. Solution: Quadrilateral $A B C D$ is divided into triangles $\triangle A B D$ and $\triangle B C D$. We will now use Hero's formula. For $\triangle A B D$ : Semiperimeter, $s=\frac{1}{2}(42+20+34)=\frac{96}{2}=48 \mathrm{~cm}$ Area of $\Delta \mathrm{ABD}=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{48(48-42)(48-34)(48-20)}$ $=\sqrt{48 \times 6 \times 14 \times 28}$ $=\sqrt{112896}$ $=336 \mathrm{~cm}^{2}...
Read More →If A is a skew-symmetric matrix of order 3 × 3,
Question: If $A$ is a skew-symmetric matrix of order $3 \times 3$, then $|A|=$________ Solution: Given:A is a skew-symmetric matrix of order 3 3 $A=-A^{T}$ Taking determinant on both sides, we get $\Rightarrow|A|=\left|-A^{T}\right|$ $\Rightarrow|A|=(-1)^{3}\left|A^{T}\right| \quad(\because$ Order of $A$ is $3 \times 3)$ $\Rightarrow|A|=(-1)^{3}|A| \quad\left(\because\left|A^{T}\right|=|A|\right)$ $\Rightarrow|A|=-|A|$ $\Rightarrow|A|+|A|=0$ $\Rightarrow 2|A|=0$ $\Rightarrow|A|=0$ Hence, $|A|=\u...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:49 x2y2+ 2xy Solution: $49-x^{2}-y^{2}+2 x y$ $=49-\left(\mathrm{x}^{2}-2 \mathrm{xy}+\mathrm{y}^{2}\right)$ $=49-\left(\mathrm{x}^{2}-2 \times \mathrm{x} \times \mathrm{y}+\mathrm{y}^{2}\right)$ $=7^{2}-(\mathrm{x}-\mathrm{y})^{2}$ $=[7-(\mathrm{x}-\mathrm{y})][7+(\mathrm{x}-\mathrm{y})]$ $=(7-\mathrm{x}+\mathrm{y})(7+\mathrm{x}-\mathrm{y})$ $=(\mathrm{x}-\mathrm{y}+7)(\mathrm{y}-\mathrm{x}+7)$...
Read More →If A is a skew-symmetric matrix of order 3 × 3,
Question: If $A$ is a skew-symmetric matrix of order $3 \times 3$, then $|A|=$________ Solution: Given:A is a skew-symmetric matrix of order 3 3 $A=-A^{T}$ Taking determinant on both sides, we get $\Rightarrow|A|=\left|-A^{T}\right|$ $\Rightarrow|A|=(-1)^{3}\left|A^{T}\right| \quad(\because$ Order of $A$ is $3 \times 3)$ $\Rightarrow|A|=(-1)^{3}|A| \quad\left(\because\left|A^{T}\right|=|A|\right)$ $\Rightarrow|A|=-|A|$ $\Rightarrow|A|+|A|=0$ $\Rightarrow 2|A|=0$ $\Rightarrow|A|=0$ Hence, $|A|=\u...
Read More →If A is a skew-symmetric matrix of order 3 × 3,
Question: If $A$ is a skew-symmetric matrix of order $3 \times 3$, then $|A|=$ Solution: Given:A is a skew-symmetric matrix of order 3 3 $A=-A^{T}$ Taking determinant on both sides, we get $\Rightarrow|A|=\left|-A^{T}\right|$ $\Rightarrow|A|=(-1)^{3}\left|A^{T}\right| \quad(\because$ Order of $A$ is $3 \times 3)$ $\Rightarrow|A|=(-1)^{3}|A| \quad\left(\because\left|A^{T}\right|=|A|\right)$ $\Rightarrow|A|=-|A|$ $\Rightarrow|A|+|A|=0$ $\Rightarrow 2|A|=0$ $\Rightarrow|A|=0$ Hence, $|A|=\underline...
Read More →The cost of fencing a square lawn at Rs 14 per metre is Rs 2800.
Question: The cost of fencing a square lawn at Rs 14 per metre is Rs 2800. Find the cost of mowing the lawn at Rs 54 per 100 cm2. Solution: Given:Costof fencing = Rs2800Rate of fencing = Rs 14Now, Perimeter $=\frac{\text { Total cost }}{\text { Rate }}=\frac{2800}{14}=200 \mathrm{~m}$ Because the lawn is square, its perimeter is $4 a$, where $a$ is the side of the square). $\Rightarrow 4 a=200 \Rightarrow a=\frac{200}{4}=50 \mathrm{~m}$ Area of the lawn $=\mathrm{Side}^{2}=50^{2}=2500 \mathrm{~m...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2+ 2ab+b2c2 Solution: $a^{2}+2 a b+b^{2}-c^{2}$ $=\left(a^{2}+2 a b+b^{2}\right)-c^{2}$ $=\left(a^{2}+2 \times a \times b+b^{2}\right)-c^{2}$ $=(a+b)^{2}-c^{2}$ $=[(a+b)-c][(a+b)+c]$ $=(a+b-c)(a+b+c)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2b2+ 2bcc2 Solution: $a^{2}-b^{2}+2 b c-c^{2}$ $=a^{2}-\left(b^{2}-2 b c+c^{2}\right)$ $=a^{2}-\left(b^{2}-2 \times b \times c+c^{2}\right)$ $=a^{2}-(b-c)^{2}$ $=[(a-(b-c)][(a+(b-c)]$ $=(a-b+c)(a+b-c)$...
Read More →In a A A B C, D is the mid-point of side
Question: In a $A A B C, D$ is the mid-point of side $A C$ such that $B D=1 / 2 A C$. Show that $\angle A B C$ is a right angle. Solution: Given In $\triangle A B C, D$ is the mid-point of $A C$ i.e., $A D=C D$ such that $B D=\frac{1}{2} A C$. To show $\angle A B C=90^{\circ}$ Proof We have, $B D=\frac{1}{2} A C$ $\ldots(i)$ Since, $D$ is the mid-point of $A C$. $\therefore$ $A D=C D=\frac{1}{2} A C$... (ii) From Eqs. (i) and (ii), $A D=C D=B D$ $\begin{array}{lll}\ln \triangle D A B_{1} A D=B D...
Read More →The adjacent sides of a ∥gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long.
Question: The adjacent sides of a ∥gmABCDmeasure 34 cm and 20 cm and the diagonalACis 42 cm long. Find the area of the ∥gm. Solution: The diagonal of a parallelogram divides it into two congruent triangles. Also, the area of the parallelogram is the sum of the areas of the triangles.We will now use Hero's formula to calculate the area of triangle ABC. Semiperimeter, $s=\frac{1}{2}(34+20+42)=\frac{1}{2}(96)=48 \mathrm{~cm}$ Area of $\Delta \mathrm{ABC}=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{48(48-42)(4...
Read More →If A = diag (1, 2, 3),
Question: If $A=\operatorname{diag}(1,2,3)$, then $|A|=$_______ Solution: Given: $A=\operatorname{diag}(1,2,3)$ If $A=\operatorname{diag}(a, b, c)$, then $|A|=a \times b \times c$. Thus, if $A=\operatorname{diag}(1,2,3)$, then $|A|=1 \times 2 \times 3=6$. Hence, $|A|=\underline{6}$....
Read More →If A = diag (1, 2, 3),
Question: If $A=\operatorname{diag}(1,2,3)$, then $|A|=$_______ Solution: Given: $A=\operatorname{diag}(1,2,3)$ If $A=\operatorname{diag}(a, b, c)$, then $|A|=a \times b \times c$. Thus, if $A=\operatorname{diag}(1,2,3)$, then $|A|=1 \times 2 \times 3=6$. Hence, $|A|=\underline{6}$....
Read More →Find the area of a trapezium whose parallel sides are 11 cm
Question: Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and non-parallel sides are 15 cm and 13 cm. Solution: We will divide the trapezium into a triangle and a parallelogram. Difference in the lengths of parallel sides $=25-11=14 \mathrm{~cm}$ We can represent this in the following figure: TrapeziumABCDis divided into parallelogramAECDand triangleCEB. 1.Consider triangleCEB. In triangleCEB, we have: $E B=25-11=14 \mathrm{~cm}$ Using Hero's theorem, we will first eva...
Read More →The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.
Question: The area of a triangle with vertices $(-3,0),(3,0)$ and $(0, k)$ is 9 sq. units. The value of $k$ will be (a) 9 (b) 3 (c) $-9$ (d) 6 Solution: Given: Area of a triangle with vertices $(-3,0),(3,0)$ and $(0, k)=9$ sq. units According to the question, $\Rightarrow\{-3(0-k)-0(3)+1(3 k)\}=\pm 18$ $\Rightarrow\{-3(-k)+1(3 k)\}=\pm 18$ $\Rightarrow\{3 k+3 k\}=\pm 18$ $\Rightarrow 6 k=\pm 18$ $\Rightarrow k=\pm 3$ Hence, the correct option is (b)....
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:25x2 10x+ 1 36y2 Solution: $25 x^{2}-10 x+1-36 y^{2}$ $=\left(25 x^{2}-10 x+1\right)-36 y^{2}$ $=\left[(5 x)^{2}-2 \times 5 x \times 1+1\right]-36 y^{2}$ $=(5 x-1)^{2}-(6 y)^{2}$ $=[(5 x-1)-6 y][(5 x-1)+6 y]$ $=(5 x-1-6 y)(5 x-1+6 y)$ $=(5 x-6 y-1)(5 x+6 y-1)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2y2+ 6y 9 Solution: $x^{2}-y^{2}+6 y-9$ $=x^{2}-\left(y^{2}-6 y+9\right)$ $=x^{2}-\left(y^{2}-2 \times y \times 3+3^{2}\right)$ $=x^{2}-(y-3)^{2}$ $=[x-(y-3)][x+(y-3)]$ $=(x-y+3)(x+y-3)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2 8ab+ 16b2 25c2 Solution: $a^{2}-8 a b+16 b^{2}-25 c^{2}$ $=\left(a^{2}-8 a b+16 b^{2}\right)-25 c^{2}$ $=\left[a^{2}-2 \times a \times 4 b+(4 b)^{2}\right]-25 c^{2}$ $=(a-4 b)^{2}-(5 c)^{2}$ $=[(a-4 b)-5 c][(a-4 b)+5 c]$ $=(a-4 b-5 c)(a-4 b+5 c)$...
Read More →Show that in a quadrilateral ABCD,
Question: Show that in a quadrilateral ABCD, AB + BC + CD + DA 2 (BD + AC) Thinking Process Firstly, draw a quadrilateral ABCD. Further use the property of a triangle that sum of two sides of a triangle is greater than third side and show the required result. Solution: Given $A B C D$ is a quadrilateral. To show $\quad A B+B C+C D+D A2(B D+A C)$ Construction Join diagonals $A C$ and $B D$. Proof in $\triangle O A B$, $O A+O BA B$$\cdots(1)$ [sum of two sides of a triangle is greater than the thi...
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