If the area of an equilateral triangle is
Question: If the area of an equilateral triangle is $81 \sqrt{3} \mathrm{~cm}^{2}$, find its height. Solution: Area of the equilateral triangle $=81 \sqrt{3} \mathrm{~cm}^{2}$ Area of an equilateral triangle $=\left(\frac{\sqrt{3}}{4} \times a^{2}\right)$, where $\mathrm{a}$ is the length of the side. $\Rightarrow 81 \sqrt{3}=\frac{\sqrt{3}}{4} \times a^{2}$ $\Rightarrow 324=a^{2}$ $\Rightarrow a=18 \mathrm{~cm}$ Height of triangle $=\frac{\sqrt{3}}{2} \times a$ $=\frac{\sqrt{3}}{2} \times 18$ $...
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Question: $x+y=5$ $y+z=3$ $x+z=4$ Solution: These equations can be written as $x+y+0 z=5$ $0 x+y+z=3$ $x+0 y+z=4$ $D=\left|\begin{array}{lll}1 1 0 \\ 0 1 1 \\ 1 0 1\end{array}\right|$ $=1(1-0)-1(0-1)+0(0-1)$ $=1(1)-1(-1)+0$ $=2$ $D_{1}=\left|\begin{array}{lll}5 1 0 \\ 3 1 1 \\ 4 0 1\end{array}\right|$ $=5(1-0)-1(3-4)+0(0-4)$ $=5(1)-1(-1)$ $=6$ $D_{2}=\left|\begin{array}{lll}1 5 0 \\ 0 3 1 \\ 1 4 1\end{array}\right|$ $=1(3-4)-5(0-1)+0(0-4)$ $=1(-1)-5(-1)$ $=4$ $D_{3}=\left|\begin{array}{lll}1 1 5...
Read More →Euclid stated that all right
Question: Euclid stated that all right angles are equal to each other in the form of (a)an axiom (b)a definition (c)a postulate (d)a proof Solution: (c) Euclid stated that all right angles are equal to each other in the form of a postulate....
Read More →Which of the following needs a proof?
Question: Which of the following needs a proof? (a)Theorems (b)Axiom (c)Definition (d)Postulate Solution: (a)The statements that were proved are called propositions or theorems....
Read More →If the area of an equilateral triangle is
Question: If the area of an equilateral triangle is $36 \sqrt{3} \mathrm{~cm}^{2}$, find its perimeter. Solution: Area of equilateral triangle $=36 \sqrt{3} \mathrm{~cm}^{2}$ Area of equilateral triangle $=\left(\frac{\sqrt{3}}{4} \times a^{2}\right)$, where $a$ is the length of the side. $\Rightarrow 36 \sqrt{3}=\frac{\sqrt{3}}{4} \times a^{2}$ $\Rightarrow 144=a^{2}$ $\Rightarrow a=12 \mathrm{~cm}$ Perimeter of a triangle = 3a $=3 \times 12$ $=36 \mathrm{~cm}$...
Read More →Pythagoras was a student of
Question: Pythagoras was a student of (a)Thales (b)Euclid (c)Both (a) and (b) (d)Archimedes Solution: (a) Pythagoras was a student of Thales....
Read More →Thales belongs to the country
Question: Thales belongs to the country (a)Babylonia (b)Egypt (c)Greece (d)Rome Solution: (c)Thales belongs to the country Greece....
Read More →Find the following product and verify the result for x = − 1,
Question: Find the following product and verify the result for x = 1, y = 2:(3x 5y) (x+y) Solution: To multiply, we will use distributive law as follows: (3x 5y) (x+y) $=3 x(x+y)-5 y(x+y)$ $=3 x^{2}+3 x y-5 x y-5 y^{2}$ $=3 x^{2}-2 x y-5 y^{2}$ $\therefore(3 x-5 y)(x+y)=3 x^{2}-2 x y-5 y^{2}$ Now, we put $x=-1$ and $y=-2$ on both sides to verify the result. LHS $=(3 x-5 y)(x+y)$ $=\{3(-1)-5(-2)\}\{-1+(-2)\}$ $=(-3+10)(-3)$ $=(7)(-3)$ $=-21$ $\mathrm{RHS}=3 x^{2}-2 x y-5 y^{2}$ $=3(-1)^{2}-2(-1)(...
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Question: $6 x+y-3 z=5$ $x+3 y-2 z=5$ $2 x+y+4 z=8$ Solution: Given: $6 x+y-3 z=5$ $D=\left|\begin{array}{ccc}6 1 -3 \\ 1 3 -2 \\ 2 1 4\end{array}\right|$ $=6(12+2)-1(4+4)-3(1-6)$ $=6(14)-1(8)-3(-5)$ $=91$ $D_{1}=\left|\begin{array}{ccc}5 1 -3 \\ 5 3 -2 \\ 8 1 4\end{array}\right|$ $=5(12+2)-1(20+16)-3(5-24)$ $=5(14)-1(36)-3(-19)$ $=91$ $D_{2}=\left|\begin{array}{ccc}6 5 -3 \\ 1 5 -2 \\ 2 8 4\end{array}\right|$ $=6(20+16)-5(4+4)-3(8-10)$ $=6(36)-5(8)-3(-2)$ $=182$ $D_{3}=\left|\begin{array}{lll}6...
Read More →Find the following product and verify the result for x = − 1,
Question: Find the following product and verify the result for x = 1, y = 2:(3x 5y) (x+y) Solution: To multiply, we will use distributive law as follows: (3x 5y) (x+y) $=3 x(x+y)-5 y(x+y)$ $=3 x^{2}+3 x y-5 x y-5 y^{2}$ $=3 x^{2}-2 x y-5 y^{2}$ $\therefore(3 x-5 y)(x+y)=3 x^{2}-2 x y-5 y^{2}$ Now, we put $x=-1$ and $y=-2$ on both sides to verify the result. LHS $=(3 x-5 y)(x+y)$ $=\{3(-1)-5(-2)\}\{-1+(-2)\}$ $=(-3+10)(-3)$ $=(7)(-3)$ $=-21$ $\mathrm{RHS}=3 x^{2}-2 x y-5 y^{2}$ $=3(-1)^{2}-2(-1)(...
Read More →Euclid belongs to the country
Question: Euclid belongs to the country (a)Babylonia (b)Egypt (c)Greece (d)India Solution: (c)Euclid belongs to the country Greece....
Read More →The height of an equilateral triangle is 6 cm. Find its area.
Question: The height of an equilateral triangle is 6 cm. Find its area.$[$ Take $\sqrt{3}=1.73]$ Solution: Let the side of the equilateral triangle bexcm. As, the area of an equilateral triangle $=\frac{\sqrt{3}}{4}(\text { side })^{2}=\frac{x^{2} \sqrt{3}}{4}$ Also, the area of the triangle $=\frac{1}{2} \times$ Base $\times$ Height $=\frac{1}{2} \times x \times 6=3 x$ So, $\frac{x^{2} \sqrt{3}}{4}=3 x$ $\Rightarrow \frac{x \sqrt{3}}{4}=3$ $\Rightarrow x=\frac{12}{\sqrt{3}}$ $\Rightarrow x=\fra...
Read More →The height of an equilateral triangle is 6 cm. Find its area.
Question: The height of an equilateral triangle is 6 cm. Find its area.$[$ Take $\sqrt{3}=1.73]$ Solution: Let the side of the equilateral triangle bexcm. As, the area of an equilateral triangle $=\frac{\sqrt{3}}{4}(\text { side })^{2}=\frac{x^{2} \sqrt{3}}{4}$ Also, the area of the triangle $=\frac{1}{2} \times$ Base $\times$ Height $=\frac{1}{2} \times x \times 6=3 x$ So, $\frac{x^{2} \sqrt{3}}{4}=3 x$ $\Rightarrow \frac{x \sqrt{3}}{4}=3$ $\Rightarrow x=\frac{12}{\sqrt{3}}$ $\Rightarrow x=\fra...
Read More →In ancient India,
Question: In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for (a)public worship (b)household rituals (c)Both (a) and (b) (d)None of these Solution: (a)In ancient India altars whose shapes were combinations of rectangles, triangles and trapeziums were used for public worship....
Read More →Greek’s emphasised on
Question: Greeks emphasised on (a)inductive reasoning (b)deductive reasoning (c)Both (a) and (b) (d)practical use of geometry Solution: (b)Greeks emphasised on deductive reasoning....
Read More →Solve the following
Question: (2xy+ 3y2) (3y2 2) Solution: To multiply, we will use distributive law as follows: (2xy+ 3y2) (3y2 2) $=2 x y\left(3 y^{2}-2\right)+3 y^{2}\left(3 y^{2}-2\right)$ $=6 x y^{3}-4 x y+9 y^{4}-6 y^{2}$ $=9 y^{4}+6 x y^{3}-6 y^{2}-4 x y$ Thus, the answer is $9 y^{4}+6 x y^{3}-6 y^{2}-4 x y$....
Read More →The number of interwoven isosceles
Question: The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is (a)seven (b)eight (b)nine (d)eleven Solution: (c)The Sriyantra (in the Atharvaveda) consists of nine interwoven isosceles triangles....
Read More →Multiply:
Question: Multiply:(2x2 1) by (4x3+ 5x2) Solution: To multiply, we will use distributive law as follows: (2x2 1) by (4x3+ 5x2) $=2 x^{2}\left(4 x^{3}+5 x^{2}\right)-1\left(4 x^{3}+5 x^{2}\right)$ $=8 x^{5}+10 x^{4}-4 x^{3}-5 x^{2}$ Thus, the answer is $8 x^{5}+10 x^{4}-4 x^{3}-5 x^{2}$....
Read More →In ancient India, the shapes
Question: In ancient India, the shapes of attars used for household rituals were (a)squares and circles (b)triangles and rectangles (c)trapeziums and pyramids (d)rectangles and squares Solution: (a)In ancient India, squares and circular altars were used for household rituals....
Read More →Each side of an equilateral triangle is 10 cm.
Question: Each side of an equilateral triangle is 10 cm. Find (i) the area of the triangle and (ii) the height of the triangle. Solution: (i) The area of the equilateral triangle $=\frac{\sqrt{3}}{4} \times \operatorname{side}^{2}$ $=\frac{\sqrt{3}}{4} \times 10^{2}$ $=\frac{\sqrt{3}}{4} \times 100$ $=25 \sqrt{3} \mathrm{~cm}^{2}$ or $25 \times 1.732=43.3 \mathrm{~cm}^{2}$ So, the area of the triangle is $25 \sqrt{3} \mathrm{~cm}^{2}$ or $43.3 \mathrm{~cm}^{2}$. (ii) As, area of the equilateral ...
Read More →It is known that,
Question: It is known that, if x + y = 10, then x+y + z = 10+ z. The Euclids axiom that illustrates this statement is (a)first axiom (b)second axiom (c)third axiom (d)fourth axiom Solution: (b)The Euclids axiom that illustrates the given statement is second axiom, according to which. If equals are added to equals, the wholes are equal....
Read More →Multiply:
Question: Multiply: $\left(3 x^{2} y-5 x y^{2}\right)$ by $\left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right)$ Solution: To multiply, we will use distributive law as follows: $\left(3 x^{2} y-5 x y^{2}\right)$ by $\left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right)$ $=\frac{1}{5} x^{2}\left(3 x^{2} y-5 x y^{2}\right)+\frac{1}{3} y^{2}\left(3 x^{2} y-5 x y^{2}\right)$ $=\frac{3}{5} x^{4} y-x^{3} y^{2}+x^{2} y^{3}-\frac{5}{3} x y^{4}$ Thus, the answer is $\frac{3}{5} x^{4} y-x^{3} y^{2}+x^{2} y^{3}-\fr...
Read More →The side faces of a pyramid are
Question: The side faces of a pyramid are (a)triangles (b)squares (c)polygons (d)trapeziums Solution: (a)The side faces of a pyramid are always triangles....
Read More →A pyramid is a solid figure,
Question: A pyramid is a solid figure, the base of which is (a) only a triangle (b) (only a square (c) only a rectangle (d) any polygon Solution: (d)A pyramid is a solid figure, the base of which is .a triangle or square or some other polygon....
Read More →Multiply:
Question: Multiply: $\left(-\frac{a}{7}+\frac{a^{2}}{9}\right) b y\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$ Solution: To multiply, we will use distributive law as follows: $\left(-\frac{a}{7}+\frac{a^{2}}{9}\right) b y\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$ $=\left(-\frac{a}{7}\right)\left(\frac{b}{2}-\frac{b^{2}}{3}\right)+\left(\frac{a^{2}}{9}\right)\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$ $=\left(-\frac{a b}{14}+\frac{a b^{2}}{21}\right)+\left(\frac{a^{2} b}{18}-\frac{a^{2} b^{2}}{27}\righ...
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