If the angles of elevation of the top of a tower form tow points at distances a and
Question: If the angles of elevation of the top of a tower form tow points at distancesaandbfrom the base and in the same straight line with it are complementary, then the height of the tower is (a) $\sqrt{\frac{a}{b}}$ (b) $\sqrt{a b}$ (c) $\sqrt{a+b}$ (d) $\sqrt{a-b}$ Solution: (b) $\sqrt{a b}$ Let $A B$ be the tower and $C$ and $D$ be the points of observation on $A C$. Let: $\angle A C B=\theta, \angle A D B=90-\theta$ and $A B=h \mathrm{~m}$ Thus, we have: $A C=a, A D=b$ and $C D=a-b$ Now, ...
Read More →The point of the form (a, – a)
Question: The point of the form (a, a) always lies on the line (a)x = a (b)y = a (c)y = x (d)x + y = 0 Solution: (d)Taking option (d), x + y = a + (-a) = a a = 0 [since, give point is of the form (a, -a)] Hence, the point (a, a) always lies on the line x + y = 0....
Read More →Find the following product:
Question: Find the following product:5a(7a 2b) Solution: To find the product, we will use distributive law as follows: $-5 a(7 a-2 b)$ $=(-5 a) \times 7 a+(-5 a) \times(-2 b)$ $=(-5 \times 7) \times(a \times a)+(-5 \times(-2)) \times(a \times b)$ $=(-35) \times\left(a^{1+1}\right)+(10) \times(a \times b)$ $=-35 a^{2}+10 a b$ Thus, the answer is $-35 a^{2}+10 a b$....
Read More →The point of the form (a, a) always lies on
Question: The point of the form (a, a) always lies on (a)X-axis (b)Y-axis (c)the line y = x (d)the line x + y =0 Solution: (c)Since, the given point (a, a) has same value of x and y-coordinates. Therefore, the point (a, a), must be lie on the line y = x....
Read More →How many linear equations in x and y
Question: How many linear equations in x and y can be satisfied by x = 1 and y = 2? (a)Only one (b)Two (c)Infinitely many (d)Three Solution: (c)Let the linear equation be ax + by + c = 0. On putting x = 1 and y = 2, in above equation we get =s a + 2b + c = 0, where a, b and c, are real number Here, different values of a, b and c satisfy a + 2b + c = 0. Hence, infinitely many linear equations in x and yean be satisfied by x = 1 and y = 2....
Read More →Find the following product:
Question: Find the following product:11a(3a+ 2b) Solution: To find the product, we will use distributive law as follows: $-11 a(3 a+2 b)$ $=(-11 a) \times 3 a+(-11 a) \times 2 b$ $=(-11 \times 3) \times(a \times a)+(-11 \times 2) \times(a \times b)$ $=(-33) \times\left(a^{1+1}\right)+(-22) \times(a \times b)$ $=-33 a^{2}-22 a b$ Thus, the answer is $-33 a^{2}-22 a b$....
Read More →Find the following product:
Question: Find the following product:2a3(3a+ 5b) Solution: To find the product, we will use distributive law as follows: $2 a^{3}(3 a+5 b)$ $=2 a^{3} \times 3 a+2 a^{3} \times 5 b$ $=(2 \times 3)\left(a^{3} \times a\right)+(2 \times 5) a^{3} b$ $=(2 \times 3) a^{3+1}+(2 \times 5) a^{3} b$ $=6 a^{4}+10 a^{3} b$ Thus, the answer is $6 a^{4}+10 a^{3} b$....
Read More →If we multiply or divide both sides of a linear
Question: If we multiply or divide both sides of a linear equation with a non-zero number, then the solution of the linear equation (a)changes (b)remains the same (c)Only changes in case of multiplication (d)Only changes in case of division Solution: (b) By property, if we multiply or divide both sidesof a linear equation with a non-zero number, then the solution of the linear equation remains the same i.e., the solution of the linear equation is remains unchanged....
Read More →Classify the following polynomials as monomials, binomials, trinomials.
Question: Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category? (i)x + y (ii) 1000 (iii)x+x2+x3+ 4y4 (iv) 7 +a+ 5b (v) 2b 3b2 (vi) 2y 3y2+ 4y3 (vii) 5x 4y+ 3x (viii) 4a 15a2 (ix)xy + yz + zt + tx (x)pqr (xi)p2q+pq2 (xii) 2p+ 2q Solution: Definitions: A polynomial ismonomialif it has exactly one term. It is calledbinomialif it has exactly two non-zero terms. A polynomialis atrinomialif it has exactly three non-zero terms. (i) The po...
Read More →Classify the following polynomials as monomials, binomials, trinomials.
Question: Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category? (i)x + y (ii) 1000 (iii)x+x2+x3+ 4y4 (iv) 7 +a+ 5b (v) 2b 3b2 (vi) 2y 3y2+ 4y3 (vii) 5x 4y+ 3x (viii) 4a 15a2 (ix)xy + yz + zt + tx (x)pqr (xi)p2q+pq2 (xii) 2p+ 2q Solution: Definitions: A polynomial ismonomialif it has exactly one term. It is calledbinomialif it has exactly two non-zero terms. A polynomialis atrinomialif it has exactly three non-zero terms. (i) The po...
Read More →The graph of the linear equation y = x
Question: The graph of the linear equation y = x passes through the point (a)(3/2, -3/2) (b)(0,3/2) (c)(1,1) (d)(-,) Solution: (c)The linear equation y = x has same value of x and y-coordinates are same. Therefore, the point (1,1) must lie on the line y = x....
Read More →The string of a kite is 100 m long and it makes an angle of 60° with the horizontal.
Question: The string of a kite is 100 m long and it makes an angle of 60 with the horizontal. If these is no slack in the string, the height of the kite from the ground is (a) $50 \sqrt{3} \mathrm{~m}$ (b) $100 \sqrt{3} \mathrm{~m}$ (c) $50 \sqrt{2} \mathrm{~m}$ (d) $100 \mathrm{~m}$ Solution: (a) $50 \sqrt{3} \mathrm{~m}$ Let $A B$ be the string of the kite and $A X$ be the horizontal line. If $B C \perp A X$, then $A B=100 \mathrm{~m}$ and $\angle B A C=60^{\circ}$. Let: $B C=h \mathrm{~m}$ In...
Read More →The graph of the linear equation 2x+ 3y = 6
Question: The graph of the linear equation 2x+ 3y = 6 is a line which meets the X-axis at the point. (a)(0, 2) (b)(2,0) (c)(3, 0) (d)(0, 3) Solution: (c)Since, the graph of linear equation 2x + 3y = 6 meets the X-axis. So, we put y = 0 in 2x + 3y = 6 = 2x + 3(0) = 6 = 2x + 0 = 6 = x = 6/2 = x = 3 Hence, the coordinate on X-axis is (3, 0)....
Read More →Identify the terms, their coefficients for each of the following expressions:
Question: Identify the terms, their coefficients for each of the following expressions: (i) 7x2yz 5xy (ii)x2+ x+ 1 (iii) 3x2y2 5x2y2z2+z2 (iv) 9 ab+bcca (v) $\frac{a}{2}+\frac{b}{2}-a b$ (vi)0.2x0.3xy +0.5y Solution: Definitions: A term in an algebraic expression can be a constant, a variable or a product of constants and variables separated by the signs of addition (+) or subtraction $(-)$. Examples: $27, x, x y z, \frac{1}{2} x^{2} y z$ etc. The number factor of the term is called its coeffici...
Read More →The positive solutions of the equation
Question: The positive solutions of the equation ax + by + c = 0 always lie in the (a)Ist quadrant (b)IInd quadrant (c)IIIrd quadrant (d)IVth quadrant Solution: (a)We know that, if a line passes through the Ist quadrant, then all solution lying on the line in first quadrant must be positive because the coordinate of all points in the Ist quadrant are positive....
Read More →Identify the terms, their coefficients for each of the following expressions:
Question: Identify the terms, their coefficients for each of the following expressions: (i) 7x2yz 5xy (ii)x2+ x+ 1 (iii) 3x2y2 5x2y2z2+z2 (iv) 9 ab+bcca (v) $\frac{a}{2}+\frac{b}{2}-a b$ (vi)0.2x0.3xy +0.5y Solution: Definitions: A term in an algebraic expression can be a constant, a variable or a product of constants and variables separated by the signs of addition (+) or subtraction $(-)$. Examples: $27, x, x y z, \frac{1}{2} x^{2} y z$ etc. The number factor of the term is called its coeffici...
Read More →Identify the terms, their coefficients for each of the following expressions:
Question: Identify the terms, their coefficients for each of the following expressions: (i) 7x2yz 5xy (ii)x2+ x+ 1 (iii) 3x2y2 5x2y2z2+z2 (iv) 9 ab+bcca (v) $\frac{a}{2}+\frac{b}{2}-a b$ (vi)0.2x0.3xy +0.5y Solution: Definitions: A term in an algebraic expression can be a constant, a variable or a product of constants and variables separated by the signs of addition (+) or subtraction $(-)$. Examples: $27, x, x y z, \frac{1}{2} x^{2} y z$ etc. The number factor of the term is called its coeffici...
Read More →If a linear equation has solutions (-2, 2),
Question: If a linear equation has solutions (-2, 2), (0, 0) and (2, 2), then it is of the form (a)y x = 0 (b)x + y = 0 Thinking Process (i)Firstly, consider a linear equation ax + by + c = 0. (ii)Secondly, substitute all points one by one and get three different equations. (iii)Further, simplify the three equations and then substitute the values of a, b and c in the considered equation. Solution: (b)Let us consider a linear equation ax + by + c = 0 (i) Since, (-2,2), (0, 0) and (2, -2) are the ...
Read More →The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30°.
Question: The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30. The height of the tower is(a) 30 m (b) $10 \sqrt{3} \mathrm{~m}$ (c) 20 m (d) $10 \sqrt{2} \mathrm{~m}$ Solution: (b) $10 \sqrt{3} \mathrm{~m}$ Let $A B$ be the tower and $O$ be the point of observation. Also, $\angle A O B=30^{\circ}$ and $O B=30 \mathrm{~m}$ Let: $A B=h \mathrm{~m}$ In $\triangle A O B$, we have: $\frac{A B}{O B}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$ $\Ri...
Read More →x = 5 and y = 2 is a solution of the linear equation
Question: x = 5 and y = 2 is a solution of the linear equation (a)x + 2y = 7 (b)5x + 2y = 7 (c)x + y = 7 (d)5x + y = 7 Solution: (c)(a)Take x + 2y, on putting x=5 and y = 2, we get 5 + 2(2) = 5+ 4 = 97. So, (5, 2) is not a solution of x + 2y = 7 (b)Take 5x + 2y, on putting x = 5 and y = 2, we get 5x 5 + 2 x2 =25+ 4 = 297 So, (5, 2) is not a solution of 5x + 2y = 7. (c)Take x + y, on putting x = 5 and y = 2, we get 5+2=7 So, (5,2) is a solution of x + y = 7. (d)Take 5x + y, on putting x = 5 and y...
Read More →The graph of y = 6 is a Line
Question: The graph of y = 6 is a Line (a)parallel to X-axis at a distance 6 units from the origin (b)parallel to Y-axis at a distance 6 units from the origin (c)making an intercept 6 on the X-axis (d)making an intercept 6 on both axes Solution: (a)Given equation of line can be written as, a . x + 1 . y = 6 To draw the graph of above equation, we need atleast two solutions. When x = 0, then y = 6 When x =2, then y = 6 Here, we find two points A (0, 6) and B (2, 6). So, draw the graph, by plottin...
Read More →The equation of X-axis is of the form
Question: The equation of X-axis is of the form (a)x = 0 (b)y = 0 (c)x + y = 0 (d)x = y Solution: (b)The equation of X-axis is of the form y = 0....
Read More →Any point on the line y = x is of the form
Question: Any point on the line y = x is of the form (a)(a, a) (b)(0, a) (c)(a, 0) (d)(a, a) Solution: (a)Every point on the line y = x has same value of x-and y-coordinates i.e., x = a and y = a. Hence, (a, a) is the required form of the solution of given linear equation....
Read More →Evaluate each of the following when x = 2, y = −1.
Question: Evaluate each of the following when x = 2, y = 1. $\left(\frac{3}{5} x^{2} y\right) \times\left(-\frac{15}{4} x y^{2}\right) \times\left(\frac{7}{9} x^{2} y^{2}\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $\left(\frac{3}{5} x^{2} y\right) \times\left(-\frac{15}{4} x y^{2}\right) \times\left(\frac{7}{9} x^{2} y^{2}\right)$ $=\left\{\frac{3}{5} \times\left(-\frac{...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The tops of two towers of heightsxandy, standing on a level groundsubtend angles of 30 and 60, respectively at the centre of the linejoining their feet. Then,x:yis(a) 1 : 2 (b) 2 : 1 (c) 1 : 3 (d) 3 : 1 Solution: Let AB and CD be the two towers such that AB =xand CD =y.We have, $\angle \mathrm{AEB}=30^{\circ}, \angle \mathrm{CED}=60^{\circ}$ and $\mathrm{BE}=\mathrm{DE}$ In $\Delta \mathrm{ABE}$, $\tan 30^{\circ}=\frac{\mathrm{AB}}{\m...
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