Summative Assessment-1 2014-2015 Class – X General

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Summative Assessment-1 2014-2015
General Instructions:
a) All questions are compulsory.
b) Question paper contains 31 questions divide into 4 sections A, B, C and D.
c) Question No. 1 to 4 are very short type questions, carrying 1 mark each. Question No. 5 to
10 are of short answer type questions, carrying 2 marks each. Question No. 11 to 20 carry
3 marks each. Question No. 21 to 31 carry 4 marks each.
d) There are no overall choices in the question paper.
e) Use of calculator is not permitted.
Section A
Question numbers 1 to 4 carry 1 mark each.
1. In ABC , D and E are points on the sides AB and AC respectively such that DE BC . If
AB=6.75cm, AC=8.50cm and EC=6.8cm, then find BD.
2. If 24 cot A=7, find the value of sin A.
3. If 45θ = ° , then find the value of 2 22cos 3secec θ θ+ .
4. What is the mean of x, x+1, x+2, x+3, x+4 and x+5?
Section B
Question number 5 to 10 carry two marks each.
5. Find the HCF of the number 31, 310 and 3100.
6. Write the decimal expansion of 2 3
1717
2 5× without actual division.
7. Find the quadratic polynomial, the sum and product of whose zeroes are -3 and 5
respectively.
8. In the figure if AC=2m, OC=3m and OD=7m, then find BD.
9. Solve the equation forθ : 2
2 2
cos 3
cot cos
Class
interval
Frequency 18 15 23 55 87 29
Section C
Question numbers 11 to 20 carry three marks each.
11. Find LCM of 36, 54 and 63 by prime factorization method. Why LCM of numbers is always
greater than or equal to each of the numbers?
12. Solve by substitution:
+ = − = −
13. Divide the polynomial 2 23 6 20 14x x x− − + by the polynomial
2 5 6x x− + and verify the division
algorithm.
14. Solve the following pair of equations by reducing them to a pair of linear equations:
1 4 2
1 3 9
− =
+ =
15. From airport two aeroplane start at the same time. If speed of first aeroplane due North is
500 km/hr and that of other due East is 650 km/hr, then find the distance of two aeroplanes
after 2 hours.
16. In the figure find CD, if it is given that AB=12cm, BC=13cm and AD=3cm.
17. Prove that: tan tan
2cos sec 1 sec 1
ec θ θ θ
θ θ θ−
19. The given distribution shows the number of wickets taken by bowlers in inter-school
competitions:
Number of
9 3 5 3 1
20. The median of the following frequency distribution is 28.5. Find the value of x and y, if sum of
frequencies is 58.
Section D
Question no. 21 to 30 carry four marks.
21. Find the greatest 5 digit number which is exactly divisible by 12, 18 and 24.
22. A group of girls and boys are made to stand in rows. If 4 students are extra in a row, there
would be two rows less. If 4 students are less in a row, there would be 4 more rows. Find the
number of students in the class. What is the motive behind making students stand rows?
23. Find all other zeroes of the polynomial 4 3 22 19 9 9x x x x− − + + , if two of its zeroes are 1 and -3.
24. Smita scored 100 marks in a test getting 2 marks for each correct answer and losing 1 mark
for each wrong answer. Had 4 marks been awarded for the each correct answer and 3 marks
been deducted for the correct answer, then she would have again scored 100 marks. How
many questions were there in the test, assuming she had attempted all the questions.
25. In the figure 90ABD XYD CDB∠ = ∠ = ∠ = ° , AB=a, XY=c and CD=b, then prove that c(a+b)=ab.
26. In the figure BED BDE∠ = ∠ and E is the middle point of BC. Prove that AF AD
CF BE =
2 2
2 2
+ + += = − − −
sin( ) 2
a) cos .cos sin .sinA B A B+
b) tan tan
1 tan .tan
2 2
cos sec
ec A A + + − = −
30. The mean of the following frequency distribution is 145. Find the missing frequencies x and
y.
Class 0-50 50-100 100-150 150-200 200-250 250-300 Total
Frequency 6 x 64 52 y 14 200
31. For one term, absentee record of students is given. If mean is 15.5, find the missing
frequencies x and y.
Number of
Section A
1. In the figure l m , 90AOD∠ = ° and 30ODB∠ = ° . Are OAC and ODB similar? If yes, by
which criterion?
2. In ABC , 90C∠ = ° , 45A∠ = ° and AB=10cm. find BC, using trigonometric ratios.
3. Find the value of 2 2sin 12 sin 78° + ° .
4. If ‘less than type ogive’ and ‘more than type ogive’ for a given data are given, then how can
you find its median?
5. Write the decimal expansion of 2 3
1717
2 5× without actual division.
6. Find the prime factorization of the denominator of rational number expressed as 6.12 in
simplest form.
7. Solve the following pair of linear equations using elimination method:
2x+3y=2 4x-3y-1
8. Two pillars of height 70m and 20m are standing 120m apart. If distance between their feet is
120m, find the distance between their tops.
9. If cos sinx a bθ θ= − and sin cosy a bθ θ= + , then prove that 2 2 2 2a b x y+ = + .
10. Form a frequency table for that the following cumulative frequency distribution.
Class limits More than
Section C
11. Show that 8n can never end with digit 0 for any natural number n.
12. If 4 3 24 7 4 7x x x x k+ − − + is completely divisible by 3x x− , then find the value of k.
13. On dividing 4 3 22 4 4 7 2x x x x− − + + by a polynomial g(x), the quadrant and the remainder
were 22 4x x− and 2x-2 respectively. Find g(x).
14. Ifα and β are zeroes of a polynomial 28 6 9x x+ + , then form a polynomial whose zeroes are
2α and 2β .
15. In ABC from any interior point O in the , OD BC⊥ and OE AC⊥ and OF AB⊥ are
drawn. Prove that 2 2 2 2 2 2 2 2 2OA OB OC OD OE OF AF BD CE+ + = + + + + +
16. A vertical pole of length 8 m costs a shadow 6 m long on the ground and at the same time a
tower casts a shadow 30 m long. Find the height of tower.
17. If tan A + cot A=2, then find the value of 2 2tan cotA A+
18. Find the value of: 2 2
2 2
sin 17 sin 73 cos31
ec ° − ° °+ ° + ° °
19. Traffic police of a city gave following distribution showing number of victims and their ages
in accidents in a year in their city. Find the median.
Age of victims
Number of
15 35 40 20 8 2
20. Weights of class IX students of a school are given in the following frequency distribution:
Weights (in kg) 35-40 40-45 45-50 50-55 55-60 60-65 65-70
Number of
Find the mode.
Section D
Question numbers 21 to 31 carry four marks each.
21. Find the HCF of 256 and 36 using Euclid’s Division Algorithm. Also find their LCM and verify
that HCF LCM× =product of the two numbers.
22. The owner of a taxi company decides to run all the taxi on CNG fuels instead of petrol/diesel.
The taxi charges in city comprises of fixed charges together with the charge for the distance
covered. For a journey of 12 km, the charge paid is Rs. 89 and for 20 km, the charge paid is
Rs. 145.
a) What will a person have a pay for travelling a distance of 30 km?
b) Why did he decide to use CNG for his taxi as a fuel?
23. Obtain all other zeroes of the polynomial 4 3 25 6 2 4x x x x− + + − , if two of its zeroes are 1 3+
and 1 3−
24. A fraction becomes 1
2 when 1 is added to the numerator and it becomes
1
3 when 1 is
subtracted from the numerator and 2 is added to the denominator. Find the fraction. Also
find the number obtained when 5 is added to numerator and 4 is subtracted from the
denominator.
25. If ABC PQR ∼ and ( ) ( )ar ABC ar PQR = then prove that ABC PQR ≅ .
26. Prove: If a line is drawn parallel to one side of a triangle intersecting other two sides at
distinct points then it divides other two sides in equal ratio
27. If 30θ = ° , verify the following:
a) 3cos3 4cos 3cosθ θ θ= −
b) sin 2 2sin cosθ θ θ=
28. If 15
3
29. If cosec A + cot A=m, show that 2
2
− = +
30. Literacy rates of 40 cities is given in the following table. If it is given that mean literacy rate is
63.5, then find the missing frequencies x and y.
Literacy
rate
(in%)
35-
50
40-
45
45-
50
50-
55
55-
60
60-
65
65-
70
70-
75
75-
80
80-
85
85-
90
Number
of cities
1 2 3 x y 6 8 4 2 3 2
31. On the annual day of a school, age – wise participation of student is given in the following
frequency distribution table:
students
Draw a ‘less than type’ ogive for the above data. Also, find median from the curve.
Section A
1. If ABC RPQ ∼ , AB=3cm, BC=5cm, AC=6cm, RP=6cm and PQ=10cm, then find QR.
2. Express cos 48 tan88ec ° + ° in terms of t-ratios of angles between 0° and 45° .
3. In PQR , if 90Q∠ = ° and 3
sin 5
R = , then find the value of cos P.
4. In the frequency distribution, if 50if =∑ and 2550i if x =∑ , then what is the mean of the
distribution?
5. Find HCF of the number 31, 310 and 3100.
6. Find the least positive integer which on adding 1 is exactly divisible by 126 and 600.
7. Find the solution of the following pair of linear equations:
3 7 5
− = + =
8. In the figure, l m and OAC OBD ∼ . If AC=5cm, OA=3cm and BD=2cm, find OB.
9. Solve the equation forθ : 2
2 2
cos 3
cot cos
10. Find the median of the data using an empirical formula, when it is given mode=35.3 and
mean=30.5.
Section C
11. Write 32875 as product of prime factors. Is this factorization unique?
12. Divide the polynomial 3 24 6 10 3x x x− − − by the polynomial 2x x+ and verify the division
algorithm.
13. Find the zeroes of the quadratic polynomial 22 5 3x x+ − and verify the relationship between
the zeros and the coefficients.
14. Solve using cross multiplication method.
4 4
− = + =
15. If in ABC , AD is median and AM BC⊥ , then prove that 2 2 21
4 AC AD BC DM BC= + × +
16. ABC is an isosceles triangle. If 90B∠ = ° , then prove that 2 22AC BC= .
17. If cos(40 ) sin 30x° − = ° , find the value of x.
18. Prove the identify:
(1 tan sec )(1 cot cos ) 2ecθ θ θ θ+ + + − =
19. The given distribution shows the number of runs scored by the batsmen in inter-school
cricket matches:
Number of
Draw a ‘more than type’ ogive for the above data.
20. In a health check-up, the number of heart beats of 40 women were recorded in the following
table:
Section D
21. Express the HCF of number 72 and 124 as a linear combination of 72 and 124.
22. Ridhi decided to use public transport to cover a distance of 300km. she travels this distance
partly by train and partly by the bus. She takes 4 hours if she travels 60 km by train and
remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10
minutes more. Find the speed of train and bus separately. Why does Ridhi decide to opt for
public transport?
23. 5 years ago, age of one sister was twice the other sister. 5 years hence their ages will be in
the ratio 2:3. Find their present ages.
24. Obtain all other zeroes of the polynomial 4 3 22 3 15 24 8x x x x+ − − − , if two of its zeroes are 2 2
and 2 2− .
25. In the figure of ABC , P is the middle point of BC and Q is middle point of AP. If extended BQ
meets AC in it, then prove that 1
3 RA CA= .
26. In a parallelogram ABCD, E is any point on side BC. Diagonal BD and AE intersect at P. Prove
that DP EP PB PA× = × .
27. If (cos ) 0A B+ = and (cos ) 3A B− = , find the value of
a) sec .tan cot .sinA B A B−
b) cos .cot sin . tanecA B A B+
28. If tan A + sin A = m and tan A – sin A = n, then prove that 2 2( ) 16m n mn− = .
29. In the adjoining figure, ABCD is a rectangle with breadth BC=7cm and 30CAB∠ = ° . Find the
length of side AB of the rectangle and length of diagonal AC. If the 60CAB∠ = ° , then what is
the size of the side AB of the rectangle (use 3 1.73= and 2 1.41= , if required)
30. During an examination, percentage of marks scored by the students are recorded and are
shown in the following table:
Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Number of
1 3 2 8 20 15 13 25 18 10
Find the mode and median for the above data.
31. In a class, heights of students are recorded as follows:
Height (in
2 5 20 40 57 75 79 80
For above data, draw a ‘less than type’ ogive and from the curve, find median. Also, verify
median by actual calculations.
Section A
1. In XYZ , A and B are points on the sides XY and XZ respectively such that AB YZ . If AY=2.2
cm, XB=3.3cm and XZ=6.6cm, then find AX.
2. Find the value of 2 24cos 60 16 tan 30ec + ° − ° .
3. If tan(3 30 ) 1x + ° = , then find the value of x.
4. Find the mean of first five prime numbers.
Section B
Question numbers 5 to 10 carry two marks each.
5. Prove that 2 2 is an irrational number.
6. Find the HCF of the numbers 520 and 468 by prime factorization method.
7. Find the zeroes of the quadratic polynomial 2 5 6x x+ + and verify the relationship between
the zeroes and their coefficients.
8. In the figure, , 120OAC OBD AOD ∠ = °∼ and 80ODB∠ = ° . Find OAC∠ and BOD∠
9. If sec tanx p qθ θ= + and tan secy p qθ θ= + , then prove that 2 2 2 2x y p q− = −
10. Draw a cumulative frequency curve for the following cumulative frequency distribution.
Class
interval
Cumulative 50 48 45 32 15 5
frequency
11. Show that square of any positive odd integer is of the form 8m+1 for some integer m.
12. Solve by elimination:
1 1 10
4 4
− = + =
15. In equilateral triangle ABC, point E lies on CA such that BE CA⊥ . Find 2 2 2AB BC CA+ + in
terms of 2BE .
16. In an isosceles triangle AB=AC. If BD AC⊥ , then show that 2 2 2 .BD CD CD AD− = .
17. If 4sec 5θ = , then evaluate: 2cos
tan cot
ec θ θ θ
− +
19. The following table gives the literacy rate of 40 cities:
Literacy
Number of
Find the modal literacy rate.
20. In the following distribution, if mean is 78, find the missing frequency (x):
Class 50-60 60-70 70-80 80-90 90-100
Frequency 8 6 12 11 x
Section D
21. If two positive integers x and y are expressible in terms of primes as 2 3x p q= and 3y p q= .
What can you say about their LCM and HCF. Is LCM a multiple of HCF? Explain.
22. Sita devi wants to make a rectangular pond on the road side for the purpose of providing
drinking water for street animals. The area of the pond will be decrease by 3 square feets if
its length is decreased by 2ft. and breadth is increased by 1 ft. Its area will be increased by 4
square feets if the length is increase by 1 ft. and breadth remains same. Find the dimension of
the pond. What motivated Sita Devi to provide water pond for street animals?
23. Pocket money of Zahira and Zohra are in the ratio 6:5 and the ratio of their expenditure are
in the ratio 4:3. If each of them saves Rs. 50 at the end of the month, find their pocket money.
24. Obtain all other zeroes of the polynomial 4 3 23 3 15 10x x x+ − − − , if two of its zeroes are 5 and
5− .
25. In right angle , 90ABC C ∠ = ° and D, E, F are three points on BC such that they divide it in
equal parts. Then prove that 2 2 2 28( ) 11 5AF AD AC AB+ = +
26. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of
their corresponding altitudes.
27. If cos(A+B)=0 and cot( ) 3A B− = , then evaluate:
a) cos .cos sin .sinA B A B−
b) cot cot
cot .cot 1
cos 1
2 2 2 2 2 2 2
cos 61 tan 29 2sin 30 3cot11 .cot 21 .cot 31 .cot 59 .cot 69 .cot 79
cos tan (90 ) tan 45 2(sin 21 sin 69 ) (cos 41 cos 49 )
ec
° − ° + ° ° ° ° ° ° °+ − ° − + ° ° + ° − ° + °
30. From a local telephone directory, some surnames were randomly picked up and the
frequency distribution of the number of letters of the English alphabets in the surnames was
obtained as follows:
Number of
6 50 40 18 5 3 1
Draw a ‘less than type’ ogive and a ‘more than type’ ogive for the above data.
31. If median salary of 100 employees of a factory is Rs.24800, then find the missing frequencies
1f and 2f in the given distribution table:
Salary (in
30/3 (SPL) 1 P.T.O.
narjmWu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wI-n¥ð >na Adí` {bIo§ & Candidates must write the Code on the
title page of the answer-book.
Series HRS H$moS> Z§. 30/3 (SPL)
Code No.
amob Z§.
Roll No.
SUMMATIVE ASSESSMENT – II
MATHEMATICS
{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 90
Time allowed : 3 hours Maximum Marks : 90
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 15 h¢ &
àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE H$moS >Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$ _wI-n¥ð> na {bI| &
H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| >34 àíZ h¢ & H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$_m§H$ Adí` {bI| &
Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU nydm©• _| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &
Please check that this question paper contains 15 printed pages.
Code number given on the right hand side of the question paper should be
written on the title page of the answer-book by the candidate.
Please check that this question paper contains 34 questions.
Please write down the Serial Number of the question before
attempting it.
15 minutes time has been allotted to read this question paper. The question
paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the
students will read the question paper only and will not write any answer on
the answer-book during this period.
30/3 (SPL) 2
gm_mÝ` {ZX}e :
(i) g^r àíZ A{Zdm`© h¢ & (ii) Bg àíZ-nÌ _| 34 àíZ h¢ Omo Mma IÊS>m| A, ~, g Am¡a X _| {d^m{OV h¢ & (iii) IÊS> A _| EH$-EH$ A§H$ dmbo 8 àíZ h¢, Omo ~hþ-{dH$ënr àíZ h¢ & IÊS> ~ _| 6 àíZ h¢
{OZ_| go àËòH$ 2 A§H$ H$m h¡ & IÊS> g _| 10 àíZ VrZ-VrZ A§H$m| Ho$ h¢ & IÊS> X _| 10 àíZ h¢ {OZ_| go àËòH$ 4 A§H$ H$m h¡ &
(iv) H¡$bHw$boQ>a H$m à`moJ d{O©V h¡ &
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A,
B, C and D.
(iii) Section A contains 8 questions of 1 mark each, which are multiple choice
type questions, Section B contains 6 questions of 2 marks each, Section C
contains 10 questions of 3 marks each and Section D contains
10 questions of 4 marks each.
(iv) Use of calculators is not permitted.
IÊS> A SECTION A
àíZ g§»`m 1 go 8 VH$ àËòH$ àíZ 1 A§H$ H$m h¡ & àíZ g§»`m 1go 8 _| àËòH$ àíZ Ho$ {bE Mma {dH$ën {XE JE h¢, {OZ_| go Ho$db EH$ ghr h¡ & ghr {dH$ën Mw{ZE & Question numbers 1 to 8 carry 1 mark each. For each of the question numbers
1 to 8, four alternative choices have been provided, of which only one is correct.
Select the correct choice.
1. PQ EH$ d¥Îm na {~ÝXþ P go JwµOaZo dmbr ñne© aoIm h¡ {OgHo$ d¥Îm H$m Ho$ÝÐ-{~ÝXþ O h¡ & `{X OPQ EH$ g_{Û~mhþ {Ì^wO h¡, Vmo OQP H$m _mZ h¡ (A) 30
(B) 45
(C) 60
(D) 90
PQ is a tangent to a circle with centre O at the point P. If OPQ is an
isosceles triangle, then OQP is equal to
(A) 30
(B) 45
(C) 60
(D) 90
30/3 (SPL) 3 P.T.O.
2. AmH¥${V 1 _|, RQ d¥Îm na EH$ ñne© aoIm h¡ {OgH$m Ho$ÝÐ-{~ÝXþ O h¡ & `{X SQ = 6 go_r VWm QR = 4 go_r h¡, Vmo OR H$s b§~mB© h¡
AmH¥${V 1
(A) 8 go_r
(B) 3 go_r
(C) 2.5 go_r
(D) 5 go_r
In figure 1, RQ is a tangent to the circle with centre O. If SQ = 6 cm and
QR = 4 cm, then OR is
Figure 1
(A) 8 cm
(B) 3 cm
(C) 2.5 cm
(D) 5 cm
30/3 (SPL) 4
3. EH$ nV§J VWm ^y{_ na EH$ {~ÝXþ Ho$ ~rM ~±Yr S>moar H$s b§~mB© 85 _r. h¡ & `{X S>moar
^y{_Vb Ho$ gmW H$moU Bg àH$ma ~Zm ahr h¡ {H$ tan = 8
15 h¡, Vmo nV§J H$s ^y{_ go
D±$MmB© {H$VZr h¡ ?
(A) 75 _r.
(B) 79.41 _r.
(C) 80 _r.
(D) 72.5 _r.
The length of a string between a kite and a point on the ground is
85 m. If the string makes an angle with the ground level such that
tan = 8
15 , then the kite is at what height from the ground ?
(A) 75 m
(B) 79.41 m
(C) 80 m
(D) 72.5 m
4. `{X EH$ d¥Îm H$s n[a{Y 8 h¡, Vmo CgH$m joÌ\$b h¡
(A) 8
(B) 16
(C) 4
(D) 32
If the circumference of a circle is 8 , then its area is
(A) 8
(B) 16
(C) 4
(D) 32
30/3 (SPL) 5 P.T.O.
5. EH$ AÀN>r Vah \|$Q>r JB© Vme Ho$ nÎmm| H$s JÈ>r _| go BªQ> Ho$ ~mXemh VWm ~oJ_ H$mo hQ>m
{X`m OmVm h¡ & {\$a ~Mo hþE nÎmm| _| go EH$ nÎmm `mÑÀN>`m N>m±Q>m OmVm h¡ & Vmo ‘{M‹S>r Ho$
~mXemh’ H$mo àmá H$aZo H$s àm{`H$Vm h¡
(A) 4
3
The king and queen of diamonds are removed from a pack of
well-shuffled playing cards. One card is selected at random from the
remaining cards. The probability of getting the ‘king of clubs’ is
(A) 4
3
6. `{X 7, x, y, – 5 EH$ g_m§Va lo‹T>r _| h¢, Vmo x – y H$m _mZ h¡
(A) 2
(B) 3
(C) 4
(D) – 4
30/3 (SPL) 6
If 7, x, y, – 5 are in A.P., then the value of x – y is
(A) 2
(B) 3
(C) 4
(D) – 4
7. `{X {~ÝXþ P(5, y), {~ÝXþAm| A(3, 5) VWm B(x, 3) H$mo {_bmZo dmbo aoImIÊS> AB H$m _Ü`-{~ÝXþ h¡, Vmo (x + y) H$m _mZ h¡
(A) 11
(B) 3
(C) 7
(D) 4
If P(5, y) is the mid-point of the line segment AB joining the points A(3, 5)
and B(x, 3), then (x + y) equals
(A) 11
(B) 3
(C) 7
(D) 4
8. EH$ nmgo H$mo EH$ ~ma CN>mbZo na, 4 go ~‹S>r g§»`m AmZo H$s àm{`H$Vm h¡
(A) 6
30/3 (SPL) 7 P.T.O.
The probability of getting a number greater than 4, when a die is rolled
once, is
(A) 6
IÊS> ~ SECTION B
àíZ g§»`m 9 go 14 VH$ àËòH$ àíZ Ho$ 2 A§H$ h¢ & Question numbers 9 to 14 carry 2 marks each.
9. PA VWm PB EH$ ~mø {~ÝXþ P go O Ho$ÝÐ dmbo d¥Îm H$s ñne© aoImE± h¢, Omo d¥Îm H$mo H«$_e:
{~ÝXþAm| A VWm B na ñne© H$aVr h¢ & Xem©BE {H$ MVw^w©O AOBP MH«$s` MVw^w©O h¡ &
PA and PB are tangents to the circle with centre O from an external
point P, touching the circle at A and B respectively. Show that the
quadrilateral AOBP is cyclic.
10. EH$ {n½Jr ~¢H$ _| 50 n¡go Ho$ 100 {g¸o$, < 1 Ho$ 50 {g¸o$, < 2 Ho$ 20 {g¸o$ VWm < 5 Ho$ 10 {g¸o$ h¢ & `{X Bg ~¢H$ H$mo CbQ>m H$aZo na àËòH$ {g¸o$ Ho$ ~mha {JaZo H$s g§^mdZm ~am~a h¡, Vmo Š`m àm{`H$Vm hmoJr {H$ {JaZo dmbm nhbm {g¸$m < 5 H$m {g¸$m Zht
h¡ ?
A piggy bank contains hundred 50 paise coins, fifty < 1 coins,
twenty < 2 coins and ten < 5 coins. If it is equally likely that one of the
coins will fall out when the bank is turned upside down, what is the
probability that the first coin will not be a < 5 coin ?
30/3 (SPL) 8
11. AmH¥${V 2 _|, ABCD EH$ Eogm MVw^w©O h¡ {Og_| D = 90 h¡ & EH$ d¥Îm {OgH$m
Ho$ÝÐ O VWm {ÌÁ`m r h¡, MVw^©wO H$s ^wOmAm| AB, BC, CD VWm DA H$mo H«$_e: P, Q, R
VWm S na ñne© H$aVm h¡ & `{X BC = 40 go_r, CD = 25 go_r VWm BP = 28 go_r h¡,
Vmo r H$m _mZ kmV H$s{OE &
AmH¥${V 2
In figure 2, ABCD is a quadrilateral such that D = 90. A circle with
centre O and radius r, touches the sides AB, BC, CD and DA at P, Q, R
and S respectively. If BC = 40 cm, CD = 25 cm and BP = 28 cm, find r.
Figure 2
12. grgo go ~Zo hþE EH$ R>mog KZ, {OgH$s ^wOm H$s _mn 44 go_r h¡, _| go {H$VZr JmobmH$ma R>mog Jmo{b`m± ~ZmB© Om gH$Vr h¢, `{X àËòH$ Jmobr H$m ì`mg 4 go_r h¡ &
[ = 7
22 br{OE ]
How many spherical solid bullets can be made out of a solid cube of
lead whose edge measures 44 cm, each bullet being 4 cm in diameter.
[ Use = 7
30/3 (SPL) 9 P.T.O.
13. EH$ g_m§Va lo‹T>r H$m Vrgam Ed§ 7dm± nX H«$_e… 13 VWm 33 h¡, Vmo BgH$m ndm± nX kmV H$s{OE &
The third and the 7th terms of an A.P. are 13 and 33 respectively. Find
the nth term of the A.P.
14. k Ho$ {H$g _mZ Ho$ {bE {ÛKmV g_rH$aU kx2 – 2 (k + 2) x + (k + 5) = 0 Ho$ _yb dmñV{dH$ VWm g_mZ hm|Jo ?
For what value of k are the roots of the quadratic equation
kx2 – 2 (k + 2) x + (k + 5) = 0 real and equal ?
IÊS> g SECTION C
15. AmH¥${V 3 _|, Xmo g§H|$Ðr` d¥Îmmo§ H$s {ÌÁ`mE± 13 go_r VWm 8 go_r h¢ & AB ~‹S>r {ÌÁ`m dmbo d¥Îm H$m ì`mg h¡ VWm BD N>moQ>o d¥Îm H$s$ {~ÝXþ D na EH$ ñne© aoIm h¡ & AD H$s b§~mB©
kmV H$s{OE &
AmH¥${V 3
30/3 (SPL) 10
In figure 3, the radii of two concentric circles are 13 cm and 8 cm. AB is a
diameter of the bigger circle and BD is a tangent to the smaller circle
touching it at D. Find the length of AD.
Figure 3
16. EH$ \$mC§Q>oZ noZ H$m ~¡ab, ~obZmH$ma h¡ VWm BgH$s b§~mB© 7 go_r d ì`mg 5 {_br_rQ>a h¡ & `{X ñ`mhr go nyao âo ~¡ab go, Am¡gVZ 330 eãX {bIo Om gH$Vo h¢, Vmo {H$gr ~moVb {Og_| EH$ brQ>a H$m nm±Mdm± ^mJ ñ`mhr âr h¡, go {H$VZo eãX {bIo Om gH|$Jo ?
The barrel of a fountain-pen, cylindrical in shape, is 7 cm long and 5 mm
in diameter. If with a full barrel of ink, on an average 330 words can be
written, then how many words would use up a bottle of ink containing
one-fifth of a litre ?
17. EH$ ^dZ H$m Am§V[aH$ ^mJ bå~-d¥Îmr` ~obZ Ho$ ê$n _| h¡ VWm CgH$m ì`mg 4.2 _r. Ed§ D±$MmB© 4 _r. h¡ & ~obZmH$ma ^mJ Ho$ D$na g_mZ ì`mg H$m EH$ e§Hw$ Amamo{nV h¡ {OgH$s D±$MmB© 2.8 _r. h¡ & ^dZ H$m ~mhar n¥ð>r` joÌ\$b kmV H$s{OE & The interior of a building is in the form of a right circular cylinder of
diameter 4.2 m and height 4 m surmounted by a cone of same diameter.
The height of the cone is 2.8 m. Find the outer surface area of the
building.
18. {H$gr jU EH$ ÜdOXÊS> H$s N>m`m H$s b§~mB©, Cg g_` H$s BgH$s N>m`m H$s b§~mB© H$s VrZ JwZr h¡, O~ gy`© H$s {H$aU| ^y{_ Ho$ gmW 60 H$m H$moU ~ZmVr h¢ & Bg jU H$m dh H$moU kmV H$s{OE Omo gy`© H$s {H$aU| y{_ Ho$ gmW ~ZmVr h¢ &
At an instant the shadow of a flag-staff is three times as long as its
shadow, when the sun-rays make an angle of 60 with the ground. Find
the angle between the sun-rays and the ground at this instant.
30/3 (SPL) 11 P.T.O.
19. dh AZwnmV kmV H$s{OE {Og_| {~ÝXþ Q (− 3, p), {~ÝXþAm| A (− 5, − 4) VWm B (− 2, 3)
H$mo {_bmZo dmbo aoImIÊS> AB H$mo {d^m{OV H$aVm h¡ & p H$m _mZ ^r kmV H$s{OE & Find the ratio in which the point Q (− 3, p) divides the line segment AB
joining the points A (− 5, − 4) and B (− 2, 3). Also, find the value of p.
20. EH$ e§Hw$ Ho$ {N>ÞH$ Ho$ XmoZm| d¥Îmr` {gam| Ho$ n[a_mn 48 go_r VWm 36 go_r h¢ & `{X {N>ÞH$
H$s D±$MmB© 11 go_r h¡, Vmo CgH$m Am`VZ kmV H$s{OE & [ = 7
22 br{OE ]
The perimeters of the two circular ends of a frustum of a cone are 48 cm
and 36 cm. If the height of the frustum is 11 cm, find its volume.
[Use = 7
22 ]
21. g_m§Va lo‹T>r 48, 42, 36, … Ho$ {H$VZo nXm| H$m `moJ\$b 216 h¡ ? Xmo CÎma AmZo Ho$ H$maU H$s ì`m»`m H$s{OE &
How many terms of the A.P. 48, 42, 36, … be taken so that the sum is
216 ? Explain the double answer.
22. 14 go_r {ÌÁ`m dmbo EH$ d¥Îm H$s EH$ Ordm, d¥Îm Ho$ Ho$ÝÐ na 60 H$m H$moU A§V[aV H$aVr h¡ & g§JV bKw VWm XrK© d¥ÎmIÊS>m| Ho$ joÌ\$b kmV H$s{OE &
[ = 7
22 VWm 3 = 1.73 br{OE]
A chord of a circle of radius 14 cm, subtends an angle of 60 at the centre.
Find the areas of the corresponding minor and major segments of the
circle. [Use = 7
22 and 3 = 1.73]
23. Ma x _| {ÛKmV g_rH$aU 12 abx2 – (9a2 – 8b2) x – 6ab = 0 Ho$ _yb kmV H$s{OE &
Find the roots of the quadratic equation 12 abx2 – (9a2 – 8b2) x – 6ab = 0
in the variable x.
24. {Ì^wO PQR H$s _mpÜ`H$mAm| RS VWm PT H$s b§~mB`m± kmV H$s{OE, O~{H$ {Ì^wO Ho$ erf© P(6, – 2), Q(6, 3) VWm R(3, 1) h¢ &
Find the lengths of the medians RS and PT of a triangle PQR whose
vertices are P(6, – 2), Q(6, 3) and R(3, 1).
30/3 (SPL) 12
IÊS> X SECTION D
25. AmH¥${V 4 _|, N>m`m§{H$V ^mJ H$m joÌ\$b kmV H$s{OE, Ohm± 12 go_r ^wOm dmbo EH$ g_~mhþ {Ì^wO OAB Ho$ erf© O H$mo Ho$ÝÐ _mZH$a 6 go_r {ÌÁ`m dmbm d¥Îmr` Mmn ItMm J`m VWm erf© B H$mo Ho$ÝÐ boH$a 6 go_r {ÌÁ`m Ho$ d¥Îm H$m EH$ {ÌÁÌÊS> ~Zm`m J`m h¡ &
AmH¥${V 4
Find the area of the shaded region in figure 4, where a circular arc of
radius 6 cm has been drawn with vertex O of an equilateral triangle OAB
of side 12 cm as centre and a sector of circle of radius 6 cm with centre B
is made.
Figure 4
30/3 (SPL) 13 P.T.O.
26. EH$ _moQ>a-Zm¡H$m, {OgH$s pñWa Ob _| Mmb 24 {H$_r à{V K§Q>m h¡, 32 {H$_r Ymam Ho$ à{VHy$b OmZo _|, dhr Xÿar Ymam Ho$ AZwHy$b OmZo H$s Anojm 1 K§Q>m A{YH$ boVr h¡ & Ymam H$s Mmb kmV H$s{OE &
A motorboat, whose speed is 24 km/h in still water, takes 1 hour more to
go 32 km upstream than to return downstream to the same spot. Find the
speed of the stream.
27. I‹S>r MÅ>mZ na I‹S>m EH$ ì`{º$ EH$ Zmd H$mo AnZo R>rH$ ZrMo {H$Zmao H$s Amoa EH$g_mZ Mmb go 30 Ho$ AdZ_Z H$moU na AmVm XoI ahm h¡ & 6 {_ZQ> Ho$ níMmV² Zmd H$m `h AdZ_Z H$moU 60 hmo OmVm h¡ & Zmd Ûmam {H$Zmao na nhþ±MZo H$m Hw$b g_` kmV H$s{OE &
A man on a cliff observes a boat at an angle of depression of 30 which is
approaching the shore to the point immediately beneath the observer
with a uniform speed. Six minutes later the angle of depression of the
boat is found to be 60. Find the total time taken by the boat to reach the
shore.
28. 32 H$mo Mma Eogo ^mJm| _| {d^m{OV H$s{OE {H$ `h Mmam| ^mJ EH$ g_m§Va lo‹T>r Ho$ Mma nX hm| VWm nhbo d Mm¡Wo nXm| Ho$ JwUZ\$b VWm Xÿgao d Vrgao nXm| Ho$ JwUZ\$b _| 7 : 15 H$m AZwnmV hmo &
Divide 32 into four parts which are the four terms of an A.P., such that
the product of the first and the fourth terms is to the product of the
second and the third terms as 7 : 15.
29. ABC {OgHo$ erf©-{~ÝXþ A(0, – 1), B(2, 1) VWm C(0, 3) h¢, H$m joÌ\$b kmV H$s{OE &
BgH$s ^wOmAm| Ho$ _Ü`-{~ÝXþAm| H$mo {_bmH$a ~ZmE JE {Ì^wO H$m joÌ\$b ^r kmV
H$s{OE & Xem©BE {H$ XmoZm| {Ì^wOm| Ho$ joÌ\$bm| _| 4 : 1 H$m AZwnmV h¡ &
Find the area of ABC with vertices A(0, – 1), B(2, 1) and C(0, 3). Also
find the area of the triangle formed by joining their mid-points. Show
that the ratio of the areas of the two triangles is 4 : 1.
30/3 (SPL) 14
30. EH$ {Ì^wO ABC H$s aMZm H$s{OE {OgH$s ^wOm BC = 7 go_r, B = 45 Am¡a A = 105 h¡ & BgHo$ níMmV² EH$ AÝ` {Ì^wO H$s aMZm H$s{OE {OgH$s ^wOmE± {Ì^wO
ABC H$s g§JV ^wOmAm| H$s 4
3 JwZm hm| &
Draw a triangle ABC with side BC = 7 cm, B = 45 and A = 105.
Then construct another triangle whose sides are 4
3 times the
corresponding sides of ABC.
31. EH$ µOma _| VrZ {^Þ a§Jm| — Zrbo, hao VWm gµ\o$X a§J Ho$ Hw$b 54 H§$Mo h¢ & EH$ Zrbo a§J Ho$
H§$Mo H$mo `mÑÀN>`m {ZH$mbZo H$s àm{`H$Vm 3
1 VWm hao a§J Ho$ H§$Mo H$mo$ `mÑÀN>`m {ZH$mbZo
H$s àm{`H$Vm 9
4 h¡ & Bg µOma _| {H$VZo gµ\o$X a§J Ho$ H§$Mo h¢ ?
A jar contains 54 marbles of three different colours — blue, green and
white. The probability of drawing a blue marble at random is 3
1 and that
4 . How many white marbles are there in the jar ?
32. 150 H$m_Jmam| H$mo EH$ H$m`© H$mo Hw$N> {XZm| _| g_má H$aZo Ho$ {bE {Z wº$ {H$`m J`m &
4 H$m_Jmam| Zo Xÿgao {XZ H$m © N>mo‹S> {X`m & 4 Am¡a H$m_Jmam| Zo Vrgao {XZ H$m`© N>mo‹S> {X`m &
Bgr àH$ma H«$_e… ha {XZ hmoVm J`m & Eogm H$aZo go H$m`© {ZYm©[aV g_` go 8 A{YH$ {XZm|
_| g_má hþAm & Vmo H$m`© Hw$b {H$VZo {XZm| _| nyam hþAm ?
150 workers were engaged to finish a piece of work in a certain number of
days. Four workers dropped the second day, four more workers dropped
the third day and so on. It took 8 more days to finish the work. Then find
the number of days in which the work was completed.
30/3 (SPL) 15 P.T.O.
33. 4 _r. ì`mg H$m EH$ Hw$Am± 14 _r. JhamB© VH$ ImoXm OmVm h¡ Am¡a ImoXZo go {ZH$br hþB© {_Å>r H$mo Hw$E± Ho$ Mmam| Amoa 4 _r. Mm¡‹S>r EH$ d¥ÎmmH$ma db` (ring) ~ZmVo hþE, g_mZ ê$n go \¡$bmH$a EH$ àH$ma H$m ~m±Y ~Zm`m OmVm h¡ & Bg ~m±Y H$s D±$MmB© kmV H$s{OE & `h àíZ {H$g _yë` H$mo Xem©Vm h¡ ?
A well of diameter 4 m is dug 14 m deep. The earth taken out of it has
been spread evenly all around it in the shape of a circular ring of width
4 m to form an embankment. Find the height of the embankment. What
value is shown in this question ?
34. {gÕ H$s{OE {H$ d¥Îm Ho$ {H$gr {~ÝXþ na ñne© aoIm, ñne© {~ÝXþ go hmoH$a OmZo dmbr {ÌÁ`m
na b§~ hmoVr h¡ &
Prove that the tangent at any point of a circle is perpendicular to the
radius through the point of contact.
30/2 (SPL) 1 P.T.O.
Series HRS H$moS> Z§. 30/2 (SPL)
Code No.
amob Z§.
Roll No.
MATHEMATICS
attempting it.
30/2 (SPL) 2
gm_mÝ` {ZX}e :
(iv) H¡$bHw$boQ>a H$m à`moJ d{O©V h¡ & General Instructions :
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.
(iii) Section A contains 8 questions of 1 mark each, which are multiple choice type questions, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 10 questions of 4 marks each.
IÊS> A SECTION A
1. AmH¥${V 1 _|, RQ d¥Îm na EH$ ñne© aoIm h¡ {OgH$m Ho$ÝÐ-{~ÝXþ O h¡ & `{X SQ = 6 go_r VWm QR = 4 go_r h¡, Vmo OR H$s b§~mB© h¡
AmH¥${V 1
(A) 8 go_r
(B) 3 go_r
(C) 2.5 go_r
(D) 5 go_r
In figure 1, RQ is a tangent to the circle with centre O. If SQ = 6 cm and
QR = 4 cm, then OR is
Figure 1
(A) 8 cm
(B) 3 cm
(C) 2.5 cm
(D) 5 cm
2. EH$ nV§J VWm ^y{_ na EH$ {~ÝXþ Ho$ ~rM ~±Yr S>moar H$s b§~mB© 85 _r. h¡ & `{X S>moar
^y{_Vb Ho$ gmW H$moU Bg àH$ma ~Zm ahr h¡ {H$ tan = 8
15 h¡, Vmo nV§J H$s ^y{_ go
D±$MmB© {H$VZr h¡ ?
(A) 75 _r.
(B) 79.41 _r.
(C) 80 _r.
(D) 72.5 _r.
30/2 (SPL) 4
The length of a string between a kite and a point on the ground is
85 m. If the string makes an angle with the ground level such that
tan = 8
15 , then the kite is at what height from the ground ?
(A) 75 m
(B) 79.41 m
(C) 80 m
(D) 72.5 m
3. `{X EH$ d¥Îm H$s n[a{Y 8 h¡, Vmo CgH$m joÌ\$b h¡
(A) 8
(B) 16
(C) 4
(D) 32
If the circumference of a circle is 8 , then its area is
(A) 8
(B) 16
(C) 4
(D) 32
4. PQ EH$ d¥Îm na {~ÝXþ P go JwµOaZo dmbr ñne© aoIm h¡ {OgHo$ d¥Îm H$m Ho$ÝÐ-{~ÝXþ O h¡ & `{X OPQ EH$ g_{Û~mhþ {Ì^wO h¡, Vmo OQP H$m _mZ h¡
(A) 30
(B) 45
(C) 60
(D) 90
30/2 (SPL) 5 P.T.O.
PQ is a tangent to a circle with centre O at the point P. If OPQ is an
isosceles triangle, then OQP is equal to
(A) 30
(B) 45
(C) 60
(D) 90
5. EH$ AÀN>r Vah \|$Q>r JB© Vme Ho$ nÎmm| H$s JÈ>r _| go BªQ> Ho$ ~mXemh VWm ~oJ_ H$mo hQ>m
{X`m OmVm h¡ & {\$a ~Mo hþE nÎmm| _| go EH$ nÎmm `mÑÀN>`m N>m±Q>m OmVm h¡ & Vmo ‘{M‹S>r Ho$
~mXemh’ H$mo àmá H$aZo H$s àm{`H$Vm h¡
(A) 4
3
The king and queen of diamonds are removed from a pack of
well-shuffled playing cards. One card is selected at random from the
remaining cards. The probability of getting the ‘king of clubs’ is
(A) 4
3
6. `{X 5, a, b VWm 11 EH$ g_m§Va lo‹T>r _| h¢, Vmo (a + b) H$m _mZ h¡
30/2 (SPL) 6
(A) 7
(B) 9
(C) 11
(D) 16
If 5, a, b and 11 are in A.P., the value of (a + b) is
(A) 7
(B) 9
(C) 11
(D) 16
7. Xmo nmgm| H$mo EH$gmW \|$H$m OmVm h¡ & XmoZm| nmgm| na AmZo dmbo A§H$m| H$m `moJ\$b A{YH$ go A{YH$ 4 AmZo H$s àm{`H$Vm h¡
(A) 6
1
Two dice are thrown simultaneously. The probability of getting a sum at
most four on the two dice is
(A) 6
30/2 (SPL) 7 P.T.O.
8. `{X {~ÝXþAm| P(5, 6) VWm Q(x, 4) H$mo {_bmZo dmbo aoImIÊS> PQ H$m _Ü`-{~ÝXþ R(4, 5)
h¡, Vmo x H$m _mZ h¡
(A) 4
(B) 3
(C) 5
(D) 6
If R(4, 5) is the mid-point of the line segment PQ joining the points P(5, 6)
and Q(x, 4), then x equals
(A) 4
(B) 3
(C) 5
(D) 6
IÊS> ~ SECTION B
9. EH$ {n½Jr ~¢H$ _| 50 n¡go Ho$ 100 {g¸o$, < 1 Ho$ 50 {g¸o$, < 2 Ho$ 20 {g¸o$ VWm
< 5 Ho$ 10 {g¸o$ h¢ & `{X Bg ~¢H$ H$mo CbQ>m H$aZo na àËòH$ {g¸o$ Ho$ ~mha {JaZo H$s g§^mdZm ~am~a h¡, Vmo Š`m àm{`H$Vm hmoJr {H$ {JaZo dmbm nhbm {g¸$m < 5 H$m {g¸$m Zht
h¡ ?
A piggy bank contains hundred 50 paise coins, fifty < 1 coins,
twenty < 2 coins and ten < 5 coins. If it is equally likely that one of the
coins will fall out when the bank is turned upside down, what is the
probability that the first coin will not be a < 5 coin ?
10. PA VWm PB EH$ ~mø {~ÝXþ P go O Ho$ÝÐ dmbo d¥Îm H$s ñne© aoImE± h¢, Omo d¥Îm H$mo H«$_e:
{~ÝXþAm| A VWm B na ñne© H$aVr h¢ & Xem©BE {H$ MVw^w©O AOBP MH«$s` MVw^w©O h¡ &
PA and PB are tangents to the circle with centre O from an external
point P, touching the circle at A and B respectively. Show that the
quadrilateral AOBP is cyclic.
30/2 (SPL) 8
11. grgo go ~Zo hþE EH$ R>mog KZ, {OgH$s ^wOm H$s _mn 44 go_r h¡, _| go {H$VZr JmobmH$ma R>mog Jmo{b`m± ~ZmB© Om gH$Vr h¢, `{X àËòH$ Jmobr H$m ì`mg 4 go_r h¡ &
[ = 7
22 br{OE ]
How many spherical solid bullets can be made out of a solid cube of
lead whose edge measures 44 cm, each bullet being 4 cm in diameter.
[ Use = 7
22 ]
12. AmH¥${V 2 _|, ABCD EH$ Eogm MVw^w©O h¡ {Og_| D = 90 h¡ & EH$ d¥Îm {OgH$m
Ho$ÝÐ O VWm {ÌÁ`m r h¡, MVw^©wO H$s ^wOmAm| AB, BC, CD VWm DA H$mo H«$_e: P, Q, R
VWm S na ñne© H$aVm h¡ & `{X BC = 40 go_r, CD = 25 go_r VWm BP = 28 go_r h¡,
Vmo r H$m _mZ kmV H$s{OE &
AmH¥${V 2
In figure 2, ABCD is a quadrilateral such that D = 90. A circle with
centre O and radius r, touches the sides AB, BC, CD and DA at P, Q, R
and S respectively. If BC = 40 cm, CD = 25 cm and BP = 28 cm, find r.
Figure 2
30/2 (SPL) 9 P.T.O.
13. k Ho$ {H$g _mZ Ho$ {bE {ÛKmV g_rH$aU (k + 2) x2 + 2kx + (k – 1) = 0 Ho$ _yb dmñV{dH$ VWm g_mZ h¢ ?
For what value of k are the roots of the quadratic equation
(k + 2) x2 + 2kx + (k – 1) = 0 real and equal ?
14. Cg g_m§Va lo‹T>r H$m ndm± nX kmV H$s{OE {OgH$m N>R>m nX – 7 h¡ VWm 10dm± nX – 19 h¡ &
Find the nth term of the A.P. whose 6th term is – 7 and 10th term is – 19.
IÊS> g SECTION C
15. EH$ e§Hw$ Ho$ {N>ÞH$ Ho$ XmoZm| d¥Îmr` {gam| Ho$ n[a_mn 48 go_r VWm 36 go_r h¢ & `{X {N>ÞH$
H$s D±$MmB© 11 go_r h¡, Vmo CgH$m Am`VZ kmV H$s{OE & [ = 7
22 br{OE ]
The perimeters of the two circular ends of a frustum of a cone are 48 cm
and 36 cm. If the height of the frustum is 11 cm, find its volume.
[Use = 7
22 ]
16. {H$gr jU EH$ ÜdOXÊS> H$s N>m`m H$s b§~mB©, Cg g_` H$s BgH$s N>m`m H$s b§~mB© H$s VrZ JwZr h¡, O~ gy`© H$s {H$aU| ^y{_ Ho$ gmW 60 H$m H$moU ~ZmVr h¢ & Bg jU H$m dh H$moU kmV H$s{OE Omo gy`© H$s {H$aU| y{_ Ho$ gmW ~ZmVr h¢ & At an instant the shadow of a flag-staff is three times as long as its
shadow, when the sun-rays make an angle of 60 with the ground. Find
the angle between the sun-rays and the ground at this instant.
17. EH$ ^dZ H$m Am§V[aH$ ^mJ bå~-d¥Îmr` ~obZ Ho$ ê$n _| h¡ VWm CgH$m ì`mg 4.2 _r. Ed§ D±$MmB© 4 _r. h¡ & ~obZmH$ma ^mJ Ho$ D$na g_mZ ì`mg H$m EH$ e§Hw$ Amamo{nV h¡ {OgH$s D±$MmB© 2.8 _r. h¡ & ^dZ H$m ~mhar n¥ð>r` joÌ\$b kmV H$s{OE & The interior of a building is in the form of a right circular cylinder of
diameter 4.2 m and height 4 m surmounted by a cone of same diameter.
The height of the cone is 2.8 m. Find the outer surface area of the
building.
30/2 (SPL) 10
18. EH$ \$mC§Q>oZ noZ H$m ~¡ab, ~obZmH$ma h¡ VWm BgH$s b§~mB© 7 go_r d ì`mg 5 {_br_rQ>a h¡ & `{X ñ`mhr go nyao âo ~¡ab go, Am¡gVZ 330 eãX {bIo Om gH$Vo h¢, Vmo {H$gr ~moVb {Og_| EH$ brQ>a H$m nm±Mdm± ^mJ ñ`mhr âr h¡, go {H$VZo eãX {bIo Om gH|$Jo ?
The barrel of a fountain-pen, cylindrical in shape, is 7 cm long and 5 mm
in diameter. If with a full barrel of ink, on an average 330 words can be
written, then how many words would use up a bottle of ink containing
one-fifth of a litre ?
19. dh AZwnmV kmV H$s{OE {Og_| {~ÝXþ Q (− 3, p), {~ÝXþAm| A (− 5, − 4) VWm B (− 2, 3)
H$mo {_bmZo dmbo aoImIÊS> AB H$mo {d^m{OV H$aVm h¡ & p H$m _mZ ^r kmV H$s{OE & Find the ratio in which the point Q (− 3, p) divides the line segment AB
joining the points A (− 5, − 4) and B (− 2, 3). Also, find the value of p.
20. AmH¥${V 3 _|, Xmo g§H|$Ðr` d¥Îmmo§ H$s {ÌÁ`mE± 13 go_r VWm 8 go_r h¢ & AB ~‹S>r {ÌÁ`m dmbo d¥Îm H$m ì`mg h¡ VWm BD N>moQ>o d¥Îm H$s {~ÝXþ D na EH$ ñne© aoIm h¡ & AD H$s b§~mB©
kmV H$s{OE &
AmH¥${V 3 In figure 3, the radii of two concentric circles are 13 cm and 8 cm. AB is a
diameter of the bigger circle and BD is a tangent to the smaller circle
touching it at D. Find the length of AD.
Figure 3
30/2 (SPL) 11 P.T.O.
21. {ÌÁ`m 4 go_r dmbo d¥Îm Ho$ EH$ {ÌÁÌÊS> H$m joÌ\$b kmV H$s{OE {OgH$m Ho$ÝÐr`
H$moU 30 h¡ & g§JV XrK© {ÌÁÌ§S> H$m joÌ\$b ^r kmV H$s{OE & ( = 3.14 br{OE)
Find the area of the sector of a circle with radius 4 cm and central angle
30. Also find the area of the corresponding major sector. (Use = 3.14)
22. {~ÝXþAm| A(– 3, 10) VWm B(x, – 8) H$mo Omo‹S>Zo dmbo aoImIÊS> AB H$mo {~ÝXþ P(– 1, 6)
{H$g AZwnmV _| {d^m{OV H$aVm h¡ ? x H$m _mZ ^r kmV H$s{OE &
Find the ratio in which the line segment AB joining the points A(– 3, 10)
and B (x, – 8) is divided by the point P (– 1, 6). Also find the value of x.
23. x Ho$ {bE hb H$s{OE :
3

24. g_m§Va lo‹T>r 7, 12, 17, … Ho$ {H$VZo nXm| H$m `moJ\$b 2542 h¡ ? Bg g_mÝVa lo‹T>r H$m
31dm± nX ^r kmV H$s{OE &
How many terms of the A.P. 7, 12, 17, … will sum up to 2542 ? Also find
the 31st term of this A.P.
30/2 (SPL) 12
IÊS> X SECTION D
25. ABC {OgHo$ erf©-{~ÝXþ A(0, – 1), B(2, 1) VWm C(0, 3) h¢, H$m joÌ\$b kmV H$s{OE &
BgH$s ^wOmAm| Ho$ _Ü`-{~ÝXþAm| H$mo {_bmH$a ~ZmE JE {Ì^wO H$m joÌ\$b ^r kmV
H$s{OE & Xem©BE {H$ XmoZm| {Ì^wOm| Ho$ joÌ\$bm| _| 4 : 1 H$m AZwnmV h¡ &
Find the area of ABC with vertices A(0, – 1), B(2, 1) and C(0, 3). Also
find the area of the triangle formed by joining their mid-points. Show
that the ratio of the areas of the two triangles is 4 : 1.
26. EH$ {Ì^wO ABC H$s aMZm H$s{OE {OgH$s ^wOm BC = 7 go_r, B = 45 Am¡a A = 105 h¡ & BgHo$ níMmV² EH$ AÝ` {Ì^wO H$s aMZm H$s{OE {OgH$s ^wOmE± {Ì^wO
ABC H$s g§JV ^wOmAm| H$s 4
3 JwZm hm| &
Draw a triangle ABC with side BC = 7 cm, B = 45 and A = 105.
Then construct another triangle whose sides are 4
3 times the
corresponding sides of ABC.
27. 150 H$m_Jmam| H$mo EH$ H$m`© H$mo Hw$N> {XZm| _| g_má H$aZo Ho$ {bE {Z wº$ {H$`m J`m &
4 H$m_Jmam| Zo Xÿgao {XZ H$m © N>mo‹S> {X`m & 4 Am¡a H$m_Jmam| Zo Vrgao {XZ H$m`© N>mo‹S> {X`m &
Bgr àH$ma H«$_e… ha {XZ hmoVm J`m & Eogm H$aZo go H$m`© {ZYm©[aV g_` go 8 A{YH$ {XZm|
_| g_má hþAm & Vmo H$m`© Hw$b {H$VZo {XZm| _| nyam hþAm ?
150 workers were engaged to finish a piece of work in a certain number of
days. Four workers dropped the second day, four more workers dropped
the third day and so on. It took 8 more days to finish the work. Then find
the number of days in which the work was completed.
30/2 (SPL) 13 P.T.O.
28. EH$ µOma _| VrZ {^Þ a§Jm| — Zrbo, hao VWm gµ\o$X a§J Ho$ Hw$b 54 H§$Mo h¢ & EH$ Zrbo a§J Ho$
H§$Mo H$mo `mÑÀN>`m {ZH$mbZo H$s àm{`H$Vm 3
1 VWm hao a§J Ho$ H§$Mo H$mo$ `mÑÀN>`m {ZH$mbZo
H$s àm{`H$Vm 9
4 h¡ & Bg µOma _| {H$VZo gµ\o$X a§J Ho$ H§$Mo h¢ ?
A jar contains 54 marbles of three different colours — blue, green and
white. The probability of drawing a blue marble at random is 3
1 and that
4 . How many white marbles are there in the jar ?
29. I‹S>r MÅ>mZ na I‹S>m EH$ ì`{º$ EH$ Zmd H$mo AnZo R>rH$ ZrMo {H$Zmao H$s Amoa EH$g_mZ Mmb go 30 Ho$ AdZ_Z H$moU na AmVm XoI ahm h¡ & 6 {_ZQ> Ho$ níMmV² Zmd H$m `h AdZ_Z H$moU 60 hmo OmVm h¡ & Zmd Ûmam {H$Zmao na nhþ±MZo H$m Hw$b g_` kmV H$s{OE &
A man on a cliff observes a boat at an angle of depression of 30 which is
approaching the shore to the point immediately beneath the observer
with a uniform speed. Six minutes later the angle of depression of the
boat is found to be 60. Find the total time taken by the boat to reach the
shore.
30. EH$ _moQ>a-Zm¡H$m, {OgH$s pñWa Ob _| Mmb 24 {H$_r à{V K§Q>m h¡, 32 {H$_r Ymam Ho$ à{VHy$b OmZo _|, dhr Xÿar Ymam Ho$ AZwHy$b OmZo H$s Anojm 1 K§Q>m A{YH$ boVr h¡ & Ymam H$s Mmb kmV H$s{OE & A motorboat, whose speed is 24 km/h in still water, takes 1 hour more to
31. 32 H$mo Mma Eogo ^mJm| _| {d^m{OV H$s{OE {H$ `h Mmam| ^mJ EH$ g_m§Va lo‹T>r Ho$ Mma nX hm| VWm nhbo d Mm¡Wo nXm| Ho$ JwUZ\$b VWm Xÿgao d Vrgao nXm| Ho$ JwUZ\$b _| 7 : 15 H$m AZwnmV hmo &
Divide 32 into four parts which are the four terms of an A.P., such that
the product of the first and the fourth terms is to the product of the
second and the third terms as 7 : 15.
30/2 (SPL) 14
32. AmH¥${V 4 _|, N>m`m§{H$V ^mJ H$m joÌ\$b kmV H$s{OE, Ohm± 12 go_r ^wOm dmbo EH$
g_~mhþ {Ì^wO OAB Ho$ erf© O H$mo Ho$ÝÐ _mZH$a 6 go_r {ÌÁ`m dmbm d¥Îmr` Mmn ItMm
J`m VWm erf© B H$mo Ho$ÝÐ boH$a 6 go_r {ÌÁ`m Ho$ d¥Îm H$m EH$ {ÌÁÌÊS> ~Zm`m J`m h¡ &
AmH¥${V 4
Find the area of the shaded region in figure 4, where a circular arc of
radius 6 cm has been drawn with vertex O of an equilateral triangle OAB
of side 12 cm as centre and a sector of circle of radius 6 cm with centre B
is made.
Figure 4
30/2 (SPL) 15 P.T.O.
33. 7 _r. ì`mg dmbm 20 _r. Jham Hw$Am± ImoXm OmVm h¡ VWm Cg_| go {ZH$br {_Å>r H$mo g_mZ ê$n go \¡$bmH$a 22 _r. 14 _r. {d_mAm| dmbm EH$ M~yVam ~Zm`m OmVm h¡ & Bg M~yVao
H$s D±$MmB© kmV H$s{OE & `h àíZ {H$g _yë` H$mo Xem©Vm h¡ ? ( = 7
22 br{OE)
A 20 m deep well, with diameter 7 m is dug out and the earth taken out
is evenly spread to form a platform 22 m 14 m. Find the height of the
platform. What value is shown in this question ? (Use = 7
22 )
34. {gÕ H$s{OE {H$ d¥Îm Ho$ {H$gr {~ÝXþ na ItMr JB© ñne© aoIm, ñne© {~ÝXþ go hmoH$a OmZo dmbr {ÌÁ`m na b§~ hmoVr h¡ &
Prove that the tangent drawn at any point of a circle is perpendicular to
the radius through the point of contact.
30/3 1 P.T.O.
Series HRS H$moS> Z§. 30/3
Code No.
amob Z§.
Roll No.
MATHEMATICS
attempting it.
30/3 2
gm_mÝ` {ZX}e :
(iv) H¡$bHw$boQ>a H$m à`moJ d{O©V h¡ &
General Instructions :
(ii) The question paper consists of 34 questions divided into four sections A,
B, C and D.
(iii) Section A contains 8 questions of 1 mark each, which are multiple choice
type questions, Section B contains 6 questions of 2 marks each, Section C
contains 10 questions of 3 marks each and Section D contains
10 questions of 4 marks each.
IÊS> A
SECTION A
1. EH$ g_H$moU {Ì^wO ABC _|, B g_H$moU h¡, BC = 12 go_r VWm AB = 5 go_r h¡ &
Bg {Ì^wO Ho$ A§VJ©V ItMo JE d¥Îm H$s {ÌÁ`m (go_r _|) h¡
(A) 4
(B) 3
(C) 2
(D) 1
30/3 3 P.T.O.
In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm.
The radius of the circle inscribed in the triangle (in cm) is
(A) 4
(B) 3
(C) 2
(D) 1
2. g§»`mAm| 1, 2, 3, ..., 15 _| go `mÑÀN>`m EH$ g§»`m MwZr JB© & MwZr JB© g§»`m Ho$ 4 H$m JwUO hmoZo H$s àm{`H$Vm h¡
(A) 15
1
The probability that a number selected at random from the numbers
1, 2, 3, ..., 15 is a multiple of 4, is
(A) 15
30/3 4
3. VrZ ~ƒm| Ho$ n[adma _|, H$_-go-H$_ EH$ b‹S>H$m hmoZo H$s àm{`H$Vm h¡
(A) 8
3
In a family of 3 children, the probability of having at least one boy is
(A) 8
3
4. EH$ 150 _r. D±$Mo _rZma Ho$ {eIa go, g‹S>H$ na I‹S>r EH$ H$ma H$m AdZ_Z H$moU 30 h¡ & _rZma go H$ma H$s Xÿar (_r. _|) h¡
(A) 50 3
(B) 150 3
(C) 150 2
30/3 5 P.T.O.
The angle of depression of a car parked on the road from the top of a
150 m high tower is 30. The distance of the car from the tower (in
metres) is
(D) 75
5. 10 go_r {ÌÁ`m Ho$ d¥Îm H$s EH$ Ordm H|$Ð na g_H$moU A§V[aV H$aVr h¡ & Bg Ordm H$s b§~mB© (go_r _|) h¡
(A) 5 2
(B) 10 2
(D) 10 3
A chord of a circle of radius 10 cm subtends a right angle at its centre.
The length of the chord (in cm) is
(A) 5 2
(B) 10 2
30/3 6
6. ABCD EH$ Am`V h¡ {OgHo$ VrZ erf© B(4, 0), C(4, 3) VWm D(0, 3) h¢ & Am`V Ho$ EH$ {dH$U© H$s b§~mB© h¡
(A) 5
(B) 4
(C) 3
(D) 25
ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3).
The length of one of its diagonals is
(A) 5
(B) 4
(C) 3
(D) 25
7. `{X k, 2k – 1 VWm 2k + 1 EH$ g_m§Va lo‹T>r Ho$ VrZ H«$_mJV nX h¢, Vmo k H$m _mZ h¡
(A) 2
(B) 3
(C) – 3
(D) 5
30/3 7 P.T.O.
If k, 2k – 1 and 2k + 1 are three consecutive terms of an A.P., the
value of k is
(A) 2
(B) 3
(C) – 3
(D) 5
8. Xmo d¥Îm nañna {~ÝXþ P na ~mø ê$n go ñne© H$aVo h¢ & d¥Îmm| H$mo {~ÝXþAm| A VWm B na ñne©
H$aVr hþB© C^`{Zð>> ñne© aoIm AB h¡ & APB H$m _mZ h¡
(A) 30
(B) 45
(C) 60
(D) 90
Two circles touch each other externally at P. AB is a common tangent to
the circles touching them at A and B. The value of APB is
(A) 30
(B) 45
(C) 60
(D) 90
30/3 8
SECTION B
9. Xmo {^Þ-{^Þ nmgm| H$mo EH$ gmW CN>mbm J`m & àm{`H$Vm kmV H$s{OE {H$ (i) XmoZm| nmgm| na AmB© g§»`mE± g_ hm| & (ii) XmoZm| nmgm| na AmB© g§»`mAm| H$m `moJ\$b 5 hmo &
Two different dice are tossed together. Find the probability
(i) that the number on each die is even.
(ii) that the sum of numbers appearing on the two dice is 5.
10. `{X EH$ R>mog AY©Jmobo H$m gånyU© n¥ð>r` joÌ\$b 462 dJ© go_r h¡, Vmo BgH$m Am`VZ kmV
H$s{OE & [ = 7
22 br{OE ]
If the total surface area of a solid hemisphere is 462 cm2, find its volume.
[ Take = 7
22 ]
11. 101 VWm 999 Ho$ ~rM 2 Am¡a 5 XmoZm| go {d^mÁ` àmH¥$V g§»`mAm| H$s g§»`m kmV H$s{OE & Find the number of natural numbers between 101 and 999 which are
divisible by both 2 and 5.
12. AmH¥${V 1 _|, H|$Ð O1 VWm O2 dmbo Xmo d¥Îmm| H$s C^`{Zð>> ñne© aoImE± AB VWm CD {~ÝXþ E na H$mQ>Vr h¢ & {gÕ H$s{OE {H$ AB = CD.
AmH¥${V 1
30/3 9 P.T.O.
In Figure 1, common tangents AB and CD to the two circles with centres
O1 and O2 intersect at E. Prove that AB = CD.
Figure 1
13. EH$ g_{Û~mhþ {Ì^wO ABC, {Og_| AB = AC h¡, Ho$ A§VJ©V ItMm J`m d¥Îm, ^wOmAm| BC, CA VWm AB H$mo H«$_e… q~XþAm| D, E VWm F na ñne© H$aVm h¡ & {gÕ H$s{OE {H$ BD = DC h¡ & The incircle of an isosceles triangle ABC, in which AB = AC, touches the
sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.
14. {ÛKmV g_rH$aU px (x – 3) + 9 = 0 _| p H$m dh _mZ kmV H$s{OE {Oggo g_rH$aU Ho$ _yb g_mZ hm| &
Find the value of p so that the quadratic equation px (x – 3) + 9 = 0 has
equal roots.
IÊS> g SECTION C
15. AmH¥${V 2 _|, O H|$Ð dmbo Xmo g§H|$Ðr` d¥Îm h¢ {OZH$s {ÌÁ`mE± 21 go_r VWm 42 go_r h¢ &
`{X AOB = 60 h¡, Vmo N>m`m§{H$V ^mJ H$m joÌ\$b kmV H$s{OE & [ = 7
22 br{OE ]
AmH¥${V 2
30/3 10
In Figure 2, two concentric circles with centre O, have radii 21 cm and
42 cm. If AOB = 60, find the area of the shaded region. [Use = 7
22 ]
Figure 2
16. 7 go_r ^wOm dmbo bH$‹S>r Ho$ EH$ R>mog KZ _| go EH$ ~‹S>o-go-~‹S>m Jmobm H$mQ>m J`m & eof
~Mr bH$‹S>r H$m Am`VZ kmV H$s{OE & [ = 7
22 br{OE ]
The largest possible sphere is carved out of a wooden solid cube of side
7 cm. Find the volume of the wood left. [Use = 7
22 ]
17. 6 _r. Mm¡‹S>r Am¡a 1.5 _r. Jhar EH$ Zha _| nmZr 4 {H$_r à{V K§Q>o H$s Mmb go ~h ahm h¡ & 10 {_ZQ > _| `h Zha {H$VZo joÌ\$b H$s qgMmB© H$a nmEJr O~{H$ qgMmB© Ho$ {bE 8 go_r Jhao nmZr H$s Amdí`H$Vm h¡ ?
Water in a canal, 6 m wide and 1.5 m deep, is flowing at a speed of
4 km/h. How much area will it irrigate in 10 minutes, if 8 cm of standing
water is needed for irrigation ?
18. AmH¥${V 3 _|, ABCD EH$ g_b§~ h¡, {OgH$m joÌ\$b 24.5 dJ© go_r h¡ & Bg_|
AD || BC, DAB = 90, AD = 10 go_r VWm BC = 4 go_r h¡ & `{X ABE EH$ d¥Îm
H$m MVwWmªe h¡, Vmo N>m`m§{H$V ^mJ H$m joÌ\$b kmV H$s{OE & [ = 7
22 br{OE ]
AmH¥${V 3
30/3 11 P.T.O.
In Figure 3, ABCD is a trapezium of area 24.5 sq. cm. In it, AD ||
BC, DAB = 90, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a
circle, find the area of the shaded region. [ Take = 7
22 ]
Figure 3
19. dh AZwnmV kmV H$s{OE {Og_| {~ÝXþAm| A(3, – 3) Am¡a B(– 2, 7) H$mo {_bmZo dmbm aoImIÊS> x-Aj go {d^m{OV hmoVm h¡ & Bg {d^mOZ {~ÝXþ Ho$ {ZX}em§H$ ^r kmV H$s{OE & Find the ratio in which the line segment joining the points A(3, – 3) and
B(– 2, 7) is divided by x-axis. Also find the coordinates of the point of
division.
20. EH$ {Ì^wO H$s aMZm H$s{OE, {OgH$s ^wOmE± 5 go_r, 5.5 go_r VWm 6.5 go_r h¢ & {\$a EH$ AÝ` {Ì^wO H$s aMZm H$s{OE, {OgH$s ^wOmE±, {XE hþE {Ì^wO H$s g§JV ^wOmAm| H$s
5
3 JwZr hm| &
Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Now construct
another triangle, whose sides are 5
3 times the corresponding sides of the
given triangle.
1–,0x; 1x

22. EH$ g_m§Va lo‹T>r H o$ àW_ 7 nXm| H$m `moJ\$b 182 h¡ & `{X Bg lo‹T>r Ho$ Mm¡Wo VWm 17d| nXm| _| 1 : 5 H$m AZwnmV h¡, Vmo g_m§Va lo‹T>r kmV H$s{OE &
The sum of the first seven terms of an AP is 182. If its 4th and the 17th
terms are in the ratio 1 : 5, find the AP.
30/3 12
23. 60 _r. D±$Mr EH$ {~pëS§>J Ho$ {eIa go EH$ _rZma Ho$ {eIa VWm nmX Ho$ AdZ_Z H$m oU H«$_e… 45 VWm 60 h¢ & _rZma H$s D±$MmB© kmV H$s{OE & [ 3 = 1.73 br{OE ]
From the top of a 60 m high building, the angles of depression of the top
and the bottom of a tower are 45 and 60 respectively. Find the height of
the tower. [ Take 3 = 1.73 ]
24. y-Aj na dh {~ÝXþ P kmV H$s{OE Omo {H$ {~ÝXþAm| A(4, 8) VWm B(– 6, 6) go g_XÿañW hmo & Xÿar AP ^r kmV H$s{OE &
Find a point P on the y-axis which is equidistant from the points A(4, 8)
and B(– 6, 6). Also find the distance AP.
IÊS> X
SECTION D
àíZ g§»`m 25 go 34 VH$ àËòH$ àíZ Ho$ 4 A§H$ h¢ &
Question numbers 25 to 34 carry 4 marks each.
25. EH$ _moQ>a-~moQ>, {OgH$s pñWa Ob _| Mmb 18 {H$_r à{V K§Q>m h¡, 24 {H$_r Ymam Ho$ à{VHy$b OmZo _|, dhr Xÿar Ymam Ho$ AZwHy$b OmZo H$s Anojm 1 K§Q>m A{YH$ boVr h¡ & Ymam H$s Mmb kmV H$s{OE &
A motorboat whose speed in still water is 18 km/h, takes 1 hour more to
26. {gÕ H$s{OE {H$ d¥Îm Ho$ {H$gr {~ÝXþ na ñne© aoIm ñne© {~ÝXþ go OmZo dmbr {ÌÁ`m na b§~ hmoVr h¡ &
Prove that the tangent at any point of a circle is perpendicular to the
radius through the point of contact.
30/3 13 P.T.O.
27. 7 go_r ì`mg Ho$ EH$ ~obZmH$ma ~V©Z, {Og_| Hw$N> nmZr h¡, _| 1.4 go_r ì`mg dmbr 150 JmobmH$ma Jmo{b`m± S>mbr JBª Omo {H$ nmZr _| nyU©V`m Sy>~ JBª & kmV H$s{OE {H$ ~V©Z _| nmZr Ho$ ñVa _| {H$VZr d¥{Õ hþB© &
150 spherical marbles, each of diameter 1.4 cm, are dropped in a
cylindrical vessel of diameter 7 cm containing some water, which are
completely immersed in water. Find the rise in the level of water in the
vessel.
28. D$na go Iwbm EH$ ~V©Z e§Hw$ Ho$ {N>ÞH$ Ho$ AmH$ma H$m h¡, {OgH$s D±$MmB© 24 go_r h¡ VWm {ZMbo VWm D$nar d¥Îmr` {gam| H$s {ÌÁ`mE± H«$_e… 8 go_r VWm 20 go_r h¢ & < 21 à{V brQ>a H$s Xa go Bg ~V©Z H$mo nyam â gH$Zo dmbo XÿY H$m _yë` kmV H$s{OE &
[ = 7
22 br{OE ]
A container open at the top, is in the form of a frustum of a cone of height
24 cm with radii of its lower and upper circular ends as 8 cm and 20 cm
respectively. Find the cost of milk which can completely fill the container
at the rate of < 21 per litre. [Use = 7
22 ]
29. ^y{_ na pñWV {~ÝXþ A go 120 _r. H$s Xÿar na pñWV EH$ _rZma Ho$ {eIa H$m CÞ`Z H$moU 45 h¡ & `{X _rZma Ho$ {eIa na bJo EH$ ÜdOXÊS> Ho$ D$nar {gao H$m {~ÝXþ A na CÞ`Z
H$moU 60 h¡, Vmo ÜdOXÊS> H$s D±$MmB© kmV H$s{OE & [ 3 = 1.73 br{OE ]
The angle of elevation of the top of a tower at a distance of 120 m from a
point A on the ground is 45. If the angle of elevation of the top of a
flagstaff fixed at the top of the tower, at A is 60, then find the height of
the flagstaff. [ Use 3 = 1.73 ]
30. EH$ {dÚmb` Ho$ N>mÌm| Zo dm`w àXÿfU H$_ H$aZo Ho$ {bE {dÚmb` Ho$ A§Xa Am¡a ~mha no‹S> bJmZo H$m {ZU©` {b`m & àËòH$ H$jm Ho$ àËòH$ AZw^mJ Ûmam AnZr H$jm H$s g§»`m Ho$
XþJwZo Ho$ ~am~a no‹S> bJmZo H$m {ZU©` {b`m & `{X {dÚmb` _| 1 go 12 VH$ H$jmE± h¢ VWm àËòH$ H$jm Ho$ Xmo AZw^mJ h¢, Vmo N>mÌm| Ûmam bJmE JE Hw$b no‹S>m| H$s g§»`m kmV H$s{OE &
Bg àíZ _| {H$g _yë` H$mo Xem©`m J`m h¡ ?
30/3 14
In a school, students decided to plant trees in and around the school to
reduce air pollution. It was decided that the number of trees, that each
section of each class will plant, will be double of the class in which they
are studying. If there are 1 to 12 classes in the school and each class has
two sections, find how many trees were planted by the students. Which
value is shown in this question ?
31. x Ho$ {bE hb H$s{OE :
7,5x; 3
7–x
6–x
5–x
4–x
32. Vme Ho$ nm±M nÎmm| – BªQ> H$m Xhbm, Jwbm_, ~oJ_, ~mXemh Am¡a B¸$m H$mo nbQ> H$a Ho$ AÀN>r àH$ma \|$Q>m OmVm h¡ & {\$a BZ_| go `mÑÀN>`m EH$ nÎmm {ZH$mbm OmVm h¡ &
(a) BgH$s Š`m àm{`H$Vm h¡ {H$ `h nÎmm EH$ ~oJ_ h¡ ?
(b) `{X ~oJ_ {ZH$b AmVr h¡, Am¡a Cgo AbJ aI H$a EH$ AÝ` nÎmm {ZH$mbm OmVm h¡, Vmo àm{`H$Vm kmV H$s{OE {H$ Xÿgam {ZH$mbm J`m nÎmm EH$ (i) B¸$m h¡ (ii) ~oJ_ h¡ &
Five cards – the ten, jack, queen, king and ace of diamonds, are well
shuffled with their faces downwards. One card is then picked up at
random.
(a) What is the probability that the drawn card is the queen ?
(b) If the queen is drawn and put aside, and a second card is drawn,
find the probability that the second card is (i) an ace (ii) a queen.
33. `{X A(4, 2), B(7, 6) VWm C(1, 4) EH$ {Ì^wO ABC Ho$ erf© h¢ VWm AD Bg {Ì^wO H$s EH$ _mpÜ`H$m h¡, Vmo {gÕ H$s{OE {H$, _mpÜ`H$m AD, {Ì^wO ABC H$mo ~am~a joÌ\$bm| dmbo Xmo {Ì^wOm| _| {d^m{OV H$aVr h¡ &
If A(4, 2), B(7, 6) and C(1, 4) are the vertices of a ABC and AD is its
median, prove that the median AD divides ABC into two triangles of
equal areas.
30/3 15 P.T.O.
34. AmH¥${V 4 _|, 4 go_r {ÌÁ`m dmbo EH$ d¥Îm Ho$ n[aJV EH$ {Ì^wO ABC Bg àH$ma ItMm J`m h¡ {H$ aoImIÊS> BD Am¡a DC H$s b§~mB`m± H«$_e… 8 go_r VWm 6 go_r h¢ & ^wOmE± AB Am¡a AC kmV H$s{OE &
AmH¥${V 4
In Figure 4, a triangle ABC is drawn to circumscribe a circle of radius
4 cm, such that the segments BD and DC are of lengths 8 cm and 6 cm
respectively. Find the sides AB and AC.
Figure 4

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Summative Assessment-1 2014-2015 Class – X General