Question Set C

CBSE 9th Standard Mathematics Introduction to Euclid's Geometry Model Questions Papers Study Material - QB365

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Questions:

Generic 2

6 x 2 = 12

In the given figure, if \(l_1||l_2\) and \(l_3||l_4\), What is y in terms of x?

Find the area of a triangle two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.

The sides of a right triangle ABC are 5 cm, 12 cm and 13 cm. Find the area of the triangle.

Observe figure and answer the following:

(a) coordinates of B
(b) point identified by the coordinates (-2, -3)
(c) abscissa of point D
(d) ordinate of point H
(e) points with the same abscissa
(f) points with the same ordinate

Factorise \(6x^2+17x+5\) by splitting the middle term and by using the Factor Theorem.

Find three rational numbers between 3/7 and 5/11.How many rational numbers can be determined lying between these numbers?

Generic 4

7 x 4 = 28

A transversal intersects two parallel lines. Prove that the bisectors of any pair of cosresponding angles so formed are parallel.

Verify if 1 and -3 are zeroes of the polynomial 3x³+5x²-11x+3. If yes, then factorize the polynomial.

Draw the graph of the linear equation 3x+4y=6. Find the points where the line representing the equation 3x+4y=6 cuts the axes of x and y.

10.

The base of an isosceles triangle measures 24 cm and its area is 60 cm², Find its perimeter.

11.

In figure the bisectors of \(\angle ABC\) and \(\angle BCA\) intersect each other at the point O.prove that \(\angle BOC=90^{ 0 }+\frac { 1 }{ 2 }\angle A\)

12.

In which quadrant, can a point have:
(i) abscissa equal to its ordinate
(ii) ordinate equal in magnitude to abscissa
(iii) ordinate equal and opposite of abscissa
(iv) abscissa twice that of the ordinate.

13.

Express \(1.\overline { 32 } +0.\overline { 35 } \) in the form \(\frac { p }{ q } \) , where p and q are integers and \(q\neq 0\)

Generic 3

9 x 3 = 27

14.

Find the values of a and b, when a+b√15 = \(\frac { \sqrt { 5 } +\sqrt { 3 } }{ \sqrt { 5 } -\sqrt { 3 } } \)

15.

Sides of a triangle in the ratio 5:12:13 and its perimeter is 120 cm. Find its area.

16.

The sides of a triangular plot are 50 m, 65 m, and 65 m. Find the cost of laying grass in this plot at the rate Rs.7 per m².

17.

Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square.

18.

Draw the graph of the equations 3x+4y=7 and 3x-2y=1 and find the point of intersection of lines representing the equations.

19.

Express the following linear equation in the form ax+by+c=0 and indicate the values of a, b and c in each case:
2x-5y

20.

(i) Plot the points M(5,-3) and N(-3,-3).
(ii) What is the length of MN?
(iii) Find the coordinates of points A, B and C lying on Mn such that MA=AB=BC=CN

21.

Plot the points A(0,3), B(5,3), C(4,0) and D(-1,0) on the graph paper.Identify the figure ABCD and find whether the point (2,2) lies inside the figure or not?

22.

Verify whether the following are zeroes of the polynomial, indicated against them.
\(p(x)=2x+1,x=\frac { 1 }{ 2 } \)

1 Marks

20 x 1 = 20

23.

In the figure below, if \(\angle A+\angle B+\angle C+\angle D+\angle E+\angle F=k\) right angles, then what is the value of k?

24.

Two supplementary angles are in ratio 2:7. Find the measure of angles.

25.

In the figure below, calculate the value of y.

26.

Write the angle which is one-fifth of its complement.

27.

What is the measure of an angle which is complement of itself?

28.

What is a surface?

29.

What is a straight line?

30.

Simplify: \(\left( x+\frac { 1 }{ 2 } \right) \left( x+\frac { 3 }{ 2 } \right) \)

31.

Explain when a system of axioms is called consistent.

32.

Write the factors of polynomial 4x²+y²+4xy+8x+4y+4

33.

Factorize: 6-x+x².

34.

Find the value of \(\sqrt[3]{625^{-2}}\)

35.

Identify an irrational number among the following numbers: 0.13, \(0.13\overline{15}\) , \(0.\overline{1315}\) , 0.3013001300013...

36.

Write the sum of \(0.\bar{3}\) and \(0.\bar{4}\)

37.

Find the decimal expansion of \(\frac{58}{1000}\)

38.

If -4 is a zero of the polynomial p(x) = x² + 11x + k, then calculate the value of k.

39.

What is the solution of the equation 3x-y=4?

40.

The equation of a line on which the point (6,2) lies is _______

41.

In a one day cricket match, Raina and Dhoni scored 198 runs. Express this as a linear equation in two carables.

42.

Is ox+oy+c=0, a linear equation?

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Answers Key:

Generic 2

6 x 2 = 12

\(x=\angle 1\) (Corresponding angles)
\(\angle 2=\angle 1\) (Corresponding angles)
\(\angle 2+2y=180^o\)
x+2y=180^o
2y=180^o-x
\(\therefore y=90^o-\frac{x}{2}\)

Here we have perimeter of the triangle = 32 cm, a = 8 cm and b = 11 cm.
Third side c = 32 cm – (8 + 11) cm = 13 cm
So, 2s = 32, i.e., s = 16 cm,
s – a = (16 – 8) cm = 8 cm,
s – b = (16 – 11) cm = 5 cm,
s – c = (16 – 13) cm = 3 cm.
Therefore, area of the triangle = \(\sqrt{s(s-a)(s-b)(s-c)}\)
\(=\sqrt{16 \times 8 \times 5 \times 3} \mathrm{~cm}^{2}=8 \sqrt{30} \mathrm{~cm}^{2}\)

30 cm^2.

(a) (2,3)
(b) A
(c) 0
(d) 0
(e) M, O, D, H; B, G
(f) M, H, O

(By splitting method) : If we can find two numbers p and q such that p + q = 17 and pq = 6 x 5 = 30, then we can get the factors.
So, let us look for the pairs of factors of 30. Some are 1 and 30, 2 and 15, 3 and 10, 5 and 6. Of these pairs, 2 and 15 will give us p + q = 17.
So, 6x² + 17x + 5 = 6x² + (2 + 15)x + 5
= 6x² + 2x + 15x + 5
= 2x(3x + 1) + 5(3x + 1)
= (3x + 1) (2x + 5)

331/770, 166/385, 333/770

Generic 4

7 x 4 = 28

Given : l||m line t is a transversal intersecting them at P and Q respectively.
To prove : PR||QS
Proof: \(\angle 5=\angle 6\) (Corresponding angles and l||m)
\(\Rightarrow \frac{1}{2}\angle5=\frac{1}{2}\angle6\)
\(\Rightarrow \angle 1=\angle3\)
PR||QS

p(x) = 3x³+5x²-11x+3
p(1)=3+5-11+3=0
\(\therefore\) 1 is a zero of p(x)
p(-3)=-81+45+33+3=0
-3 is a zero of p(x)
(x-1)(x+3)=x²+2x-3 is a factor of p(x)
\(\frac { p(x) }{ { x }^{ 2 }+2x-3 } =3x-1,\) when we divide physically
Hence p(x)=(x-1)(x+3)(3x-1)

Given equation is 3x+4y=6
(i) When it cuts x-axis then
put y=0, i.e 3x=6
x=2
Hence point on x-axis is (2,0)
(ii) when it cuts y-axis then
put x=0 i.e., 4y=6
y=3/2
hence point on y-axis is \((0,{3\over2})\)
3x+4y=6
\(\Rightarrow y=\frac{6-3x}{4}\)

x	2	-2	6
y	0	3	-3

10.

Area = \(=\frac { a }{ 4 } \sqrt { 4{ b }^{ 2 }-{ a }^{ 2 } } \)
\(\Rightarrow 60=\frac { 24 }{ 4 } \sqrt { 4{ b }^{ 2 }-{ (24) }^{ 2 } } \)
\(\Rightarrow 10=\sqrt { 4{ b }^{ 2 }-576 } \)
\(\Rightarrow 100=4{ b }^{ 2 }-576\) | Squaring
\(\Rightarrow 4{ b }^{ 2 }=676\)
\(\\ \Rightarrow { b }^{ 2 }=\frac { 676 }{ 4 } =169\)
\(\Rightarrow b=\sqrt { 169 } \) = 13 cm
\(\therefore\) Perimeter = a + b + b
= 24 + 13 + 13 = 50 cm

11.

\(\therefore \) BO is the bisector of \(\angle ABC\)
\(\therefore \angle OBC=\frac { 1 }{ 2 } \angle ABC=\frac { 1 }{ 2 } \angle B\)
\(\therefore \) CO is the bisector of \(\angle ACB\)
\(\angle OCB=\frac { 1 }{ 2 } \angle ACB=\frac { 1 }{ 2 } \angle C\)
In OBC ,\(\angle BOC+\angle OBC+\angle OCB=180^{ 0 }\)
| \(\therefore \) The sum of the three angles of a is \(180^{ 0 }\)
\(\Rightarrow \angle BOC+\frac { 1 }{ 2 } \angle B+\angle C)=180^{ 0 }\)
\(\Rightarrow \angle BOC=180^{ 0 }-\frac { 1 }{ 2 } (\angle B+\angle C)\)
In \(\triangle \) ABC
\(\angle A+\angle B+\angle C=180^{ 0 }\)
| The sum of the three angles of a triangle is \(180^{ 0 }\)
\(\Rightarrow \angle B+\angle C=180^{ 0 }-\angle A\)
\(\Rightarrow \frac { 1 }{ 2 } (\angle B+\angle C)=\frac { 180^{ 0 }-\angle A }{ 2 } \)
=\(90^{ 0 }-\frac { 1 }{ 2 } \angle A\)
From (3) and (4) we have
\(\angle BOC=180^{ 0 }-90^{ 0 }-\frac { 1 }{ 2 } \angle A=90^{ 0 }+\frac { 1 }{ 2 } \angle A\)

12.

(i) I, III
(ii) I, II, III, IV
(iii) II, IV
(iv) I, III

13.

Let x = \(1.\overline { 32 } \)
x = 1.3222...
10x = 13.222... ...(1)
100x = 132.222... ....(2)
Subtracting (1) from (2), we get
90x = 119
x = \(\frac { 119 }{ 90 } \) ....(3)
Let x = \(0.\overline { 35 } \)
Then x = 0.35 35 35...(4)
100x = 35.35 35 35...(5)
Subtracting (4) from (5), we get
99x = 35
x = 35/99
\(1.\overline { 32 } +0.\overline { 35 } =\frac { 119 }{ 90 } +\frac { 35 }{ 99 } \)
\(\\ =\frac { 1309+350 }{ 990 } =\frac { 1659 }{ 990 } =\frac { 553 }{ 330 } \)
Here, p = 553, q =3 30(\(\neq 0\))

Generic 3

9 x 3 = 27

14.

\(\frac { \sqrt { 5 } +\sqrt { 3 } }{ \sqrt { 5 } -\sqrt { 3 } } =\frac { \sqrt { 5 } +\sqrt { 3 } }{ \sqrt { 5 } -\sqrt { 3 } } \times \frac { \sqrt { 5 } +\sqrt { 3 } }{ \sqrt { 5 } +\sqrt { 3 } } \)
\(=\frac { 8+2\sqrt { 15 } }{ 2 } =4+\sqrt { 15 } \)
\(a+b\sqrt { 15 } =4+\sqrt { 15 } \)
a = 4, b = 1

15.

480 cm²

16.

Rs.10500

17.

(i) Parallel lines. Lines which do not intersect anywhere are called parallel lines.
(ii) Perpendicular lines. Two lines which are at a right angle to each other are called perpendicular lines.
(iii) Line segment. It is a terminated line.
(iv) Radius. The length of the line segment joining the centre of a circle to any point on its circumference is called its radius.
(v) Square. A quadrilateral with all the four sides equal and all the four angles of measure 90° each is called a square.

18.

3x+4y=7

x	1	-3
y	1	4

3x-2y=1

x	1	-1
y	1	-2

From the graph, point of intersection is (1,1).

19.

2x=-5y
2x+5y+0=0
Comparing with ax+by+c=0, we get
a=2, b=5, c=0

20.

(i)

(ii) 8 units
(iii) \(A\rightarrow \left( 3,-3 \right)\)
\(\\ B\rightarrow \left( 1,-3 \right)\)
\(\\ C\rightarrow \left( -1,-3 \right) \)

21.

Parallelogram; yes

22.

\(p\left( \frac { 1 }{ 2 } \right) =2\left( \frac { 1 }{ 2 } \right) +1=1+1=2\neq 0\)
\(\therefore \frac { 1 }{ 2 } \)is not a zero of p(x).

1 Marks

20 x 1 = 20

23.

()
In \(\triangle ABC,\)
\(\angle A+\angle B+\angle C=180^o\)
Also, in \(\triangle DEF,\)
\(\angle D+\angle E+\angle F=180^o\)
\(\therefore \angle A+\angle B+\angle C+\angle D+\angle E+\angle F=360^o\)
=4 X 90^o
Hence,k=4 right angles.

24.

()
2x+7x=180^o\(\Rightarrow\)x=20^o
So the angles are
2x=2X20^o
=40^o
7x=7X20^o
=140^o
So two angles are 40^o and 140^o

25.

()
Here, 40^o+3y+2y=180^o(\(\because\) Straight line makes an angle of 180^o)
5y=140^o
y=28^o

26.

()
Let the angle be x, then
By given condition, \(x=\frac{1}{5}(90^o-x)\Rightarrow6x=90^0\)
\(\Rightarrow x=15^0\)

27.

()
Let the angle be x, then
Angle x= Complement of x
\(\Rightarrow\) x=90^o-x\(\Rightarrow\)x=45^o

28.

()
A surface is that which has length and breadth.

29.

()
Two planes intersect each other to form a straight line.

30.

()
\(\left( x+\frac { 1 }{ 2 } \right) \left( x+\frac { 3 }{ 2 } \right) \)\(={ x }^{ 2 }+\frac { 3 }{ 2 } x+\frac { 1 }{ 2 } x+\frac { 3 }{ 4 } \)
\(={ x }^{ 2 }+2x+\frac { 3 }{ 4 } \)

31.

()
A system of axioms is called consistent, when it is impossible to deduce from these axioms, a statement that contradicts any axiom or previously proved statement.

32.

()
4x²+y²+4xy+8x+4y+4 = (2x)²+(y)²+(2)²+2\(\times\)2x\(\times\)y+2\(\times\)2x\(\times\)2+2\(\times\)y\(\times\)2
= (2x+y+2)².
Factor is 2x + y + 2.

33.

()
6x-x-x² = 6-3x+2x-x²
= 3(2-x)+x(2-x)
= (2-x)(3+x)

34.

()
\(\sqrt[3]{625^{-2}}\) = (625^-2)^1/4= (625^-2x1/4)
= (625^-1/2)
= \((\frac{1}{625})^{1/2}=\frac{1}{25}\)

35.

()
0.13 is a terminating number. So, it is not an irrational number.
\(0.13\overline{15}\) = 0.131515...,15 is representing continuously, so it is not an irrational number.
\(0.\overline{1315}\) = 0.13151315..., is representing continuously, so it is not an irrational number.
0.3013001300013..., non-terminating and non-recurring decimal. Hence, it is an irrational number. So, 0.3013001300013 is an irrational number.

36.

()
\(0.\bar{3}+0.\bar{4}\)=(0.333...) + (0.444...)
= 0.777...
Let x = 0.777...
10x = 7.777...
⇒ 10x - x = (7.777...) - (0.777...)
⇒ 9x = 7.0
⇒ x = \(\frac{7}{9}\)

37.

()
\(\frac{58}{1000}\)= 0.058 (Decimal point is shifted three places to the left)

38.

()
Given, p(x) = x² + 11x + k
Since -4 is a zero of polynomial
p(-4) = 0
\(\Rightarrow\) (-4)² + 11 \(\times\) (-4) + k = 0
\(\Rightarrow\) 16 - 44 + k = 0
\(\therefore\) k = 28