A rectangular plot measures 125 m by 78 m.
Question: A rectangular plot measures 125 m by 78 m. It has gravel path 3 m wide all around on the outside. Find the area of the path and the cost of gravelling it at Rs 75 per m2. Solution: The plot with the gravel path is shown in the figure. Area of the rectangular plot $=l \times b$ Area of the rectangular plot $=125 \times 78=9750 \mathrm{~m}^{2}$ Length of the park including the path = 125 + 6 = 131 mBreadth of the park including the path = 78 + 6 = 84 mArea of the plot including the path ...
Read More →In the figure, ∠1 = 60° and ∠6 = 120°.
Question: In the figure, 1 = 60 and 6 = 120. Show that the lines m and n are parallel. Solution: Given In the figure 1 = 60 and 6 = 120 To show m||n Proof Since, 1 = 60 and 6 = 120 Here, 1 = 3 [vertically opposite angles] 3 = 1 = 60 Now, 3 + 6 = 60 + 120 = 3 + 6 = 180 We know that, if the sum of two interior angles on same side of l is 180, then lines are parallel. Hence, m || n...
Read More →In the figure, OD is the bisector of ∠AOC,
Question: In the figure, OD is the bisector of AOC, OE is the bisector of BOC and OD OE. Show that the points A, 0 and B are collinear. Thinking Process For showing collinearity of A, O and B, we have to show that AOB =180. Solution: Given In the figure, OD OE, OD and OE are the bisectors of AOC and BOC. To show Points A, O and B are collinear i.e., AOB is a straight line. Proof Since, OD and OE bisect angles AOC and BOC, respectively. AOC =2 DOC (i) and COB = 2 COE (ii) On adding Eqs. (i) and (...
Read More →A rectangular park 35 m long 18 m wide is to be covered with grass, leaving 2.5 m uncovered all around it.
Question: A rectangular park 35 m long 18 m wide is to be covered with grass, leaving 2.5 m uncovered all around it. Find the area to be laid with grass. Solution: The field is planted with grass, with 2.5 m uncovered on its sides.The field is shown in the given figure. Thus, we have; Length of the area planted with grass $=35-(2.5+2.5)=35-5=30 \mathrm{~m}$ Width of the area planted with grass $=18-(2.5+2.5)=18-5=13 \mathrm{~m}$ Area of the rectangular region planted with grass $=30 \times 13=39...
Read More →Two lines l and m are perpendicular
Question: Two lines l and m are perpendicular to the same line n. Are l and m perpendicular to each other? Give reason for your answer. Solution: No, since, lines l and m are perpendicular to the line n. 1 = 2 = 90 [l n and min] It implies that these are corresponding angles. Hence, l|| m....
Read More →Simplify the following using the formula:
Question: Simplify the following using the formula: (ab)(a+b) =a2b2: (i) (82)2 (18)2 (ii) (467)2 (33)2 (iii) (79)2 (69)2 (iv) 197 203 (v) 113 87 (vi) 95 105 (vii) 1.8 2.2 (viii) 9.8 10.2 Solution: Here, we will use the identity $(a-b)(a+b)=a^{2}-b^{2}$ (i) Let us consider the following expression: $(82)^{2}-(18)^{2}$ $=(82+18)(82-18)$ $=100 \times 64$ $=6400$ (ii) Let us consider the following expression: $(467)^{2}-(33)^{2}$ $=(467+33)(467-33)$ $=500 \times 434$ $=217000$ (iii) Let us consider ...
Read More →In the figure, which of the two
Question: In the figure, which of the two lines are parallel and why? Solution: In Fig. (i) sum of two interior angles 132 + 48 = 180 [ equal to 180] Here, we see that the sum of two interior angles on the same side of n is 180, then they are the parallel lines. In Fig. (ii), the sum of two interior angles 73 + 106 = 179 180. Here, we see that the sum of two interior angles on same side of r is not equal to 180, then they are not the parallel lines....
Read More →The area of a rectangle is 192 cm2 and its perimeter is 56 cm.
Question: The area of a rectangle is 192 cm2and its perimeter is 56 cm. Find the dimensions of the rectangle. Solution: Area of the rectangle = 192 cm2Perimeter of the rectangle = 56 cm Perimeter $=2$ (length $+$ breadth $)$ $\Rightarrow 56=2(l+b)$ $\Rightarrow l+b=28$ $\Rightarrow l=28-b$ Area $=$ length $\times$ breadth $\Rightarrow 192=(28-b) x b$ $\Rightarrow 192=28 b-b^{2}$ $\Rightarrow b^{2}-28 b+192=0$ $\Rightarrow(b-16)(b-12)=0$ $\Rightarrow b=16$ or 12 Thus, we have; $l=28-b$ $\Rightarr...
Read More →If one of the angles formed by two
Question: If one of the angles formed by two intersecting lines is a right angle, what can you say about the other three angles? Give reason for your answer. Solution: Let two intersecting lines l and m makes a one right angle, then it means that lines I and m are perpendicular each other. By using linear pair axiom aniom, other three angles will be a right angle....
Read More →Two adjacent angles are equal.
Question: Two adjacent angles are equal. Is it necessary that each of these angles will be a right angle? Justify your answer. Solution: No, because each of these will be a right angle only when they form a linear pair....
Read More →A 36 m-long, 15 broad verandah is to be paved with stones, each measuring 6 dm by 5 cm.
Question: A 36 m-long, 15 broad verandah is to be paved with stones, each measuring 6 dm by 5 cm. How many stones will be required? Solution: Area of the verandah $=$ Length $\times$ Breadth $=36 \times 15=540 \mathrm{~m}^{2}$ Length of the stone = 6 dm = 0.6 mBreadth of the stone = 5 dm = 0.5 m Area of one stone $=0.6 \times 0.5=0.3 \mathrm{~m}^{2}$ Number of stones required $=\frac{\text { Area of the verendah }}{\text { Area of the stone }}$ $=\frac{540}{0.3}$ $=1800$ Thus, 1800 stones will b...
Read More →In the figure, find the value of x for which the lines l and m are parallel.
Question: In the figure, find the value of x for which the lines l and m are parallel. Solution: In the given figure, l || m and we know that, if a transversal intersects two parallel lines, then sum of interior angles on the same side of a transversal is supplementary. x + 44 = 180 x = 180-44 = x = 136 ....
Read More →How many triangles can be drawn
Question: How many triangles can be drawn having its angles as 53, 64 and 63? Give reason for your answer. Solution: Infinitely many triangles, The sum of given angles = 53 + 64 + 63 = 180 Here, we see that sum of all interior angles of triangle is 180, so infinitely many triangles can be drawn....
Read More →How many triangles can be drawn
Question: How many triangles can be drawn having its angles as 45, 64 and 72? Give reason for your answer. Solution: None, the sum of given angles = 45 + 64 + 72 = 181 180. Hence, we see that sum of all three angles is not equal to 180. So, no triangle can be drawn with the given angles....
Read More →The floor of a rectangular hall is 24 m long and 18 m wide.
Question: The floor of a rectangular hall is 24 m long and 18 m wide. How many carpets, each of length 2.5 m and breadth 80 cm, will be required to cover the floor of the hall? Solution: The length and breadth of floor of a rectangular hall is $24 \mathrm{~m}$ and $18 \mathrm{~m}$ respectively. The area of rectangular floor is $24 \times 18=432 \mathrm{~m}^{2}$. The length and breadth of carpet is $2.5 \mathrm{~m}$ and $80 \mathrm{~cm}$ or $0.8 \mathrm{~m}$ respectively. The area of carpet is $2...
Read More →Can a triangle have two obtuse angles?
Question: Can a triangle have two obtuse angles? Give reason for your answer. Solution: No, because if the triangle have two obtuse angles i.e., more than 90 angle, then the sum of all three angles of a triangle will not be equal to 180....
Read More →Solve this
Question: $3 x-y+2 z=3$ $2 x+y+3 z=5$ $x-2 y-z=1$ Solution: Given: $3 x-y+2 z=3$ $2 x+y+3 z=5$ $x-2 y-z=1$ $D=\left|\begin{array}{ccc}3 -1 2 \\ 2 1 3 \\ 1 -2 -1\end{array}\right|$ $=3(-1+6)+1(-2-3)+2(-4-1)$ $=0$ $D_{1=}\left|\begin{array}{ccc}3 -1 2 \\ 5 1 3 \\ 1 -2 -1\end{array}\right|$ $=3(-1+6)+1(-5-3)+2(-10-1)$ $=-15$ $D_{2}=\left|\begin{array}{ccc}3 3 2 \\ 2 5 3 \\ 1 1 -1\end{array}\right|$ $=3(-5-3)-3(-2-3)+2(2-5)$ $=-15$ $D_{3}=\left|\begin{array}{ccc}3 -1 3 \\ 2 1 5 \\ 1 -2 1\end{array}\...
Read More →Can a triangle have all angles
Question: Can a triangle have all angles less than 60? Give reason for your answer. Solution: No, a triangle cannot have all angles less than 60, because if all angles will be less than 60, then their sum will not be equal to 180. Hence, it will not be a triangle....
Read More →The floor of a rectangular hall is 24 m long and 18 m wide.
Question: The floor of a rectangular hall is 24 m long and 18 m wide. How many carpets, each of length 2.5 m and breadth 80 cm, will be required to cover the floor of the hall? Solution: The length and breadth of floor of a rectangular hall is $24 \mathrm{~m}$ and $18 \mathrm{~m}$ respectively. The area of rectangular floor is $24 \times 18=432 \mathrm{~m}^{2}$. The length and breadth of carpet is $2.5 \mathrm{~m}$ and $80 \mathrm{~cm}$ or $0.8 \mathrm{~m}$ respectively. The area of carpet is $2...
Read More →For what value of x + y in figure will ABC be a line?
Question: For what value of x + y in figure will ABC be a line? Justify your answer. Solution: For ABC to be a line, the sum of the two adjacent angles must be 180 i.e.,x + y = 180....
Read More →A room is 16 m long and 13.5 m broad.
Question: A room is 16 m long and 13.5 m broad. Find the cost of covering its floorwith 75-m-wide carpet at₹60 per metre. Solution: As, the area of the floor $=$ length $\times$ breadth $=16 \times 13.5$ $=216 \mathrm{~m}^{2}$ And, the width of the carpet $=75 \mathrm{~m}$ So, the length of the carpet required $=\frac{\text { Area of the floor }}{\text { Width of the carpet }}$ $=\frac{216}{75}$ $=2.88 \mathrm{~m}$ Now, the cost of the carpet required $=2.88 \times 60=₹ 172.80$ Hence, the cost o...
Read More →Angles of a triangle are in the ratio 2:4:3.
Question: Angles of a triangle are in the ratio 2:4:3. The smallest angle of the triangle is (a) 60 (b) 40 (c) 80 (d) 20 Thinking Process Use the concept, the sum of all angles in a triangle is 180. Further, simplify it and get the smallest angle. Solution: (b) Given, the ratio of angles of a triangle is 2 : 4 : 3. Let the angles of a triangle be A, B and C. A = 2x, B = 4x C = 3x , A+B+ C= 180 [sum of all the angles of a triangle is 180] 2x + 4x + 3x = 180 9x = 180 x=180/9 =20 A=2x=2 x 20 = 40 B...
Read More →Using the formula for squaring a binomial, evaluate the following:
Question: Using the formula for squaring a binomial, evaluate the following: (i) (102)2 (ii) (99)2 (iii) (1001)2 (iv) (999)2 (v) (703)2 Solution: (i) Here, we will use the identity $(a+b)^{2}=a^{2}+2 a b+b^{2}$ $(102)^{2}$ $=(100+2)^{2}$ $=(100)^{2}+2 \times 100 \times 2+2^{2}$ $=10000+400+4$ $=10404$ (ii) Here, we will use the identity $(a-b)^{2}=a^{2}-2 a b+b^{2}$ $(99)^{2}$ $=(100-1)^{2}$ $=(100)^{2}-2 \times 100 \times 1+1^{2}$ $=10000-200+1$ $=9801$ (iii) Here, we will use the identity $(a+...
Read More →In the figure, if OP || RS, ∠OPQ = 110°
Question: In the figure, if OP || RS, OPQ = 110 and QRS = 130, then PQR is equal to (a) $40^{\circ}$ (b) $50^{\circ}$ (c) $60^{\circ}$ (d) $70^{\circ}$ Solution: (c) In the given figure, producing $O P$, to intersect $R Q$ at $X$. Since, $O P \| R S$ and $R X$ is a transversal. So, $\quad \angle R X P=\angle X R S \quad$ [alternate angles] $\Rightarrow$ $\angle R X P=130^{\circ}$ $\left[\because \angle Q R S=130^{\circ}\right.$ (given)]...(i) Now, $R Q$ is a line segment. So, $\angle P X Q+\angl...
Read More →A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3.
Question: A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. The area of the lawn is 3375 m2. Find the cost of fencing the lawn at Rs 65 per metre. Solution: Let the length and breadth of the rectangular lawn be 5xm and 3xm, respectively.Given: Area of the rectangular lawn $=3375 \mathrm{~m}^{2}$ $\Rightarrow 3375=5 x \times 3 x$ $\Rightarrow 3375=15 x^{2}$ $\Rightarrow \frac{3375}{15}=x^{2}$ $\Rightarrow 225=x^{2}$ $\Rightarrow x=15$ Thus, we have: $l=5 x=5 \times 15=75 \m...
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