The least number by which 72 be
Question: The least number by which 72 be multiplied to make it a perfect cube, is________ Solution: 3 Resolving 72 into prime factors, we get 72=2 x 2 x 2 x 3 x 3 Grouping the factors in triplets of equal factors, we get 72 = (2 x 2 x 2) x 3 x 3 We find that 2 occurs as a prime factor of 72 thrice, but 3 occurs as a prime factor only twice. Thus, if we multiply 72 by 3, 3 will also occurs as a prime factor thrice and the product will be 2 x 2 x 2 x 3 x 3 x 3, which is a perfect cube. Hence, the...
Read More →In which of the following quadrants does the point P (3, 6) lie?
Question: In which of the following quadrants does the pointP(3, 6) lie? (a) I (b) II (c) III (d) IV Solution: (a) I...
Read More →Fill in the blanks to make the given statements true.
Question: Fill in the blanks to make the given statements true. (i) A point in which thex-coordinate is 0 andy-coordinate is 5 will lie on ..... . (ii) All point withy-coordinate as 0 lie on ...... . (iii) Thex-coordinate of any point lying on they-axis is always ..... . (iv) The point where the two axes intersect is called the ..... . (v) For the point (5, 2), the distance from thex-axis is ...... units. (vi) The distance of any point from they-axis is the ...... coordinate. (vii) The coordinat...
Read More →Write the coordinates of the vertices of each of the figures given below:
Question: Write the coordinates of the vertices of each of the figures given below: Solution: From the graph Co-ordinate of A =(1,1)Co-ordinate of B =(3,0)Co-ordinate of C =(4,2)Co-ordinate of D =(2,3)Co-ordinate of E =(5,1)Co-ordinate of F =(6,3)Co-ordinate of G =(5,5)Co-ordinate of H =(4,3)Co-ordinate of I =(1,2)Co-ordinate of J =(2,4)Co-ordinate of K =(0,5)Co-ordinate of L =(2,5)Co-ordinate of M =(1,5)Co-ordinate of N=(2,6)Co-ordinate of O =(3,6)Co-ordinate of P =(4,5)Co-ordinate of Q =(4,4)...
Read More →The least number by which 125 be
Question: The least number by which 125 be multiplied to make it a perfect square, is________ Solution: 5 Resolving 125 into prime factors, we get $125=5 \times 5 \times 5$ Grouping the factors in triplets of equal factors, we get $125=(5 \times 5) \times 5$ We observe that 5 is without a pair. Thus, we must multiply the number by 5 , so that the product is a perfect square. Hence, the least number, which should be multiplied with 125 to make it a perfect square, is 5 ....
Read More →Solve this
Question: $\sqrt{1+4 \sqrt{-3}}$ Solution: Let, $(a+i b)^{2}=1+4^{\sqrt{3}} i$ Now using, $(a+b)^{2}=a^{2}+b^{2}+2 a b$ $\Rightarrow a^{2}+(b i)^{2}+2 a b i=1+4^{\sqrt{3}} i$ Since $i^{2}=-1$ $\Rightarrow a^{2}-b^{2}+2 a b i=1+4^{\sqrt{3}} i$ Now, separating real and complex parts, we get $\Rightarrow a^{2}-b^{2}=1 \ldots \ldots \ldots \ldots \ldots$ eq. 1 $\Rightarrow 2 \mathrm{ab}=4^{\sqrt{3}} \ldots \ldots . \mathrm{eq} .2$ $\Rightarrow a=\frac{2 \sqrt{3}}{b}$ Now, using the value of a in eq....
Read More →Write the coordinates of all points in the graph given below:
Question: Write the coordinates of all points in the graph given below: Solution: From the graphCo-ordinate of A =(4,7)Co-ordinate of B =(7,4)Co-ordinate of C =(4,1)Co-ordinate of D =(1,4)Co-ordinate of E =(3,5)Co-ordinate of F =(5,5)Co-ordinate of G =(5,3)Co-ordinate of H =(3,3)Co-ordinate of I =(4,5)Co-ordinate of J =(5,4)Co-ordinate of K =(4,3)Co-ordinate of L =(3,4)...
Read More →From the given figure, choose the letters that indicate the location of the following points:
Question: From the given figure, choose the letters that indicate the location of the following points: (a) (2, 1) (b) (0,5) (c) (2,0) Also, mention the coordinates of the pointsAandF. Solution: (a) Location of (2,1) is shown by E. (b) Location of (0,5) is shown by B. (C) Location of (2,0) is shown by G. Co-ordinates of point A is (4,5) and F is (5.5,0)...
Read More →The cube root of a number x
Question: The cube root of a number x is denoted by________ Solution: $\sqrt[3]{x}$ or $x^{1 / 3}$ The cube root of a number $x$ is denoted by $\sqrt[3]{x}$ or $x^{1 / 3}$....
Read More →Plot the points A (2, 4) and B (4, 2) on a graph sheet and draw a line passing through these two points.
Question: Plot the points A (2, 4) and B (4, 2) on a graph sheet and draw a line passing through these two points. Extend the line on both sides to meet the coordinate axes. What are the coordinates of the points where this line meets thex-axis and they-axis? Solution: From the graph, we can see that after extending line AB, it cutsx-axis at P(6,0) andy-axis at Q(0,6)....
Read More →Plot the points A (2, 4) and B (4, 2) on a graph sheet and draw a line passing through these two points.
Question: Plot the points A (2, 4) and B (4, 2) on a graph sheet and draw a line passing through these two points. Extend the line on both sides to meet the coordinate axes. What are the coordinates of the points where this line meets thex-axis and they-axis? Solution:...
Read More →Locate the points P (3, 4), Q (1, 0), R (0, 4) and S (4, 1)
Question: Locate the pointsP(3, 4),Q(1, 0),R(0, 4) and S (4, 1) on a graph sheet and write the coordinates of the points of intersection of line segmentsPQandRS. Solution: From the graph, we can see that point of intersection of line segment PQ and RS is O(2.2,2.3)...
Read More →The cube of an odd number is always
Question: The cube of an odd number is always an________number. Solution: odd We know that, the cubes of all odd natural numbers are odd....
Read More →Locate the points A (1, 2), B (3, 4) and C (5, 2) on a graph sheet taking suitable axes.
Question: Locate the pointsA(1, 2),B(3, 4) andC(5, 2) on a graph sheet taking suitable axes. Write the coordinates of the fourth pointDin order to complete the rhombusABCD. Measure the diagonals of this rhombus and check whether they are equal or not. Solution: Co-ordinates of D =(3,0)Distance AC = 4 unitdistance BD = 4 unit Hence, we can see that diagonals of the given rhombus are of equal length i.e, 4 units....
Read More →Solve this
Question: $\sqrt{-2+2 \sqrt{3} i}$ Solution: Let, $(a+i b)^{2}=-2+2^{\sqrt{3}} i$ Now using, $(a+b)^{2}=a^{2}+b^{2}+2 a b$ $\Rightarrow a^{2}+(b i)^{2}+2 a b i=-2+2^{\sqrt{3}} i$ Since $i^{2}=-1$ $\Rightarrow a^{2}-b^{2}+2 a b i=-2+2^{\sqrt{3}} i$ Now, separating real and complex parts, we get $\Rightarrow a^{2}-b^{2}=-2 \ldots \ldots \ldots \ldots \ldots e q \cdot 1$ $\Rightarrow 2 \mathrm{ab}=2^{\sqrt{3}} \ldots \ldots \ldots . \mathrm{eq} .2$ $\Rightarrow \mathrm{a}=\frac{\sqrt{3}}{b}$ Now, u...
Read More →Locate the points A (1, 2), B (4, 2) and C (1, 4) on a graph sheet taking suitable axes.
Question: Locate the pointsA(1, 2),B(4, 2) andC(1, 4) on a graph sheet taking suitable axes. Write the coordinates of the fourth pointDin order to complete the rectangleABCD. Solution: Co-ordinates of D = (4,4)...
Read More →(1.2)3=________
Question: (1.2)3=________ Solution: $1.728$ We have, $(1.2)^{3}=\left(\frac{12}{10}\right)^{3}=\frac{12}{10} \times \frac{12}{10} \times \frac{12}{10}=\frac{1728}{1000}=1.728$...
Read More →Plot the given points on a graph sheet and check if they lie on a straight line.
Question: Plot the given points on a graph sheet and check if they lie on a straight line. If not, name the shape they form when joined in the given order. (i) A(0, 2),B(0, 3.5),C(0, 5) andD(0, 6) (ii)P(1, 3),Q(1, 5),R(3, 3) andS(3, 5) (iii)E(4, 5),F(5, 5),G(5, 7) andH(6, 5) (iv)L(2, 0),M(2, 3),N(0, 3) andO(0, 0) (v)J(4, 3),K(6, 1),L(6, 5) andM(4, 7) Solution: (i) After joining the given points, we see that they form a straight line.(ii) After joining the given points, we see that they form a sq...
Read More →Plot each of the following points on a graph sheet.
Question: Plot each of the following points on a graph sheet. Verify that they lie on a line. (i) A(4, 0),B(4, 2),C(4, 2.5), andD(4, 6) (ii)P(1, 1),Q(2, 2),R(3, 3), andS(4,4) (iii)L(6, 2),M(5, 3),N(3, 5), andO(2, 6) Solution: (i) After joining points A, B, C, D, we can see that the given points forms a straight line (ii) After joining points P, Q ,R, S, we can see that the given points forms a straight line (iii) After joining points L, M, N, O, we can see that the given points forms a straight ...
Read More →prove that
Question: 1.4 We have, $\sqrt{1.96}=\sqrt{\frac{196}{100}}=\sqrt{\frac{14 \times 14}{10 \times 10}}=\frac{14}{10}=1.4$ Solution: $1.728$ We have, $(1.2)^{3}=\left(\frac{12}{10}\right)^{3}=\frac{12}{10} \times \frac{12}{10} \times \frac{12}{10}=\frac{1728}{1000}=1.728$...
Read More →On a graph paper, plot each of the following points:
Question: On a graph paper, plot each of the following points: (i) A (4, 3) (ii) B (2, 5) (iii) C (0, 4) (iv) D (7, 0) (v) E (3, 5) (vi) F (5, 3) (vii) G (5, 5) (viii) H (0, 0) Solution:...
Read More →The sides of a right angled triangle
Question: The sides of a right angled triangle whose hypotenuse is 17cm, are________and________ Solution: 8,15 As, hypotenuse of right angled triangle is $17 \mathrm{~cm}$. So, $17^{2}=a^{2}+b^{2}$ [by Pythagoras theorem] where, $a$ and $b$ are other sides of riaht analed trianale. $=8^{2}+15^{2} \quad\left[\because 17^{2}=8^{2}+15^{2}\right]$ Hence, $a=8$ and $b=15$. As, hypotenuse of right angled triangle is 17 cm....
Read More →Write the ordinate of each of the following points:
Question: Write the ordinate of each of the following points: (i) (4, 0) (ii) (5, 2) (iii) (1, 4) (iv) (10, 7) Solution: (i) Ordinate of (4,0) = 0 (ii) Ordinate of (5,2) =2 (iii) Ordinate of (1,4) = 4 (iv) Ordinate of (10,7) = 7...
Read More →Solve this
Question: $\sqrt{-7+24 i}$ Solution: Let, $(a+i b)^{2}=-7+24 i$ Now using, $(a+b)^{2}=a^{2}+b^{2}+2 a b$ $\Rightarrow a^{2}+(b i)^{2}+2 a b i=-7+24 i$ Since $i^{2}=-1$ $\Rightarrow a^{2}-b^{2}+2 a b i=-7+24 i$ Now, separating real and complex parts, we get $\Rightarrow a^{2}-b^{2}=-7 \ldots \ldots \ldots \ldots \ldots$ eq. 1 $\Rightarrow 2 \mathrm{ab}=24 \ldots \ldots . . \mathrm{eq} .2$ $\Rightarrow a=\frac{12}{b}$ Now, using the value of a in eq.1, we get $\Rightarrow\left(\frac{12}{b}\right)^...
Read More →Write the abscissa of each of the following points:
Question: Write the abscissa of each of the following points: (i) (0, 5) (ii) (3, 7) (iii) (2, 4) (iv) (6, 3) Solution: (i) Abscissa of (0,5) = 0 (ii) Abscissa of (3,7) =3 (iii) Abscissa of (2,4) = 2 (iv) Abscissa of (6,3) = 6...
Read More →