9th Standard Mathematics All Chapter Important Creative Questions-II-2019-2020

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Questions:

Generic 3

14 x 3 = 42

When a dice is rolled, find the probability to get the number greater than 4?

(i) If cosec A = sec 34⁰, then find A
(ii) If tan B = cot 47⁰, then find B.

The sides of a triangular park are in the ratio 9:10:11 and its perimeter is 300 m. Find the area of the triangular park.

Verify A -(BUC) = (A-B)∩(A-C) using Venn diagrams.

If A = {2,5,6,7} and B = {3,5,7,8}, then verify the commutative property of union sets

In (5x+4) a factor of 5x³+ 14x²- 32x -32

Express the following surds in its simple form \(\sqrt [ 4 ]{ 324 } \)

Find the Median of the given data: 36, 44, 86, 31, 37, 44, 86, 35, 60, 51

Express in scientific notation:
(i) 9768854
(ii) 0.04567891
(iii) 72006865.48

10.

In the figure find x⁰ and y⁰.

11.

Three vertices of a rectangle are (3, 2), (-4, 2) and (-4, 5). Plot the points and find the coordinates of the fourth vertex.

12.

Show that the following points taken in order form the vertices of a parallelogram.
A(–3, 1), B(–6, –7), C (3, –9) and D(6, –1)

13.

Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial
p(x) = 7x²+ 6x -1 q(x) = 6x - 9

14.

Consider the given pairs of triangles and say whether each pair is that of congruent triangles. If the triangles are congruent, say ‘how’; if they are not congruent say ‘why’ and also say if a small modification would make them congruent:

Generic 2

25 x 2 = 50

15.

Two unbiased coins are tossed simultaneously find the probability of getting
(i) two heads
(ii) one head
(iii) at least one head
(iv) at most one head

16.

From the given figure, find all the trigonometric ratios of angle \(\theta\).

17.

Find the surface area of a cube whose edge is
(i) 27 cm
(ii) 3 cm
(iii) 6 cm
(iv) 2.1 cm

18.

A car travels, at an uniform speed. At 2 pm it is at a distance of 5 km at 6 pm it is at a distance of 120 km. Using section formula, find at what distance it will reach 2 midnight.

19.

Solve using the method of substitution.
5x - y = 5, 3x +y = 11

20.

The dimensions of a match box are 6 cm × 3.5 cm × 2.5 cm. Find the volume of a packet containing 12 such match boxes.

21.

Frame two problems in calculating probability, based on the spinner shown here.

22.

Find the value of the following:
\(\left( \frac { cos47° }{ sin43° } \right) +\left( \frac { sin72° }{ cos18° } \right) -2\cos^{ 2 }45°\)

23.

Find the coordinates of the point which divides the line segment joining the points A(4,−3) and B(9,7) in the ratio 3:2.

24.

Find any 3 irrational numbers between 0.12 and 0.13.

25.

Express the following in the form 3ⁿ: \(\sqrt { 27 } \)

26.

Find the value of X^o

27.

The radius of a circle 15 cm and the length of one of its chord is 24 cm. Find the distance of the chord from the centre.

28.

In a week, temperature of a certain place is measured during winter are as follows 26^oC, 24^oC, 28^oC, 31^oC, 30^oC, 26^oC, 24^oC. Find the mean temperature of the week.

29.

Find the mode for the set of values 17, 18, 20, 20, 21, 21, 22, 22.

30.

In the given figure, AB and CD are the parallel chords of a circle with centre O. Such that AB = 8cm and CD = 6cm. If OM ⊥ AB and OL⊥CD distance between LM is 7cm. Find the radius of the circle?

31.

In the given figure, ABCD is a cyclic quadrilateral where diagonals intersect at P such that ㄥDBC = 40° and ㄥBAC = 60°

find (i) ㄥCAD (ii) ㄥBCD

32.

Expand the following: (x+2y+3z)²

33.

Express the following in the form 2ⁿ:
\(\sqrt{8}\)

34.

Without actual division classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.
\({17\over 200}\)

35.

Let U= {x : -3 : < x < 4} A = {-1,2,3} B = {0,1,2,3} C = {-3;-2,-1,0,1,2}. Find (i) A' UB' (ii) (A ∩ B)' (iii) (A ⋂ C)'

36.

Write the following in "Roster" form?
(a) A = set of the months having 31 days.
(b) B = {x : x is a natural number of 2 digits divisible by 13}
(c) C = {set of vowels in the word "father"}
(d) D = {x : 5 < x < 10 ; x ∈ N}
(e) E = {x: x is a square natural number less than 16}

37.

Using the given Venn diagram, write the elements of A∪B

38.

If S = {square, rectangle, circle, rhombus, triangle}, list the elements of the following subset of S.
The set of shapes in which the sum of all interior angles is 180⁰.

39.

Write the coefficient of x²and x in each of the following polynomials \(6-{ 2x }^{ 2 }+3x^{ 3 }-\sqrt { 7 } x\)

Multiple Choice Question

40 x 1 = 40

40.

The total surface area of a cuboid with dimension 10 cm × 6 cm × 5 cm is _______.

(a)

280 cm²

(b)

300 cm²

(c)

360 cm²

(d)

600 cm²

41.

If the lateral surface area of a cube is 600 cm², then the total surface area is _______.

(a)

150 cm²

(b)

400 cm²

(c)

900 cm²

(d)

1350 cm²

42.

If A is any event in S and its complement is A' then, P(A′) is equal to _______.

(a)

(b)

(c)

1-A

(d)

1-P(A)

43.

The probability of an event cannot be _______.

(a)

Equal to zero

(b)

Greater than zero

(c)

Equal to one

(d)

Less than zero

44.

The value of \(\frac { sin{ 29 }^{ 0 }31' }{ cos{ 60 }^{ 0 }29' } \) is

(a)

(b)

(c)

(d)

-1

45.

The value of tan72° tan18° is ________.

(a)

(b)

(c)

18⁰

(d)

72⁰

46.

Which of the following is a linear equation.

(a)

\(x+\frac { 1 }{ x } =2\)

(b)

x(x −1) = 2

(c)

3x+5=\(\frac { 2 }{ 3 } \)

(d)

x³ - x =5

47.

In what ratio does the y-axis divides the line joining the points (−5, 1) and (2, 3) internally ______.

(a)

1 :3

(b)

2 :5

(c)

3 :1

(d)

5 :2

48.

\(n(A\cup B\cup C)\)=________

(a)

n(A) + n(B) + n(C)

(b)

n(A) + n(b) + n(C) -\(n\left( A\cap B\cap C \right) \)

(c)

\(n\left( A\cap B\cap C \right) \)

(d)

n(A) + n(B) + n(C) -\(n\left( A\cap B \right) \)-\(n\left( B\cap A \right) -n\left( A\cap C \right) +n\left( A\cap B\cap C \right) \)

49.

\(\left[ n\left( A\cup B\cup C \right) ^{ ' } \right] \)=_______

(a)

\(n\left( A\cap B\cap C \right) \)

(b)

\(n\left( U \right) -n\left( A\cup B\cup C \right) \)

(c)

n(U)

(d)

\(\Phi \)

50.

Degree of the linear polynomial is ________

(a)

(b)

(c)

(d)

51.

The polynomial whose factors are (x+2)(x+3)is_____

(a)

x²+ 5x + 6

(b)

x²- 4

(c)

x²- 9

(d)

x²+ 6x + 5

52.

If x - 2 is a factor of q(x), then the remainder is___________

(a)

q(-2)

(b)

x - 2

(c)

(d)

-2

53.

Rationalising the denominator \(\cfrac { 1 }{ \sqrt [ 3 ]{ 3 } } \) ___________

(a)

(b)

\(\cfrac { { 3 }^{ \frac { 2 }{ 3 } } }{ 3 } \)

(c)

\(\sqrt { 3 } \)

(d)

\(\sqrt [ 3 ]{ 3 } \)

54.

\(\sqrt [ 4 ]{ 405 } =h\sqrt [ 4 ]{ 5 } \), then h = ____________

(a)

(b)

(c)

(d)

55.

The angle subtend by equal chords of a circle at the centre is________

(a)

Complementary

(b)

Supplementary

(c)

equal

(d)

unequal

56.

The mean of the square of first 11 natural number is ___________

(a)

(b)

(c)

(d)

57.

The mean of a, b, c, d and e is 28. If the mean of a, c and e is 24, then mean of b and d is _______________

(a)

(b)

(c)

(d)

58.

The algebraic sum of the deviations of a set of n values from their mean is ___________

(a)

(b)

n-1

(c)

(d)

n+1

59.

The mean of set of seven number is 81. If one of the nimbers is discarded,the mean of remaining number is 78. The value of discarded number is

(a)

101

(b)

100

(c)

(d)

60.

Find the mean of the prime factors of 165.

(a)

(b)

(c)

(d)

61.

The median of the first 10 whole numbers is ______________

(a)

(b)

4.5

(c)

(d)

5.5

62.

In the given figure, If OP = 17cm PQ = 30, cm and OS is perpendicular to PQ, then RS is ________.

(a)

10 cm

(b)

6 cm

(c)

7 cm

(d)

9 cm

63.

In the figure, ABCD is a cyclic quadrilateral in which DC produced to E and CF is drawn parallel to AB such that ㄥADC = 80° and ㄥECF = 20°, then ㄥBAD = ?

(a)

100°

(b)

20°

(c)

120°

(d)

110°

64.

The roots of the polynominal equation \({ x }^{ 2 }+2x=0\) are_______________

(a)

x = 0, 2

(b)

x = 1, 2

(c)

x = 1, -2

(d)

x = 0, -2

65.

ABCD is a parallelogram as shown. Find x and y.

(a)

1, 7

(b)

2, 6

(c)

3, 5

(d)

4, 4

66.

In a parallelogram \(\angle{A}:\angle{B}=1:2\) Then ㄥA ............

(a)

30°

(b)

60°

(c)

45°

(d)

90°

67.

What is the name of a regular polygon of six sides?

(a)

Square

(b)

Equilateral triangle

(c)

Regular hexagon

(d)

Regular octagon

68.

The set does not have a proper subset is __________

(a)

Finite set

(b)

Infinite set

(c)

Null set

(d)

Singleton set

69.

The rational number lying between \(\frac { 1 }{ 5 } \) and \(\frac { 1 }{ 2 } \) is___________.

(a)

\(\frac { 7 }{ 20 } \)

(b)

\(\frac { 2 }{ 10 } \)

(c)

\(\frac { 2 }{ 7 } \)

(d)

\(\frac { 3 }{ 10 } \)

70.

The centre of a circle is (0, 0). One end point of a diameter is (5, -1), then ______________

(a)

\(\sqrt{24}\)

(b)

\(\sqrt{37}\)

(c)

\(\sqrt{26}\)

(d)

\(\sqrt{17}\)

71.

The distance between the points (a, 0) and (0, b) is____________

(a)

a unit

(b)

b unit

(c)

\(\sqrt{a^2+{b^2}}\ unit\)

(d)

\(\sqrt{a^2-{b^2}}\ unit\)

72.

A point which lies in the III quadrant is__________________

(a)

(5, 4)

(b)

(5, - 4)

(c)

(-5, - 4)

(d)

(-5,4)

73.

On which quadrant does the point (- 4, 3) lie?

(a)

(b)

(c)

III

(d)

74.

The distance between the point ( 5, –1 ) and the origin is _______.

(a)

\(\sqrt { 24 } \)

(b)

\(\sqrt { 37 } \)

(c)

\(\sqrt { 26 } \)

(d)

\(\sqrt { 17 } \)

75.

An irrational number between 2 and 2.5 is ________.

(a)

\(\sqrt { 11 } \)

(b)

\(\sqrt { 5 } \)

(c)

\(\sqrt { 2.5 } \)

(d)

\(\sqrt { 8 } \)

76.

Which one of the following has a terminating decimal expansion?

(a)

\(\frac { 5 }{ 64 } \)

(b)

\(\frac { 8 }{ 9 } \)

(c)

\(\frac { 14 }{ 15 } \)

(d)

\(\frac { 1 }{ 12 } \)

77.

The root of the polynomial equation 2x + 3 = 0 is ________.

(a)

\(\frac{1}{3}\)

(b)

\(-\frac{1}{3}\)

(c)

\(-\frac{3}{2}\)

(d)

\(-\frac{2}{3}\)

78.

From the adjacent diagram n[P(AΔB)] is ________.

(a)

(b)

(c)

(d)

79.

If A∪B = A∩B, then ________.

(a)

A ≠ B

(b)

A = B

(c)

A ⊂ B

(d)

B ⊂ A

Generic 5

9 x 5 = 45

80.

A farmer has a field in the shape of a rhombus. The perimeter of the field is 400 m and one of its diagonal is 120 m. He wants to divide the field into two equal parts to grow two different types of vegetables. Find the area of the field.

81.

In a school, 80 students like Maths,90 students like Science 82 students like History, 21 like both Maths and Science 19 like both science and History 20 like both Maths and History and 8 liked all the three subjects. If each student like atleast one subject, then find
(i) the number of students in the school
(ii)the number of students who like only one subject.

82.

Factorise 2x³- x² - 12x - 9 into linear factors

83.

The following are the scored by the students in the Summative Assessment exam

Class	0-10	10-20	20-30	30-40	40-50	50-60
No.of students	2	7	15	10	11	5

84.

In the class, weight of students is measured for the class records. Caculate mean weight of the students using direct method.

Weight in kg	15-25	25-35	35-45	45-55	55-65	56-75
No.of students	4	11	19	14	0	2

85.

Represent \(-\frac { 2 }{ 11 } ,-\frac { 5 }{ 11 } and-\frac { 9 }{ 11 } \)on the number line.

86.

Find any three rational numbers between \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 5 } \)

87.

Diagonal AC of a parallelogram ABCD bisects ㄥA. Show that
(i) it bisects ㄥC also
(ii) ABCD is a rhombus.

88.

Show that the point A (3,7) B (6, 5) and C (15, -1) are collinear.

Generic 8

1 x 8 = 8

89.

Construct the centroid of \(\triangle\)PQR such that PQ = 9 cm, PQ = 7cm, RP = 8 cm.

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Answers Key:

Generic 3

14 x 3 = 42

Sample space S = {1, 2, 3, 4, 5, 6}
Let E be the event of getting a number greater than 4
E = {5, 6}
\(P(E)=\frac { Number\ of\ favourable\ outcomes }{ Total\ number\ of\ outcomes } \)
\(P(E)=\frac { n(E) }{ n(S) } =\frac { 2 }{ 6 } =0.333...\)

(i) We know that cosec A = sec(90⁰ A)
sec(90⁰ - A) sec(34⁰)
90⁰ - A =34⁰
We get A = 90° − 34°
A = 56⁰
(ii) We know that tan B = cot(90⁰ - B)
cot(90⁰ - B) = cot 47⁰
90⁰ - B = 47⁰
We get B = 90° − 47°
B = 43⁰

Given the sides are in the ratio 9:10:11, let the sides be 9k, 10k, 11k
The perimeter of the triangular park = 300 m
9k +10k +11k = 300m
30k = 300
k = 10m
Therefore, the sides are a = 90 m, b = 100 m, c = 110 m
s=\(\frac{1+b+c}{2}=\frac{90+100+110}{2}=\frac{300}{2}\)= 150 m
Hence, Area of triangular park = \(\sqrt { s(s-a)(s-b)(s-c) } \)
= \(\sqrt { 150\times (150-90)(150-100)(150-110) } \)
= \(\sqrt { 150\times 60\times 50\times 40 } \)
= \(\sqrt { 3\times 50\times 20\times 3\times 50\times 2\times 20 } \)
= 50 \(\times\) 20 \(\times\) 3\(\sqrt{2}\)
= 3000 \(\times\)1.414 = 4242 m²

From (1) and (2), we get A -(BUC) = (A-B)∩(A-C). Hence it is verified.

Given, A = {2,5,6,7} and B = {3,5,7,8}
\(A\cup B\) = {2,3,5,6,7,8} ...........(1)
\(B\cup A\) = {2,3,5,6,7,8} ..............(2)

Let p(x) = 5x³+ 14x²- 32x - 32
By factor theorem, 5x + 4 is a factor, if \(p\left( \cfrac { -4 }{ 5 } \right) \)= 0
\(p\left( \cfrac { -4 }{ 5 } \right) =5\left( \cfrac { -4 }{ 5 } \right) ^{ 3 }+14\left( \cfrac { -4 }{ 5 } \right) ^{ 2 }-\left( 32\left( \cfrac { -4 }{ 5 } \right) \right) -32\)
= \(5\left( \cfrac { -64 }{ 125 } \right) +14\left( \cfrac { 16 }{ 25 } \right) +32\left( \cfrac { 4 }{ 5 } \right) -32\)
=\(\cfrac { -64 }{ 25 } +\cfrac { 224 }{ 25 } +\cfrac { 128 }{ 5 } -32\) = \(\cfrac { -65 }{ 25 } +\cfrac { 224 }{ 25 } +\cfrac { 640 }{ 25 } -\cfrac { 800 }{ 25 } \)
=\(\cfrac { -65+224+640-800 }{ 25 } \) = 0
\(p\left( \cfrac { -4 }{ 5 } \right) \) = 0
Therefore,5x+4 is a factor 5x³+ 14x²- 32x - 32

\(\sqrt [ 4 ]{ 324 } \) =\(\sqrt [ 4 ]{ 81\times 4 } =\sqrt [ 4 ]{ { 3 }^{ 4 }\times 4 } =\sqrt [ 4 ]{ { 3 }^{ 4 } } \times \sqrt [ 4 ]{ 4 } \) \(\left[ \because \sqrt [ n ]{ a^{ n } } \times \sqrt [ n ]{ b } =\sqrt [ n ]{ ab } \right] \)
= \(3\times \sqrt [ 4 ]{ 4 } \) \(\left[ \because \sqrt [ n ]{ a^{ n } } =a \right] \)
order = 4; radicand = 4; Coefficient = 3

Values in ascending order
31, 35, 36, 37, 44, 44, 51, 60, 86, 86
Number of terms = 10 which is even
Median = Average of \(\left[\frac{N}{2}\right]^{\text {th }} \text { term }+\left(\frac{n}{2}+1\right)^{\text {th }} \text { term }\)
= Average of [5^th term + 6^th term]
\(=\frac{44+44}{2}=44\)
Median = 44

(i)
The decimal point is to be moved six places to the left. Therefore n = 6.
(ii)
The decimal point is to be moved two places to the right. Therefore n = −2
(iii)
The decimal point is to be moved seven places to the left. Therefore n = 7

10.

ㄥACD = ㄥA + ㄥB
(An exterior angle of a triangle is sum of its interior opposite angles)
120° = 50° + x⁰
x⁰ = 120° - 50°
= 70°
In the the triangle ABC
ㄥA + ㄥB + ㄥACB = 180° (Sum of the angles of a Δ)
50° +x + ㄥACB = 180°
50° + 70° + ㄥACB = 180°
ㄥACB = 180° - 120°
y = 60° (OR)
ㄥACD + ㄥACB = 180⁰(Angles of a linear pair)
ㄥACB = 180° - 120°
= 60°
The value ofx = 70° andy = 60°.

11.

(3, 5)

12.

Distance =\(\sqrt { ({ x }_{ 2 }-{ x }_{ 1 })^{ 2 }+(y_{ 2 }-y_{ 1 })^{ 2 } } \)
AB =\(\sqrt { (-6+3)^{ 2 }+(-7-1)^{ 2 } } \)
= \(\sqrt { (-3)^{ 2 }+(-8)^{ 2 } } =\sqrt { 9+64 } =\sqrt { 73 } \)
BC = \(\sqrt { (3+6)^{ 2 }+(-9+7)^{ 2 } } \)
= \(\sqrt { { 9 }^{ 2 }+(-2)^{ 2 } } =\sqrt { 81+4 } =\sqrt { 85 } \)
CD = \(\sqrt { (6-3)^{ 2 }+(-1+9)^{ 2 } } \)
= \(\sqrt { (3)^{ 2 }+({ 8 })^{ 2 } } =\sqrt { 9+64 } =\sqrt { 73 } \)
AD = \(\sqrt { (6+3)^{ 2 }+(-1-1)^{ 2 } } \)
= \(\sqrt { (9)^{ 2 }+(-2)^{ 2 } } =\sqrt { 81+4 } =\sqrt { 85 } \)
AB = CD =\(\sqrt { 73 } \) and BC = AD = \(\sqrt { 85 } \)(Opposite sides sre equal)
∴ ABCD is a parallelogram.

13.

p(x) - q(x) = 7x² + 6x - 1 - (6x - 9)
= 7x² + 6x - 1 - 6x + 9
= 7x² + 6x - 6x - 1 + 9
= 7x²+8
The degree of the polynomial is 2

14.

In the given diagram
AB = CD (Given)
BD is common.
\(\angle ABD=\angle BDC\) (alternate angles)
(Since AC and CD are parallel and BD is the transversal)
By SAS congruency
\(\therefore\triangle ABD\cong\triangle CDB\)

Generic 2

25 x 2 = 50

15.

S = {HH, HT, TH, TT}
(i) probability of two heads = \(\frac { 1 }{ 4 } \)
(ii) probability of one head = \(\frac { 1 }{ 2 } \)
(iii) probability of at least one head = \(\frac { 3 }{ 4 } \)
(iv) probability of at most one head = \(\frac { 3 }{ 4 } \).

16.

\(x=\sqrt { { 13 }^{ 2 }-{ 12 }^{ 2 } } \)
\(\sqrt { 169-144 } =\sqrt { 25 } =5\)
\(sin\theta =\cfrac { 5 }{ 13 } ;cos\theta =\cfrac { 12 }{ 13 } ;tan\cfrac { 5 }{ 12 } ;cosec\theta =\cfrac { 13 }{ 5 } ;\)
\(sec\theta =\cfrac { 13 }{ 12 } ;cot\theta =\cfrac { 12 }{ 5 } \)

17.

(i) TSA = 6a² = 6 \(\times\) 27² = 6 \(\times\) 729 = 4374 cm²
(ii) TSA = 6a² = 6 \(\times\)3² = 54 cm²
(iii) TSA = 6a² = 6 \(\times\) 6²= 6 \(\times\) 36 = 216 cm²
(iv) TSA = 6a² = 6 \(\times\) 2.1² = 6 \(\times\) 4.41 = 26.46 cm²

18.

\(120=\frac { 8(50)+4(y) }{ 12 } \)
1440 = 400 + 4y
4y = 1040
\(y=\frac { 1040 }{ 4 } =260\ km\)

19.

5x - y = 5 -------------(1)
3x +y = 11 ----------(2)
(1) \(\Rightarrow\) y = 5x-5
(2) \(\Rightarrow\) 3x + 5x - 5 = 11
8x = 11+5
8x = 16
\(x=\cfrac { 16 }{ 8 } =2\)
(1) \(\Rightarrow\) 5 (2) - y = 5 \(\Rightarrow\) 10 - 5 =y \(\Rightarrow\) y = 5
\(\therefore\) Solution is x = 2, y = 5

20.

Dimensions of a match box = 6 cm \(\times\) 3.5 cm \(\times\) 2.5 cm
V = (6 \(\times\) 3.5 \(\times\) 2.5) cm³ = 52.5 cm³
Volume of 12 such boxes = 12 \(\times\) 52.5 cm³ = 630 cm³.

21.

(i) What is the probability that the spinner will not land on a multiple of 2?
(ii) What is the probability that the spinner will land on an odd number?

22.

\(\left( \frac { cos47° }{ sin43° } \right) +\left( \frac { sin72° }{ cos18° } \right) -2cos^{ 2 }45°\)
= \(\left( \cfrac { cos\left( { 90 }^{ 0 }-{ 43 }^{ 0 } \right) }{ { sin43 }^{ 0 } } \right) ^{ 2 }+\left( \cfrac { sin\left( { 90 }^{ 0 }-{ 18 }^{ 0 } \right) }{ cos18 } \right) ^{ 2 }-2{ cos }^{ 0 }{ 45 }^{ 0 }\)
= \(\left( \cfrac { sin43 }{ sin43 } \right) ^{ 2 }+\left( \cfrac { cos18 }{ cos18 } \right) ^{ 2 }-2\left( \cfrac { 1 }{ \sqrt { 2 } } \right) ^{ 2 }\)

23.

x₁ y₁ x₂ y₂ m : n
A (4, -3), B (9, 7), 3 : 2
By section formula \(P\left( \frac { { mx }_{ 2 }+{ nx }_{ 1 } }{ m+n } ,\frac { { my }_{ 2 }+{ ny }_{ 1 } }{ m+n } \right) =P(x,y) \)

\(P(x,y)=\left( \frac { 3(9)+2(4) }{ 3+2 } ,\frac { 3(7)+2(-3) }{ 3+2 } \right) \)
\(=\left( \frac { 27+8 }{ 5 } ,\frac { 21-6 }{ 5 } \right) =\left( \frac { 35 }{ 5 } ,\frac { 15 }{ 5 } \right) =(7,3)\)

24.

Three irrational numbers between 0.12 and 0.13 are 0.12010010001…, 0.12040040004…, 0.12070070007…

25.

\(\sqrt { 27 } \) =\(\sqrt { 3 } \times \sqrt { 3 } \times \sqrt { 3 } =\left( { 3 }^{ \cfrac { 1 }{ 2 } } \right) ^{ 3 }=\left( 3 \right) ^{ \left( \cfrac { 3 }{ 2 } \right) }\)

26.

x^o=\(\cfrac { 1 }{ 2 } \angle BOC\)
=\(\cfrac { 1 }{ 2 } \) (50^o+30^o)
=\(\cfrac { 1 }{ 2 } \) \(\times\) 80 =40^o

27.

Distance of the chord from the centre
= \(\sqrt { { 15 }^{ 2 }-{ 12 }^{ 2 } } \)
= \(\sqrt { 225-144 } \)
= \(\sqrt { 89 } \)
= 9 cm

28.

Mean \(\bar { x } =\cfrac { \Sigma x }{ n } \)
= \(\cfrac { 26+24+28+31+30+26+24 }{ 7 } =\cfrac { 189 }{ 7 } \)
Mean temperature of the week = 27⁰C

29.

In this example, three values 20, 21, 22 occur two times each. There are three modes for the given data!

30.

In the figure

LM = 7cm
Let OM = (7-x)cm
MB = \(\cfrac { 8 }{ 2 } \) = 4cm
OB =\(\sqrt { { 4 }^{ 2 }+\left( 7-x \right) ^{ 2 } } \)
OD =\(\sqrt { { 3 }^{ 2 }+{ x }^{ 2 } } \)
\(\sqrt { 16+\left( 7-x \right) ^{ 2 } } =\sqrt { { 3 }^{ 2 }+{ x }^{ 2 } } \)
Squaring both sides
16 + (7-x)²= 9 + x²

14x = 65 - 9
14x = 56 = x\(\cfrac { 54 }{ 14 } \)= 4
\(\therefore\) Radius OD =\(\sqrt { { 3 }^{ 2 }+{ 4 }^{ 2 } } =\sqrt { 9+16 } =\sqrt { 25 } =5cm\)

31.

\(\angle\)DBC = 40⁰
\(\angle\)BAC = 60⁰
(i) \(\angle\)CAD = 60⁰
(\(\therefore \) angles in the same segment are equal)
(ii) \(\angle\)BCD+\(\angle\)BAD = 180⁰
(\(\therefore\) In cyclic quadrilateral opposite angles are supplementary)

\(\angle\)BCD+ (\(\angle\)BAD +\(\angle\)CAD) =180⁰
\(\angle\)BCD+ (60⁰+ 40⁰) = 180⁰
\(\angle\)BCD = 180⁰-100⁰= 80⁰

32.

(a+b+c)²= a²+ b²+ c²+ 2ab + 2bc + 2ca
ஃ (x + 2y + 3z)²= (x)²+(2y)²+(3z)²+2(x)(2y)+2(2y)(3z)+2(3z)(x)
= x²+ 4y²+ 9z²+ 4xy + 12yz + 6xz

33.

\(\sqrt { 8 } =\sqrt { 2 } \times \sqrt { 2 } \times \sqrt { 2 } =\left( 2^{ \frac { 1 }{ 2 } } \right) ^{ 3 }\)
which may be written as \({ 2 }^{ \frac { 3 }{ 2 } }\).

34.

\({17\over 200}={17\over 2^3\times 5^2}\)
\(\therefore {17\over 200}\) has a terminating decimal expansion.

35.

(i) {-3,-2,-1,0,1,4} (ii) {-3,-2,-1,0,1,4} (iii) {-3,-2,0,1,3,4}

36.

(a) A = {Jan, March, May, July, Aug, Oct, Dec}
(b) B = {13,26,39,52,65,78,91}
(c) C = {a,e}
(d) D = {6,7,8,9,10}
(e) E = {1,4,9}

37.

{2, 3, 4, 6, 7, 8, 9, 10, 11}

38.

{triangle}

39.

Coefficient of x²is-2 and Coefficient of x is -\(\sqrt 7\)

Multiple Choice Question

40 x 1 = 40

40.

(a)
280 cm²

41.

42.

(d)
1-P(A)

43.

(d)
Less than zero

44.

45.

(b)
1

46.

47.

(d)
5 :2

48.

(d)
n(A) + n(B) + n(C) -\(n\left( A\cap B \right) \)-\(n\left( B\cap A \right) -n\left( A\cap C \right) +n\left( A\cap B\cap C \right) \)

49.

(b)
\(n\left( U \right) -n\left( A\cup B\cup C \right) \)

50.

(a)
1

51.

(a)
x²+ 5x + 6

52.

53.

(b)
\(\cfrac { { 3 }^{ \frac { 2 }{ 3 } } }{ 3 } \)

54.

(d)
3

55.

56.

(b)
46

57.

(d)
34

58.

(a)
0

59.

(b)
100

60.

(d)
55

61.

(b)
4.5

62.

(d)
9 cm

63.

64.

(d)
x = 0, -2

65.

66.

(b)
60°

67.

68.

69.

(a)
\(\frac { 7 }{ 20 } \)

70.

71.

72.

73.

(b)
II

74.

75.

(b)
\(\sqrt { 5 } \)

76.

(a)
\(\frac { 5 }{ 64 } \)

77.

78.

79.

(b)
A = B

Generic 5

9 x 5 = 45

80.

Let ABCD be the rhombus.
Its perimeter = 4 × side = 400 m
Therefore, each side of the rhombus = 100 m
Given the length of the diagonal AC = 120 m
In \(\triangle\)ABC, let a =100 m, b =100 m, c = 120 m

s = \(\frac{a+b+c}{2}=\frac{100+100+120}{2}\) = 160 m
Area of \(\triangle\)ABC =\(\sqrt{160(160-100)(160-100)(160-120)}\)
= \(\sqrt{160 \times 60\times 60 \times40}\)
= \(\sqrt{40 \times 2 \times \times2\times60\times60\times40}\)
= 40 × 2 × 60 = 4800 m²
Therefore, Area of the field ABCD = 2 × Area of \(\triangle\)ABC = 2 × 4800 = 9600 m²

81.

Let M.S and H represent sets students who like Maths,Science and history respectively.
Then, n(M) = 80,n(S) = 90,n(H) = 82,\(n\left( M\cap S \right) \) = 21,\(n\left( S\cap H \right) \) = 19,\(n\left( M\cap H \right) \) = 20,\(n\left( M\cap S\cap H \right) \)= 8
Let us represents the given data in a venn diagram

(i) The number of student in the school = 52 + 59 + 55 + 12 + 11 + 8 + 8 = 205
(ii) The number of students who like only one subject = 52 + 59 + 55 = 166

82.

Let p (x) 2x³ - x² - 12x - 9
Sum of the co-efficients = 2 - 1- 12- 9 = -20 \(\neq \) 0
Hence x-1 is not a factor
Sum of co-efficients of even powers with constant = -1 - 9 = -10
Sum of co-efficients of odd powers = 2 - 12= -10
Hence x + 1 is a factor of x.
Now we use synthetic division to find the other factors.

Then p (x) = (x + 1)(2x² - 3x - 9)
Now 2x² - 3x - 9 = 2x² - 6x + 3x - 9 = 2x (x - 3) + 3 (x - 3)
= (x - 3)(2x + 3)
Hence 2x³ - x²- 12x - 9 (x + 1) (x - 3) (2x + 3)

83.

Calculate the median

Class	Number of students	Cummulative frequency(cf)
0-10	2	2
10-20	7	9
20-30	15	24
30-40	10	34
40-50	11	45
50-60	5	50

N = 50
Median Class = \(\left(\cfrac{N}{2}\right)^{th}\)Value
=\(\left(\cfrac{50}{2}\right)^{th}\) Value = 25^th Value
Median value = 30 - 40
\(\cfrac{N}{2}\)=25, l = 30
m = 24, c =10, f =10
\(\therefore\) Median = \(\cfrac{l+\left(\cfrac{N}{2}-m\right)}{f}\times{c} \)
=\(\cfrac{30+25-24}{10}\times{10}=31\)

84.

Weight in kgs(x)	Number of students(f)	Mid value of x	fx
15-25	4	20	80
25-35	11	30	33
35-45	19	40	760
45-55	14	50	700
55-65	0	60	0
65-75	2	70	140
	\(\Sigma{f}=50\)		2010

85.

To represent \(-\frac { 2 }{ 11 } ,-\frac { 5 }{ 11 } and-\frac { 9 }{ 11 } \)on the number line we make 11 markings each being equal distance \(\frac { 1 }{ 11 } \) on the left of 0.
The point A represents \(\left( -\frac { 2 }{ 11 } \right) \) , the point B represents\(\left( -\frac { 2 }{ 11 } \right) \) and the point C represents \(\left( -\frac { 9 }{ 11 } \right) \)

86.

Relational between \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 5 } \) = \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
= \(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } +\frac { 1 }{ 5 } \right) \)
Rational numbers between\(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \)
=\(\frac { 1 }{ 2 } \left( \frac { 10+7 }{ 20 } \right) \)
=\(\frac { 17 }{ 40 } \)
Rational number between \(\frac { 1 }{ 2 } \) and \(\frac { 7 }{ 20 } \) =\(\frac { 1 }{ 2 } \left( \frac { 1 }{ 2 } \times \frac { 17 }{ 40 } \right) \)
=\(\frac { 1 }{ 2 } \left( \frac { 20+17 }{ 40 } \right) \)
=\(\frac { 37 }{ 80 } \)
Thus the rational numbers are \(\frac { 7 }{ 20 } ,\frac { 17 }{ 40 } and\frac { 37 }{ 80 } \)

87.

We have a parallelogram ABCD in which diagonals AC bisect ㄥA.
ㄥDAC = ㄥBAC
(i) To prove that AC bisects LC
∵ ABCD is a parallelogram
∴ AB II DC and AC is a transversal
∴ ㄥ1 = ㄥ3 (Alternate interior angle) .........(1)
.Also BC II AD and AC is a transversal.
∴ ㄥ2 = ㄥ4 (Alternate interior angle) ...........(2)
But AC bisects ㄥA
∴ ㄥ1 = ㄥ2
From (1), (2) and (3) we get
ㄥ3 = ㄥ4
∴ AC bisects ㄥC.
(ii) To prove that ABCD is a rhombus.
In ΔABC, we have ㄥ1 = ㄥ4 [ ∵ ㄥ1 = ㄥ2 = ㄥ4]
∴ BC = AB (side opposite to equal angles are equal) (4)
Similarly AD = DC ....... (5)
But ABCD is a parallelogram AB = DC (Opposite sides of a parallelogram) .......(6)
From (4), (5) and (6) we have AB = BC = CD = DA.
Thus ABCD is a rhombus.

88.

Distance = \(\sqrt{(x_2-x_2)^2+(y_2-y_1)^2}\)
AB =\(\sqrt{(6-3)^2+(5-7)^2}\)
\(\sqrt{3^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}\)
BC=\(\sqrt{(15-6)^2+(-1-5)^2}=\sqrt{(9)^2+(-6)^2}\)
\(\sqrt{81+36}=\sqrt{117}\)
\(\sqrt{9\times 13}-3\sqrt{13}\)
AC =\(\sqrt{(15-3)^2+(-1-7)^2}\)
\(\sqrt{12^2+(-8)^2}=\sqrt{144+64}\)
\(=\sqrt{208}=\sqrt{16\times 13}=4\sqrt{13}\)

AB + BC =AC\(\Rightarrow \sqrt{13}+3\sqrt{13}=4\sqrt{13}\)
\(\therefore\)The points A,B,C are collinear.

Generic 8

1 x 8 = 8

89.

In \(\triangle\)PQR, PQ = 5 cm, PR = 6 cm, \(\angle\)QPR = 60°

Construction:
Step 1: Draw \(\triangle\) PQR using the given measurements PQ = 9 cm, QR = 7 cm and RP = 8 cm and construct the perpendicular bisector of any two sides (PQ and QR) to find the mid-points M and N of PQ and QR respectively.
Step 2: Draw the medians PN and RM and let them meet at G. The point G is the centroid of the given \(\triangle\)PQR.