CBSE 9th
Mathematics
Sample Paper 2
Time: 3hour Total Marks: 80
General Instructions:
SectionA (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 $p=2a$, 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)


9. 
If two opposite angles of a parallelogram are $(633 x)^{\circ}$ and $(4 x7)^{\circ} .$ 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.


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: $\frac{73 \times 73 \times 73+27 \times 27 \times 27}{73 \times 7373 \times 27+27 \times 27}$ 

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:


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 $B C=D E$.


OR 

$\mathrm{AB}$ is a line segment and $\mathrm{P}$ is its midpoint. 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 $\angle \mathrm{BAC}$.


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:
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 $x^{2}+x yy^{2}$ 

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 $15 \mathrm{x}$. 

25. 
Draw the graph of two lines whose equations are $x+y6=0$ and $xy2=0$, 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 $\angle \mathrm{BOC}=90^{\circ}\frac{1}{2} \angle \mathrm{BAC}$


27. 
In the adjoining figure, $\mathrm{P}$ is the point in the interior of a parallelogram $\mathrm{ABCD}$. Show that $\operatorname{ar}(\triangle \mathrm{APB})+\operatorname{ar}(\triangle \mathrm{PCD})=\frac{1}{2} \operatorname{ar}( \mathrm{gm} \mathrm{ABCD})$


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:
Construct a histogram to represent the data above. 

OR 

Construct a frequency polygon for the following data:

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