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Questions:

Multiple Choice Question

12 x 1 = 12

The area of the quadrilateral is

(a)

6 cm²

(b)

12 cm²

(c)

3 cm²

(d)

8 cm²

The volume of a cylinder of base radius r and height h is

(a)

2\(\pi\) rh

(b)

\(\pi\)r²h

(c)

2\(\pi\)r(r+h)

(d)

\(\frac { 1 }{ 3 } \)\(\pi\)r²h

If 10² = 100, then the square root of 100 is

(a)

(b)

(c)

100

(d)

1000

How many nonsquare numbers lie between the pair of numbers 36² and 37²?

(a)

(b)

(c)

(d)

The coefficient of xy²z in -7x²y³z is

(a)

-7xy

(b)

7xy

(c)

-xy

(d)

xy.

The factorisation of ax²y + bxy²+ cxyz is

(a)

xy(ax + by + cz)

(b)

axy(ax + by + cz)

(c)

bxy(ax + by + cz)

(d)

cxy(ax + by + cz),

Which of the following is the co-efficient of y in -5 xyz?

(a)

-5xz

(b)

-5x

(c)

-5y

(d)

-5

Which of the following is the number of faces of a solid sphere?

(a)

(b)

(c)

many

(d)

none of these

A square board has an area of 121 square units. How long is each side of the board?

(a)

11 units

(b)

12 units

(c)

13 units

(d)

14 units

10.

The cube of 0.08 is

(a)

0.000512

(b)

0.0512

(c)

0.512

(d)

5.12

11.

Multiplicative inverse of a negative rational number is

(a)

(b)

-1

(c)

a negative rational number

(d)

a positive rational number

12.

Which of the following is not true?

(a)

\(\frac{10}{11}+\frac{11}{12}=\frac{11}{12}+\frac{10}{11}\)

(b)

\(\frac{10}{11}\times \frac{11}{12}=\frac{11}{12}\times \frac{10}{11}\)

(c)

\(\frac{10}{11}+ \frac{11}{12}=\frac{11}{12}\div \frac{10}{11}\)

(d)

\(\frac{10}{11}\div \frac{11}{12}=\frac{11}{12}\times \frac{10}{11}\)

Fill up / 1 Marks

5 x 1 = 5

13.

5² = 25, then square root of 25 =______.

14.

The perimeter of a circle and its diameter vary___________with each other.

15.

The large number of observations are usually organised in groups of equal width called _______.

16.

In a ______________ opposite sides are equal, opposite angles are equal and diagonals bisect one another.

17.

Numbers	Associative for
Numbers	Addition	Subtraction	Multiplication	Division
Rational numbers	___________	____________	___________	No

Generic 2

5 x 2 = 10

18.

Is \(\frac{-8}{9} \times (\frac{-4}{7}) =\frac{-4}{7}\times (\frac{-8}{9})?\)

19.

Write each of the following in standard form: 150,000,000,000

20.

Express 2 yr in seconds.

21.

Can we say whether the following numbers are perfect squares? How do we know? 2061
Write five numbers which you can decide by looking at their units digit that they are not square numbers.

22.

Write the additive inverse of the following \(19\over-6\)

Generic 3

10 x 3 = 30

23.

Without finding the prime factors, estimate the cube roots of the following cubes:12167

24.

Is 1008 a perfect square? If not, find the smallest multiple of 1008 which is a perfect square and then find the square root of the new number.

25.

Numbers I to 10 are written on ten separate cards such that one number on one slip. These are mixed well and one slip is chosen from the box without looking into it. What is the probability of
(i) getting a card on which 7 is written?
(ii) getting a card having two-digit number on it?
(iii) getting a number less than 5?
(iv) getting a number more than 5?

26.

Study the following frequency table and answer the questions given below.
(i) What is the class interval?
(ii) Which class has the lowest frequency?
(iii) Which class has the highest frequency?
(iv) Which two classes have the same frequency?

Class interval (Dailv income in rupees)	Frequency (Number of workers)
200-225	25
225-250	38
250-275	42
275-300	45
300-325	17
325-350	23
350-375	45
375-400	120
Total	365

(v) What is the lower limit of the class interval 300-325?
(vi) What is the upper limit of the class interval 275-300?

27.

Divide the following algebraic expressions: r(r⁸ - 1)\(\div\)(r³-r)

28.

Find the cube root of 614125 through estimation.

29.

Solve the following:
\((\frac{2}{3})^{-2}\times(\frac{2}{3})^{5}\)

30.

For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also, find the square root of the square number so obtained.1620

31.

Find the values of the letters in each of the following and give reasons for the steps involved.\(\begin{matrix} \quad \quad 1\quad 2\quad A \\ +\quad 6\quad A\quad B \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad A\quad 0\quad 9 \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)

32.

Solve \(-5+{7\over10}+{3\over7}+(-3)+{5\over14}+{-4\over5}\)

1 Marks

4 x 1 = 4

33.

Regroup the terms and factorise: z - 19 + 19xy - xyz

34.

Without calculating square roots, find the number of digits in the square root of the number 36481.

35.

Read the following circle graphs and answer the questions given below:
(a) The time spent by a child during a day:
(i) On which activity maximum number of hours are spent?
(ii) On which two activities does he spend equal number of hours?
(iii) Find the central angles for each sector of activities

(b) Age group of people in a town:
(i) In which age group are the maximum number of people?
(ii) How many people are there is the '0-14 years' group?
(iii) Find the central angle of the sector corresponding to the age group 15-60 years'.

36.

Write the additive inverse of the following: \(\frac { -21 }{ 112 } \)

Generic 4

1 x 4 = 4

37.

Take a clock and fix its minute hand at 12.
Record the angle turned through by the minute hand from its original position and the time that has passed, in the following table:

Time Passed (T) (in minutes)	(T₁) 15	(T₂) 15	(T₃) 45	(T₄) 60
Angle turned (A) (in degree)	(A₁) 90	(A₂) __	(A₃) __	(A₄) __
\(T\over A\)	-	-	-	-

What do you observe about T and A? Do they increase together? Is \(T\over A\) same every time?

Is the angle turned through by the minute hand directly proportional to the time that has passed? Yes; From the above table, you can also see
T₁ : T₂ = A₁ A₂, because
T₁ : T₂ = 15 30= 1 :2
A₁ : A₂ = 90 180 = 1 :2
Check if T₂: T₃ = A₂ A₃ and T₃ : T₄ = A₃ : A₄
You can repeat this activity by choosing your own time interval.

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Answers Key:

Multiple Choice Question

12 x 1 = 12

Area = \(\frac { 4\times (1+2) }{ 2 } \)=6 cm²

(b)
\(\pi\)r²h

\(\sqrt{100}=10\)

2 x 36 = 72.

- 7x²y³z = (- 7xy)(xy²z).

ax²y + bxy² + cxyz = xy (ax + by + cz)

(a)
-5xz

(a)
1

(a)
11 units

10.

11.

12.

Fill up / 1 Marks

5 x 1 = 5

13.

()
5

14.

()
Directly

15.

For large number of observations, groups of equal width i.e. class interval are organised.

16.

()
Parallelogram

17.

()

Numbers

Associative for

Addition

Subtraction

Multiplication

Division

Rational numbers

Yes \(e.g {-1\over2} + [{3\over7}+{-4\over3}]\\ =[{-1\over2}+{3\over7}]+({-4\over3}) \\ \Rightarrow {-1\over2}+({9-28\over21}) \\ = ({-7+6\over14}) -{4\over3} \\ \Rightarrow - {1\over2}-{19\over21}={-1\over14}-{4\over3} \\ \Rightarrow{-59\over42}={-59\over42},\)
which is true

No
\(e.g {-2\over3} -({-4\over5}-{1\over2}) \\ \neq [{-2\over3}-({-4\over5})]-{1\over2}\\ \Rightarrow{-2\over3} - ({-8-5\over10})\neq [{-10+12\over15}]-{1\over2} \\ \Rightarrow -{2\over3}+{13\over10}\neq {2\over15}-{1\over2}\\ \Rightarrow {-20+39\over30}\neq {4-15\over30}\Rightarrow {19\over30}\neq {-11\over30}\)
which is not true.

Yes e.g
\({2\over3} \times({-6\over7}\times{4\over5})\\ =({2\over3}\times {-6\over7})\times {4\over5}\\ \Rightarrow {2\over3} \times ({-24\over35}) \\ {-12\over21} \times{4\over5} \\ \Rightarrow {-16\over35}={-16\over35},\)
which is true.

No
\({1\over2}\div[{-1\over3}\div {2\over5}]\neq [{1\over2}\div({-1\over3})] \div {2\over5}\\ \Rightarrow {1\over2}\div [-{1\over3}\times {5\over2}] \\ \neq [ {1\over2}\times(-{3\over1})]\div {2\over5}\\ \Rightarrow {1\over2}\times (-{6\over5})\neq [{-3\over2}] \times{5\over2}\\ \Rightarrow {-3\over5} \neq {-15\over4}\)

which is not true

Generic 2

5 x 2 = 10

18.

\(\frac{-8}{9} \times (\frac{-4}{7}) = \frac{32}{63}\)
\(\frac{-4}{7} \times (\frac{-8}{9})=\frac{32}{63}\)
So, \(\frac{-8}{9}\times (\frac{-4}{7})=\frac{-4}{7} \times (\frac{-8}{9})\)

19.

1.5 \(\times\) 10¹¹

20.

6.3072 x 10⁷ s

21.

We know that, a number ends with 2, 3, 7 or 8 is neverapeIfea square.
The number 2061 ends with 1, so it may or may not be a perfect square.
But 45²= 45\(\times\)45 = 2025, 46² = 46\(\times\)46 = 2116
So, 2061 is not a perfect square
Five numbers for which we can decide by looking at their one's digit that they are not square numbers are 268, 43112, 87547, 1233 and 193.

22.

we have, \(19\over-6\) so, additive inverse of \(19\over-6\) is \(19\over6\)

Generic 3

10 x 3 = 30

23.

24.

We have 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7
As the prime factor 7 has no pair.
\(\therefore\)1008 is not a perfect square.
Obviously, if 7 gets a pair, then the number will become a perfect square.
\(\therefore\)1008 x 7
= [2 x 2 x 2 x 2 x 3 x 3 x 7] x 7
or 7056 = 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7
Thus, 7056 is the required multiple of 1008
which is a perfect square.
Now, \(\sqrt{7056}\) = 2 x 2 x 3 x 7 = 84

25.

Total number of outcomes = 10
(i) ∴ One card is having 7 on it
∴ Number of favourable outcome =
⇒ Probability of getting a card having 7 on it = \(1\over10\)
(ii) ∵ Only one card IS having two digit number (i.e., 10)
∴ Number of favourable outcome =
⇒ Probability of getting a card having 10
(two diizgiit number) on it = 1/10
(iii) ∴ The numbers less than 5 are I, 2, 3,
and 4
∴ Number of favourable outcomes = 4
⇒ Probability of getting a number less then \(5={4\over10}\ or\ {2\over5}\)
(iv) The numbers more than 5 are 6, 7, 8, 9 and 10.
∴ Number of favourable outcomes = 5
⇒ Probability of getting a number more than \(5={5\over10}\ or\ {1\over5}\)

26.

(i) The class interval = [Upper limit of a class] - [Lower limit of the same class] = 225 - 200 = 25
(ii) The class 300 - 325 is having the lowest frequency (which is 17).
(iii) The class 375 - 400 is having the highest frequency (which is 120).
(iv) The class intervals 275-300 and 350 - 375 have the same frequency (which is 45).
(v) The lower limit of the class interval 300 - 325 is 300.
(vi) The upper limit of the class interval 275 - 300 is 300.

27.

(r² +1)(r⁴+1)

28.

Given, number is 614125.
So, groups of 614125 are

In the first group, the number 125 ends with 5. We know that, 5 comes at the unit's place of a number only when its cube ends in 5.
So, 5 will come at unit's place.
Now, in the second group, the number is 614
∵ (8)³ = 512 and (9)³=729
and 512 < 614 < 729
So, ten's place of required cube root is 8.
Hence, \(\sqrt [ 3 ]{ 614125 } \) = 85

29.

\((\frac{2}{3})^{-2}\times(\frac{2}{3})^{5}\)=\((\frac{2}{3})^{-2+5}\)=\((\frac{2}{3})^{3}=\frac{(2)^{3}}{(3)^{3}}\)
=\(\frac{2\times2\times2}{3\times3\times3}=\frac{8}{27}\)

30.

The prime factorisation of 1620 is
1620 = 2\(\times\)2\(\times\)3\(\times\)3\(\times\)3\(\times\)3\(\times\)5
By pairing the prime factors, we get
1620 =\(\underline { 2\times 2 } \)\(\times\)\(\underline { 3\times 3 } \)\(\times\)\(\underline { 3\times 3 } \)\(\times\)5
We see that the prime factor 5 has no pair. So, if we divide 1620 by 5, then we get
1620 ÷ 5 =\(\underline { 2\times 2 } \)\(\times\)\(\underline { 3\times 3 } \)\(\times\)\(\underline { 3\times 3 } \)
Now each factor has a pair. Therefore, 1620 ÷ 5 = 324 is a perfect square. Thus, the required smallest number is 5
Hence, \(\sqrt { 324 } \)= 2\(\times\)3\(\times\)3= 18

31.

\(\begin{matrix} \quad \quad 1\quad 2\quad A \\ +\quad 6\quad A\quad B \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad A\quad 0\quad 9 \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)
Here, we have two letters A and B whose values are to be found.
Studying the addition in ten's column. We have 2 + A and we get 0 from this. Therefore, A must be 8, since A will be one digit number, then the puzzle becomes
\(\begin{matrix} \quad \quad 1\quad 2\quad 8 \\ +\quad 6\quad 8\quad B \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad 8\quad 0\quad 9 \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)
Now, studying the addition in one's column. We have 8 + B and we get 9 from this. So, B must be 1.
Thus, the puzzle is solved as shown below:
\(\begin{matrix} \quad \quad 1\quad 2\quad 8 \\ +\quad 6\quad 8\quad 1 \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad 8\quad 0\quad 9 \\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \end{matrix}\)
Hence, A = 8 and B = 1.

32.

\(-256\over35\)

1 Marks

4 x 1 = 4

33.

()
(xy-1)(19-z)

34.

()
3

35.

()
(i) Sleep (ii) Play and others
(iii) Sleep 120°, school 90°, home work 60°; play 45°, others 45°

36.

()
The additive inverse of\(\frac { 21 }{ 112 } \) is \(\frac { -21 }{ 112 } \)

Generic 4

1 x 4 = 4

37.

Time Passed (T) (in minutes)	(T₁) 15	(T₂) 15	(T₃) 45	(T₄) 60
Angle turned (A) (in degree)	(A₁) 90	(A₂) __	(A₃) __	(A₄) __
\(T\over A\)	\({15\over 90}={1\over6}\)	\({30\over 180}={1\over6}\)	\({45\over 270}={1\over6}\)	\({60\over 360}={1\over6}\)

We observe about T and A that they increase together and is \(T\over A\) same every time.
Yes; The angle turned by the minute hand is directly proportional to the time that has passed.
On checking, we find that
T₂ : T₃ = A₁ : A₃ = 2: 3
and T₃ : T₄ = A₃ : A₄ = 3 : 4

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