The sides of a triangle are in the ratio 5 : 12 : 13,
Question: The sides of a triangle are in the ratio 5 : 12 : 13, and its perimeter is 150 m. Find the area of the triangle Solution: Let the sides of a triangle be 5xm ,12xm and 13xm.Since, perimeter is the sum of all the sides, $5 x+12 x+13 x=150$ $\Rightarrow 30 x=150$ or, $x=\frac{150}{30}=5$ The lengths of the sides are: $a=5 \times 5=25 \mathrm{~m}$ $b=12 \times 5=60 \mathrm{~m}$ $c=13 \times 5=65 \mathrm{~m}$ Semiperimeter $(\mathrm{s})$ of the triangle $=\frac{\text { Perimeter }}{2}=\frac...
Read More →Multiply: (2x + 8) by (x − 3)
Question: Multiply:(2x+ 8) by (x 3) Solution: To multiply the expressions, we will use the distributive law in the following way: $(2 x+8)(x-3)$ $=2 x(x-3)+8(x-3)$ $=(2 x \times x-2 x \times 3)+(8 x-8 \times 3)$ $=\left(2 x^{2}-6 x\right)+(8 x-24)$ $=2 x^{2}-6 x+8 x-24$ $=2 x^{2}+2 x-24$ Thus, the answer is $2 x^{2}+2 x-24$....
Read More →Multiply: (5x + 3) by (7x + 2)
Question: Multiply:(5x+ 3) by (7x+ 2) Solution: To multiply, we will use distributive law as follows: $(5 x+3)(7 x+2)$ $=5 x(7 x+2)+3(7 x+2)$ $=(5 x \times 7 x+5 x \times 2)+(3 \times 7 x+3 \times 2)$ $=\left(35 x^{2}+10 x\right)+(21 x+6)$ $=35 x^{2}+10 x+21 x+6$ $=35 x^{2}+31 x+6$ Thus, the answer is $35 x^{2}+31 x+6$....
Read More →The linear equation that converts Fahrenheit
Question: The linear equation that converts Fahrenheit (F) to Celsius (C) is given by the relation, $C=\frac{5 F-160}{9}$ (i)If the temperature is 86F, what is the temperature in Celsius? (ii)If the temperature is 35C, what is the temperature in Fahrenheit? (iii)If the temperature is 0C, what is the temperature in Fahrenheit and if the temperature is 0F, what is the temperature in Celsius? (iv)What is the numerical value of the temperature which is same in both the scales? Solution: Given relati...
Read More →Simplify:
Question: Simplify: (i) 2x2(x3x) 3x(x4+ 2x) 2(x4 3x2) (ii)x3y(x2 2x) + 2xy(x3x4) (iii) 3a2+ 2(a+ 2) 3a(2a+ 1) (iv)x(x+ 4) + 3x(2x2 1) + 4x2+ 4 (v)a(bc) b(ca) c(ab) (vi)a(bc) +b(ca) +c(ab) (vii) 4ab(ab) 6a2(bb2) 3b2(2a2a) + 2ab(ba) (viii)x2(x2+ 1) x3(x+ 1) x(x3x) (ix) 2a2+ 3a(1 2a3) +a(a+ 1) (x)a2(2a 1) + 3a+a3 8 (xi) $\frac{3}{2} x^{2}\left(x^{2}-1\right)+\frac{1}{4} x^{2}\left(x^{2}+x\right)-\frac{3}{4} x\left(x^{3}-1\right)$ (xii)a2b(ab2)+ab2(4ab 2a2) a3b(1 2b) (xiii)a2b(a3a+ 1) ab(a4 2a2+ 2a)...
Read More →Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm.
Question: Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm. Also, find the height corresponding to the smallest side. Solution: Let the sides of triangle be a= 18 cm,b= 24 cm andc= 30 cm.Let s be the semi-perimeter of the triangle. $s=\frac{1}{2}(a+b+c)$ $s=\frac{1}{2}(18+24+30)$ $s=36 \mathrm{~cm}$ Area of a triangle $=\sqrt{s(s-a)(s-b)(s-c)}$ $=\sqrt{36(36-18)(36-24)(36-30)}$ $=\sqrt{36 \times 18 \times 12 \times 6}$ $=\sqrt{46656}$ $=216 \mathrm{~cm}^{2}$ The smallest sid...
Read More →Find the area of the triangle whose sides are 42 cm, 34 cm and 20 cm in length
Question: Find the area of the triangle whose sides are 42 cm, 34 cm and 20 cm in length. Find the height corresponding to the longest side. Solution: Let the sides of the triangle be a = 20 cm, b = 34 cm and c = 42 cm.Let s be the semi-perimeter of the triangle. $s=\frac{1}{2}(a+b+c)$ $s=\frac{1}{2}(20+34+42)$ $s=48 \mathrm{~cm}$ Area of the triangle $=\sqrt{s(s-a)(s-b)(s-c)}$ $\Rightarrow \sqrt{48(48-20)(48-34)(48-42)}$ $\Rightarrow \sqrt{48 \times 28 \times 14 \times 6}$ $\Rightarrow \sqrt{1...
Read More →Draw the graph of the linear equation 3x + 4y = 6.
Question: Draw the graph of the linear equation 3x + 4y = 6. At what points, the graph cuts X and K-axes? Solution: The given equation is $3 x+4 y=6$. To draw the graph of this equation, we need atleast two points lying on the graph of $4 y=6-3 x$ $\Rightarrow$ $y=\frac{6-3 x}{4}$ When $x=2$, then $y=\frac{6-3 \times 2}{4}=\frac{6-6}{4}=0$ When $x=0$, then $y=\frac{6-3 \times 0}{4}=\frac{6}{4}=\frac{3}{2}$ Here, we find two points $A\left(0, \frac{3}{2}\right)$ and $B(2,0)$. Now, plot the points...
Read More →Find the area of the triangle whose base measures 24 cm and the corresponding height measures 14.5 cm.
Question: Find the area of the triangle whose base measures 24 cm and the corresponding height measures 14.5 cm. Solution: Given: base $=24 \mathrm{~cm}$, correponding height $=14.5 \mathrm{~cm}$ Area of a triangle $=\frac{1}{2} \times$ base $\times$ coresponding height $=\frac{1}{2} \times 24 \times 14.5$ $=174 \mathrm{~cm}^{2}$...
Read More →Multiply the monomial by the binomial and find the value of each for x = −1,
Question: Multiply the monomial by the binomial and find the value of each forx= 1,y= 0.25 andz= 0.05: (i) 15y2(2 3x) (ii) 3x(y2+z2) (iii)z2(xy) (iv)xz(x2+y2) Solution: (i) To find the product, we will use distributive law as follows: $15 y^{2}(2-3 x)$ $=15 y^{2} \times 2-15 y^{2} \times 3 x$ $=30 y^{2}-45 x y^{2}$ Substituting $x=-1$ and $y=0.25$ in the result, we get: $30 y^{2}-45 x y^{2}$ $=30(0.25)^{2}-45(-1)(0.25)^{2}$ $=30 \times 0.0625-\{45 \times(-1) \times 0.0625\}$ $=30 \times 0.0625-\...
Read More →The following observed values of
Question: The following observed values of $x$ and $y$ are thought to satisfy a linear equation. Write the linear equation Draw the graph, using the values of x, y as given in the above table. At what points the graph of the linear equation (i)cuts the X-axis? (ii)cuts the Y- axis? Solution: Given, points are $(6,-2)$ and $(-6,6)$. Let the linear equation $y=m x+c$ is satisfied by the points $(6,-2)$ and $(-6,6)$, then at point $(6,-2)$ $-2=6 m+c$$\ldots$ (i) and at point $(-6,6)$, $6=-6 m+c$ .....
Read More →Solve the following
Question: Multiply $-\frac{3}{2} x^{2} y^{3}$ by $(2 x-y)$ and verify the answer for $x=1$ and $y=2$. Solution: To find the product, we will use distributive law as follows: $-\frac{3}{2} x^{2} y^{3} \times(2 x-y)$ $=\left(-\frac{3}{2} x^{2} y^{3} \times 2 x\right)-\left(-\frac{3}{2} x^{2} y^{3} \times y\right)$ $=\left(-3 x^{2+1} y^{3}\right)-\left(-\frac{3}{2} x^{2} y^{3+1}\right)$ $=-3 x^{3} y^{3}+\frac{3}{2} x^{2} y^{4}$ Substitutingx= 1 andy = 2 in the result, we get: $-3 x^{3} y^{3}+\frac{...
Read More →Solve the following
Question: Multiply $-\frac{3}{2} x^{2} y^{3}$ by $(2 x-y)$ and verify the answer for $x=1$ and $y=2$. Solution: To find the product, we will use distributive law as follows: $-\frac{3}{2} x^{2} y^{3} \times(2 x-y)$ $=\left(-\frac{3}{2} x^{2} y^{3} \times 2 x\right)-\left(-\frac{3}{2} x^{2} y^{3} \times y\right)$ $=\left(-3 x^{2+1} y^{3}\right)-\left(-\frac{3}{2} x^{2} y^{3+1}\right)$ $=-3 x^{3} y^{3}+\frac{3}{2} x^{2} y^{4}$ Substitutingx= 1 andy = 2 in the result, we get: $-3 x^{3} y^{3}+\frac{...
Read More →Find the product −
Question: Find the product 3y(xy+y2) and find its value forx= 4 andy= 5. Solution: To find the product, we will use distributive law as follows: $-3 y\left(x y+y^{2}\right)$ $=-3 y \times x y+(-3 y) \times y^{2}$ $=-3 x y^{1+1}-3 y^{1+2}$ $=-3 x y^{2}-3 y^{3}$ Substitutingx= 4 andy = 5 in the result, we get: $-3 x y^{2}-3 y^{3}$ $=-3(4)(5)^{2}-3(5)^{3}$ $=-3(4)(25)-3(125)$ $=-300-375$ $=-675$ Thus, the product is $\left(-3 x y^{2}-3 y^{3}\right)$, and its value for $x=4$ and $y=5$ is $(-675)$....
Read More →Find the product
Question: Find the product 24x2(1 2x) and evaluate its value forx= 3. Solution: To find the product, we will use distributive law as follows: $24 x^{2}(1-2 x)$ $=24 x^{2} \times 1-24 x^{2} \times 2 x$ $=24 x^{2}-48 x^{1+2}$ $=24 x^{2}-48 x^{3}$ Substituting x = 3 in the result, we get: $24 x^{2}-48 x^{3}$ $=24(3)^{2}-48(3)^{3}$ $=24 \times 9-48 \times 27$ $=216-1296$ $=-1080$ Thus, the product is $\left(24 x^{2}-48 x^{3}\right)$ and its value for $x=3$ is $(-1080)$....
Read More →Show that the points A (1, 2), B (-1, -16)
Question: Show that the points A (1, 2), B (-1, -16) and C (0, -7) lie on the graph of the linear equation y = 9x 7. Thinking Process (i)Firstly, make a table for the equation y = 9x 7. (ii)Secondly, plot the obtained points from the table on a graph and join them to get a straight line. (iii)Further, we plot the given points on a graph paper and find that whether these points lie on the line or not. Solution: Firstly, to draw the graph of equation $y=9 x-7$ When $x=2$, then $y=9 \times 2-7$ $=1...
Read More →Let y varies directly as x. If y = 12 when x = 4,
Question: Let y varies directly as x. If y = 12 when x = 4, then write a linear equation. What is value of y when x = 5? Thinking Process (i)Firstly, write the given condition as y x, then remove their proportionality sign by considering arbitrary constant k. (ii)Secondly, substitute the value of x and y in the obtained equation and determine the value of k. (iii)Further, substitute the value of k in obtained equation, to get the required equation. Solution: Given that, $y$ varies directly as $x...
Read More →An observer 1.5 m tall 28.5 away from a tower and the angle of elevation of the top of the tower form the eye of the observer is 45°.
Question: An observer 1.5 m tall 28.5 away from a tower and the angle of elevation of the top of the tower form the eye of the observer is 45. The height of the tower is(a) 27 m(b) 30 m(c) 28.5 m(d) none of these Solution: (b) 30 m Let $A B$ be the observer and $C D$ be the tower. Draw $B E \perp C D$, Let $C D=h$ metres. Then, $A B=1.5 \mathrm{~m}, B E=A C=28.5 \mathrm{~m}$ and $\angle E B D=45^{\circ}$. $D E=(C D-E C)=(C D-A B)=(h-1.5) \mathrm{m} .$ In right $\Delta B E D$, we have: $\frac{D E...
Read More →For what value of c,
Question: For what value of c, the linear equation 2x + cy = 8 has equal values of x and y for its solution? Solution: The given linear equation is 2x + cy= 8. (i) Now, by condition, x and y-coordinate of given linear equation are same, i.e., x = y. Put y = x in Eq. (i), we get $2 x+c x=8$ $\Rightarrow$$c x=8-2 x$ $\Rightarrow$ $c=\frac{8-2 x}{x}, x \neq 0$ Hence, the required value of $c$ is $\frac{8-2 x}{x}$....
Read More →Find the following product:
Question: Find the following product: $\frac{4}{3} a\left(a^{2}+b^{2}-3 c^{2}\right)$ Solution: To find the product, we will use distributive law as follows: $\frac{4}{3} a\left(a^{2}+b^{2}-3 c^{2}\right)$ $=\frac{4}{3} a \times a^{2}+\frac{4}{3} a \times b^{2}-\frac{4}{3} a \times 3 c^{2}$ $=\frac{4}{3} a^{1+2}+\frac{4}{3} a b^{2}-4 a c^{2}$ $=\frac{4}{3} a^{3}+\frac{4}{3} a b^{2}-4 a c^{2}$ Thus, the answer is $\frac{4}{3} a^{3}+\frac{4}{3} a b^{2}-4 a c^{2}$....
Read More →Find the solution of the linear equation x+2y = 8
Question: Find the solution of the linear equation x+2y = 8 which represents a point on (i)X-axis (ii)Y-axis Solution: We have, x + 2y = 8 ,..(i) (i)When the point is on the X-axis, then put y = 0 in Eq. (i), we get x+2 (0)=8 = x = 8 Hence, the required point is (8, 0). (ii) When the point is on the Y-axis, then put x = 0 in Eq. (i), we get 0 + 2y = 8 = y = 8/2 = 4 Hence, the required point is (0, 4)....
Read More →Find the following product:
Question: Find the following product: $\frac{7}{5} x^{2} y\left(\frac{3}{5} x y^{2}+\frac{2}{5} x\right)$ Solution: To find the product, we will use distributive law as follows: $\frac{7}{5} x^{2} y\left(\frac{3}{5} x y^{2}+\frac{2}{5} x\right)$ $=\frac{7}{5} x^{2} y \times \frac{3}{5} x y^{2}+\frac{7}{5} x^{2} y \times \frac{2}{5} x$ $=\frac{21}{25} x^{2+1} y^{1+2}+\frac{14}{25} x^{2+1} y$ $=\frac{21}{25} x^{3} y^{3}+\frac{14}{25} x^{3} y$ Thus, the answer is $\frac{21}{25} x^{3} y^{3}+\frac{14...
Read More →How many solutions) of the equation 2x + 1 = x – 3
Question: How many solutions) of the equation 2x + 1 = x 3 are there on the (ii) Cartesian plane? Solution: 2x + 1 = x 3 2x-x = -3-1 x = 4 ..(i) and it can be written as 1.x + 0. y = 4 ..(ii) (i)Number line represent the all real values of x on the X-axis. Therefore, x = 4 is exactly one point which lies on the number line. (ii)Whereas the equation x + 4 = 0 represent a straight line parallel to Y-axis and infinitely many points lie on a line in the cartesian plane....
Read More →If the elevation of the sun changes form 30° to 60°, then the difference between the lengths of shadows of a pole 15 m high, is
Question: If the elevation of the sun changes form 30 to 60, then the difference between the lengths of shadows of a pole 15 m high, is(a) 7.5 m(b) 15 m (c) $10 \sqrt{3} \mathrm{~m}$ (d) $5 \sqrt{3} \mathrm{~m}$ Solution: (c) $10 \sqrt{3} \mathrm{~m}$ Let $A B$ be the pole and $A C$ and $A D$ be its shadows. We have: $\angle A C B=30^{\circ}, \angle A D B=60^{\circ}$ and $A B=15 \mathrm{~m}$ In $\triangle A C B$, we have: $\frac{A C}{A B}=\cot 30^{\circ}=\sqrt{3}$ $\Rightarrow \frac{A C}{15}=\sq...
Read More →Find the following product:
Question: Find the following product: $250.5 x y\left(x z+\frac{y}{10}\right)$ Solution: To find the product, we will use distributive law as follows: $250.5 x y\left(x z+\frac{y}{10}\right)$ $=250.5 x y \times x z+250.5 x y \times \frac{y}{10}$ $=250.5 x^{1+1} y z+25.05 x y^{1+1}$ $=250.5 x^{2} y z+25.05 x y^{2}$ Thus, the answer is $250.5 x^{2} y z+25.05 x y^{2}$....
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