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CBSE CLASS 9 MATHS SAMPLE PAPER - 2

CBSE 9th

Mathematics

Sample Paper 2

 

Time: 3-hour                                                                                                                  Total Marks: 80

 

General Instructions:

 

  • All questions are compulsory. 
  • The question paper consists of 30 questions divided into four sections A, B, C and D. 
  • Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
  • There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. 
  • Use of calculators is not permitted.

 

 

Section-A

(Question numbers 1 to 6 carry 1 mark each)

 

Q. No.

   

1.

If $\mathrm{x}^{\mathrm{a} / \mathrm{b}}=1$, then find the value of ‘a’.

 

2.

If $\mathrm{p}(\mathrm{x})=2 \mathrm{x}^{3}+5 \mathrm{x}^{2}-3 \mathrm{x}-2$ is divided by $\mathrm{x}-1$, then find the remainder.

 

3.

The distance of the point $(0,-1)$ from the origin is

 

4.

If the vertical angle of an isosceles triangle is $100^{\circ}$, then find the measures of its base angles.

 

5.

The ratio of the whole surface area of a solid sphere and a solid hemisphere is

 

6.

There are 60 boys and 40 girls in a class. A student is selected at random. Find the probability that student is a girl.

 
 

Section B

(Question numbers 7 to 12 carry 2 marks each)

 

7.

If $p=2-a$, then prove that $a^{3}+6$ a $p+p^{3}-8=0$.

 

8.

In the adjoining figure, we have $\mathrm{AB}=\mathrm{BC}, \mathrm{BX}=\mathrm{BY}$. Show that $\mathrm{AX}=\mathrm{CY}$ (using appropriate Euclid’s axiom)

Diagram Description automatically generated

 

9.

If two opposite angles of a parallelogram are $(63-3 x)^{\circ}$ and $(4 x-7)^{\circ} .$ Find all the angles of the parallelogram. 

 

10.

Three Schools situated at $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ in the figure are equidistant from each other as shown in the figure. Find $\angle$ QOR.

Diagram Description automatically generated

 

11.

The diameter of the two right circular cones are equal if their slant heights are in the

ratio $3: 2$, then what is the ratio of their curved surface areas?

 

12.

A batsman in his $11^{\text {th }}$ innings makes a score of 68 runs and there by increases his average score by 2 . What is his average score after the $11^{\text {th }}$ innings.

 
 

Section C

(Question numbers 13 to 22 carry 3 marks each)

 

13.

Represent $\sqrt{10}$ on the number line.

 

14.

Simplify: $\frac{73 \times 73 \times 73+27 \times 27 \times 27}{73 \times 73-73 \times 27+27 \times 27}$

 

15.

Determine the point on the graph of the linear equation $2 \mathrm{x}+5 \mathrm{y}=19$, whose ordinate is $1 \frac{1}{2}$ times its abscissa.

 

16.

Locate the points $(3,0),(-2,3),(2,-3),(-5,4)$ and $(-2,-4)$ in Cartesian plane. Also find the quadrant in which they lie.

 
 

OR

 
 

Observe the fig. given below and answer the following:

A picture containing antenna Description automatically generated

 

i.

The coordinates of $\mathrm{B}$.

 

ii.

The coordinates of $\mathrm{C}$.

 

iii.

The point identified by the coordinate $(-3,-5)$. 

 

iv.

The coordinates of $\mathrm{H}$.

 

v.

The coordinates of origin

 

vi.

The abscissa of the point $\mathrm{D}$.

 

17.

In figure, $A C=A E, A B=A D$ and $\angle B A D=\angle E A C .$ Show that $B C=D E$.

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OR

 
 

Diagram Description automatically generated

$\mathrm{AB}$ is a line segment and $\mathrm{P}$ is its mid-point. D and $\mathrm{E}$ are points on the same side of $\mathrm{AB}$ such that $\angle \mathrm{BAD}=\angle \mathrm{ABE}$ and $\angle \mathrm{EPA}=\angle \mathrm{DPB}$. Show that:

 

i.

$\triangle \mathrm{DAP} \cong \triangle \mathrm{EBP}$

 

ii.

$\mathrm{AD}=\mathrm{BE}$

 

18.

Show that the area of a rhombus is half the product of the lengths of its diagonals.

 

19.

$\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are the four points on a circle. AC and BD intersect at point $\mathrm{E}$ such that $\angle$ $\mathrm{BEC}=130^{\circ}$ and $\angle \mathrm{ECD}=20^{\circ} .$ Find $\angle \mathrm{BAC}$.

Diagram Description automatically generated

 
 

OR

 
 

Prove that equal chords of a circle subtend equal angles at the centre. 

 

20.

Sides of a triangle are in the ratio $12: 17: 25$ and its perimeter is $540 \mathrm{~cm}$. Find its area.

 

21.

The diameter of a garden roller is $14 \mathrm{~m}$ and it is $2 \mathrm{~m}$ long. How much area will it cover in 10 revolutions?

 
 

OR

 
 

The sum of height and radius of the base of a solid cylinder is $37 \mathrm{~cm}$. If the total surface area of the cylinder is $1628 \mathrm{~cm}^{2}$, then find its volume.

 

22.

Fifty seeds were selected at random from each 5 bags seeds and were kept under standardized conditions favorable to germination. After days, the number of seeds which

had germinated in each collection were counted and recorded as follows:

Bag

1

2

3

4

5

Number of seeds generated

40

48

42

39

38

What is the probability of germination of

 

i.

More than 40 seeds in a bag

 

ii.

49 seeds in a bag

 

iii.

More than 35 seeds in a bag

 
 

Section D

(Question numbers 23 to 30 carry 4 marks each)

 

23.

If $\mathrm{x}=\frac{6-\sqrt{3} 2}{2}$, then find the value of $\left(x^{3}+\frac{1}{x^{3}}\right)-6\left(x^{2}+\frac{1}{x^{2}}\right)+\left(x+\frac{1}{x}\right)$.

 
 

OR

 
 

If $x=\frac{\sqrt{3}+1}{\sqrt{3}-1}, y=\frac{\sqrt{3}-1}{\sqrt{3}+1}$, find the value of $x^{2}+x y-y^{2}$

 

24.

Determine the value of $\mathrm{b}$ ‘ for which the polynomial $5 \mathrm{x}^{3}-\mathrm{x}^{2}+4 \mathrm{x}+\mathrm{b}$ is divisible by $1-5 \mathrm{x}$.

 

25.

Draw the graph of two lines whose equations are $x+y-6=0$ and $x-y-2=0$, on the same graph paper. Find the area of triangle formed by the two lines and y axis.

 
 

OR

 
 

The force exerted to pull a cart is directly proportional to the acceleration produced in the cart. Express the statement as a linear equation in two variables and draw the graph

for the same by taking the constant mass equal to $6 \mathrm{~kg}$.

 

26.

In figure the sides $A B$ and AC of are produced to points $E$ and D respectively. If bisectors $\mathrm{BO}$ and $\mathrm{CO}$ of $\angle \mathrm{CBE}$ and $\angle \mathrm{BCD}$ respectively meet at point $\mathrm{O}$, then prove that $\angle \mathrm{BOC}=90^{\circ}-\frac{1}{2} \angle \mathrm{BAC}$

Diagram Description automatically generated

 

27.

In the adjoining figure, $\mathrm{P}$ is the point in the interior of a parallelogram $\mathrm{ABCD}$. Show that $\operatorname{ar}(\triangle \mathrm{APB})+\operatorname{ar}(\triangle \mathrm{PCD})=\frac{1}{2} \operatorname{ar}(|| \mathrm{gm} \mathrm{ABCD})$

Diagram Description automatically generated

 

28.

Construct a right angled triangle whose base is $5 \mathrm{~cm}$ and sum of its hypotenuse and other side is $8 \mathrm{~cm}$.

 

29.

The floor of a rectangular hall has a perimeter $300 \mathrm{~cm}$. Let the cost of painting of four walls at the rate of Rs.12 per $\mathrm{cm}^{2}$ is Rs. 24,000 , then find the height of the hall.

 

30.

The marks obtained (out of 100) by a class of 80 students are given below:

Marks

10-20

20-30

30-50

50-70

70-100

No. of students

6

17

15

16

26

Construct a histogram to represent the data above.

 
 

OR

 
 

Construct a frequency polygon for the following data:

Ages

(in years)

0-2

2-4

4-6

6-8

8-10

Frequency

4

7

12

5

2

 
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