ML Aggarwal Archives - Noon Academy

ML Aggarwal

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...

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...

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(...

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...

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{...

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...

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{...

If A is an acute angle and sin A = 3/5, find all other trigonometric ratios of angle A (using trigonometric identities)

Maths Trigonometric identities

sin A = 3/5 and A is an acute angle So, in ∆ABC we have ∠B = 90o And, AC = 5 and BC = 3 By Pythagoras theorem, AB = √(AC2 – BC2) = √(52 – 32) = √(25 – 9) = √16 = 4 Now, cos A = AB/AC = 4/5 tan A =...

The hypotenuse of a right-angled triangle is
$1$
m less than twice the shortest side. If the third side is
$1$
m more than the shortest side, find the sides of the triangle.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let’s consider the length of the shortest side = x m Length of hypotenuse = \[2x\text{ }\text{ }1\] And third side = \[x\text{ }+\text{ }1\] Now according to the given condition in the problem, we...

A farmer wishes to grow a
$100{{m}^{2}}$
rectangular vegetable garden. Since he has with him only
$30$
m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side fence. Find the dimensions of his garden.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given, Area of rectangular garden = \[100{{m}^{2}}\] Length of barbed wire = \[30\] m Let’s assume the length of the side opposite to wall to be x And the length of other side = \[\left( 30\text{...

The length of a rectangular garden is
$12$
m more than its breadth. The numerical value of its area is equal to
$4$
times the numerical value of its perimeter. Find the dimensions of the garden.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let’s assume the breadth of the rectangular garden as x m Then, length = \[(x+12)\] m So, Area = l × b \[{{m}^{2}}\] = \[x\text{ }\times \text{ }\left( x\text{ }+\text{ }12 \right)\text{...

Two squares have sides x cm and
$(x+4)$
cm. The sum of their areas is
$656$
sq. cm. Express this as an algebraic equation and solve it to find the sides of the squares.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

We have, Side of first square = x cm And the side of second square = \[(x+4)\] cm Now according to the given conditions in the problem, we have \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{\left(...

Two natural numbers are in the ratio
$3\text{ }:\text{ }4$
. Find the numbers if the difference between their squares is
$175$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given, ratio of two natural numbers is \[3\text{ }:\text{ }4\] Let the numbers be taken as \[3x\] and \[4x\] Then, according to the conditions in the problem, we have \[\begin{array}{*{35}{l}}...

Divide
$16$
into two parts such that the twice the square of the larger part exceeds the square of the smaller part by
$164$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let the larger part be considered as x Then, the smaller part will be \[\left( 16\text{ }\text{ }x \right)\] According to the conditions given in the problem, we have \[\begin{array}{*{35}{l}}...

Find two natural numbers which differ by
$3$
and whose squares have the sum
$117$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let the first natural number be x Then, second the natural number will be \[x+3\] According to the condition given in the problem, \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{\left( x\text{...

Find the value(s) of k for which each of the following quadratic equation has equal roots: (i)
$\mathbf{3k}{{\mathbf{x}}^{\mathbf{2}}}~=\text{ }\mathbf{4}\left( \mathbf{kx}\text{ }\text{ }\mathbf{1} \right)\text{ }\left( \mathbf{ii} \right)\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{4} \right){{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$
Also, find the roots for that value(s) of k in each case.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

For a quadratic equation to have equal roots, discriminant (D) = 0 (i) \[3k{{x}^{2}}~=\text{ }4\left( kx\text{ }\text{ }1 \right)\] Let’s rearrange the given equation, \[\begin{array}{*{35}{l}}...

Find the values of m so that the quadratic equation
$\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{2m}\text{ }=\text{ }\mathbf{0}$
has two distinct real roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given quadratic equation, \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{2m}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }3,\text{ }b\text{...

Find the values of k so that the quadratic equation
$\left( \mathbf{4}\text{ }\text{ }\mathbf{k} \right){{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}\left( \mathbf{k}\text{ }+\text{ }\mathbf{2} \right)\mathbf{x}\text{ }+\text{ }\left( \mathbf{8k}\text{ }+\text{ }\mathbf{1} \right)\text{ }=\text{ }\mathbf{0}$
has equal roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given quadratic equation, \[\left( \mathbf{4}\text{ }\text{ }\mathbf{k} \right){{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}\left( \mathbf{k}\text{ }+\text{ }\mathbf{2} \right)\mathbf{x}\text{...

Discuss the nature of roots of the following equations: (iii)
$\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{6}\surd \mathbf{5x}\text{ }+\text{ }\mathbf{9}\text{ }=\text{ }\mathbf{0}$
(iv)
$\surd \mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2x}\text{ }\text{ }\surd \mathbf{3}\text{ }=\text{ }\mathbf{0}$
In case the real roots exist, then find them.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(iii) \[5{{x}^{2}}~\text{ }6\surd 5x\text{ }+\text{ }9\text{ }=\text{ }0\] Let us consider, \[a\text{ }=\text{ }5,\text{ }b\text{ }=\text{ }-6\surd 5,\text{ }c\text{ }=\text{ }9\] By using the...

Solve for x using the quadratic formula. Write your answer correct to two significant figures:
${{\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)}^{\mathbf{2}}}~\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given quadratic equation, \[\begin{array}{*{35}{l}} {{\left( x\text{ }\text{ }1 \right)}^{2}}~\text{ }3x\text{ }+\text{ }4\text{ }=\text{ }0 \\ {{x}^{2}}~\text{ }2x\text{ }\text{ }3x\text{ }+\text{...

(i)
$ (1) <img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b7baf3f835bd16bb0eed2d5efad8bb23_l3.png" height="37" width="380" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align} & {{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\left( \mathbf{4}\text{ }\text{ }\mathbf{3a} \right)\mathbf{x}\text{ }\text{ }\mathbf{12a}\text{ }=\text{ }\mathbf{0} \\ \end{align}" title="Rendered by QuickLaTeX.com"/>$
(ii)
$10a{{x}^{2}}~\text{ }6x\text{ }+\text{ }15ax\text{ }\text{ }9\text{ }=\text{ }0,\text{ }a\text{ }\ne \text{ }0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\left( \mathbf{4}\text{ }\text{ }\mathbf{3a} \right)\mathbf{x}\text{ }\text{ }\mathbf{12a}\text{ }=\text{ }\mathbf{0}\] Let us consider,...

(i)
$\left( \mathbf{3x}\text{ }\text{ }\mathbf{4} \right)/\mathbf{7}\text{ }+\text{ }\mathbf{7}/\left( \mathbf{3x}\text{ }\text{ }\mathbf{4} \right)\text{ }=\text{ }\mathbf{5}/\mathbf{2},\text{ }\mathbf{x}\text{ }\ne \text{ }\mathbf{4}/\mathbf{3}$
(ii)
$~\mathbf{4}/\mathbf{x}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{5}/\left( \mathbf{2x}\text{ }+\text{ }\mathbf{3} \right),\text{ }\mathbf{x}\text{ }\ne \text{ }\mathbf{0},\text{ }-\mathbf{3}/\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\left( 3x\text{ }\text{ }4 \right)/7\text{ }+\text{ }7/\left( 3x\text{ }\text{ }4 \right)\text{ }=\text{ }5/2\] Taking L.C.M, we get \[\begin{align} & \begin{array}{*{35}{l}} [{{\left(...

A model of a ship is made to a scale of 1: 250 calculate:
(i) The length of the ship, if the length of model is 1.6 m.
(ii) The area of the deck of the ship, if the area of the deck of model is 2.4 m2.
(iii) The volume of the model, if the volume of the ship is 1 km3.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, a model of a ship is made to a scale of 1 : 250 (i) Given, the length of the model is 1.6 m Then, length of the ship = (1.6 × 250)/1 = 400 m (ii)...

(i)
$\left( \mathbf{2x}\text{ }+\text{ }\mathbf{5} \right)/\left( \mathbf{3x}\text{ }+\text{ }\mathbf{4} \right)\text{ }=\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{1} \right)/\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)$
(ii)
$\mathbf{2}/\left( \mathbf{x}\text{ }+\text{ }\mathbf{2} \right)\text{ }\text{ }\mathbf{1}/\left( \mathbf{x}\text{ }+\text{ }\mathbf{1} \right)\text{ }=\text{ }\mathbf{4}/\left( \mathbf{x}\text{ }+\text{ }\mathbf{4} \right)\text{ }\text{ }\mathbf{3}/\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\left( \mathbf{2x}\text{ }+\text{ }\mathbf{5} \right)/\left( \mathbf{3x}\text{ }+\text{ }\mathbf{4} \right)\text{ }=\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{1} \right)/\left(...

In the adjoining figure, ABCD is a parallelogram. E is mid-point of BC. DE meets the diagonal AC at O and meet AB (produced) at F. Prove that

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a parallelogram. E is mid-point of BC. DE meets the diagonal AC at O. (i) Now consider the ∆AOD and ∆EDC, ∠AOD = ∠EOC … [because Vertically...

(i)
$\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}\text{ }\left( \mathbf{ii} \right)\text{ }\mathbf{x}\left( \mathbf{3x}\text{ }+\text{ }{\scriptscriptstyle 1\!/\!{ }_2} \right)\text{ }=\text{ }\mathbf{6}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i)\[~2{{x}^{2}}~\text{ }3x\text{ }\text{ }1\text{ }=\text{ }0\] Let us consider, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }-3,\text{ }c\text{ }=\text{ }-1\] So, by using the formula,...

(i)
$\surd \left( \mathbf{x}\text{ }+\text{ }\mathbf{15} \right)\text{ }=\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{3}\text{ }\left( \mathbf{ii} \right)\text{ }\surd (\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2x}\text{ }\text{ }\mathbf{1})\text{ }=\text{ }\mathbf{2x}\text{ }\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\surd \left( \mathbf{x}\text{ }+\text{ }\mathbf{15} \right)\text{ }=\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{3}\] Let us simplify the given expression, \[\begin{array}{*{35}{l}} x\text{...

In the given figure, ABCD is a trapezium in which AB || DC. If 2AB = 3DC, find the ratio of the areas of ∆AOB and ∆COD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a trapezium in which AB || DC. If 2AB = 3DC. So, 2AB = 3DC AB/DC = 3/2 Now, consider ∆AOB and ∆COD ∠AOB = ∠COD … [because vertically opposite...

(i)
$\mathbf{x}\left( \mathbf{x}\text{ }+\text{ }\mathbf{1} \right)\text{ }+\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{2} \right)\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)\text{ }=\text{ }\mathbf{42}$
(ii)
$\mathbf{6}/\mathbf{x}\text{ }\text{ }\mathbf{2}/\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)\text{ }=\text{ }\mathbf{1}/\left( \mathbf{x}\text{ }\text{ }\mathbf{2} \right)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{x}\left( \mathbf{x}\text{ }+\text{ }\mathbf{1} \right)\text{ }+\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{2} \right)\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)\text{...

(i)
$\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{ax}\text{ }\text{ }\mathbf{a}{}^\text{2}\text{ }=\text{ }\mathbf{0}$
(ii)
$\surd 3x{}^\text{2}\text{ }+\text{ }10x\text{ }+\text{ }7\surd 3\text{ }=\text{ }0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{ax}\text{ }\text{ }\mathbf{a}{}^\text{2}\text{ }=\text{ }\mathbf{0}\] Let us factorize the given expression, \[\begin{array}{*{35}{l}}...

In the adjoining figure, the diagonals of a parallelogram intersect at O. OE is drawn parallel to CB to meet AB at E, find area of ∆AOE : area of parallelogram ABCD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the given figure, The diagonals of a parallelogram intersect at O. OE is drawn parallel to CB to meet AB at E. In the figure four triangles have equal area. So, area of ∆OAB = ¼ area...

(i)
$\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{6x}\text{ }\text{ }\mathbf{16}\text{ }=\text{ }\mathbf{0}~$
(ii)
$\mathbf{3x}{}^\text{2}\text{ }+\text{ }\mathbf{11x}\text{ }+\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{6x}\text{ }\text{ }\mathbf{16}\text{ }=\text{ }\mathbf{0}~\] Let us factorize the given expression, \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{...

In the adjoining figure, D is a point on BC such that ∠ABD = ∠CAD. If AB = 5 cm, AC = 3 cm and AD = 4 cm, find
(i) BC
(ii) DC
(iii) area of ∆ACD : area of ∆BCA.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∠ABD = ∠CAD AB = 5 cm, AC = 3 cm and AD = 4 cm Now, consider the ∆ABC and ∆ACD ∠C = ∠C … [common angle for both triangles] ∠ABC = ∠CAD … [from the...

If the areas of two similar triangles are 360 cm² and 250 cm² and if one side of the first triangle is 8 cm, find the length of the corresponding side of the second triangle.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, the areas of two similar triangles are 360 cm² and 250 cm². one side of the first triangle is 8 cm So, PQR and XYZ are two similar triangles, So, let...

In a ∆ABC, D and E are points on the sides AB and AC respectively such that AD = 5.7cm, BD = 9.5cm, AE = 3.3cm and AC = 8.8cm. Is DE || BC? Justify your answer.

CBSE, Class 10, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, In a ∆ABC, D and E are points on the sides AB and AC respectively. AD = 5.7cm, BD = 9.5cm, AE = 3.3cm and AC = 8.8cm Consider the ∆ABC, EC = AC – AE =...

The hypotenuse of grassy land in the shape of a right triangle is
$1$
metre more than twice the shortest side. If the third side is
$7$
metres more than the shortest side, find the sides of the grassy land.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let’s consider the shortest side to be ‘x’ cm Hypotenuse = \[2x\text{ }+\text{ }1\] And third side = \[x\text{ }+\text{ }7\] Now, by Pythagoras theorem we have \[\begin{array}{*{35}{l}} {{\left(...

If the sum of two smaller sides of a right-angled triangle is
$17$
cm and the perimeter is
$30$
cm, then find the area of the triangle.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given, The perimeter of the triangle = \[30\] cm Let’s assume the length of one of the two small sides as x cm Then, the other side will be \[\left( 17\text{ }\text{ }x \right)\]cm Now, length of...

In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm, find BD and CE.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, In a ∆ABC, D and E are points on the sides AB and AC respectively. DE || BC AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm Consider the ∆ABC, Given,...

If the perimeter of a rectangular plot is
$68$
m and the length of its diagonal is
$26$
m, find its area.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given, Perimeter = \[68\] m and diagonal = \[26\] m So, Length + breadth = Perimeter/\[2\] = \[68/2\] = \[34\] m Let’s consider the length of the rectangular plot to be ‘x’ m Then, breadth =...

The lengths of the parallel sides of a trapezium are
$(x+9)$
cm and
$(2x-3)$
cm and the distance between them is
$(x+4)$
cm. If its area is
$540c{{m}^{2}}$
, find x.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

We know that, Area of a trapezium = \[{\scriptscriptstyle 1\!/\!{ }_2}\] × (sum of parallel sides) × (height) Given, the length of parallel sides are \[(x+9)\] and \[(2x-3)\] And height = \[(x+4)\]...

The perimeter of a rectangular plot is
$180$
m and its area is
$1800$

${{m}^{2}}$
. Take the length of the plot as x m. Use the perimeter
$180$
m to write the value of the breadth in terms of x. Use the values of length, breadth and the area to write an equation in x. Solve the equation to calculate the length and breadth of the plot. (1993)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given, The perimeter of a rectangular field = \[180\]m And area = \[1800\] \[{{m}^{2}}\] Let’s assume the length of the rectangular field as ‘x’ m We know that, Perimeter of rectangular field =...

The length of a rectangle exceeds its breadth by
$5$
m. If the breadth was doubled and the length reduced by
$9$
m, the area of the rectangle would have increased by
$140{{m}^{2}}$
. Find its dimensions.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(ii) In first case: Let us consider length of the rectangle be ‘x’ meter Width = \[(x-5)\] meter Area = lb = \[x\left( x\text{ }\text{ }5 \right)\]sq.m In second case: Length = \[\left( x\text{...

A rectangular garden
$10$
m by
$16$
m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is
$120$
square meters, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x. (1992)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given: Length of garden = \[16\]cm Width = \[10\]cm Let the width of walk be ‘x’ meter Outer length = \[16+2x\] Outer width = \[10+2x\] So according to the question, \[\begin{array}{*{35}{l}} \left(...

In the adjoining figure, 2 AD = BD, E is mid-point of BD and F is mid-point of AC and EC || BH. Prove that:
(i) DF || BH
(ii) AH = 3 AF.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, 2 AD = BD, EC || BH (i) Given, E is mid-point of BD 2DE = BD … [equation (i)] 2AD = BD … [equation (ii)] From equation (i) and equation (ii) we get,...

A two digit positive number is such that the product of its digits is
$6$
. If
$9$
is added to the number, the digits interchange their places. Find the number. (2014)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let us consider \[2\]-digit number be ‘xy’ = \[10x\text{ }+\text{ }y\] Reversed digits = yx = \[10y\text{ }+\text{ }x\] So according to the question, \[10x\text{ }+\text{ }y\text{ }+\text{ }9\text{...

A two digit number contains the bigger at ten’s place. The product of the digits is
$27$
and the difference between two digits is
$6$
. Find the number.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let us consider unit’s digit be ‘x’ Ten’s digit = \[x+6\] Number = \[x\text{ }+\text{ }10\left( x+6 \right)\] \[\begin{array}{*{35}{l}} =\text{ }x\text{ }+\text{ }10x\text{ }+\text{ }60 \\ =\text{...

In the figure given below, CD = ½ AC, B is mid-point of AC and E is mid-point of DF. If BF || AG, prove that :
(i) CE || AG
(ii) 3 ED = GD

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, CD = ½ AC BF || AG (i) We have to prove that, CE || AG Consider, CD = ½ AC AC = 2BC … [because from the figure B is mid-point of AC] So, CD = ½ (2BC)...

In the figure given below. ∠AED = ∠ABC. Find the values of x and y.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the figure it is given that, ∠AED = ∠ABC Consider the ∆ABC and ∆ADE ∠AED = ∠ABC … [from the figure] ∠A = ∠A … [common angle for both triangles] Therefore, ∆ABC ~ ∆ADE … [by AA axiom]...

In the adjoining figure, AB = AC. If PM ⊥ AB and PN ⊥ AC, show that PM x PC = PN x PB.

CBSE, Class 10, Maths, ML Aggarwal, Similarity

Solution:- From the given figure, AB = AC. If PM ⊥ AB and PN ⊥ AC We have to show that, PM x PC = PN x PB Consider the ∆ABC, AB = AC … [given] ∠B = ∠C Then, consider ∆CPN and ∆BPM ∠N = ∠M … [both...

In the adjoining figure, ∠1 = ∠2 and ∠3 = ∠4. Show that PT x QR = PR x ST.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∠1 = ∠2 and ∠3 = ∠4 We have to prove that, PT x QR = PR x ST Given, ∠1 = ∠2 Adding ∠6 to both LHS and RHS we get, ∠1 + ∠6 = ∠2 + ∠6 ∠SPT = ∠QPR...

A model of a ship is made to a scale of 1 : 200.
(i) If the length of the model is 4 m, find the length of the ship.
(ii) If the area of the deck of the ship is 160000 m², find the area of the deck of the model.
(iii) If the volume of the model is 200 liters, find the volume of the ship in m³. (100 liters = 1 m³)

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, a model of a ship is made to a scale of 1 : 200 (i) Given, the length of the model is 4 m Then, length of the ship = (4 × 200)/1 = 800 m (ii) Given,...

The model of a building is constructed with the scale factor 1 : 30. (i) If the height of the model is 80 cm, find the actual height of the building in metres. (ii) If the actual volume of a tank at the top of the building is 27 m³, find the volume of the tank on the top of the model.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, The model of a building is constructed with the scale factor 1 : 30 So, Height of the model/Height of actual building = 1/30 (i) Given, the height of...

In a certain positive fraction, the denominator is greater than the numerator by
$3$
. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by
$1/14$
. Find the fraction.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let the denominator be ‘x’ So the numerator will be ‘\[8-x\]’ The obtained fraction is \[8-x/x\] So according to the question, By cross multiplying, \[\begin{array}{*{35}{l}} 35\left( 4x\text{...

On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD has the following measurements AB = 12 cm and BG = 16 cm. Calculate:
(i) the distance of a diagonal of the plot in km.
(ii) the area of the plot in sq. km.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, Map drawn to a scale of 1: 25000 AB = 12 cm, BG = 16 cm Consider the ∆ABC, From the Pythagoras theorem, AC2 = AB2 + BC2 AC = √(AB2 + BC2) = √((12)2 +...

On a map drawn to a scale of 1 : 250000, a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and ∠ABC = 90°. Calculate
(i) the actual length of AB in km.
(ii) the area of the plot in sq. km:

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, Map drawn to a scale of 1: 250000 AB = 3 cm, BC = 4 cm and ∠ABC = 90o (i) We have to find the actual length of AB in km. Let us assume scale factor K =...

(i) Find three successive even natural numbers, the sum of whose squares is
$308$
. (ii) Find three consecutive odd integers, the sum of whose squares is
$83$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) Find three successive even natural numbers, the sum of whose squares is \[308\]. Let us consider first even natural number be ‘\[2x\]’ Second even number be ‘\[2x+2\]’ Third even number be...

There are three consecutive positive integers such that the sum of the square of the first and the product of the other two is
$154$
. What are the integers?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let us consider the first integer be ‘x’ Second integer be ‘\[x+1\]’ Third integer be ‘\[x+2\]’ So, according to the question, \[{{x}^{2}}~+\text{ }\left( x\text{ }+\text{ }1 \right)\text{ }\left(...

Two isosceles triangles have equal vertical angles and their areas are in the ratio 7: 16. Find the ratio of their corresponding height.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- Consider the two isosceles triangle PQR and XYZ, ∠P = ∠X … [from the question] So, ∠Q + ∠R = ∠Y + ∠Z ∠Q = ∠R and ∠Y = ∠Z [because opposite angles of equal sides] Therefore, ∠Q = ∠Y and ∠R...

Sum of two natural numbers is
$8$
and the difference of their reciprocals is
$2/15$
. Find the numbers.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let us consider two numbers as ‘x’ and ‘y’ So according to the question, \[1/x\text{ }\text{ }1/y\text{ }=\text{ }2/15\]….. (i) It is given that, \[x\text{ }+\text{ }y\text{ }=\text{ }8\] So,...

Five times a certain whole number is equal to three less than twice the square of the number. Find the number.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Let us consider the number be ‘x’ So according to the question, \[\begin{array}{*{35}{l}} 5x\text{ }=\text{ }2{{x}^{2}}~\text{ }3 \\ 2{{x}^{2}}~\text{ }3\text{ }\text{ }5x\text{ }=\text{ }0 \\...

(iii) Find two consecutive even natural numbers such that the sum of their squares is
$340$
. (iv) Find two consecutive odd integers such that the sum of their squares is
$394$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(iii) Find two consecutive even natural numbers such that the sum of their squares is \[340\]. Let us consider first positive even integer number be ‘\[2x\]’ Second even integer number be ‘\[2x+2\]’...

(i) If the product of two positive consecutive even integers is
$288$
, find the integers. (ii) If the product of two consecutive even integers is
$224$
, find the integers.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) If the product of two positive consecutive even integers is \[288\], find the integers. Let us consider first positive even integer number be ‘\[2x\]’ Second even integer number be ‘\[2x+2\]’ So...

(i) Find two consecutive natural numbers such that the sum of their squares is
$ (1) <img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3db298f6c229faade1da540be3c4a14c_l3.png" height="21" width="29" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align} & 61 \\ \end{align}" title="Rendered by QuickLaTeX.com"/>$
. (ii) Find two consecutive integers such that the sum of their squares is
$61$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) Find two consecutive natural numbers such that the sum of their squares is \[61\]. Let us consider first natural number be ‘x’ Second natural number be ‘\[x+1\]’ So according to the question,...

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that:
(i) ∆ADE ~ ∆ACB.
(ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.
(iii) Find, area of ∆ADE : area of quadrilateral BCED.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∠ABC = 90° AB and DE is perpendicular to AC (i) Consider the ∆ADE and ∆ACB, ∠A = ∠A … [common angle for both triangle] ∠B = ∠E … [both angles are equal...

In the figure given below, ∠ABC = ∠DAC and AB = 8 cm, AC = 4 cm, AD = 5 cm.
(i) Prove that ∆ACD is similar to ∆BCA
(ii) Find BC and CD
(iii) Find the area of ∆ACD : area of ∆ABC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∠ABC = ∠DAC AB = 8 cm, AC = 4 cm, AD = 5 cm (i) Now, consider ∆ACD and ∆BCA ∠C = ∠C … [common angle for both triangles] ∠ABC = ∠CAD … [from the...

In the figure given below, ABCD is a trapezium in which DC is parallel to AB. If AB = 9 cm, DC = 6 cm and BB = 12 cm., find
(i) BP
(ii) the ratio of areas of ∆APB and ∆DPC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, DC is parallel to AB AB = 9 cm, DC = 6 cm and BB = 12 cm (i) Consider the ∆APB and ∆CPD ∠APB = ∠CPD … [because vertically opposite angles are equal]...

In the figure (ii) given below, AB || DC and AB = 2 DC. If AD = 3 cm, BC = 4 cm and AD, BC produced meet at E, find
(i) ED
(ii) BE
(iii) area of ∆EDC : area of trapezium ABCD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, AB || DC AB = 2 DC, AD = 3 cm, BC = 4 cm Now consider ∆EAB, EA/DA = EB/CB = AB/DC = 2DC/DC = 2/1 (i) EA = 2, DA = 2 × 3 = 6 cm Then, ED = EA – DA = 6 –...

Find the values of p for which the equation
$\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{px}\text{ }+\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}$
has real roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given: \[\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{px}\text{ }+\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }-p,\text{ }c\text{...

Find the least positive value of k for which the equation
$\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}$
has real roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given: \[\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }k,\text{ }c\text{...

In the figure (i) given below, DE || BC and the ratio of the areas of ∆ADE and trapezium DBCE is 4 : 5. Find the ratio of DE : BC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, DE || BC The ratio of the areas of ∆ADE and trapezium DBCE is 4 : 5 Now, consider the ∆ABC and ∆ADE ∠A = ∠A … [common angle for both triangles] ∠D = ∠B...

Find the value(s) of p for which the equation
$\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{p}\text{ }=\text{ }\mathbf{0}$
has real roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given: \[\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{p}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }3,\text{ }c\text{...

Find the value(s) of p for which the quadratic equation
$\left( \mathbf{2p}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}{}^\text{2}\text{ }\text{ }\left( \mathbf{7p}\text{ }+\text{ }\mathbf{2} \right)\mathbf{x}\text{ }+\text{ }\left( \mathbf{7p}\text{ }\text{ }\mathbf{3} \right)\text{ }=\text{ }\mathbf{0}$
has equal roots. Also find these roots.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given: \[\left( 2p\text{ }+\text{ }1 \right)x{}^\text{2}\text{ }\text{ }\left( 7p\text{ }+\text{ }2 \right)x\text{ }+\text{ }\left( 7p\text{ }\text{ }3 \right)\text{ }=\text{ }0\] Let us compare...

Find the values of k for which each of the following quadratic equation has equal roots: (i)
$\mathbf{9x}{}^\text{2}\text{ }+\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{x}{}^\text{2}\text{ }\text{ }\mathbf{2kx}\text{ }+\text{ }\mathbf{7k}\text{ }\text{ }\mathbf{12}\text{ }=\text{ }\mathbf{0}$
Also, find the roots for those values of k in each case.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{9x}{}^\text{2}\text{ }+\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }9,\text{ }b\text{ }=\text{ }k,\text{ }c\text{...

Find the value(s) of m for which each of the following quadratic equation has real and equal roots: (i)
$\left( \mathbf{3m}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{2}\left( \mathbf{m}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}\text{ }+\text{ }\mathbf{m}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{2}\left( \mathbf{m}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{x}\text{ }+\text{ }\left( \mathbf{m}\text{ }+\text{ }\mathbf{5} \right)\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\left( \mathbf{3m}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{2}\left( \mathbf{m}\text{ }+\text{ }\mathbf{1} \right)\mathbf{x}\text{ }+\text{...

Find the value (s) of k for which each of the following quadratic equation has equal roots: (i)
$\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{4kx}\text{ }+\text{ }({{\mathbf{k}}^{\mathbf{2}}}~\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2})\text{ }=\text{ }\mathbf{0}$
(ii)
$\left( \mathbf{k}\text{ }\text{ }\mathbf{4} \right){{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}\left( \mathbf{k}\text{ }\text{ }\mathbf{4} \right)\mathbf{x}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{x}{}^\text{2}\text{ }+\text{ }\mathbf{4kx}\text{ }+\text{ }({{\mathbf{k}}^{\mathbf{2}}}~\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2})\text{ }=\text{ }\mathbf{0}\] Let us consider,...

Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots: (i)
$\mathbf{px}{}^\text{2}\text{ }\text{ }\mathbf{4x}\text{ }+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{x}{}^\text{2}\text{ }+\text{ }\left( \mathbf{p}\text{ }\text{ }\mathbf{3} \right)\mathbf{x}\text{ }+\text{ }\mathbf{p}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{px}{}^\text{2}\text{ }\text{ }\mathbf{4x}\text{ }+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }p,\text{ }b\text{ }=\text{ }-4,\text{ }c\text{...

In the adjoining figure, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2 and DP produced meets AB produced at Q. If area of ∆CPQ = 20 cm², find
(i) area of ∆BPQ.
(ii) area ∆CDP.
(iii) area of parallelogram ABCD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a parallelogram. BP: PC = 1: 2 area of ∆CPQ = 20 cm² Construction: draw QN perpendicular CB and Join BN. Then, area of ∆BPQ/area of ∆CPQ =...

Find the nature of the roots of the following quadratic equations: (i)
$\mathbf{x}{}^\text{2}\text{ }~\mathbf{1}/\mathbf{2x}~~\mathbf{1}/\mathbf{2}~=\text{ }\mathbf{0}$
(ii)
$\mathbf{x}{}^\text{2}\text{ }\text{ }\mathbf{2}\surd \mathbf{3x}\text{ }\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$
If real roots exist, find them.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{x}{}^\text{2}\text{ }~\mathbf{1}/\mathbf{2x}~~\mathbf{1}/\mathbf{2}~=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-1/2,\text{ }c\text{ }=\text{...

Discuss the nature of the roots of the following quadratic equations: (iii)
$\text{ }\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$
(iv)
$\mathbf{2}\surd \mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }+\text{ }\surd \mathbf{3}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(iii) \[\text{ }2x{}^\text{2}\text{ }+\text{ }x\text{ }+\text{ }1\text{ }=\text{ }0\] Let us consider, \[a\text{ }=\text{ }-2,\text{ }b\text{ }=\text{ }1,\text{ }c\text{ }=\text{ }1\] By using the...

Discuss the nature of the roots of the following quadratic equations: (i)
$\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{4}\surd \mathbf{3x}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}$
(ii)
$x{}^\text{2}\text{ }~1/2x~+\text{ }4\text{ }=\text{ }0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{4}\surd \mathbf{3x}\text{ }+\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{...

Find the discriminate of the following equations and hence find the nature of roots: (iii)
$16x{}^\text{2}\text{ }\text{ }40x\text{ }+\text{ }25\text{ }=\text{ }0$
(iv)
$\mathbf{2x}{}^\text{2}\text{ }+\text{ }\mathbf{15x}\text{ }+\text{ }\mathbf{30}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(iii) \[16x{}^\text{2}\text{ }\text{ }40x\text{ }+\text{ }25\text{ }=\text{ }0\] Let us consider, \[a\text{ }=\text{ }16,\text{ }b\text{ }=\text{ }-40,\text{ }c\text{ }=\text{ }25\] By using the...

In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find
(i) EF : AD
(ii) area of ∆BEF : area of ∆ABD In the figure
(iii) given below, ABCD is a parallelogram
(iv) area of ∆FEO : area of ∆OBC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O. AE : EB = 2 : 3 (i) We have to find EF : AD So, AB/BE = AD/EF EF/AD =...

Find the discriminate of the following equations and hence find the nature of roots: (i)
$\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{2x}{}^\text{2}\text{ }\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{3x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }3,\text{ }b\text{ }=\text{ }-5,\text{ }c\text{...

In the figure (ii) given below, ABCD is a parallelogram. AM ⊥ DC and AN ⊥ CB. If AM = 6 cm, AN = 10 cm and the area of parallelogram ABCD is 45 cm², find
(i) AB
(ii) BC
(iii) area of ∆ADM : area of ∆ANB.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a parallelogram, AM ⊥ DC and AN ⊥ CB AM = 6 cm AN = 10 cm The area of parallelogram ABCD is 45 cm² Then, area of parallelogram ABCD = DC × AM =...

In the figure (i) given below, ABCD is a trapezium in which AB || DC and AB = 2 CD. Determine the ratio of the areas of ∆AOB and ∆COD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a trapezium in which AB || DC and AB = 2 CD, Then, ∠OAB = ∠OCD … [because alternate angles are equal] ∠OBA = ∠ODC Then, ∆AOB ~ ∆COD So, area of...

In ∆ABC, AP : PB = 2 : 3. PO is parallel to BC and is extended to Q so that CQ is parallel to BA. Find:
(i) area ∆APO : area ∆ABC.
(ii) area ∆APO : area ∆CQO.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, PB = 2: 3 PO is parallel to BC and is extended to Q so that CQ is parallel to BA. (i) we have to find the area ∆APO: area ∆ABC, Then, ∠A = ∠A … [common...

In the given figure, AB and DE are perpendicular to BC.
(i) Prove that ∆ABC ~ ∆DEC
(ii) If AB = 6 cm: DE = 4 cm and AC = 15 cm, calculate CD.
(iii) Find the ratio of the area of ∆ABC : area of ∆DEC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- (i) Consider the ∆ABC and ∆DEC, ∠ABC = ∠DEC … [both angles are equal to 90o] ∠C = ∠C … [common angle for both triangles] Therefore, ∆ABC ~ ∆DEC … [by AA axiom] (ii) AC/CD = AB/DE...

In the given figure, DE || BC.
(i) Prove that ∆ADE and ∆ABC are similar.
(ii) Given that AD = ½ BD, calculate DE if BC = 4.5 cm.
(iii) If area of ∆ABC = 18cm2, find the area of trapezium DBCE

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- (i) From the question it is given that, DE || BC We have to prove that, ∆ADE and ∆ABC are similar ∠A = ∠A … [common angle for both triangles] ∠ADE = ∠ABC … [because corresponding angles...

In the figure (ii) given below, DE || BC and AD : DB = 1 : 2, find the ratio of the areas of ∆ADE and trapezium DBCE.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, DE || BC and AD : DB = 1 : 2, ∠D = ∠B, ∠E = ∠C … [corresponding angles are equal] Consider the ∆ADE and ∆ABC, ∠A = ∠A … [common angles for both...

In the figure (i) given below, DE || BC. If DE = 6 cm, BC = 9 cm and area of ∆ADE = 28 sq. cm, find the area of ∆ABC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, DE || BC, DE = 6 cm, BC = 9 cm and area of ∆ADE = 28 sq. cm From the fig, ∠D = ∠B and ∠E = ∠C … [corresponding angles are equal] Now consider the ∆ADE...

In the figure (ii) given below, AB || DC. AO = 10 cm, OC = 5cm, AB = 6.5 cm and OD = 2.8 cm. (i) Prove that ∆OAB ~ ∆OCD. (ii) Find CD and OB. (iii) Find the ratio of areas of ∆OAB and ∆OCD.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, AB || DC. AO = 10 cm, OC = 5cm, AB = 6.5 cm and OD = 2.8 cm (i) We have to prove that, ∆OAB ~ ∆OCD So, consider the ∆OAB and ∆OCD ∠AOB = ∠COD …...

In the figure, (i) given below, PB and QA are perpendiculars to the line segment AB. If PO = 6 cm, QO = 9 cm and the area of ∆POB = 120 cm², find the area of ∆QOA.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, PO = 6 cm, QO = 9 cm and the area of ∆POB = 120 cm² From the figure, Consider the ∆AOQ and ∆BOP, ∠OAQ = ∠OBP … [both angles are equal to 90o] ∠AOQ =...

The area of two similar triangles are 36 cm² and 25 cm². If an altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other triangle.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, The area of two similar triangles are 36 cm² and 25 cm². Let us assume ∆PQR ~ ∆XYZ, PM and XN are their altitudes. So, area of ∆PQR = 36 cm2 Area of...

∆ABC ~ ∆DEF. If BC = 3 cm, EF = 4 cm and area of ∆ABC = 54 sq. cm. Determine the area of ∆DEF.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∆ABC ~ ∆DEF BC = 3 cm, EF = 4 cm Area of ∆ABC = 54 sq. cm. We know that, Area of ∆ABC/ area of ∆DEF = BC2/EF2 54/area of ∆DEF = 32/42 54/area of ∆DEF =...

Solve the equation
$ (1) <img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-06da6114d4206adf7d19d29c0a04ed2e_l3.png" height="29" width="229" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align} & \mathbf{5x}{}^\text{2}\text{ }\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0} \\ \end{align}" title="Rendered by QuickLaTeX.com"/>$
and give your answer correct to 3 significant figures:

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given equation: \[\mathbf{5x}{}^\text{2}\text{ }\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{4}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }5,\text{ }b\text{ }=\text{ }-3,\text{...

∆ABC ~ DEF. If area of ∆ABC = 9 sq. cm., area of ∆DEF =16 sq. cm and BC = 2.1 cm., find the length of EF.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ∆ABC ~ DEF Area of ∆ABC = 9 sq. cm Area of ∆DEF =16 sq. cm We know that, area of ∆ABC/area of ∆DEF = BC2/EF2 area of ∆ABC/area of ∆DEF = BC2/EF2 9/16 =...

Given that ∆s ABC and PQR are similar. Find: (i) The ratio of the area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3. (ii) the ratio of their corresponding sides if area of ∆ABC : area of ∆PQR = 25 : 36.

CBSE, Class 10, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, (i) The area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3 Then, ∆ABC ~ ∆PQR area of ∆ABC/area of ∆PQR = BC2/QR2 So,...

In the figure (2) given below AD is bisector of ∠BAC. If AB = 6 cm, AC = 4 cm and BD = 3cm, find BC

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, AD is bisector of ∠BAC AB = 6 cm, AC = 4 cm and BD = 3cm Construction, from C draw a straight line CE parallel to DA and join AE ∠1 = ∠2 … [equation...

Solve the following equations and give your answer correct to two significant figures. (i)
${{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{4x}\text{ }\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{x}\text{ }-\text{ }\mathbf{18}/\mathbf{x}\text{ }=\text{ }\mathbf{6}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) Given equation: \[{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{4x}\text{ }\text{ }\mathbf{8}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{...

Solve the following equations by using quadratic formula and give your answer correct to 2 decimal places: (i)
$\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{2x}\text{ }~\mathbf{1}/\mathbf{x}~=\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }4,\text{ }b\text{ }=\text{ }-5,\text{...

Solve the following quadratic equations for x and give your answer correct to 2 decimal places: (i)
$\mathbf{x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0}$
(ii)
${{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{7x}\text{ }=\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{x}{}^\text{2}\text{ }\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{10}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }1,\text{ }b\text{ }=\text{ }-5,\text{ }c\text{...

Solve for x:
$2\left( \frac{2x-1}{x+3} \right)-3\left( \frac{x+3}{2x-1} \right)=5,x\ne -3,1/2$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

So the equation becomes, \[2x\text{ }\text{ }3/x\text{ }=\text{ }5\] By taking LCM \[\begin{array}{*{35}{l}} 2{{x}^{2}}~\text{ }3\text{ }=\text{ }5x \\ 2{{x}^{2}}~\text{ }5x\text{ }\text{ }3\text{...

(i)
$x\text{ }-\text{ }1/x\text{ }=\text{ }3,\text{ }x\text{ }\ne \text{ }0$
(ii)
$1/\text{ }x\text{ }+\text{ }1/\left( x\text{ }-\text{ }2 \right)\text{ }=\text{ }3,\text{ }x\text{ }\ne \text{ }0,\text{ }2$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i)\[x\text{ }-\text{ }1/x\text{ }=\text{ }3,\text{ }x\text{ }\ne \text{ }0\] Let us simplify the given expression, By taking LCM \[\begin{array}{*{35}{l}} {{x}^{2}}~\text{ }1\text{ }=\text{ }3x \\...

(i)
$a\text{ }\left( x{}^\text{2}\text{ }+\text{ }1 \right)\text{ }=\text{ }\left( a{}^\text{2}\text{ }+\text{ }1 \right)\text{ }x,\text{ }a\text{ }\ne \text{ }0$
(ii)
$\mathbf{4x}{}^\text{2}\text{ }\text{ }\mathbf{4ax}\text{ }+\text{ }\left( \mathbf{a}{}^\text{2}\text{ }\text{ }\mathbf{b}{}^\text{2} \right)\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) a (x² + 1) = (a² + 1) x, a ≠ 0 Let us simplify the expression, \[\begin{array}{*{35}{l}} a{{x}^{2}}~+\text{ }a\text{ }\text{ }{{a}^{2}}x\text{ }+\text{ }x\text{ }=\text{ }0 \\ a{{x}^{2}}~\text{...

In the figure (1) given below, AB || CR and LM || QR.
(i) Prove that BM/MC = AL/LQ
(ii) Calculate LM : QR, given that BM : MC = 1 : 2.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, AB || CR and LM || QR (i) We have to prove that, BM/MC = AL/LQ Consider the ∆ARQ LM || QR … [from the question] So, AM/MR = AL/LQ … [equation (i)] Now,...

. (i)
$\frac{x-2}{x+2}+\frac{x+2}{x-2}=4$
(ii)
$\frac{x+1}{x+3}=\frac{3x+2}{2x+3}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

\[\begin{array}{*{35}{l}} D\text{ }=\text{ }{{b}^{2}}~\text{ }4ac \\ =\text{ }{{\left( 0 \right)}^{2}}~\text{ }4\left( 1 \right)\text{ }\left( -12 \right) \\ =\text{ }0\text{ }+\text{ }48 \\...

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at O. Using Basic Proportionality theorem, prove that AO/BO = CO/DO

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, ABCD is a trapezium in which AB || DC and its diagonals intersect each other at O Now consider the ∆OAB and ∆OCD, ∠AOB = ∠COD [because vertically...

(i)
$\mathbf{2x}{}^\text{2}\text{ }+\text{ }\surd \mathbf{5x}\text{ }\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}$
(ii)
$\surd \mathbf{3x}{}^\text{2}\text{ }+\text{ }\mathbf{10x}\text{ }\text{ }\mathbf{8}\surd \mathbf{3}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{2x}{}^\text{2}\text{ }+\text{ }\surd \mathbf{5x}\text{ }\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }\surd 5,\text{...

(i)
$\mathbf{256x}{}^\text{2}\text{ }\text{ }\mathbf{32x}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$
(ii)
$\mathbf{25x}{}^\text{2}\text{ }+\text{ }\mathbf{30x}\text{ }+\text{ }\mathbf{7}\text{ }=\text{ }\mathbf{0}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[\mathbf{256x}{}^\text{2}\text{ }\text{ }\mathbf{32x}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}\] Let us consider, \[a\text{ }=\text{ }256,\text{ }b\text{ }=\text{ }-32,\text{...

In the adjoining given below, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. show that BC || QR.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- Consider the ∆POQ AB || PQ … [given] So, OA/AP = OB/BQ … [equation (i)] Then, consider the ∆OPR AC || PR OA/AP = OC/CR … [equation (ii)] Now by comparing both equation (i) and equation...

(i)
$2x{}^\text{2}\text{ }\text{ }7x\text{ }+\text{ }6\text{ }=\text{ }0$
(ii)
$2x{}^\text{2}\text{ }\text{ }6x\text{ }+\text{ }3\text{ }=\text{ }0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

(i) \[2x{}^\text{2}\text{ }\text{ }7x\text{ }+\text{ }6\text{ }=\text{ }0\] Let us consider, \[a\text{ }=\text{ }2,\text{ }b\text{ }=\text{ }-7,\text{ }c\text{ }=\text{ }6\] So, by using the...

In the give figure, ∠D = ∠E and AD/BD = AE/EC. Prove that BAC is an isosceles triangle.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the given figure, ∠D = ∠E and AD/BD = AE/EC, We have to prove that, BAC is an isosceles triangle So, consider the ∆ADE ∠D = ∠E … [from the question] AD = AE … [sides opposite to...

(a) In the figure (i) given below, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the given figure, CD || LA and DE || AC, Consider the ∆BCA, BE/BC = BD/BA By using the corollary of basic proportionality theorem, BE/(BE + EC) = BD/AB 4/(4 + 2) = BD/AB … [equation...

In figure (ii) given below, AB || DE and BD || EF. Prove that DC² = CF x AC.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the figure it is given that, AB || DE and BD || EF. We have to prove that, DC² = CF x AC Consider the ∆ABC, DC/CA = CE/CB … [equation (i)] Now, consider ∆CDE CF/CD = CE/CB …...

In figure (i) given below, DE || BC and BD = CE. Prove that ABC is an isosceles triangle.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the question it is given that, DE || BC and BD = CE So, we have to prove that ABC is an isosceles triangle. Consider the triangle ABC, AD/DB = AE/EC Given, DB = EC … [equation (i)]...

A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.

CBSE, Class 10, Maths, ML Aggarwal, Similarity

Solution:- From the dimensions given in the question, Consider the ∆PQR So, PQ/PA = 12.5/5 = 2.5/1 PR/PB = (PB + BR)/PB = (4 + 6)/4 = 10/4 = 2.5 By comparing both the results, 2.5 = 2.5 Therefore,...

If x = 3 is a solution of the equation
$\left( \mathbf{k}\text{ }+\text{ }\mathbf{2} \right){{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0}$
, find the value of k. Hence, find the other root of the equation.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given equation: \[\left( \mathbf{k}\text{ }+\text{ }\mathbf{2} \right){{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0}\] And x = 3 is a solution...

If x = p is a solution of the equation
$\mathbf{x}\left( \mathbf{2x}\text{ }+\text{ }\mathbf{5} \right)\text{ }=\text{ }\mathbf{3}$
, then find the value of p.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Quadratic Equations in One Variable

Given that, x = p is a solution of the equation \[\mathbf{x}\left( \mathbf{2x}\text{ }+\text{ }\mathbf{5} \right)\text{ }=\text{ }\mathbf{3}\] Then, upon substituting x = p in must satisfy the...

E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR: of a ∆PQR. For each of the following cases, state whether EF || QR:PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm.

CBSE, Class 10, maths, Maths, ML Aggarwal, Similarity

Solution:- From the dimensions given in the question, Consider the ∆PQR So, PQ/PE = 1.28/0.18 = 128/18 = 64/9 Then, PR/PF = 2.56/0.36 = 256/36 = 64/9 By comparing both the results, 64/9 = 64/9...