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# ICSE CLASS 9 MATHS SAMPLE PAPER - 2

ICSE Board Class IX

Mathematics Sample Paper – 2

Time:  2½ hrs                                                                                                               Total Marks:  80

General Instructions:

• Answers to this paper must be written on the  paper provided  separately.
• You will  NOT  be  allowed  to  write  during  the  first  15  minutes.  This  time  is  to  be  spent  in reading  the question paper.
• The time given  at  the head  of  this paper is the time allowed  for  writing  the answers.
• This question paper is divided into two  Sections.  Attempt all questions from  Section  A and any four questions from  Section  B.
•  Intended  marks  for  questions  or  parts  of  questions  are  given  in  brackets  along  the questions.
• All  working,  including  rough  work,  must  be  clearly  shown  and  should  be  done  on  the same  sheet  as  the  rest  of  the  answer.  Omission  of  essential  working  will  result  in  loss  of marks
• Mathematical tables are  provided.

SECTION  –  A (40  Marks)

(Answer  all  questions  from  this  Section)

 Q. No. Question Marks 1. a) Show that $\sqrt{5}$ is an irrational number 3 b) Express $0.001$ as a fraction in the simplest form. 3 c) Find the median of the following set of numbers:$7,80,5,81,18,27,4,48,12,47,9,15$ 4 2. a) Use congruency of triangles to find the value of x and y. 3 b) Two concentric circles are of radii 7 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 3 c) The compound interest on a certain sum of money at 7% p.a. for 5 years is Rs. 300. Find the sum. 4 3. a) If $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=x-y \sqrt{77}$, find the values $x$ and $y$. 3 b) Given $\log x=\mathrm{a}+\mathrm{b}$ and $\log \mathrm{y}=\mathrm{a}-\mathrm{b}$, find the value of $\log \frac{10 x}{y^{2}}$ in terms of ‘ $\mathrm{a}$ ‘ and ‘ $\mathrm{b}$ ‘. 3 c) Sonu and Suraj had adjacent triangular fields with a common boundary of 25 m.  The other two sides of Sonu’s field were 52 m and 63 m, while Suraj’s were 114 m and101 m. If the cost of fertilization is Rs 20 per sq m, then find the total cost of fertilization for both of Sonu and Suraj together. 4 4. a) Evaluate $\frac{4}{\tan ^{2} 60^{\circ}}+\frac{1}{\cos ^{2} 30^{\circ}}-\tan ^{2} 45^{\circ}$ 3 b) The bisector of $\angle \mathrm{A}$ of a $\triangle \mathrm{ABC}$ meets $\mathrm{BC}$ at $\mathrm{D}$ and $\mathrm{BC}$ is produced to $\mathrm{E}$. prove that $\angle \mathrm{ABC}+\angle \mathrm{ACE}=2 \angle \mathrm{ADC}$ 3 c) The difference between the exterior angle of a regular polygon of $\mathrm{n}$ sides and a regular polygon of $(\mathrm{n}+2)$ sides is 6 . Find the number of sides. 4 SECTION – B (40 Marks)(Answer any four questions from this Section) 5. a) Graphically solve the simultaneous equations: $x-2 y=1 ; x+y=4$ 4 b) How many sides does a regular polygon have, each angle of which is of measure $108^{\circ}$ ? 3 c) Prove that the interior angle of a regular pentagon is three times the exterior angle of a regular decagon. 3 6. a) In the given figure, $\mathrm{m} \angle \mathrm{PSR}=90^{\circ}, \mathrm{PQ}=10 \mathrm{~cm}, \mathrm{QS}=6 \mathrm{~cm}, \mathrm{RQ}=9 \mathrm{~cm} .$ Calculate the length of PR. 3 b) A and B together can do a piece of work in 15 days. If A’s one day’s work is $\frac{3}{2}$ times B’s one day’s work; in how many days can $A$ and $B$ do the work alone? 3 c) Prove that: $9^{\log 4}=16^{\log 3}$ 4 7. a) In the figure, $Q$ is a point on side of $\Delta$ PSR such that $P Q=P R$. Prove that $P S>P Q$. 4 b) The amount at compound interest which is calculated yearly on a certain sum of money is Rs. 1250 in one year and Rs. 1375 in two years. Calculate the rate of interest. 3 c) The dimensions of a rectangular field are $120 \mathrm{~cm} \times 70 \mathrm{~cm}$. The field is to be changed into garden, leaving a path way of $5 \mathrm{~m}$ width around the garden. Find the expenses that are met when the cost per square meter is Rs. 10 . 3 8. a) In a rectangle PQRS, prove that $\mathrm{PR}^{2}+\mathrm{QS}^{2}=\mathrm{PQ}^{2}+\mathrm{QR}^{2}+\mathrm{RS}^{2}+\mathrm{SP}^{2}$ 4 b) Use graph paper for this question: 6 i. Draw the graph of $3 x-y-2=0$ and $2 x+y-8=0 .$ Take $1 \mathrm{~cm}=1$ unit on both the axes and plot only three points per line. ii. Write down the co-ordinates of the point of intersection. 9. a) In a river, a boat covers $8 \mathrm{~km}$ in $40 \mathrm{~min}$ while travelling $\underline{\text { downstream, but takes }} 60 \mathrm{~min}$ for the return journey. If the speed of the boat and the flow of the river are uniform, find the speed of the boat in still water and speed of the stream. 4 b) A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was Rs. 1500 after 4 years of service and Rs. 1800 after 10 years of his service, what was his starting salary and what is the annual increment? 3 c) Solve: $3 p-2 q=5, q-1=3 p$ 3 10. a) In the given figure, $A B C D$ is a parallelogram in which $X$ and $Y$ are the midpoints of $A D$ and BC respectively, Prove that: $\mathrm{AE}=\mathrm{EF}=\mathrm{FC}$. 4 b) Two chords $\mathrm{AB}$ and CD of lengths $5 \mathrm{~cm}$ and $11 \mathrm{~cm}$ respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between $A B$ and CD is 6 $\mathrm{cm}$, then find the radius of the circle. 3 c) 3 equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of three cubes. 3 11. a) In the given figure, area of $\Delta \mathrm{PQR}=44.8 \mathrm{~cm}^{2}, \mathrm{PL}=\mathrm{LR}$ and $\mathrm{QM}=\mathrm{MR}$. Find the area of Find the area of DLMR. 4 b) Factorize: $x^{3}-3 x^{2}-x+3$ 3 c) The mean of 5 numbers is $20 .$ If one number is excluded the mean of the remaining numbers becomes $23 .$ Find the excluded number. 3

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