Class 10 Archives - Noon Academy

Class 10

A motor boat whose speed in still water is $178 \mathrm{~km} / \mathrm{hr}$ , takes 1 hour more to go $24 \mathrm{~km}$ upstream than to return to the same spot. Find the speed of the stream.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given: Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / h r$ Speed upstream...

A train covers a distance of $90 \mathrm{~km}$ at a uniform speed. Had the speed been $15 \mathrm{~km} / \mathrm{hr}$ more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the original speed of the train be $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\frac{90}{x}-\frac{90}{(x+15)}=\frac{1}{2}$ $\begin{array}{l} \Rightarrow \frac{90(x+15)-90...

A train travels at a certain average speed for a distanced of $54 \mathrm{~km}$ and then travels a distance of 63 $\mathrm{km}$ at an average speed of $6 \mathrm{~km} / \mathrm{hr}$ more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the first speed of the train be $x \mathrm{~km} / \mathrm{h}$. Time taken to cover $54 \mathrm{~km}=\frac{54}{x} h .$ New speed of the train $=(x+6) \mathrm{km} / \mathrm{h}$ $\therefore$ Time...

A train covers a distance of $480 \mathrm{~km}$ at a uniform speed. If the speed had been $8 \mathrm{~km} / \mathrm{hr}$ less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

Find the domain and range of the relation
R = {(-1, 1), (1, 1), (-2, 4), (2, 4)}.

CBSE, Class 12, Excercise 1A, Maths, Relations, RS Aggarwal

Solution: Set of all the first elements or $x$-coordinates of the ordered pairs is called Domain. Set of all the second elements or $y$-coordinates of the ordered pairs is called Range. Therefore,...

If
$x=\frac{\sqrt[3]{a+1}+\sqrt[3]{a-1}}{\sqrt[3]{a+1}-\sqrt[3]{a-1}}$
prove that :
${{x}^{3}}-3a{{x}^{2}}+3x-a=0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[\begin{array}{*{35}{l}} {{x}^{3}}~+\text{ }3x\text{ }=\text{ }3a{{x}^{2}}~+\text{ }a \\ {{x}^{3}}~\text{ }3a{{x}^{2}}~+\text{ }3x\text{ }\text{ }a\text{...

Find a from the equation
$\frac{a+x+\sqrt{{{a}^{2}}-{{x}^{2}}}}{a+x-\sqrt{{{a}^{2}}-{{x}^{2}}}}=\frac{b}{x}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that

If
$x=\frac{pab}{a+b}$
, prove that
$\frac{x+pa}{x-pa}-\frac{x+pb}{x-pb}=\frac{2({{a}^{2}}-{{b}^{2}})}{ab}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that = RHS

If
$x=\frac{2mab}{a+b}$
, find the value of
$\frac{x+ma}{x-ma}+\frac{x+mb}{x-mb}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that

If
$(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{11}:\text{ }\mathbf{9}$
, find the value of
$\frac{3{{x}^{4}}+25{{y}^{4}}}{3{{x}^{4}}-25{{y}^{4}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{...

If a: b =
$9:10$
, find the value of (i)
$\frac{5a+3b}{5a-3b}$
(ii)
$\frac{2{{a}^{2}}-3{{b}^{2}}}{2{{a}^{2}}+3{{b}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a: b = \[9:10\] So we get a/b = \[9/10\] = \[5\]

If
$\frac{x}{b+c-a}=\frac{y}{c+a-b}=\frac{z}{a+b-c}$
prove that each ratio’s equal to:
$\frac{x+y+z}{a+b+c}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider So we get x = k (b + c – a) y = k (c + a – b) z = k (a + b – a) Here = k Therefore, it is proved.

If x: a = y: b, prove that
$\frac{{{x}^{4}}+{{a}^{4}}}{{{x}^{3}}+{{a}^{3}}}+\frac{{{y}^{4}}+{{b}^{4}}}{{{y}^{3}}+{{b}^{3}}}=\frac{{{(x+y)}^{4}}+{{(a+b)}^{4}}}{{{(x+y)}^{3}}+{{(a+b)}^{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

We know that x/a = y/b = k So we get x = ak, y = bk Here Here LHS = RHS Therefore, it is proved.

If x/a = y/b = z/c, prove that
$\frac{3{{x}^{3}}-5{{y}^{3}}+4{{z}^{3}}}{3{{a}^{3}}-5{{b}^{3}}+4{{c}^{3}}}={{\left( \frac{3x-5y+4z}{3a-5b+4c} \right)}^{3}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x/a = y/b = z/c = k So we get x = ak, y = bk, z = ck Here = \[{{k}^{3}}\] Hence, LHS = RHS.

If a/b = c/d = e/f, prove that each ratio is (i)
$\sqrt{\frac{3{{a}^{2}}-5{{c}^{2}}+7{{e}^{2}}}{3{{b}^{2}}-5{{d}^{2}}+7{{f}^{2}}}}$
(ii)
${{\left[ \frac{2{{a}^{3}}+5{{c}^{2}}+7{{e}^{2}}}{2{{b}^{3}}+5{{d}^{3}}+7{{f}^{3}}} \right]}^{\frac{1}{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a/b = c/d = e/f = k So we get a = k, c = dk, e = fk Therefore, it is proved. = k Therefore, it is proved.

If q is the mean proportional between p and r, prove that:
${{p}^{2}}-3{{q}^{2}}+{{r}^{2}}={{q}^{4}}(\frac{1}{{{p}^{2}}}-\frac{3}{{{q}^{2}}}+\frac{1}{{{r}^{2}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that q is the mean proportional between p and r q2 = pr Here LHS = \[{{p}^{2}}~\text{ }3{{q}^{2}}~+\text{ }{{r}^{2}}\] We can write it as \[=\text{ }{{p}^{2}}~\text{ }3pr\text{ }+\text{...

Find two numbers whose mean proportional is
$16$
and the third proportional is
$128$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x and y as the two numbers Mean proportion = \[16\] Third proportion = \[128\] \[\begin{array}{*{35}{l}} \surd xy\text{ }=\text{ }16 \\ xy\text{ }=\text{ }256 \\ \end{array}\] Here...

If a, b, c, d, e are in continued proportion, prove that:
$\mathbf{a}:\text{ }\mathbf{e}\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{4}}}:\text{ }{{\mathbf{b}}^{\mathbf{4}}}$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d, e are in continued proportion We can write it as a/b = b/c = c/d = d/e = k \[d\text{ }=\text{ }ek,\text{ }c\text{ }=\text{ }e{{k}^{2}},\text{ }b\text{ }=\text{...

If
$\mathbf{2},\text{ }\mathbf{6},\text{ }\mathbf{p},\text{ }\mathbf{54}$
and q are in continued proportion, find the values of p and q.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{2},\text{ }\mathbf{6},\text{ }\mathbf{p},\text{ }\mathbf{54}\] and q are in continued proportion We can write it as \[2/6\text{ }=\text{ }6/p\text{ }=\text{ }p/54\text{...

If
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right)$
are in continued proportion, prove that b is the mean proportional between a and c.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left(...

What number must be added to each of the numbers
$\mathbf{15},\text{ }\mathbf{17},\text{ }\mathbf{34}\text{ }\mathbf{and}\text{ }\mathbf{38}$
to make them proportional?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be added to each number So the numbers will be \[15\text{ }+\text{ }x,\text{ }17\text{ }+\text{ }x,\text{ }34\text{ }+\text{ }x\text{ }and\text{ }38\text{ }+\text{ }x\] Based on the...

In an examination, the number of those who passed and the number of those who failed were in the ratio of
$3:1$
. Had
$8$
more appeared, and
$6$
less passed, the ratio of passed to failures would have been
$2:1$
. Find the number of candidates who appeared.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider the number of passed = \[3x\] Number of failed = x So the total candidates appeared = \[3x\text{ }+\text{ }x\text{ }=\text{ }4x\] In the second case Number of candidates appeared =...

The ratio of the pocket money saved by Lokesh and his sister is
$5:6$
. If the sister saves Rs
$30$
more, how much more the brother should save in order to keep the ratio of their savings unchanged?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider \[5x\] and \[6x\] as the savings of Lokesh and his sister. Lokesh should save Rs y more Based on the problem \[\left( 5x\text{ }+\text{ }y \right)/\text{ }\left( 6x\text{ }+\text{ }30...

The ratio of the shorter sides of a right angled triangle is
$5:12$
. If the perimeter of the triangle is
$360$
cm, find the length of the longest side.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider the two shorter sides of a right-angled triangle as \[5x\] and \[12x\] So the third longest side = \[13x\] It is given that \[5x\text{ }+\text{ }12x\text{ }+\text{ }13x\text{ }=\text{...

If a: b =
$3:5$
, find
$\left( \mathbf{3a}\text{ }+\text{ }\mathbf{5b} \right):\text{ }\left( \mathbf{7a}\text{ }\text{ }\mathbf{2b} \right)$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a: b = \[3:5\] We can write it as a/b = \[3/5\] Here \[\left( 3a\text{ }+\text{ }5b \right):\text{ }\left( 7a\text{ }\text{ }2b \right)\] Now dividing the terms by b Here \[\left(...

If
$\left( \mathbf{7p}\text{ }+\text{ }\mathbf{3q} \right):\text{ }\left( \mathbf{3p}\text{ }\text{ }\mathbf{2q} \right)\text{ }=\text{ }\mathbf{43}:\text{ }\mathbf{2}$
, find p: q.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\left( 7p\text{ }+\text{ }3q \right):\text{ }\left( 3p\text{ }\text{ }2q \right)\text{ }=\text{ }43:\text{ }2\] We can write it as \[\left( 7p\text{ }+\text{ }3q \right)/\text{...

Find the compound ratio of
${{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{2}}}:\text{ }{{\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right)}^{\mathbf{2}}},\text{ }({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{a}}^{\mathbf{4}}}~\text{ }{{\mathbf{b}}^{\mathbf{4}}}):\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{4}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} {{\left( a\text{ }+\text{ }b \right)}^{2}}:\text{ }{{\left( a\text{ }\text{ }b \right)}^{2}} \\ ({{a}^{2}}~\text{ }{{b}^{2}}):\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}) \\...

If
$\frac{x+y}{ax+by}=\frac{y+z}{ay+bz}=\frac{z+x}{az+bx}$
, prove that each of these ratio is equal to
$\frac{2}{a+b}$
unless x+y+z=0

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that If \[x\text{ }+\text{ }y\text{ }+\text{ }z\text{ }\ne \text{ }0\] Therefore, it is proved.

Using the properties of proportion, solve the following equation for x; given
$\frac{{{x}^{3}}+3x}{3{{x}^{2}}+1}=\frac{341}{91}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[\begin{array}{*{35}{l}} 6x\text{ }\text{ }6\text{ }=\text{ }5x\text{ }+\text{ }5 \\ 6x\text{ }\text{ }5x\text{ }=\text{ }5\text{ }+\text{ }6 \\ ...

Given
$\frac{{{x}^{3}}+12x}{6{{x}^{2}}+8}=\frac{{{y}^{3}}+27y}{9{{y}^{2}}+27}$
Using componendo and dividendo find x y.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By further calculation \[\begin{array}{*{35}{l}} 2x/4\text{ }=\text{ }2y/3 \\ x/2\text{ }=\text{ }y/3 \\ \end{array}\] By cross multiplication \[x/y\text{ }=\text{ }2/3\] Hence,...

Given that
$\frac{{{a}^{3}}+3a{{b}^{2}}}{{{b}^{3}}+3{{a}^{2}}b}=\frac{63}{62}$
. Using componendo and dividendo find a : b.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[a\text{ }+\text{ }b\text{ }=\text{ }5a\text{ }\text{ }5b\] We can write it as \[\begin{array}{*{35}{l}} 5a\text{ }\text{ }a\text{ }\text{ }5b\text{ }\text{...

Given
$x=\frac{\sqrt{{{a}^{2}}+{{b}^{2}}}+\sqrt{{{a}^{2}}-{{b}^{2}}}}{\sqrt{{{a}^{2}}+{{b}^{2}}}-\sqrt{{{a}^{2}}-{{b}^{2}}}}$
use componendo and dividend to prove that
${{b}^{2}}=\frac{2{{a}^{2}}x}{{{x}^{2}}+1}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

If
$x=\frac{\sqrt{a+x}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}}$
, using properties of proportion, show that
${{x}^{2}}-2ax+1=0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that We get \[\begin{array}{*{35}{l}} 2ax\text{ }=\text{ }{{x}^{2}}~+\text{ }1 \\ {{x}^{2}}~\text{ }2ax\text{ }+\text{ }1\text{ }=\text{ }0 \\ \end{array}\] Therefore, it is...

Solve for x:
$16{{(\frac{a-x}{a+x})}^{3}}=\frac{a+x}{a-x}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

So we get \[\begin{array}{*{35}{l}} 3x\text{ }=\text{ }a \\ x\text{ }=\text{ }a/3 \\ \end{array}\] So we get x = \[3a\] Therefore, x= \[a/3,3a\].

Solve
$\frac{1+x+{{x}^{2}}}{1-x+{{x}^{2}}}=\frac{62(1+x)}{63(1-x)}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

x = \[1/5\]

Using properties of proportion solve for x. Given that x is positive. (i)
$\frac{3x+\sqrt{9{{x}^{2}}-5}}{3x-\sqrt{9{{x}^{2}}-5}}=5$
(ii)
$\frac{2x+\sqrt{4{{x}^{2}}-1}}{2x-\sqrt{4{{x}^{2}}-1}}=4$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 81{{x}^{2}}~\text{ }45\text{ }=\text{ }36{{x}^{2}} \\ 81{{x}^{2}}~\text{ }36{{x}^{2}}~=\text{ }45 \\ \end{array}\] So we get...

Using properties of properties, find x from the following equations: (v)
$\frac{3x+\sqrt{9{{x}^{2}}+5}}{3x-\sqrt{9{{x}^{2}}+5}}=5$
(vi)
$\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=\frac{c}{d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 81{{x}^{2}}~\text{ }45\text{ }=\text{ }36{{x}^{2}} \\ 81{{x}^{2}}~\text{ }36{{x}^{2}}~=\text{ }45 \\ \end{array}\] So we get 45x2 = 45...

Using properties of properties, find x from the following equations: (iii)
$\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{a}{b}$
(iv)
$\frac{\sqrt{12x+1}+\sqrt{2x-3}}{\sqrt{12x+1}-\sqrt{2x-3}}=\frac{3}{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 50x\text{ }\text{ }75\text{ }=\text{ }12x\text{ }+\text{ }1 \\ 50x\text{ }\text{ }12x\text{ }=\text{ }1\text{ }+\text{ }75 \\ \end{array}\] So we...

Using properties of properties, find x from the following equations: (i)
$\frac{\sqrt{2-x}+\sqrt{2+x}}{\sqrt{2-x}-\sqrt{2+x}}=3$
(ii)
$\frac{\sqrt{x+4}+\sqrt{x-10}}{\sqrt{x+4}-\sqrt{x-10}}=\frac{5}{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[8\text{ }+\text{ }4x\text{ }=\text{ }2\text{ }\text{ }x\] So we get \[\begin{array}{*{35}{l}} 4x\text{ }+\text{ }x\text{ }=\text{ }2\text{ }\text{ }8 \\ 5x\text{ }=\text{...

If
$x=\frac{4\sqrt{6}}{\sqrt{2}+\sqrt{3}}$
find the value of
$\frac{x+2\sqrt{2}}{x-2\sqrt{2}}+\frac{x+2\sqrt{3}}{x-2\sqrt{3}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

If
$x=\frac{8ab}{a+b}$
find the value of
$\frac{x+4a}{x-4a}+\frac{x+4b}{x-4b}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} =\text{ }2\left( a\text{ }\text{ }b \right)/\text{ }\left( a\text{ }\text{ }b \right) \\ =\text{ }2 \\ \end{array}\]

If
$x=\frac{2ab}{a+b}$
find the value of
$\frac{x+a}{x-a}+\frac{x+b}{x-b}$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} =\text{ }2\left( a\text{ }\text{ }b \right)/\text{ }\left( a\text{ }\text{ }b \right) \\ =\text{ }2 \\ \end{array}\]

If
$(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})$
, prove that a: b :: c: d.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{...

If (ma + nb): b :: (mc + nd): d, prove that a, b, c, d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that (ma + nb): b :: (mc + nd): d We can write it as (ma + nb)/ b = (mc + nd)/ d By cross multiplication mad + nbd = mbc + nbd Here mad = mbc ad = bc By further calculation a/b = c/d...

If (pa + qb): (pc + qd) :: (pa – qb): (pc – qd) prove that a: b :: c: d.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that (pa + qb): (pc + qd) :: (pa – qb): (pc – qd) We can write it as Therefore, it is proved that a: b :: c: d.

If
$~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d} \right)\text{ }\left( \mathbf{4c}\text{ }+\text{ }\mathbf{5d} \right)$
, prove that a, b, c, d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d}...

(i) If
$\frac{5x+7y}{5u+7v}=\frac{5x-7y}{5u-7v}$
, show that
$\frac{x}{y}=\frac{u}{v}$
(ii)
$\frac{8a-5b}{8c-5d}=\frac{8a+5b}{8c+5d}$
, prove that
$\frac{a}{b}=\frac{c}{d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Therefore, it is proved. Therefore, it is proved.

If a: b :: c: d, prove that (iii)
$\left( \mathbf{2a}\text{ }+\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }+\text{ }\mathbf{3d} \right)$
(iv) (la + mb): (lc + mb) :: (la – mb): (lc – mb)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) We know that If a: b :: c: d we get a/b = c/d By multiplying \[2/3\] \[2a/3b\text{ }=\text{ }2c/3d\] By applying componendo and dividendo \[\left( 2a\text{ }+\text{ }3b \right)/\text{ }\left(...

If a: b :: c: d, prove that (i)
$\frac{2a+5b}{2a-5b}=\frac{2c+5d}{2c-5d}$
(ii)
$\frac{5a+11b}{5c+11d}=\frac{5a-11b}{5c-11d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) We know that If a: b :: c: d we get a/b = c/d By multiplying \[2/5\] \[2a/5b\text{ }=\text{ }2c/5d\] By applying componendo and dividendo \[\left( 2a\text{ }+\text{ }5b \right)/\text{ }\left(...

If a, b, c, d are in continued proportion, prove that: (V)
${{\left( \frac{a-b}{c}+\frac{a-c}{b} \right)}^{2}}-{{\left( \frac{d-b}{c}+\frac{d-c}{b} \right)}^{2}}={{(a-d)}^{2}}\left( \frac{1}{{{c}^{2}}}-\frac{1}{{{b}^{2}}} \right)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c, d are in continued proportion, prove that: (iii)
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{d} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{d} \right)\text{ }=\text{ }{{\left( \mathbf{b}\text{ }\text{ }\mathbf{c} \right)}^{\mathbf{2}}}$
(iv) a: d = triplicate ratio of (a – b): (b – c)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c, d are in continued proportion, prove that: (i)
$\frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{{{b}^{3}}+{{c}^{3}}+{{d}^{3}}}=\frac{a}{d}$
(ii)
$({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }({{\mathbf{c}}^{\mathbf{2}}}~\text{ }{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }{{({{\mathbf{b}}^{\mathbf{2}}}~\text{ }{{\mathbf{c}}^{\mathbf{2}}})}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c are in continued proportion, prove that: (v)
$\mathbf{abc}\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{bc}\text{ }+\text{ }\mathbf{ca} \right)}^{\mathbf{3}}}$
(vi)
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k (v) LHS = \[abc\text{ }{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)}^{3}}\] We can write it as \[=\text{...

If a, b, c are in continued proportion, prove that: (iii)
$\mathbf{a}:\text{ }\mathbf{c}\text{ }=\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})$
(iv)
$~{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}~({{\mathbf{a}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{-\mathbf{4}}})\text{ }=\text{ }{{\mathbf{b}}^{-\mathbf{2}}}~({{\mathbf{a}}^{\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{\mathbf{4}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k (iii) \[~a:\text{ }c\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}):\text{ }({{b}^{2}}~+\text{ }{{c}^{2}})\] We...

If a, b, c are in continued proportion, prove that: (i)
$\frac{a+b}{b+c}=\frac{{{a}^{2}}(b-c)}{{{b}^{2}}(a-b)}$
(ii)
$\frac{1}{{{a}^{3}}}+\frac{1}{{{b}^{3}}}+\frac{1}{{{c}^{3}}}=\frac{a}{{{b}^{2}}{{c}^{2}}}+\frac{b}{{{c}^{2}}{{a}^{2}}}+\frac{c}{{{a}^{2}}{{b}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k Therefore, LHS = RHS. Therefore, LHS = RHS.

If a, b, c are in continued proportion, prove that:
$\frac{p{{a}^{2}}+qab+r{{b}^{2}}}{p{{b}^{2}}+qbc+r{{c}^{2}}}=\frac{a}{c}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion \[\frac{p{{a}^{2}}+qab+r{{b}^{2}}}{p{{b}^{2}}+qbc+r{{c}^{2}}}=\frac{a}{c}\] Consider a/b = b/c = k So we get a = bk and b = ck ….. (1) From...

If x, y, z are in continued proportion, prove that:
${{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y} \right)}^{\mathbf{2}}}/\text{ }{{\left( \mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{2}}}~=\text{ }\mathbf{x}/\mathbf{z}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x, y, z are in continued proportion Consider x/y = y/z = k So we get y = kz \[x\text{ }=\text{ }yk\text{ }=\text{ }kz\text{ }\times \text{ }k\text{ }=\text{ }{{k}^{2}}z\] Therefore,...

If a, b, c and d are in proportion, prove that: (vii)
$\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}=\frac{ab+ad-bc}{bc+cd-ad}$
(viii)
$abcd\left[ \frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{{{d}^{2}}} \right]={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk Therefore, LHS = RHS. So we get = d2 (1 + k2) + b2 (1 + k2) = (1 + k2) (b2 + d2) RHS = a2 + b2 + c2 + d2 We can...

If a, b, c and d are in proportion, prove that: (v)
$\frac{{{(a+c)}^{3}}}{{{(b+d)}^{3}}}=\frac{a{{(a-c)}^{2}}}{b{{(b-d)}^{2}}}$
(vi)
$\frac{{{a}^{2}}+ab+{{b}^{2}}}{{{a}^{2}}-ab+{{b}^{2}}}=\frac{{{c}^{2}}+cd+{{d}^{2}}}{{{c}^{2}}-cd+{{d}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk Therefore, LHS = RHS. Therefore, LHS = RHS.

If a, b, c and d are in proportion, prove that: (iii)
$({{\mathbf{a}}^{\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{\mathbf{4}}}):\text{ }({{\mathbf{b}}^{\mathbf{4}}}~+\text{ }{{\mathbf{d}}^{\mathbf{4}}})\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}:\text{ }{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{d}}^{\mathbf{2}}}$
(iv)
$\frac{{{a}^{2}}+ab}{{{c}^{2}}+cd}=\frac{{{b}^{2}}-2ab}{{{d}^{2}}-2cd}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk (iii) \[({{a}^{4}}~+\text{ }{{c}^{4}}):\text{ }({{b}^{4}}~+\text{ }{{d}^{4}})\text{ }=\text{...

If a, b, c and d are in proportion, prove that: (i)
$\left( \mathbf{5a}\text{ }+\text{ }\mathbf{7v} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{5c}\text{ }+\text{ }\mathbf{7d} \right)\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)$
(ii) (ma + nb): b = (mc + nd): d

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk (i) LHS = \[\left( 5a\text{ }+\text{ }7b \right)\text{ }\left( 2c\text{ }\text{ }3d \right)\] Substituting the...

18. If ax = by = cz; prove that
$\frac{{{x}^{2}}}{yz}+\frac{{{y}^{2}}}{zx}+\frac{{{z}^{2}}}{xy}=\frac{bc}{{{a}^{2}}}+\frac{ca}{{{b}^{2}}}+\frac{ab}{{{c}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider ax = by = cz = k It can be written as x = k/a, y = k/b, z = k/c

If a/b = c/d = e/f prove that:
$\frac{{{a}^{2}}}{{{b}^{2}}}+\frac{{{c}^{2}}}{{{d}^{2}}}+\frac{{{e}^{2}}}{{{f}^{2}}}=\frac{ac}{bd}+\frac{ce}{df}+\frac{ae}{df}$
(iv)
$bdf{{\left[ \frac{a+b}{b}+\frac{c+d}{d}+\frac{c+f}{f} \right]}^{3}}=27(a+b)(c+d)(e+f)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk Therefore, LHS = RHS. So we get \[=\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1\text{ }+\text{ }k\text{ }+\text{ }1\text{ }+\text{...

If a/b = c/d = e/f prove that: (i)
$({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{d}}^{\mathbf{2}}}~+\text{ }{{\mathbf{f}}^{\mathbf{2}}})\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }{{\mathbf{e}}^{\mathbf{2}}})\text{ }=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{cd}\text{ }+\text{ }\mathbf{ef} \right)}^{\mathbf{2}}}$
(ii)
$\frac{{{({{a}^{3}}+{{c}^{3}})}^{2}}}{{{({{b}^{3}}+{{d}^{3}})}^{2}}}=\frac{{{e}^{6}}}{{{f}^{6}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk (i) LHS = \[({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{e}^{2}})\] We can write it...

16. If x/a = y/b = z/c, prove that (iii)
$\frac{ax-by}{(a+b)(x-y)}+\frac{by-cz}{(b+c)(y-z)}+\frac{cz-ax}{(c+a)(z-x)}=3$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Therefore, LHS = RHS.

16. If x/a = y/b = z/c, prove that (i)
$\frac{{{x}^{3}}}{{{a}^{2}}}+\frac{{{y}^{3}}}{{{b}^{2}}}+\frac{{{z}^{3}}}{{{c}^{2}}}=\frac{{{(x+y+z)}^{3}}}{{{(a+b+c)}^{2}}}$
(ii)
${{\left[ \frac{{{a}^{2}}{{x}^{2}}+{{b}^{2}}{{y}^{2}}+{{c}^{2}}{{z}^{2}}}{{{a}^{3}}x+{{b}^{3}}y+{{c}^{3}}z} \right]}^{3}}=\frac{xyz}{abc}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x/a = y/b = z/c We can write it as x = ak, y = bk and z = ck Therefore, LHS = RHS. Therefore, LHS = RHS.

If a + c = mb and
$\mathbf{1}/\mathbf{b}\text{ }+\text{ }\mathbf{1}/\mathbf{d}\text{ }=\text{ }\mathbf{m}/\mathbf{c}$
, prove that a, b, c and d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a + c = mb and \[\mathbf{1}/\mathbf{b}\text{ }+\text{ }\mathbf{1}/\mathbf{d}\text{ }=\text{ }\mathbf{m}/\mathbf{c}\] a + c = mb Dividing the equation by b a/b + c/d = m ……. (1)...

If y is mean proportional between x and z, prove that
$\mathbf{xyz}\text{ }{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{xy}\text{ }+\text{ }\mathbf{yz}\text{ }+\text{ }\mathbf{zx} \right)}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that y is mean proportional between x and z We can write it as \[{{y}^{2}}~=\text{ }xz\]…… (1) Consider LHS = \[xyz\text{ }{{\left( x\text{ }+\text{ }y\text{ }+\text{ }z \right)}^{3}}\]...

If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between
$({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that b is the mean proportional between a and c \[{{b}^{2}}~=\text{ }ac\]…. (1) Here (ab + bc) is the mean proportional between \[({{\mathbf{a}}^{\mathbf{2}}}~+\text{...

If b is the mean proportional between a and c, prove that a, c,
${{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~\mathbf{and}\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~$
are proportional.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Solution: It is given that b is the mean proportional between a and c We can write it as b2 = a × c b2 = ac ….. (1) We know that a, c, \[{{\mathbf{a}}^{\mathbf{2}}}~+\text{...

Find two numbers such that the mean proportional between them is
$28$
and the third proportional to them is
$224$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a and b as the two numbers It is given that \[28\] is the mean proportional \[a:\text{ }28\text{ }::\text{ }28:\text{ }b\] We get \[ab\text{ }=\text{ }{{28}^{2}}~=\text{ }784\] Here...

What number must be added to each of the numbers
$\mathbf{16},\text{ }\mathbf{26}\text{ }\mathbf{and}\text{ }\mathbf{40}$
so that the resulting numbers may be in continued proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be added to each number \[16\text{ }+\text{ }x\text{ },\text{ }26\text{ }+\text{ }x\text{ }and\text{ }40\text{ }+\text{ }x\] are in continued proportion It can be written as \[\left(...

If
$\mathbf{x}\text{ }+\text{ }\mathbf{5}$
is the mean proportion between
$\mathbf{x}\text{ }+\text{ }\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}$
, find the value of x.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{x}\text{ }+\text{ }\mathbf{5}\] is the mean proportion between \[\mathbf{x}\text{ }+\text{ }\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}\]...

If
$\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{ }\text{ }\mathbf{3}$
are in proportion, find k.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{...

What number should be subtracted from each of the numbers
$\mathbf{23},\text{ }\mathbf{30},\text{ }\mathbf{57}\text{ }\mathbf{and}\text{ }\mathbf{78}$
so that the remainders are in proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be subtracted from each term \[23\text{ }\text{ }x,\text{ }30\text{ }\text{ }x,\text{ }57\text{ }\text{ }x\text{ }and\text{ }78\text{ }\text{ }x\] are proportional It can be written as...

What number must be added to each of the numbers
$\mathbf{5},\text{ }\mathbf{11},\text{ }\mathbf{19}\text{ }\mathbf{and}\text{ }\mathbf{37}$
so that they are in proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x to be added to \[\mathbf{5},\text{ }\mathbf{11},\text{ }\mathbf{19}\text{ }\mathbf{and}\text{ }\mathbf{37}\] to make them in proportion \[5\text{ }+\text{ }x:\text{ }11\text{ }+\text{...

If a,
$12,16$
and b are in continued proportion find a and b.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, \[12,16\] and b are in continued proportion \[a/12\text{ }=\text{ }12/16\text{ }=\text{ }16/b\] We know that \[a/12\text{ }=\text{ }12/16\] By cross multiplication \[a/12\text{...

Find the mean proportion of: (iii)
$\mathbf{8}.\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{2}.\mathbf{5}$
(iv)
$\left( a\text{ }\text{ }b \right)\text{ }and\text{ }({{a}^{3}}~\text{ }{{a}^{2}}b),\text{ }a>b$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) Consider x as the mean proportion of \[8.1\text{ }and\text{ }2.5\] \[8.1:\text{ }x\text{ }::\text{ }x:\text{ }2.5\] It can be written as \[\begin{array}{*{35}{l}} {{x}^{2}}~=\text{ }8.1\text{...

Find the mean proportion of: (i)
$\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{80}$
(ii)
$\mathbf{1}/\mathbf{12}\text{ }\mathbf{and}\text{ }\mathbf{1}/\mathbf{75}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider x as the mean proportion of 5 and 80 \[5:\text{ }x\text{ }::\text{ }x:\text{ }80\] It can be written as \[\begin{array}{*{35}{l}} {{x}^{2}}~=\text{ }5\text{ }\times \text{ }80\text{...

Find the third proportional to (iii)
$\mathbf{Rs}.\text{ }\mathbf{3},\text{ }\mathbf{Rs}.\text{ }\mathbf{12}$
(iv)
$\mathbf{5}\text{ }{\scriptscriptstyle 1\!/\!{ }_4}\text{ }\mathbf{and}\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) Consider x as the third proportional to \[~Rs.\text{ }3\text{ }and\text{ }Rs.\text{ }12\] \[3:\text{ }12\text{ }::\text{ }12:\text{ }x\] It can be written as \[\begin{array}{*{35}{l}} 3\text{...

Find the third proportional to (i)
$5,10$
(ii)
$\mathbf{0}.\mathbf{24},\text{ }\mathbf{0}.\mathbf{6}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider x as the third proportional to \[5,10\] \[5:\text{ }10\text{ }::\text{ }10:\text{ }x\] It can be written as \[\begin{array}{*{35}{l}} 5\text{ }\times \text{ }x\text{ }=\text{ }10\text{...

Find the fourth proportional to (iii)
$\mathbf{1}.\mathbf{5},\text{ }\mathbf{2}.\mathbf{5},\text{ }\mathbf{4}.\mathbf{5}$
(iv)
$\mathbf{9}.\mathbf{6}\text{ }\mathbf{kg},\text{ }\mathbf{7}.\mathbf{2}\text{ }\mathbf{kg},\text{ }\mathbf{28}.\mathbf{8}\text{ }\mathbf{kg}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[1.5,\text{ }2.5,\text{ }4.5\] Consider x as the fourth proportional to \[1.5,\text{ }2.5,\text{ }4.5\] \[1.5:\text{ }2.5\text{ }::\text{ }4.5:\text{ }x\] We can write it as \[1.5\text{...

Find the fourth proportional to (i)
$\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}$
(ii)
$\mathbf{1}/\mathbf{3},\text{ }\mathbf{1}/\mathbf{4},\text{ }\mathbf{1}/\mathbf{5}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}\] Consider x as the fourth proportional to \[\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}\] \[3:\text{ }12\text{ }::\text{ }15:\text{...

Find the value of x in the following proportions: (iii)
$\mathbf{2}.\mathbf{5}:\text{ }\mathbf{1}.\mathbf{5}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{3}$
(iv)
$\mathbf{x}:\text{ }\mathbf{50}\text{ }::\text{ }\mathbf{3}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii)\[~2.5:\text{ }1.5\text{ }=\text{ }x:\text{ }3\] We can write it as \[1.5\text{ }\times \text{ }x\text{ }=\text{ }2.5\text{ }\times \text{ }3\] So we get \[\begin{array}{*{35}{l}} x\text{...

Find the value of x in the following proportions: (i)
$\mathbf{10}:\text{ }\mathbf{35}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{42}$
(ii)
$\mathbf{3}:\text{ }\mathbf{x}\text{ }=\text{ }\mathbf{24}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i)\[\mathbf{10}:\text{ }\mathbf{35}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{42}\] We can write it as \[35\text{ }\times \text{ }x\text{ }=\text{ }10\text{ }\times \text{ }42\] So we get...

In an examination, the ratio of passes to failures was
$4:1$
. If
$30$
less had appeared and
$20$
less passed, the ratio of passes to failures would have been
$5:1$
. How many students appeared for the examination.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider number of passes = \[4x\] Number of failures = x Total number of students appeared = \[4x\text{ }+\text{ }x\text{ }=\text{ }5x\] In case \[2\] Number of students appeared = \[5x\text{...

(i) The monthly pocket money of Ravi and Sanjeev are in the ratio
$5:7$
. Their expenditures are in the ratio
$3:5$
. If each saves Rs
$80$
per month, find their monthly pocket money. (ii) In class X of a school, the ratio of the number of boys to that of the girls is
$4:3$
. If there were
$20$
more boys and
$12$
less girls, then the ratio would have been
$2:1$
. How many students were there in the class?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider the monthly pocket money of Ravi and Sanjeev as \[5x\] and \[7x\] Their expenditure is \[3y\] and \[5y\] respectively. \[5x\text{ }\text{ }3y\text{ }=\text{ }80\] …… (1) \[7x\text{...

(i) In a mixture of
$45$
litres, the ratio of milk to water is
$13:2$
. How much water must be added to this mixture to make the ratio of milk to water as
$3:1$
? (ii) The ratio of the number of boys to the numbers of girls in a school of
$560$
pupils is
$5:3$
. If
$10$
new boys are admitted, find how many new girls may be admitted so that the ratio of the number of boys to the number of girls may change to
$3:2$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Mixture of milk to water = \[45\] litres Ratio of milk to water = \[13:2\] Sum of ratio = \[13\text{ }+\text{ }2\text{ }=\text{ }15\] Here the quantity of milk = \[\left(...

(i) A certain sum was divided among A, B and C in the ratio
$7:5:4$
. If B got Rs
$500$
more than C, find the total sum divided. (ii) In a business, A invests Rs
$50000$
for
$6$
months, B Rs
$60000$
for
$4$
months and C Rs
$80000$
for
$5$
months. If they together earn Rs
$18800$
find the share of each.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Ratio between A, B and C = \[7:\text{ }5:\text{ }4\] Consider A share = \[7x\] B share = \[5x\] C share = \[4x\] So the total sum =\[~7x\text{ }+\text{ }5x\text{ }+\text{...

Three numbers are in the ratio
$\mathbf{1}/\mathbf{2}:\text{ }\mathbf{1}/\mathbf{3}:\text{ }{\scriptscriptstyle 1\!/\!{ }_4}$
. If the sum of their squares is
$244$
, find the numbers.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that Ratio of three numbers \[=\text{ }1/2:\text{ }1/3:\text{ }1/4\] \[\begin{array}{*{35}{l}} =\text{ }\left( 6:\text{ }4:\text{ }3 \right)/\text{ }12 \\ =\text{ }6:\text{ }4:\text{...

(i) The sides of a triangle are in the ratio
$7:5:3$
and its perimeter is
$30$
cm. Find the lengths of sides. (ii) If the angles of a triangle are in the ratio
$2:3:4$
, find the angles.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Perimeter of triangle = \[30\] cm Ratio among sides = \[7:5:3\] Here the sum of ratios = \[7\text{ }+\text{ }5\text{ }+\text{ }3\text{ }=\text{ }15\] We know that Length of...

(i) A woman reduces her weight in the ratio
$7:5$
. What does her weight become if originally it was
$91$
kg. (ii) A school collected Rs 2100 for charity. It was decided to divide the money between an orphanage and a blind school in the ratio of 3: 4. How much money did each receive?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Ratio of original and reduced weight of woman = \[7:5\] Consider original weight = \[7x\] Reduced weight = \[5x\] Here original weight = \[91\] kg So the reduced weight = \[~\left( 91\text{...

(i) Find two numbers in the ratio of
$\mathbf{8}:\text{ }\mathbf{7}$
such that when each is decreased by
$\mathbf{12}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$
, they are in the ratio
$\mathbf{11}:\text{ }\mathbf{9}$
. (ii) The income of a man is increased in the ratio of
$\mathbf{10}:\text{ }\mathbf{11}$
. If the increase in his income is Rs
$\mathbf{600}$
per month, find his new income.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Ratio = \[\mathbf{8}:\text{ }\mathbf{7}\] Consider the numbers as \[8x\] and \[7x\] Using the condition \[\left[ 8x\text{ }\text{ }25/2 \right]/\text{ }\left[ 7x\text{ }\text{ }25/2...

(iii) If
$\left( \mathbf{x}\text{ }+\text{ }\mathbf{2y} \right):\text{ }\left( \mathbf{2x}\text{ }\text{ }\mathbf{y} \right)$
is equal to the duplicate ratio of
$\mathbf{3}:\text{ }\mathbf{2}$
, find x: y.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\left( x\text{ }+\text{ }2y \right)/\text{ }\left( 2x\text{ }\text{ }y \right)\text{ }=\text{ }{{3}^{2}}/\text{ }{{2}^{2}}\] So we get \[\left( x\text{ }+\text{ }2y \right)/\text{ }\left(...

(i) If
$\left( \mathbf{x}\text{ }\text{ }\mathbf{9} \right):\text{ }\left( \mathbf{3x}\text{ }+\text{ }\mathbf{6} \right)$
is the duplicate ratio of
$\mathbf{4}:\text{ }\mathbf{9}$
, find the value of x. (ii) If
$\left( \mathbf{3x}\text{ }+\text{ }\mathbf{1} \right):\text{ }\left( \mathbf{5x}\text{ }+\text{ }\mathbf{3} \right)$
is the triplicate ratio of
$\mathbf{3}:\text{ }\mathbf{4}$
, find the value of x.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{ }+\text{ }6 \right)\text{ }=\text{ }{{\left( 4/9 \right)}^{2}}\] So we get \[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{...

(i) If
$(\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{xy}):\text{ }(\mathbf{3xy}\text{ }\text{ }{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{12}:\text{ }\mathbf{5}$
, find
$\left( \mathbf{x}\text{ }+\text{ }\mathbf{2y} \right):\text{ }\left( \mathbf{2x}\text{ }+\text{ }\mathbf{y} \right)$
(ii) If
$\mathbf{y}\text{ }\left( \mathbf{3x}\text{ }\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\text{ }\left( \mathbf{4x}\text{ }+\text{ }\mathbf{y} \right)\text{ }=\text{ }\mathbf{5}:\text{ }\mathbf{12}$
. Find
$({{\mathbf{x}}^{\mathbf{2}}}~+\text{ }{{\mathbf{y}}^{\mathbf{2}}}):\text{ }{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y} \right)}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[(\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{xy}):\text{ }(\mathbf{3xy}\text{ }\text{ }{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{12}:\text{ }\mathbf{5}\] We can write it...

(i) If
$\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}$
,find x: y. (ii) If
$\mathbf{a}:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{3}:\text{ }\mathbf{11}$
, find
$\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}\] By cross multiplication \[9x\text{ }+\text{ }15y\text{ }=\text{...

(i) If
$\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{1}/\mathbf{4}:\text{ }\mathbf{1}/\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{1}/\mathbf{7}:\text{ }\mathbf{1}/\mathbf{6}$
, find A: B: C. (ii) If
$\mathbf{3A}\text{ }=\text{ }\mathbf{4B}\text{ }=\text{ }\mathbf{6C}$
, find A: B: C

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) We know that \[\begin{array}{*{35}{l}} A:\text{ }B\text{ }=\text{ }1/4\text{ }\times \text{ }5/1\text{ }=\text{ }5/4 \\ B:\text{ }C\text{ }=\text{ }1/7\text{ }\times \text{ }6/1\text{ }=\text{...

(i) If
$\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{C}:\text{ }\mathbf{D}\text{ }=\text{ }\mathbf{6}:\text{ }\mathbf{7}$
, find A: D. (ii) If
$\mathbf{x}:\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{y}:\text{ }\mathbf{z}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{7}$
, find x: y: z.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that \[\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{5}\text{...

Arrange the following ratios in ascending order of magnitude:
$\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{17}:\text{ }\mathbf{21},\text{ }\mathbf{11}:\text{ }\mathbf{14}\text{ }\mathbf{and}\text{ }\mathbf{5}:\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{17}:\text{ }\mathbf{21},\text{ }\mathbf{11}:\text{ }\mathbf{14}\text{ }\mathbf{and}\text{ }\mathbf{5}:\text{ }\mathbf{7}\] We can...

Find the reciprocal ratio of (iii)
$\mathbf{1}/\mathbf{9}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\mathbf{1}/\mathbf{9}:\text{ }\mathbf{2}\] We know that Reciprocal ratio of \[1/9:\text{ }2\text{ }=\text{ }2:\text{ }1/9\text{ }=\text{ }18:\text{ }1\]

Find the reciprocal ratio of (i)
$\mathbf{4}:\text{ }\mathbf{7}$
(ii)
${{\mathbf{3}}^{\mathbf{2}}}:\text{ }{{\mathbf{4}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{4}:\text{ }\mathbf{7}\] We know that Reciprocal ratio of \[4:\text{ }7\text{ }=\text{ }7:\text{ }4\] (ii) \[{{\mathbf{3}}^{\mathbf{2}}}:\text{ }{{\mathbf{4}}^{\mathbf{2}}}\] We know...

Find the sub-triplicate ratio of (iii)
$\mathbf{27}{{\mathbf{a}}^{\mathbf{3}}}:\text{ }\mathbf{64}{{\mathbf{b}}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[27{{a}^{3}}:\text{ }64{{b}^{3}}\] We know that Sub-triplicate ratio of \[27{{a}^{3}}:\text{ }64{{b}^{3}}~=\text{ }{{[{{\left( 3a \right)}^{3}}]}^{1/3}}:\text{ }{{[{{\left( 4b...

Find the sub-triplicate ratio of (i)
$\mathbf{1}:\text{ }\mathbf{216}$
(ii)
$\mathbf{1}/\mathbf{8}:\text{ }\mathbf{1}/\mathbf{125}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{1}:\text{ }\mathbf{216}\] We know that Sub-triplicate ratio of \[1:\text{ }216\text{ }=\sqrt[3]{1}:\sqrt[3]{216}\] By further calculation \[\begin{array}{*{35}{l}} =\text{...

Find the sub-duplicate ratio of (iii)
$\mathbf{9}{{\mathbf{a}}^{\mathbf{2}}}:\text{ }\mathbf{49}{{\mathbf{b}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[9{{a}^{2}}:\text{ }49{{b}^{2}}\] We know that Sub-duplicate ratio of \[9{{a}^{2}}:\text{ }49{{b}^{2}}~=\text{ }\surd 9{{a}^{2}}:\text{ }\surd 49{{b}^{2}}~=\text{ }3a:\text{ }7b\]

Find the sub-duplicate ratio of (i)
$\mathbf{9}:\text{ }\mathbf{16}$
(ii)
${\scriptscriptstyle 1\!/\!{ }_4}:\text{ }\mathbf{1}/\mathbf{9}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{9}:\text{ }\mathbf{16}\] We know that Sub-duplicate ratio of \[9:\text{ }16\text{ }=\text{ }\surd 9:\text{ }\surd 16\text{ }=\text{ }3:\text{ }4\] (ii) \[{\scriptscriptstyle 1\!/\!{...

Find the triplicate ratio of (iii)
${{\mathbf{1}}^{\mathbf{3}}}:\text{ }{{\mathbf{2}}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii)\[~{{1}^{3}}:\text{ }{{2}^{3}}\] We know that Triplicate ratio of \[{{1}^{3}}:\text{ }{{2}^{3}}~=\text{ }{{({{1}^{3}})}^{3}}:\text{ }{{({{2}^{3}})}^{3}}~=\text{ }{{1}^{3}}:\text{...

Find the triplicate ratio of (i)
$\mathbf{3}:\text{ }\mathbf{4}$
(ii)
${\scriptscriptstyle 1\!/\!{ }_2}:\text{ }\mathbf{1}/\mathbf{3}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3}:\text{ }\mathbf{4}\] We know that Triplicate ratio of \[3:\text{ }4\text{ }=\text{ }{{3}^{3}}:\text{ }{{4}^{3}}~=\text{ }27:\text{ }64\] (ii) \[{\scriptscriptstyle 1\!/\!{...

Find the duplicate ratio of (iii)
$\mathbf{5a}:\text{ }\mathbf{6b}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

iii) \[5a:\text{ }6b\] We know that Duplicate ratio of \[5a:\text{ }6b\text{ }=\text{ }{{\left( 5a \right)}^{2}}:\text{ }{{\left( 6b \right)}^{2}}~=\text{ }25{{a}^{2}}:\text{ }36{{b}^{2}}\]

Find the duplicate ratio of (i)
$\mathbf{2}:\text{ }\mathbf{3}$
(ii)
$\surd \mathbf{5}:\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{2}:\text{ }\mathbf{3}\] We know that Duplicate ratio of \[2:\text{ }3\text{ }=\text{ }{{2}^{2}}:\text{ }{{3}^{2}}~=\text{ }4:\text{ }9\] (ii) \[\surd \mathbf{5}:\text{ }\mathbf{7}\] We...

Find the compounded ratio of: (iii)
$\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right):\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right),\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{2}}}:\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{a}}^{\mathbf{4}}}~\text{ }{{\mathbf{b}}^{\mathbf{4}}}):\text{ }{{({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}})}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right):\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right),\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}...

Find the compounded ratio of: (i)
$\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{4}:\text{ }\mathbf{9}$
(ii)
$\mathbf{4}:\text{ }\mathbf{5},\text{ }\mathbf{5}:\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{9}:\text{ }\mathbf{11}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{4}:\text{ }\mathbf{9}\] We know that Compound ratio \[\begin{array}{*{35}{l}} ~=\text{ }2/3\text{ }\times \text{ }4/9 \\...

An alloy consists of
$\mathbf{27}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$
kg of copper and
$\mathbf{2}\text{ }{\scriptscriptstyle 3\!/\!{ }_4}$
kg of tin. Find the ratio by weight of tin to the alloy.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that Copper = \[\mathbf{27}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\] kg = \[55/2\]kg Tin = \[\mathbf{2}\text{ }{\scriptscriptstyle 3\!/\!{ }_4}\] kg = \[11/4\] kg We know that Total...

If one root of the equation $3 x^{2}-10 x+3=0$ is $\frac{1}{3}$ then the other root is
(a) $\frac{-1}{3}$
(b) $\frac{1}{3}$
(c) -3
(d) $3$

CBSE, Class 10, Exercise 10f, Maths, Quadratic Equations, RS Aggarwal

Answer is (d)3 Given: $3 x^{2}-10 x+3=0$ One root of the equation is $\frac{1}{3}$. Let the other root be $\alpha$. Product of the roots $=\frac{c}{a}$ $\begin{array}{l} \Rightarrow \frac{1}{3}...

If the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$ then the value of $\mathrm{k}$ is
(a) $-2$
(b) $-8$
(c) 8
(d) 12

CBSE, Class 10, Exercise 10f, Maths, Quadratic Equations, RS Aggarwal

Answer is (c) 8 It is given that the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$. The equation can be rewritten as: $x^{2}-3 x+(k-10)=0$ Product of the roots of a quadratic...

If one root of the equation $2 x^{2}+a x+6=0$ is 2 then $\mathrm{a}=$ ?
(a) 7
(b) $-7$
(c) $\frac{7}{2}$
(d) $\frac{-7}{2}$

CBSE, Class 10, Exercise 10f, Maths, Quadratic Equations, RS Aggarwal

Answer is (b) $-7$ It is given that one root of the equation $2 x^{2}+a x+6=0$ is 2 . $\begin{array}{l} \therefore 2 \times 2^{2}+a \times 2+6=0 \\ \Rightarrow 2 a+14=0 \\ \Rightarrow a=-7...

Which of the following is not a quadratic equation?
(a) $3 x-x^{2}=x^{2}+5$
(b) $(x+2)^{2}=2\left(x^{2}-5\right)$
(c) $(\sqrt{2} x+3)^{2}=2 x^{2}+6$
(d) $(x-1)^{2}=3 x^{2}+x-2$

CBSE, Class 10, Exercise 10f, Maths, Quadratic Equations, RS Aggarwal

Answer is (c) $(\sqrt{2} x+3)^{2}=2 x^{2}+6$ $\begin{array}{l} \because(\sqrt{2} x+3)^{2}=2 x^{2}+6 \\ \Rightarrow 2 x^{2}+9+6 \sqrt{2} x=2 x^{2}+6 \end{array}$ $\Rightarrow 6 \sqrt{2} x+3=0$, which...

The area of right -angled triangle is 165 sq meters. Determine its base and altitude if the latter exceeds the former by 7 meters.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) m$. Area of a triangle $=\frac{1}{2} \times$ Base $\times$ Altitude $\begin{array}{l} \therefore \frac{1}{2} \times x...

The area of a right triangle is $600 \mathrm{~cm}^{2}$ . If the base of the triangle exceeds the altitude by $10 \mathrm{~cm}$ , find the dimensions of the triangle.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the altitude of the triangle be $x \mathrm{~cm}$ Therefore, the base of the triangle will be $(x+10) \mathrm{cm}$ $\begin{array}{l} \text { Area of triangle }=\frac{1}{2} x(x+10)=600 \\...

The length of a rectangle is thrice as long as the side of a square. The side of the square is $4 \mathrm{~cm}$ more than the width of the rectangle. Their areas being equal, find the dimensions.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...

A rectangular filed in $16 \mathrm{~m}$ long and $10 \mathrm{~m}$ wide. There is a path of uniform width all around it, having an area of $120 \mathrm{~m}^{2}$ . Find the width of the path

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...

The length of a hall is 3 meter more than its breadth. If the area of the hall is 238 sq meters, calculate its length and breadth.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the breath of the rectangular hall be $x$ meter. Therefore, the length of the rectangular hall will be $(x+3)$ meter. According to the question: $\begin{array}{l} x(x+3)=238 \\ \Rightarrow...

The length of rectangle is twice its breadth and its areas is $288 \mathrm{~cm}^{2}$ . Find the dimension of the rectangle.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the length and breadth of the rectangle be $2 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $2 x \times x=288$ $\Rightarrow 2 x^{2}=288$ $\Rightarrow x^{2}=144$...

Two pipes running together can fill a tank in $11 \frac{1}{9}$ minutes. If on pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the time taken by one pipe to fill the tank be $x$ minutes. $\therefore$ Time taken by the other pipe to fill the tank $=(x+5) \min$ Suppose the volume of the tank be $V$. Volume of the tank...

A takes 10 days less than the time taken by $\mathrm{B}$ to finish a piece of work. If both $\mathrm{A}$ and $\mathrm{B}$ together can finish the work in 12 days, find the time taken by $\mathrm{B}$ to finish the work.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let B takes $x$ days to complete the work. Therefore, A will take $(x-10)$ days. $\begin{array}{l} \therefore \frac{1}{x}+\frac{1}{(x-10)}=\frac{1}{12} \\ \Rightarrow...

The speed of a boat in still water is $8 \mathrm{~km} / \mathrm{hr}$ . It can go $15 \mathrm{~km}$ upstream and $22 \mathrm{~km}$ downstream is 5 hours. Fid the speed of the stream

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Speed of the boat in still water $=8 \mathrm{~km} / \mathrm{hr}$. Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Speed...

The distance between Mumbai and Pune is $192 \mathrm{~km}$ . Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two train differ by $20 \mathrm{~km} / \mathrm{hr}$ .

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the speed of the Deccan Queen be $x \mathrm{~km} / \mathrm{hr}$. According to the question: Speed of another train $=(x-20) \mathrm{km} / \mathrm{hr}$ $\begin{array}{l} \therefore...

A train covers a distance of $90 \mathrm{~km}$ at a uniform speed. Had the speed been $15 \mathrm{~km} / \mathrm{hr}$ more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the original speed of the train be $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\frac{90}{x}-\frac{90}{(x+15)}=\frac{1}{2}$ $\begin{array}{l} \Rightarrow \frac{90(x+15)-90...

By using ruler and compass only, construct a quadrilateral ABCD in which AB = 6.5 cm, AD = 4 cm and ∠DAB = 750. C is equidistant to from the sides if AB and AD, if also C is equidistant from the points A and B.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Construct a line segment \[AB\text{ }=\text{ }6.5\text{ }cm.\] (ii) At the point A, construct a ray which makes an angle 750 and cut off \[AD\text{...

Without using set square or protractor, construct the parallelogram ABCD in which AB = 5.1 cm, the diagonal AC = 5.6 cm and diagonal BD = 7 cm. Locate the point P on DC, which is equidistant from AB and BC

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Consider \[AB\text{ }=\text{ }5.1\text{ }cm.\] (ii) At the point A, radius \[=\text{ }5.6/2\text{ }=\text{ }2.8\text{ }cm\] At the point B, radius...

Construct a rhombus PQRS whose diagonals PR, QS are 8 cm and 6 cm respectively. Find by construction a point X equidistant from PQ, PS and equidistant from R, S. Measure XR.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Take \[PR\text{ }=\text{ }8\text{ }cm\] and construct the perpendicular bisector of PR which intersects it at point O. (ii) From the point O, cut off...

Two straight lines PQ and PK cross each other at P at an angle of 750. S is a stone on the road PQ, 800 m from P towards Q. By drawing a figure to scale 1 cm = 100 m, locate the position of a flagstaff X, which is equidistant from P and S, and is also equidistant from the road.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

We know that \[\begin{array}{*{35}{l}} 1\text{ }cm\text{ }=\text{ }100\text{ }cm \\ 800\text{ }m\text{ }=\text{ }8\text{ }cm \\ \end{array}\] Following are the steps of Construction: (i) Construct...

AB and CD are two intersecting lines. Find the position of a point which is at a distance of 2 cm from AB and 1.6 cm from CD.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of construction: (i) AB and CD are two intersecting lines which intersect each other at the point O. (ii) Construct a line EF which is parallel to AB and GH which is parallel...

Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is (i) at a distance of 3 cm from AB. (ii) at a distance of 5 cm from the point M. Mark the points P, Q, R, S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Construct a line AB = 12 cm. (ii) Take M as the midpoint of line AB. (iii) Construct straight lines CD and EF which is parallel to AB at 3 cm distance....

Draw a line segment AB of length 7 cm. Construct the locus of a point P such that area of triangle PAB is 14 cm2.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

According to ques, Length of \[AB\text{ }=\text{ }7\text{ }cm\] (base) Area of triangle PAB \[=\text{ }14\text{ }c{{m}^{2}}\] We know that Height = (area × 2)/ base Substituting the values...

A point P is allowed to travel in space. State the locus of P so that it always remains at a constant distance from a fixed point C.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

According to ques, A point P is allowed to travel in space and is at a constant distance from a fixed point C. hence, its locus is a sphere.

Draw a straight line AB of length 8 cm. Draw the locus of all points which are equidistant from A and B. Prove your statement.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of construction: (i) Construct a line segment AB = 8 cm. (ii) Construct the perpendicular bisector of AB which intersects AB at the point D. now, every point P on it will be...

Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment. (i) Construct a triangle ABC, in which BC = 6 cm, AB = 9 cm and ∠ABC = 600. (ii) Construct the locus of all points, inside Δ ABC, which are equidistant from B and C.(iii) Construct the locus of the vertices of the triangle with BC as base, which are equal in area to Δ ABC. (iv) Mark the point Q, in your construction, which would make Δ QBC equal in area to Δ ABC and isosceles. (v) Measure and record the length of CQ.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Construct \[AB\text{ }=\text{ }9\text{ }cm\] (ii) At the point B construct an angle of 600 and cut off \[BC\text{ }=\text{ }6\text{ }cm.\] (iii) Now join...

Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length 6 cm and 5 cm respectively. (i) Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction. (ii) Construct the locus of points, inside the circle, that are equidistant from AB and AC.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Taking O as centre and 4 cm radius construct a circle. (ii) Mark a point A on this circle. (iii) Taking A and centre and 6 cm radius construct an arc...

Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Construct a line \[AB\text{ }=\text{ }6\text{ }cm.\] (ii) Taking A and B as centre and 4 cm radius, construct two arcs which intersect each other at the...

While boarding an aeroplane, a passengers got hurt. The pilot showing promptness and concern, made arrangements to hospitalize the injured and so the plane started late by 30 minutes. To reach the destination, $1500 \mathrm{~km}$ away, in time, the pilot increased the speed by $100 \mathrm{~km} /$ hour. Find the original speed of the plane. Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the original speed of the plane be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Actual speed of the plane $=(x+100) \mathrm{km} / \mathrm{h}$ Distance of the journey $=1500 \mathrm{~km}$ Time...

The sum of the ages of a boy and his brother is 25 years, and the product of their ages in years is 126 . Find their ages.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the present ages of the boy and his brother be $x$ years and $(25-x)$ years. According to the question: $\begin{array}{l} x(25-x)=126 \\ \Rightarrow 25 x-x^{2}=126 \\ \Rightarrow x^{2}-(18-7)...

One year ago, man was 8 times as old as his son. Now, his age is equal to the square of his son’s age. Find their present ages.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the present age of the son be $x$ years. $\therefore$ Present age of the $\operatorname{man}=x^{2}$ years One year ago, Age of the son $=(x-1)$ years Age of the man $=\left(x^{2}-1\right)$ years...

A man buys a number of pens for Rs. 180 . If he had bought 3 more pens for the same amount, each pen would have cost him Rs. 3 less. How many pens did he buy?

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the total number of pens be $x$. According to the question: $\begin{array}{l} \frac{80}{x}-\frac{80}{x+4}=1 \\ \Rightarrow \frac{80(x+4)-80 x}{x(x+4)}=1 \\ \Rightarrow \frac{80+320-80 x}{x^{2}+4...

A person on tour has Rs. 10800 for his expenses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs. 90 . Find the original duration of the tour.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the original duration of the tour be $x$ days. $\therefore \text { Original daily expenses }=\text { γ } \frac{10,800}{x}$ If he extends his tour by 4 days, then his new daily expenses...

Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, ∠B = 600, AD = 5 cm and D is equidistant from AB and BC. Measure CD.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of Construction: (i) Construct \[AB\text{ }=\text{ }6\text{ }cm.\] (ii) At point B, construct angle 600 and cut off \[BC\text{ }=\text{ }5\text{ }cm\] (iii) Construct the...

By using ruler and compasses only, construct an isosceles triangle ABC in which BC = 5 cm, AB = AC and ∠BAC = 900. Locate the point P such that: (i) P is equidistant from the sides BC and AC. (ii) P is equidistant from the points B and C.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Steps of Construction: (i) Construct \[BC\text{ }=\text{ }5\text{ }\] cm and bisect it at point D. (ii) Taking BC as diameter, construct a semicircle. (iii) At the point D, construct a...

Without using set-squares or protractor construct: (i) Triangle ABC, in which AB = 5.5 cm, BC = 3.2 cm and CA = 4.8 cm. (ii) Draw the locus of a point which moves so that it is always 2.5 cm from B. (iii) Draw the locus of a point which moves so that it is equidistant from the sides BC and CA. (iv) Mark the point of intersection of the loci with the letter P and measure PC.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Steps of Construction: (i) Construct \[BC\text{ }=\text{ }3.2\text{ }cm\] long. (ii) Taking B as centre and 5.5 cm radius and C as centre and 4.8 cm radius construct arcs intersecting each other at...

Without using set square or protractor, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ∠ABC. Find the point R on AD such that RB = RC. Measure the length of AR.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the steps of construction, (i) Construct \[AB\text{ }=\text{ }4\text{ }cm\] . (ii) Taking A as centre, construct an arc of radius 5 cm and with B as centre construct another arc of 4...

Without using set square or protractor, construct the quadrilateral ABCD in which ∠BAD = 450, AD = AB = 6 cm, BC = 3.6 cm and CD = 5 cm. (i) Measure ∠BCD. (ii) Locate the point P on BD which is equidistant from BC and CD.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

following are the steps of construction, (i) Consider \[AB\text{ }=\text{ }6\] cm long. (ii) At point A, construct the angle of 450 and cut off \[AD\text{ }=\text{ }6\text{ }cm.\] (iii) Taking D as...

Draw two intersecting lines to include an angle of 300. Use ruler and compasses to locate points which are equidistant from these lines and also 2 cm away from their point of intersection. How many such points exist?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) AB and CD are the two lines which intersect each other at the point O. (ii) Construct the bisector of ∠BOD and ∠AOD. (iii) Taking O as centre and 2 cm radius mark points on the bisector of...

Points A, B and C represent position of three towers such that AB = 60 mm, BC = 73 mm and CA = 52 mm. Taking a scale of 10 m to 1 cm, make an accurate drawing of Δ ABC. Find by drawing, the location of a point which is equidistant from A, B and C and its actual distance from any of the towers.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

According to ques, \[AB\text{ }=\text{ }60\text{ }mm\text{ }=\text{ }6\text{ }cm\] \[BC\text{ }=\text{ }73\text{ }mm\text{ }=\text{ }7.3\text{ }cm\] \[CA\text{ }=\text{ }52\text{ }mm\text{ }=\text{...

In the diagram, A, B and C are fixed collinear points; D is a fixed point outside the line. Locate:(v) Are the points P, Q, R collinear? (vi) Are the points P, Q, S collinear?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Here the points A, B and C are collinear and D is any point which is outside AB. (v) No, the points P, Q, R are not collinear. (vi) Yes, the points P, Q, S are collinear.

In the diagram, A, B and C are fixed collinear points; D is a fixed point outside the line. Locate:(iii) the points R on AB such that DR = 4 cm. How many such points are possible? (iv) the points S such that CS = DS and S is 4 cm away from the line CD. How many such points are possible?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Here the points A, B and C are collinear and D is any point which is outside AB. (iii) Taking D as centre and 4 cm radius construct an arc which intersects AB at R and R’ now R and R’ are the...

In the diagram, A, B and C are fixed collinear points; D is a fixed point outside the line. Locate: (i) the point P on AB such that CP = DP. (ii) the points Q such that CQ = DQ = 3 cm. How many such points are possible?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Here the points A, B and C are collinear and D is any point which is outside AB. (i) Join CD. Construct the perpendicular bisector of CD which meets AB in P. now, P is the required point such...

Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 1050. Hence: (i) Construct the locus of points equidistant from BA and BC. (ii) Construct the locus of points equidistant from B and C. (iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the Steps of Construction: Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 1050. (i) The points which are equidistant from BA and BC lies on the bisector of ∠ABC. (ii)...

Use ruler and compasses only for this question.(i) Construct the locus of points inside the triangle which are equidistant from B and C. (ii) Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and record the length of PB.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Since, In Δ ABC, AB = 3.5 cm, BC = 6 cm and ∠ABC = 600 (i) Construct a perpendicular bisector of BC which intersects BY at point P. (ii) It is given that point P is equidistant from AB, BC and also...

Use ruler and compasses only for this question. (i) Construct Δ ABC, where AB = 3.5 cm, BC = 6 cm and ∠ABC = 600. (ii) Construct the locus of points inside the triangle which are equidistant from BA and BC.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

since, In Δ ABC, AB = 3.5 cm, BC = 6 cm and ∠ABC = 600 Following are the Steps of Construction: (i) Construct a line segment BC = 6 cm. At the point B construct a ray BX which makes an angle 600 and...

A straight line AB is 8 cm long. Locate by construction the locus of a point which is: (i) Equidistant from A and B. (ii) Always 4 cm from the line AB. (iii) Mark two points X and Y, which are 4 cm from AB and equidistant from A and B. Name the figure AXBY.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Following are the Steps of Construction, (i) Construct a line segment AB = 8 cm. (ii) Using compasses and ruler, construct a perpendicular bisector l of AB which intersects AB at the point O. (iii)...

Construct triangle ABC, with AB = 7 cm, BC = 8 cm and ∠ABC = 600. Locate by construction the point P such that: (i) P is equidistant from B and C and (ii) P is equidistant from AB and BC (iii) Measure and record the length of PB.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) Consider, BC = 8 cm as the long line segment. now, At the point B construct a ray BX making an angle of 600 with BC Now cut off BA = 7 cm and join AC. Construct the perpendicular bisector of BC....

Using ruler and compasses construct: (i) a triangle ABC in which AB = 5.5 cm, BC = 3.4 cm and CA = 4.9 cm. (ii) the locus of points equidistant from A and C.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) Construct BC = 3.4 cm and mark the arcs 5.5 and 4.9 cm from the points B and C. Now , join A, B and C where ABC is the required triangle. (ii) Construct a perpendicular bisector of AC. (iii)...

Describe completely the locus of points in each of the following cases: centre of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with centre O.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(1) Now, if the circle with 2 cm as radius touches the given circle externally then the locus of the centre of circle will be a concentric circle of radius 3 + 2 = 5 cm (2) If the circle with 2 cm...

Describe completely the locus of points in each of the following cases: (v) centre of a circle of varying radius and touching two arms of ∠ADC. (vi) centre of a circle of varying radius and touching a fixed circle, centre O, at a fixed point A on it.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(v) Construct the bisector BX of ∠ABC. hence, this bisector of an angle is the locus of the centre of a circle having different radii. now, any point on BX is equidistant from BA and BC which are...

Describe completely the locus of points in each of the following cases:(iii) point in a plane equidistant from a given line. (iv) point in a plane, at a constant distance of 5 cm from a fixed point (in the plane).

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(iii) Since, AB is the given line and P is a point in the plane. From the point P, construct a line CD and another line EF from P’ parallel to AB. Hence, CD and EF are the lines which are the locus...

Describe completely the locus of points in each of the following cases: (i) mid-point of radii of a circle. (ii) centre of a ball, rolling along a straight line on a level floor.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) The locus of midpoints of the radii of a circle is another concentric circle with radius which is half of radius of given circle. (ii) Consider, AB as a straight line on the ground and the ball...

Draw and describe the locus in each of the following cases:(v) The locus of a point in rhombus ABCD which is equidistant from AB and AD. (vi) The locus of a point in the rhombus ABCD which is equidistant from points A and C.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(v) In a rhombus ABCD, join AC. now, AC is the diagonal of rhombus ABCD since, AC bisects ∠A hence, any point on AC is the locus which is equidistant from AB and AD. (vi) In a rhombus ABCD, join BD....

Draw and describe the locus in each of the following cases:(iii) The locus of points inside a circle and equidistant from two fixed points on the circle. (iv) The locus of centres of all circles passing through two fixed points.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(iii) 1. Construct a circle with O as centre. 2. Take points A and B on it then join them. 3. Construct a perpendicular bisector of AB which passes from point O and meets the circle at C. now, CE ,...

Draw and describe the locus in each of the following cases: (i) The locus of points at a distance 2.5 cm from a fixed line. (ii) The locus of vertices of all isosceles triangles having a common base.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) 1. Construct a line AB. 2. Construct lines l and m parallel to AB at a distance of 2.5 cm. now, lines l and m are the locus of point P at a distance of 2.5 cm. (ii) According to ques, Δ ABC is...

(i) AB is a fixed line. State the locus of the point P so that ∠APB = 900. (ii) A, B are fixed points. State the locus of the point P so that ∠APB = 900.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) According to ques, AB is a fixed line and P is a point , hence ∠APB = 900 Now, the locus of point P will be the circle where AB is the diameter. Also, the angle in a semi-circle is equal to...

P is a fixed point and a point Q moves such that the distance PQ is constant, what is the locus of the path traced out by the point Q?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

let us consider, P as a fixed point and Q as a moving point which is always at an equidistant from point P. Now, P is the center of the path of Q which is a circle. Since, the distance between the...

A point P moves so that its perpendicular distance from two given lines AB and CD are equal. State the locus of the point P.

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

(i) Since, if two lines AB and CD are parallel, then the locus of point P which is equidistant from AB and CD is a line (l) in the midway of lines AB and CD and is parallel to them. (ii) If both AB...

A point moves such that its distance from a fixed line AB is always the same. What is the relation between AB and the path travelled by P?

Class 10, Exercise 1, ICSE, Locus, Maths, ML Aggarwal

Considering point P , it is at a fixed distance from the fixed line AB. now this is a set of two lines l and m which are parallel to AB, drawn on either side at an equal distance from it....

Find the range of values of a, which satisfy 7 ≤ – 4x + 2 < 12, x ∈ R. Graph these values of a on the real number line.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to ques,

Find three smallest consecutive natural numbers such that the difference between one-third of the largest and one-fifth of the smallest is at least 3.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

Let the first least natural number as x Then, second number \[=\text{ }x\text{ }+\text{ }1\] And third number \[=\text{ }x\text{ }+\text{ }2\] Therefore, according the conditions given in the...

Find positive integers which are such that if 6 is subtracted from five times the integer then the resulting number cannot be greater than four times the integer.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

Let the positive integer be x Then according to the problem, we get \[5a\text{ }\text{ }6\text{ }<\text{ }4x\] \[5a\text{ }\text{ }4x\text{ }<\text{ }6\] \[\Rightarrow x\text{ }<\text{ }6\]...

If x ∈ R, solve 2x – 3 ≥ x + (1 – x)/3 > 2x/5. Also represent the solution on the number line.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[2x\text{ }\text{ }3\text{ }\ge \text{ }x\text{ }+\text{ }\left( 1\text{ }\text{ }x \right)/3\text{ }>\text{ }2x/5\] Therefore, we get \[2x\text{ }\text{ }3\text{...

Solve the inequation: $\frac{5x+1}{7}-4(\frac{x}{7}+\frac{2}{5})\le 1\frac{3}{5}+\frac{3x-1}{7},x\in R$

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[\begin{array}{*{35}{l}} \left( 5x\text{ }+\text{ }1 \right)/7\text{ }\text{ }4\text{ }\left( 5x\text{ }+\text{ }14 \right)/35 \\ \le \text{ }8/5\text{ }+\text{ }\left(...

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15}$ . Find the fraction.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the denominator of the required fraction be $x$. Numerator of the required fraction $=x-3$ $\therefore$ Original fraction $=\frac{x-3}{x}$ If 1 is added to the denominator, then the new fraction...

If x ∈ R (real numbers) and -1 < 3 – 2x ≤ 7, find solution set and present it on a number line.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[-1\text{ }<\text{ }3\text{ }\text{ }2x\text{ }\le \text{ }7\] \[-1\text{ }\text{ }3\text{ }<\text{ }-2x\text{ }\le \text{ }7\text{ }\text{ }3\] \[-4\text{ }<\text{...

Find the solution set of the inequation x + 5 ≤ 2x + 3; x ∈ R Graph the solution set on the number line.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[x\text{ }+\text{ }5\text{ }\le \text{ }2x\text{ }+\text{ }3\] \[x\text{ }\text{ }2x\text{ }\le \text{ }3\text{ }\text{ }5\] \[-x\text{ }\le \text{ }-2\] \[x\text{ }\ge...

A two-digit number is such that the product of its digits is 14 . If 45 is added to the number, the digit interchange their places. Find the number.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the digits at units and tens places be $x$ and $y$, respectively. $\therefore x y=14$ $\Rightarrow y=\frac{14}{x}$ According to the question: $\begin{array}{l} (10 y+x)+45=10 x+y \\ \Rightarrow...

Solve the inequation: 6x – 5 < 3x + 4, x ∈ I

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[6x\text{ }\text{ }5\text{ }<\text{ }3x\text{ }+\text{ }4\] \[6x\text{ }\text{ }3x\text{ }<\text{ }4\text{ }+\text{ }5\] \[3x\text{ }<\text{ }9\] \[x\text{...

Solve the inequation: 5x – 2 ≤ 3 (3 – x) where x ∈ {-2, -1, 0, 1, 2, 3, 4}. Also represent its solution on the number line.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

According to question, \[5x\text{ }\text{ }2\text{ }\le \text{ }3\text{ }\left( 3\text{ }\text{ }x \right)\] \[5x\text{ }\text{ }2\text{ }\le \text{ }9\text{ }\text{ }3x\] \[5x\text{ }+\text{...

Three consecutive positive integers are such that the sum of the square of the first and product of the other two is 46 . Find the integers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the three consecutive positive integers be $x, x+1$ and $x+2$. According to the given condition, $\begin{array}{l} x^{2}+(x+1)(x+2)=46 \\ \Rightarrow x^{2}+x^{2}+3 x+2=46 \\ \Rightarrow 2...

Divide two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the two natural numbers be $x$ and $y$. According to the question: $\begin{array}{l} x^{2}+y^{2}=25(x+y) \quad \ldots \ldots(i) \\ x^{2}+y^{2}=50(x-y) \end{array}$ From (i) and (ii), we get:...

Divide 27 into two parts such that the sum of their reciprocal is $\frac{3}{20}$ .

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the two parts be $x$ and $(27-x)$. According to the given condition, $\begin{array}{l} \frac{1}{x}+\frac{1}{27-x}=\frac{3}{20} \\ \Rightarrow \frac{27-x+x}{x(27-x)}=\frac{3}{20} \\ \Rightarrow...

One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole.

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

Let’s assume the length of the shortest pole = x metre Now, Length of the pole which is buried in mud = x/3 Length of the pole which is in the water = x/6 Then according to the given condition, we...

The sum of natural number and its reciprocal is $\frac{65}{8}$ . Find the number.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the natural number be $x$. According to the given condition, $\begin{array}{l} x+\frac{1}{x}=\frac{65}{8} \\ \Rightarrow \frac{x^{2}+1}{x}=\frac{65}{8} \\ \Rightarrow 8 x^{2}+8=65 x \\...

The difference of two natural numbers is 5 and the difference of heir reciprocals is $\frac{5}{14}$ . Find the numbers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the required natural numbers be $x$ and $(x+5)$. Now, $x<x+5$ $\therefore \frac{1}{x}>\frac{1}{x+5}$ According to the given condition, $\begin{array}{l}...

Find the greatest integer which is such that if 7 is added to its double, the resulting number becomes greater than three times the integer

Class 10, Exercise 1, ICSE, Linear Inequations, Maths, ML Aggarwal

Let the greatest integer to be x Then according to the given condition, we get \[2x\text{ }+\text{ }7\text{ }>\text{ }3x\] \[2x\text{ }\text{ }3x\text{ }>\text{ }-7\] \[-x\text{ }>\text{...

The sum of two natural numbers is 15 and the sum of their reciprocals is $\frac{3}{10}$ . Find the numbers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the required natural numbers be $x$ and $(15-x)$. According to the given condition, $\begin{array}{l} \frac{1}{x}+\frac{1}{15-x}=\frac{3}{10} \\ \Rightarrow \frac{15-x+x}{x(15-x)}=\frac{3}{10}...

Find the two consecutive positive even integers whose product is 288.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the two consecutive positive even integers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x(x+2)=288 \\ \Rightarrow x^{2}+2 x-288=0 \\ \Rightarrow x^{2}+18 x-16 x-288=0...

Find two consecutive multiples of 3 whose product is 648.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the required consecutive multiples of 3 be $3 x$ and $3(x+1)$. According to the given condition, $\begin{array}{l} 3 x \times 3(x+1)=648 \\ \Rightarrow 9\left(x^{2}+x\right)=648 \\ \Rightarrow...

The product of two consecutive positive integers is 306 . Find the integers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the two consecutive positive integers be $x$ and $(x+1)$. According to the given condition, $\begin{array}{l} x(x+1)=306 \\ \Rightarrow x^{2}+x-306=0 \\ \Rightarrow x^{2}+18 x-17 x-306=0 \\...

The sum of the squares to two consecutive positive odd numbers is 514 . Find the numbers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the two consecutive positive odd numbers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x^{2}+(x+2)^{2}=514 \\ \Rightarrow x^{2}+x^{2}+4 x+4=514 \\ \Rightarrow 2 x^{2}+4...

The sum of two natural number is 28 and their product is 192 . Find the numbers.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the required number be $x$ and $(28-x)$. According to the given condition, $\begin{array}{l} x(28-x)=192 \\ \Rightarrow 28 x-x^{2}=192 \\ \Rightarrow x^{2}-28 x+192=0 \\ \Rightarrow x^{2}-16...

The sum of natural number and its positive square root is $132 .$ Find the number.

CBSE, Class 10, Exercise 10e, Maths, Quadratic Equations, RS Aggarwal

Let the required natural number be $x$. According to the given condition, $x+\sqrt{x}=132$ Putting $\sqrt{x}=y$ or $x=y^{2}$, we get $y^{2}+y=132$ $\Rightarrow y^{2}+y-132=0$ $\Rightarrow y^{2}+12...

If the roots of the equations $a x^{2}+2 b x+c=0$ and $b x^{2}-2 \sqrt{a c x}+b=0$ are simultaneously real then prove that $b^{2}=a c$

CBSE, Class 10, Exercise 10D, Maths, Quadratic Equations, RS Aggarwal

It is given that the roots of the equation $a x^{2}+2 b x+c=0$ are real. $\begin{array}{l} \therefore D_{1}=(2 b)^{2}-4 \times a \times c \geq 0 \\ \Rightarrow 4\left(b^{2}-a c\right) \geq 0 \\...

Find the value of $k$ for which the roots of $9 x^{2}+8 k x+16=0$ are real and equal

CBSE, Class 10, Exercise 10D, Maths, Quadratic Equations, RS Aggarwal

Given: $\begin{array}{l} 9 x^{2}+8 k x+16=0 \\ \text { Here, } \\ a=9, b=8 k \text { and } c=16 \end{array}$ It is given that the roots of the equation are real and equal; therefore, we have:...

Find the value of $p$ for which the quadratic equation $2 x^{2}+p x+8=0$ has real roots.

CBSE, Class 10, Exercise 10D, Maths, Quadratic Equations, RS Aggarwal

$2 x^{2}+p x+8=0$ Here, $a=2, b=p$ and $c=8$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 2 \times 8 \\ =\left(p^{2}-64\right) \end{array}$ If $D...