Time allowed: 3 hours Maximum Marks: 90

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1 SUMMATIVE ASSESSMENT-I, CLASS-X, MATHEMATICS LYSVYI3 Time allowed: 3 hours Maximum Marks: 90 General Instructions: 1. This question are compulsory.. The question paper is divided into four sections Section A: 4 question (1mark each) Section B: 6 question ( marks each) Section C: 10 questions (3 mark each) Section D: 11 questions (4 mark each). 3. There is no overall choice in this question paper. 4. Use of calculators is not permitted. Question number 1 to 4 carry one mark each Section A Q. 1 If the corresponding medians of two similar triangles are in the ratio 5:7, then find the ratio of their corresponding sides. Q. If 4 cota 7, Find the value of sina Q.3 Simplify : cosec A 1 cot A Q.4 If the point of intersection of two gives is (18, 54), then find the value of median. Question number 5 to 10 carry two mark each Q.5 Show that 5 6 is an irrational number Section B Q.6 Express 5050 as product of its prime factors. Is it unique? Q.7 The taxi charges in a city consists of a fixed change together with the charge for the distance covered. For a distance of 6 km, the charges paid are Rs 58 while for a journey of 10 km, the charges paid are Rs. 90. Find the charge per km and the fixed charge. Q.8 State which of the two triangles given in the figure are similar. Also state the similarity criterion used. Q.9 Prove that: sec 1 1 cos ec 1 Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

2 Q.10 Show that the mode of the series obtained by combining the two series S1 and S give below is different from that of S 1 and S taken separately: S 1 : 3,5,8,8,9,1,13,9,9 S : 7,4,7,8,7,8,13 Section C Question number 11 to 0 carry three mark each Q.11 Pens are sold in pack of 8 and notepads are sold in pack of 1. Find the least number of pack of each type that one should buy so that there are equal number of pen and notepads. Q.1 Solve using cross multiplication method : u 7v 1 4u 3v 15 Q.13 If p x x 5x, what is the value of p 3 q? Q.14 For what value of K, will the following system of equations have no solution? 3k 1 x 3y ; k 1 x k y 5 Q.15 In ABC, X is middle point of AC. If XY AB then prove that Y is middle point of AB Q.16 In the figure, ABCD is a rectangle. If in ADE and ABE, E F, then prove that AD AB. AE AF Q.17 If 7sin A _ 3cos A 4, show that 1 tan A. 3 Q.18 Prove that: sin cos 3 sin tan 3 cos Q.19 In a class test, marks scored by students are given in the following frequency distribution : Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

3 Marks Number of students Find the mean and median of the data. Q.0 Some surnames were picked up from a local telephone directory and the frequentation distribution of the number of letters of the English alphabets was obtained as follows: Number of letters Number of surnames x 1 8 If it is given that mode of the distribution is 8, then find the missing frequency (x). Section D Question number 1 to 31 carry four mark each Q.1 What is the HCF and LCM of two prime numbers a and b? Three alarm clocks ring at intervals of 6, 9 and 15 minutes respectively. If they start ringing together, after what time will they next ring together. Q. Draw the graph of the following pair of linear equation: x 3y 6 and x 3y 1 Find the ratio of the areas of the two triangles formed by first line, x 0, y 0 and second line x 0, y Q.3 If the polynomial x x 8 1x 18 is divided by another polynomial x 5 remainder comes out to be px q, find the values of p and q. Q.4 Ram s mother has given him money to buy some boxes from the market at the rate of 4 3 4x 3x. The total amount of money is represented by8x 14x x 7x 8. Out of this money he donated some amount to a child who was studying in the light of street Iamp. Find how much amount of money he donated and purchased how many boxes from the market? Why Ram did so? o Q.5 In a right angled triangle ABC, A 90 and AD BC. Prove that: (i) (ii) AB BD BC AD BD DC, the Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

4 (iii) AC BC CD Q.6 Find the length of the diagonal of the rectangle BCDE, If BCE DCF, AC=6 m and CF=1m. Q.7 Evaluate Q.8 Prove that o o o o tan1 tan tan 3...tan89 b x a y a b, if : (i) x a sec, y b tan (ii) x a coec, y bcot Q.9 Prove that: cot cosec 1. cot cosec 1.tan Q.30 Cost of Living Index fox some period is given in the following frequency distribution : Index Number of weeks Q.31 Following is the ages of asthmatic patients admitted during a year in a hospital. Find the mean age of the patients. Age (in years) Number of weeks Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

5 SUMMATIVE ASSESSMENT-I, CLASS-X, MATHEMATICS GG-RO-103 Time allowed: 3 hours Maximum Marks: 80 General Instructions: 1. All questions are compulsory.. Question paper contains 31 questions divided into 4 section A, B, C & D. 3. Section-A comprises of 4 questions carrying 1 mark each, Section-B comprises of 6 questions carrying marks each, Section-C comprises of 10 questions carrying 3 marks each, Section-D comprises of 11 questions carrying 4 marks each, 4. Al questions in section-a are very short answer questions. 5. There are no overall choices in the question paper. 6. Use of calculator is not permitted. 7. If required Graph papers will be provided. Question numbers 1 to 4 carry 1 mark each. Section A Q. 1 Find the value of tan 30 cot 60 o o Q. What is the altitude of an equilateral triangle of each side 6 cm. Q. 3 If the sum of zeroes of quadratic polynomial Q. 4 Find the class marks of class Question numbers 5 to 10 carry marks each. Section B Q. 5 In a right isosceles triangle ABC right angled at C prove that Q. 6 Use Euclid s division algorithm to find the HCF of 870 and 55. 3x kx 6 is 3, then fine the value of k. AB AC Q. 7 Find the mode of the following distribution of marks obtained by 80 students: Marks obtained No. of students Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

6 Q. 8 Find the value of k. If the pair of linear equation : 3x 4y k 9x 1y 6 Has infinitely many solutions. Q. 9 If the LCM of a and 18 is 36 and the HCF of a and 18 is, then find the value of a. Q. 10 Find a quadratic polynomial, the sum and product of whose zeroes are -3 and respectively. Section C Question numbers 11 to 0 carry 3 marks each. Q. 11 Use Euclid s division Lemma, to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m. Q. 1 Solve the following pairs of linear equations by Cross-multiplication method : 4x y 14 5x 6y 7 Q. 13 If the areas of two similar triangles are equal, then prove that they are congruent. 1 sin A 1 sin A Q. 14 sec A tan A Q. 15 Find the mean of the following distribution, using step deviation method : Class Frequency Q. 16 Find the zeroes of the quadratic polynomial the zeroes and the coefficients. 6x 3 7x and verify the relationship between Q. 17 In triangle PQR right angled at Q, PR+QR=5cm and PQ=5cm determine the value of sin P, cos P and tan P. Q. 18 In figure PQ CD and PR CB, Prove that AQ AR QD RB Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

7 Q. 19 Find the median of the following frequency distribution table: marks No. of students , Q. 0 If sin A B cos A B A B 90, A B, Find A and B. Section D Question numbers 1 to 31 carry 4 marks each. Q. 1 Show that: 5 3 is an irrational number Q. Find all the zeroes of the polynomials x 4 x 3 9x 3x 18. If it is given that two of its zeroes are 3 and 3 Divide OR 3 3x x 3x 5 by x 1 x and verify the division algorithm. Q. 3 Prove that in a Right Angled Triangle, the square of the Hypotenuse is equal to the sum of the square of the others two sides OR Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians. tan cot Q. 4 Prove that: 1_ sec cosec 1 cot 1 tan OR Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

8 sin cos 3 sin tan 3 cos Q. 5 Solve the following system of Linear Equation Graphically: X Y = 1, X + Y = 8. Shade the area bounded by these two lines and Y-axis. Q. 6 The median of the following data is 50. Find the values of p and q if the sum of all the frequencies is 90. marks No. of students P Q Q. 7 If 1 3 sin Show that 3 cosb 4Cos B 0 Q. 8 During the medical check-up of the 35 students of a class their weights were recorded as follows: Weight in KG No. of students Draw a less than type give for the above data Q. 9 Solve the pair of linear equation:, 3x y 3x y x y 3x y 8 Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

9 OR The Sum of the digits of a two digit number is 9, Also nine times this number is twice the number obtain by reversing the order the digits. Find the number. o o o 7cos70 3 cos55 cosec35 Q. 30 Evaluate: sin 0 tan5 tan 45 tan85 tan65 tan 5 o o o o o o Q. 31 Mr. Balwant Singh has a triangular field ABC. He has three sons. He wants to divide the field into four equal and identical parts, so that he may give three parts to his three sons and retain the fourth part with him. i) Is it possible to divide the field into four parts which are equal and identical? ii) If yes, explain the method of division. iii) By doing so, which values is depicted by Mr. Balwant Singh. Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

10 Summative Assessment Mathematics Class X Time allowed: 3:00 hours Maximum Marks: 90 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 marks each. Question No. 11 to 0 carry 3 marks each. Question No. 1 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 XYZ, A and B are points on the sides XY and XZ respectively such that AB YZ. If AY=.cm, XB=3.3cm and XZ=6.6cm, then find AX.. If tanθ + cotθ =, then find the value of tan θ + cot θ. 3. Ifθ = 45, then find the value of sin θ + 3cos ec θ? 4. Life time of electric bulbs are given in the following frequency distribution: Life time (in hours) Number of bulbs Find the class mark of the modal class interval. Section B Question numbers 5 to 10 are two marks each. 5. Find whether decimal expansion of 13 is a terminating or non-terminating decimal. If it 64 terminates, find the number of decimal places its decimal expansion has Write the decimal expansion of without actual division Ifα and βare the zeroes of a polynomial9y + 1y + 4, then find the value of α + β + αβ. 8. Are the given figures similar? Give reason. Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

11 9. Simplify: (1-sin A)(tan A + sec A) 10. The following distribution shows the daily pocket allowance of children of a locality: Daily pocket allowance (in Rs.) Number of children Find the median of the data. Section C Question numbers 11 to 0 carry three marks each. 11. Prove that is an irrational number. 1. Solve for x and y: x+4y=7xy x+y=1xy 13. Determine graphically whether the following pair of linear equations x-3y=8 4x-6y=16 has a) A unique solution, b) Infinitely many solution or c) No solution If 4x + 7x 4x 7x + kis completely divisible by x 3 x, then find the value of k. 15. As shown in the figure, a 6m long ladder is placed at A. if it is placed along wall PQ, it reaches a height of 4m whereas it reaches a height of 10m if it is placed against wall RS. Find the distance between the walls. 16. If in ABC, AD is median and AM BC, then prove that 1 AB + AC = AD + BC sin A cos A 17. Prove that: + = sec A cos ec A cos A sin A In ABC, right angled at C, if tan A =, show that sin A. cos B + cos A. sin B=1 3 Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

12 19. In a study on asthmatic patients, the following frequency distribution was obtained. Find the average (mean) age at the detection. Age at detection (in years) Number of patients For the following distribution, draw a less than type ogive and from the curve, find the median. Marks Less Less Less Less Less Less Less Less Less obtained than than than than than than than than than Number of students Section D Question numbers 1 to 31 carry four marks each. 1. Dhudnath has two vessels containing 70 ml and 405 ml of milk respectively. Milk from these containers is poured into glasses of equal capacity to their brim. Find the minimum number of glasses that can be filled.. The ratio of incomes of two persons A and B is 9:7and the ratio of their expenditure is 4:3. If their savings are Rs. 00 per month, find their monthly incomes. Why is it necessary to save money? Find all the zeroes of x 5x + x + 15x 1, if it is given that two of its zeroes are 1 and A boat goes 30 km upstream and 0 km downstream in 7 hours. In 6 hours, it can go 18 km upstream and 30 km downstream. Determine the speed of the stream and that of the boat in still water. 5. In ABC, AD BCand D lies on BC such that 4DB=CD, then proves that 5AB = 5AC 3BC 6. ABC is an isosceles triangle in which B = 90 and AC = 3 m. Two equilateral triangles ACP and ABQ are drawn on the sides AC and AB. Find the ratio of area ( ABQ) and area ( ACP). 7. In the adjoining figure, ABCD is a rectangle with breadth BC=7cm and CAB = 30. Find the length of side AB of the rectangle and length of diagonal AC. If the CAB = 60, then what is the size of the side AB of the rectangle (use 3 = 1.73and = 1.41, if required) 8. If acosθ bsinθ = c, then prove that a sinθ + b cosθ = ± a + b c Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

13 9. Given that sin( A B) = sin A cos B cos A sin B. Find the value ofsin15 in two ways. a) Taking A = 60, B = 45, and b) Taking A = 45, B = A class test in mathematics was conducted for class VI of a school. Following distribution gives marks (out of 60) of students: Marks Number of students Find the mean of the marks obtained. 31. In an examination, 150 students appeared, and their marks (out of 00) are given in the following distribution. Find the missing frequencies x and y, when it is given that mean marks is 103. Marks Number of students 10 x 30 y Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

14 Series RLH H$moS> Z. 30/ Code No. amob Z. Roll No. narjmwu H$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. H $n`m Om±M H$a b {H$ Bg àíz-nì _o _w{ðv n ð> 11 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 _ >31 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 _ ~Oo {H$`m OmEJm & ~Oo go ~Oo VH$ 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 11 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 31 questions. g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS SET- Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at a.m. From a.m. to a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 90 Time allowed : 3 hours Maximum Marks : 90 30/ 1 P.T.O.

15 gm_mý` {ZX}e : (i) g^r àíz A{Zdm` h & (ii) Bg àíz-nì _ 31 àíz h Omo Mma IÊS>m A, ~, g Am a X _ {d^m{ov h & (iii) IÊS> A _ EH$-EH$ A H$ dmbo 4 àíz h & IÊS> ~ _ 6 àíz h {OZ_ go àë`oh$ A H$ H$m h & IÊS> g _ 10 àíz VrZ-VrZ A H$m Ho$ h & IÊS> X _ 11 àíz h {OZ_ go àë`oh$ 4 A H$ H$m h & (iv) H $bhw$boq>a H$m à`moj d{o V h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections A, (iii) (iv) B, C and D. Section A contains 4 questions of 1 mark each. Section B contains 6 questions of marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. Use of calculators is not permitted. àíz g»`m 1 go 4 VH$ àë`oh$ àíz 1 A H$ H$m h & IÊS> A SECTION A Question numbers 1 to 4 carry 1 mark each. 1. AmH ${V 1 _, O H $Ð dmbo d Îm H$s PQ EH$ Ordm h VWm PT EH$ ñne aoim h & `{X QPT = 60 h, Vmo PRQ kmv H$s{OE & AmH ${V 1 30/

16 In Figure 1, PQ is a chord of a circle with centre O and PT is a tangent. If QPT = 60, find PRQ. Figure 1. `{X {ÛKmV g_rh$au px 5 px + 15 = 0 Ho$ Xmo g_mz _yb hm, Vmo p H$m _mz kmv H$s{OE & If the quadratic equation px then find the value of p. 5 px + 15 = 0 has two equal roots, 3. AmH ${V _, EH$ _rzma AB H$s D±$MmB 0 _rq>a h Am a BgH$s ^y{_ na nan>mb BC H$s bå~mb 0 3 _rq>a h & gy` H$m CÞVm e kmv H$s{OE & AmH ${V In Figure, a tower AB is 0 m high and BC, its shadow on the ground, is 0 3 m long. Find the Sun s altitude. Figure 30/ 3 P.T.O.

17 4. Xmo {^Þ nmgm H$mo EH $gmw CN>mbm J`m & XmoZm nmgm Ho$ D$nar Vbm na AmB g»`mam H$m JwUZ\$b 6 AmZo H$s àm{`h$vm kmv H$s{OE & Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is 6. IÊS> ~ SECTION B àíz g»`m 5 go 10 VH$ àë`oh$ àíz A H H$m h & Question numbers 5 to 10 carry marks each. 5. `{X {~ÝXþ A(x, y), B( 5, 7) VWm C( 4, 5) ñ maoir` hm, Vmo x VWm y _ gå~ýy kmv H$s{OE & Find the relation between x and y if the points A(x, y), B( 5, 7) and C( 4, 5) are collinear. 6. EH$ g_m Va lo T>r Ho$ àw_ n nxm Ho$ `moj\$b H$mo S n Ûmam Xem `m OmVm h & Bg lo T>r _ `{X S 5 + S 7 = 167 VWm S 10 = 35 h, Vmo g_m Va lo T>r kmv H$s{OE & In an AP, if S 5 + S 7 = 167 and S 10 = 35, then find the AP, where S n denotes the sum of its first n terms. 7. AmH ${V 3 _, Xmo ñne aoime± RQ VWm RP d Îm Ho$ ~mø {~ÝXþ R go ItMr JB h & d Îm H$m Ho$ÝÐ O h & `{X PRQ = 10 h, Vmo {gõ H$s{OE {H$ OR = PR + RQ. AmH ${V 3 30/ 4

18 In Figure 3, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If PRQ = 10, then prove that OR = PR + RQ. Figure 3 8. AmH ${V 4 _, 3 go_r {ÌÁ`m dmbo EH$ d Îm Ho$ n[ajv EH$ {Ì^wO ABC Bg àh$ma ItMm J`m h {H$ aoimiês> BD VWm DC H$s b ~mb`m± H«$_e 6 go_r VWm 9 go_r h & `{X ABC H$m joì\$b 54 dj go_r h, Vmo ^womam AB VWm AC H$s bå~mb`m± kmv H$s{OE & AmH ${V 4 In Figure 4, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. If the area of ABC is 54 cm, then find the lengths of sides AB and AC. Figure 4 30/ 5 P.T.O.

19 9. {ZåZ {ÛKmV g_rh$au H$mo x Ho$ {be hb H$s{OE : 4x + 4bx (a b ) = 0 Solve the following quadratic equation for x : 4x + 4bx (a b ) = `{X A(4, 3), B( 1, y) VWm C(3, 4) EH$ g_h$mou {Ì^wO ABC Ho$ erf h, {Og_ A na g_h$mou h, Vmo y H$m _mz kmv H$s{OE & If A(4, 3), B( 1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y. IÊS> g SECTION C àíz g»`m 11 go 0 VH$ àë`oh$ àíz 3 A H$ H$m h & Question numbers 11 to 0 carry 3 marks each. 11. AMmZH$ ~m T> AmZo na, Hw$N> H$ë`mUH$mar g ñwmam Zo {_b H$a gah$ma H$mo Cgr g_` 100 Q> Q> bjdmzo Ho$ {be H$hm VWm Bg na AmZo dmbo IM H$m 50% XoZo H$s noeh$e H$s & `{X àë`oh$ Q> Q> H$m {ZMbm ^mj ~obzmh$ma h {OgH$m ì`mg 4. _r. h VWm D±$MmB 4 _r. h VWm D$nar ^mj Cgr ì`mg H$m e Hw$ h {OgH$s D±$MmB. 8 _r. h, Am a Bg na bjzo dmbo H $Zdg H$s bmjv < 100 à{v dj _r. h, Vmo kmv H$s{OE {H$ BZ g ñwmam H$mo {H$VZr am{e XoZr hmojr >& BZ g ñwmam Ûmam {H$Z _yë`m H$m àxe Z {H$`m J`m? [ = 7 br{oe ] Due to sudden floods, some welfare associations jointly requested the government to get 100 tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of the form of a cylinder of diameter 4. m and height 4 m with the conical upper part of same diameter but of height. 8 m, and the canvas to be used costs < 100 per sq. m, find the amount, the associations will have to pay. What values are shown by these associations? [Use = 7 ] 30/ 6

20 1. YamVb Ho$ EH$ {~ÝXþ A go EH$ hdmb OhmµO H$m CÞ`Z H$moU 60 h & 15 goh$ês H$s C S>mZ Ho$ nímmv², CÞ`Z H$moU 30 H$m hmo OmVm h & `{X hdmb OhmµO EH$ {ZpíMV D±$MmB _rq>a na C S> ahm hmo, Vmo hdmb OhmµO H$s J{V {H$bmo_rQ>a/K Q>m _ kmv H$s{OE & The angle of elevation of an aeroplane from a point A on the ground is 60. After a flight of 15 seconds, the angle of elevation changes to 30. If the aeroplane is flying at a constant height of m, find the speed of the plane in km/hr. 13. EH$ AÕ Jmobr` ~V Z H$m AmÝV[aH$ ì`mg 36 go_r h & `h Vab nxmw go ^am h & Bg Vab H$mo 7 ~obzmh$ma ~movbm _ S>mbm J`m h & `{X EH$ ~obzmh$ma ~movb H$m ì`mg 6 go_r hmo, Vmo àë`oh$ ~movb H$s D±$MmB kmv H$s{OE, O~{H$ Bg {H«$`m _ 10% Vab {Ja OmVm h & A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 7 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer. 14. EH$ Oma _ Ho$db bmb, Zrbr VWm Zma Jr a J H$s J X h & `mñàn>`m EH$ bmb a J H$s J X Ho$ {ZH$mbZo H$s àm{`h$vm 4 1 h & Bgr àh$ma Cgr Oma go `mñàn>`m EH$ Zrbr J X Ho$ {ZH$mbZo H$s àm{`h$vm 3 1 h & `{X Zma Jr a J H$s Hw$b J X 10 h, Vmo ~VmBE {H$ Oma _ Hw$b {H$VZr J X h & The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is 4 1. The probability of selecting a blue ball at random from the same jar is 3 1. If the jar contains 10 orange balls, find the total number of balls in the jar go_r ^wom dmbo EH$ KZmH$ma ãbm H$ Ho$ D$na EH$ AY Jmobm aim hþam h & AY Jmobo H$m A{YH$V_ ì`mg Š`m hmo gh$vm h? Bg àh$ma ~Zo R>mog Ho$ g nyu n ð>r` joì H$mo n Q> H$admZo H$m < 5 à{v 100 dj go_r H$s Xa go ì`` kmv H$s{OE & [ = br{oe ] A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, at the rate of < 5 per 100 sq. cm. [ Use = ] 30/ 7 P.T.O.

21 16. `{X (, ) VWm (, 4) H«$_e {~ÝXþ A VWm B Ho$ {ZX}em H$ h, Vmo {~ÝXþ P Ho$ {ZX}em H$ kmv H$s{OE O~{H$ P aoimiês> AB na h VWm AP = 7 3 AB. If the coordinates of points A and B are (, ) and (, 4) respectively, find the coordinates of P such that AP = 7 3 AB, where P lies on the line segment AB go_r ì`mg VWm 3 go_r D±$Mo 504 e Hw$Am H$mo {nkbmh$a EH$ YmpËdH$ Jmobm ~Zm`m J`m & Jmobo H$m ì`mg kmv H$s{OE & AV BgH$m n ð>r` joì\$b kmv H$s{OE & [ = 7 br{oe ] 504 cones, each of diameter 3. 5 cm and height 3 cm, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area. [Use = 7 ] 18. EH$ g_mvw^w O Ho$ g^r erf EH$ d Îm na pñwv h & `{X Bg d Îm H$m joì\$b 156 dj go_r h, Vmo g_mvw^w O H$m joì\$b kmv H$s{OE & [ = br{oe ] All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if the area of the circle is 156 cm. [ Use = ] 19. x Ho$ {be hb H$s{OE : x x 60 = 0 Solve for x : x x 60 = 0 0. EH$ g_mýva lo T>r H$m 16dm± nx BgHo$ Vrgao nx H$m nm±m JwZm h & `{X BgH$m 10dm± nx 41 h, Vmo BgHo$ àw_ 15 nxm H$m `moj\$b kmv H$s{OE & The 16 th term of an AP is five times its third term. If its 10 th term is 41, then find the sum of its first fifteen terms. 30/ 8

22 IÊS> X SECTION D àíz g»`m 1 go 31 VH$ àë`oh$ àíz 4 A H$ H$m h & Question numbers 1 to 31 carry 4 marks each. 1. {gõ H$s{OE {H$ d Îm H$s {H$gr Mmn Ho$ _Ü`-{~ÝXþ na ItMr JB ñne aoim, Mmn Ho$ A Ë` {~ÝXþþAm H$mo {_bmzo dmbr Ordm Ho$ g_m Va hmovr h & Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.. EH$ Prb _ nmzr Ho$ Vb go 0 _rq>a D±$Mo {~ÝXþ A go, EH$ ~mxb H$m CÞ`Z H$moU 30 h & Prb _ ~mxb Ho$ à{v{~å~ H$m A go AdZ_Z H$moU 60 h & A go ~mxb H$s Xÿar kmv H$s{OE & At a point A, 0 metres above the level of water in a lake, the angle of elevation of a cloud is 30. The angle of depression of the reflection of the cloud in the lake, at A is 60. Find the distance of the cloud from A. 3. AÀN>r Vah go \ $Q>r JB EH$ Vme H$s JÈ>r go EH$ nîmm `mñàn>`m {ZH$mbm J`m & àm{`h$vm kmv H$s{OE {H$ {ZH$mbm J`m nîmm (i) (ii) (iii) (iv) hþhw$_ H$m nîmm h `m EH$ B $m h & EH$ H$mbo a J H$m ~mxemh h & Z Vmo Jwbm_ h VWm Z hr ~mxemh h & `m Vmo ~mxemh h `m ~oj_ h & A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is (i) (ii) (iii) (iv) a card of spade or an ace. a black king. neither a jack nor a king. either a king or a queen. 30/ 9 P.T.O.

23 4. AmH ${V 5 _, PQRS EH$ djm H$ma bm Z h {OgH$s ^wom PQ = 4 _rq>a h & Xmo d ÎmmH$ma \y$bm H$s Š`m[a`m± ^wom PS VWm QR na h {OZH$m Ho$ÝÐ Bg dj Ho$ {dh$um] H$m à{vàn>oxz {~ÝXþ O h & XmoZm \y$bm H$s Š`m[a`m (N>m`m {H$V ^mj) H$m Hw$b joì\$b kmv H$s{OE & AmH ${V 5 In Figure 5, PQRS is a square lawn with side PQ = 4 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts). Figure 5 5. EH$ R>mog YmVw Ho$ ~obz Ho$ XmoZmo {H$Zmam go Cgr ì`mg Ho$ AÕ Jmobo Ho$ ê$n _ YmVw {ZH$mbr JB & ~obz H$s D±$MmB 10 go_r VWm BgHo$ AmYma H$s {ÌÁ`m 4. go_r h & eof ~obz H$mo {nkbmh$a 1. 4 go_r _moq>r ~obzmh$ma Vma ~ZmB JB & Vma H$s bå~mb kmv H$s{OE & [ = 7 br{oe ] From each end of a solid metal cylinder, metal was scooped out in hemispherical form of same diameter. The height of the cylinder is 10 cm and its base is of radius 4. cm. The rest of the cylinder is melted and converted into a cylindrical wire of 1. 4 cm thickness. Find the length of the wire. [Use = ] 7 30/ 10

24 6. EH$ Am`VmH$ma IoV H$m {dh$u BgH$s N>moQ>r ^wom go 16 _rq>a A{YH$ h & `{X BgH$s ~ S>r ^wom N>moQ>r ^wom go 14 _rq>a A{YH$ h, Vmo IoV H$s ^womam H$s bå~mb`m± kmv H$s{OE & The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field. 7. g_m Va lo T>r 8, 10, 1,... H$m 60dm± nx kmv H$s{OE, `{X Cg_ Hw$b 60 nx h & AV Bg lo T>r Ho$ A {V_ 10 nxm H$m `moj\$b kmv H$s{OE & Find the 60 th term of the AP 8, 10, 1,..., if it has a total of 60 terms and hence find the sum of its last 10 terms. 8. EH$ ~g nhbo 75 {H$bmo_rQ>a H$s Xÿar {H$gr Am gv Mmb go MbVr h VWm CgHo$ ~mx H$s 90 {H$bmo_rQ>a H$s Xÿar nhbo go 10 {H$bmo_rQ>a à{v K Q>m A{YH$ H$s Am gv Mmb go MbVr h & `{X Hw$b Xÿar 3 K Q>o _ nyar hmovr h, Vmo ~g H$s nhbr Mmb kmv H$s{OE & A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/h more than the first speed. If it takes 3 hours to complete the total journey, find its first speed. 9. {gõ H$s{OE {H$ d Îm Ho$ {H$gr {~ÝXþ na ñne aoim ñne {~ÝXþ go OmZo dmbr {ÌÁ`m na b ~ hmovr h & Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 30. EH$ g_h$mou {Ì^wO ABC H$s amzm H$s{OE, {Og_ AB = 6 go_r, BC = 8 go_r VWm B = 90 h & B go AC na b ~ BD It{ME & {~ÝXþAm B, C VWm D go hmoh$a OmZo dmbm EH$ d Îm It{ME VWm A go Bg d Îm na ñne aoimam H$s amzm H$s{OE & Construct a right triangle ABC with AB = 6 cm, BC = 8 cm and B = 90. Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle. 31. k Ho$ _mz kmv H$s{OE {OZgo (k+1, 1), (4, 3) VWm (7, k) erfm] dmbo {Ì^wO H$m joì\$b 6 dj BH$mB hmo &$ Find the values of k so that the area of the triangle with vertices (k+1, 1), (4, 3) and (7, k) is 6 sq. units. 30/ 11 P.T.O.

25 Series RLH/ H$moS> Z. 30//1 Code No. amob Z. Roll No. SET-1 narjmwu H$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. H $n`m Om±M H$a b {H$ Bg àíz-nì _o _w{ðv n ð> 1 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 _ >31 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 _ ~Oo {H$`m OmEJm & ~Oo go ~Oo VH$ 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 1 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 31 questions. Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at a.m. From a.m. to a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. g H${bV narjm II SUMMATIVE ASSESSMENT II J{UV MATHEMATICS {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 90 Time allowed : 3 hours Maximum Marks : 90 30//1 1 P.T.O.

26 gm_mý` {ZX}e : (i) (ii) (iii) (iv) g^r àíz A{Zdm` h & Bg àíz-nì _ 31 àíz h Omo Mma IÊS>m A, ~, g Am a X _ {d^m{ov h & IÊS> A _ EH$-EH$ A H$ dmbo 4 àíz h & IÊS> ~ _ 6 àíz h {OZ_ go àë`oh$ A H$m H$m h & IÊS> g _ 10 àíz VrZ-VrZ A H$m Ho$ h Am a IÊS> X _ 11 àíz h {OZ_ go àë`oh$ 4 A H$m H$m h & H $bhw$boq>a H$m à`moj d{o V h & General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections A, B, C and D. (iii) (iv) Section A contains 4 questions of 1 mark each. Section B contains 6 questions of marks each. Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. Use of calculators is not permitted. àíz g»`m 1 go 4 VH$ àë`oh$ àíz 1 A H$ H$m h & IÊS> A SECTION A Question numbers 1 to 4 carry 1 mark each. 1. g_m Va lo T>r 5, 5, 0, 5,... H$m 5dm± nx kmv H$s{OE & Find the 5 th term of the A.P. 5, 5 5, 0,,... 30//1

27 . O~ gy` H$m CÞ`Z H$moU 60 h, Vmo EH$ Iå^o H$s ^y{_ na N>m`m H$s b ~mb 3 _rq>a h & Iå^o H$s D±$MmB kmv H$s{OE & A pole casts a shadow of length elevation is 60. Find the height of the pole. 3 m on the ground, when the sun s 3. g `moj Ho$ EH$ Iob _ EH$ Vra H$mo Kw_m`m OmVm h, Omo éh$zo na g»`mam 1,, 3, 4, 5, 6, 7, 8 _ go {H$gr EH$ g»`m H$mo B {JV H$aVm h & `{X `h g^r n[aum_ g_àm{`h$ hm, Vmo Vra Ho$ 8 Ho$ {H$gr EH$ JwUZIÊS> na éh$zo H$s àm{`h$vm kmv H$s{OE & A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1,, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. Find the probability that the arrow will point at any factor of {ÌÁ`mE± a VWm b (a > b) Ho$ Xmo g Ho$ÝÐr` d Îm {XE JE h & ~ S>>o d Îm H$s Ordm, Omo N>moQ>o d Îm H$s ñne aoim h, H$s bå~mb kmv H$s{OE & Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. IÊS> ~ SECTION B àíz g»`m 5 go 10 VH$ àë`oh$ àíz Ho$ A H$ h & Question numbers 5 to 10 carry marks each. 5. AmH ${V 1 _, d Îm H$m Ho$ÝÐ O h & PT VWm PQ Bg d Îm na ~mø {~ÝXþ P go Xmo ñne -aoime± h & `{X TPQ = 70 h, Vmo TRQ kmv H$s{OE & AmH ${V 1 30//1 3 P.T.O.

28 In Figure 1, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If TPQ = 70, find TRQ. Figure 1 6. AmH ${V _, 5 go_r {ÌÁ`m dmbo d Îm _ Ordm PQ H$s bå~mb 8 go_r h & P VWm Q na ñne -aoime± nañna {~ÝXþ T na {_bvr h & TP VWm TQ H$s bå~mb`m± kmv H$s{OE & AmH ${V In Figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ. Figure 30//1 4

29 7. x Ho$ {be hb H$s{OE : Solve for x : x ( 3 +1) x + 3 = 0 x ( 3 +1) x + 3 = 0 8. EH$ g_m Va lo T>r H$m Mm Wm nx 11 h & Bg g_m Va lo T>r Ho$ nm±md VWm gmvd nxm H$m `moj\$b 34 h & BgH$m gmd AÝVa kmv H$s{OE & The fourth term of an A.P. is 11. The sum of the fifth and seventh terms of the A.P. is 34. Find its common difference. 9. {gõ H$s{OE {H$ {~ÝXþ (a, a), ( a, a) VWm ( 3 a, 3 a) EH$ g_~mhþ {Ì^wO Ho$ erf {~ÝXþ h & Show that the points (a, a), ( a, a) and ( 3 a, 3 a) are the vertices of an equilateral triangle. 10. k Ho$ {H$Z _mzm Ho$ {be {~ÝXþ (8, 1), (3, k) VWm (k, 5) g aoir` h? For what values of k are the points (8, 1), (3, k) and (k, 5) collinear? IÊS> g SECTION C àíz g»`m 11 go 0 VH$ àë`oh$ àíz 3 A H$m H$m h & Question numbers 11 to 0 carry 3 marks each. 11. {~ÝXþ A, {~ÝXþAm P(6, 6) VWm Q( 4, 1) H$mo {_bmzo dmbo aoimiês> PQ na Bg àh$ma pñwv h {H$ PA PQ _mz kmv H$s{OE & 5 & `{X {~ÝXþ P aoim 3x + k (y + 1) = 0 na ^r pñwv hmo, Vmo k H$m Point A lies on the line segment PQ joining P(6, 6) and Q( 4, 1) in PA such a way that. If point P also lies on the line 3x + k (y + 1) = 0, PQ 5 find the value of k. 30//1 5 P.T.O.

30 1. x Ho$ {be hb H$s{OE : x + 5x (a + a 6) = 0 Solve for x : x + 5x (a + a 6) = `{X EH$ g_m Va lo T>r H$m 1dm± nx 13 h VWm BgHo$ àw_ Mma nxm H$m `moj\$b 4 h, Vmo BgHo$ àw_ Xg nxm H$m `moj\$b kmv H$s{OE & In an A.P., if the 1 th term is 13 and the sum of its first four terms is 4, find the sum of its first ten terms. 14. EH$ W bo _ 18 J X o h {OZ_ x bmb J X h & (i) (ii) `{X W bo _ go EH$ J X `mñàn>`m {ZH$mbr OmE, Vmo BgHo$ bmb J X Ho$ Z hmozo H$s àm{`h$vm Š`m h? `{X W bo _ bmb J X Am a S>mb Xr OmE±, Vmo bmb J X Ho$ AmZo H$s àm{`h$vm, nhbr AdñWm _ bmb J X Ho$ AmZo H$s àm{`h$vm H$s 8 9 JwZm h & x H$m _mz kmv H$s{OE & A bag contains 18 balls out of which x balls are red. (i) (ii) If one ball is drawn at random from the bag, what is the probability that it is not red? If more red balls are put in the bag, the probability of drawing a 9 red ball will be times the probability of drawing a red ball in the 8 first case. Find the value of x. 30//1 6

31 _rq>a D±$Mo Q>mda Ho$ {eia go EH$ Iå^o Ho$ erf VWm nmx Ho$ AdZ_Z H$moU H«$_e: 30 VWm 45 h & kmv H$s{OE (i) Q>mda Ho$ nmx go Iå^o Ho$ nmx H$s Xÿar, (ii) Iå^o H$s D±$MmB & ( 3 = 1 73 H$m à`moj H$s{OE) From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pole are 30 and 45 respectively. Find (i) how far the pole is from the bottom of a tower, (ii) the height of the pole. (Use 3 = 1 73) 16. EH$ K S>r H$s ~ S>r gwb VWm N>moQ>r gwb H«$_e: 6 go_r VWm 4 go_r bå~r h & gwb `m H$s ZmoH$m Ûmam 4 K Q>m _ V` Xÿ[a`m H$m `moj\$b kmv H$s{OE & ( = H$m à`moj H$s{OE) The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 4 hours. (Use = 3. 14) 17. EH$ hr YmVw Ho$ Xmo Jmobm H$m ^ma 1 {H$bmoJ«m_ VWm 7 {H$bmoJ«m_ h & N>moQ>o Jmobo H$s {ÌÁ`m 3 go_r h & XmoZmo Jmobm H$mo {nkbm H$a EH$ ~ S>m Jmobm ~Zm`m J`m & ZE Jmobo H$m ì`mg kmv H$s{OE & Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere. 18. YmVw H$o EH$ ~obz H$s {ÌÁ`m 3 go_r VWm D±$MmB 5 go_r h & Bg H$m ^ma H$_ H$aZo Ho$ {be ~obz _ EH$ e ŠdmH$ma N>oX {H$`m J`m & Bg e ŠdmH$ma N>oX H$s {ÌÁ`m 3 go_r VWm JhamB 8 go_r h & eof ~Mo ~obz H$s YmVw Ho$ Am`VZ H$m e ŠdmH$ma N>oX H$aZo hovw 9 {ZH$mbr JB YmVw Ho$ Am`VZ go AZwnmV kmv H$s{OE & A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of 3 cm and its depth is 9 8 cm. Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape. 30//1 7 P.T.O.

32 19. AmH ${V 3 _, ABCD EH$ g_b ~ h {Og_ AB DC h, AB = 18 go_r, DC = 3 go_r Am a AB VWm DC Ho$ ~rm H$s Xÿar 14 go_r h & `{X A, B, C VWm D àë`oh$ H$mo H $Ð _mz H$a g_mz {ÌÁ`m 7 go_r H$s Mmn {ZH$mbr JB h, Vmo N>m`m {H$V ^mj H$m joì\$b kmv H$s{OE & AmH ${V 3 In Figure 3, ABCD is a trapezium with AB DC, AB = 18 cm, DC = 3 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A, B, C and D, then find the area of the shaded region. Figure 3 0. nmzr go nyam ^ao 60 go_r {ÌÁ`m VWm 180 go_r D±$MmB dmbo EH$ b ~d Îmr` ~obz _, 60 go_r D±$MmB VWm 30 go_r {ÌÁ`m dmbm EH$ R>mog b ~d Îmr` e Hw$ S>mbm J`m & ~obz _ ~Mo nmzr H$m Am`VZ KZ _rq>am _ kmv H$s{OE & [ = 7 H$m à`moj H$s{OE ] A solid right-circular cone of height 60 cm and radius 30 cm is dropped in a right-circular cylinder full of water of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres. [Use = 7 ] 30//1 8

33 IÊS> X SECTION D àíz g»`m 1 go 31 VH$ àë`oh$ àíz 4 A H$m H$m h & Question numbers 1 to 31 carry 4 marks each. 1. `{X x =, g_rh$au 3x + 7x + p = 0 H$m EH$ _yb h, Vmo k Ho$ dh _mz kmv H$s{OE, {H$ g_rh$au x + k (4x + k 1) + p = 0 Ho$ _yb g_mz hm & If x = is a root of the equation 3x + 7x + p = 0, find the values of k so that the roots of the equation x + k (4x + k 1) + p = 0 are equal.. VrZ-A H$m dmbr CZ g^r g»`mam, {OZH$mo 4 go ^mj H$aZo na 3 eof AmVm h, go ~Zr lo T>r H$m _Ü` nx kmv H$s{OE & _Ü` nx Ho$ XmoZm Amoa AmZo dmbr g^r g»`mam H$m AbJ-AbJ `moj\$b ^r kmv H$s{OE & Find the middle term of the sequence formed by all three-digit numbers which leave a remainder 3, when divided by 4. Also find the sum of all numbers on both sides of the middle term separately. 3. EH$ H$n S>>o H$s Hw$N> b ~mb H$s Hw$b bmjv <$ 00 h & `{X H$n S>m 5 _rq>a A{YH$ bå~m hmo VWm àë`oh$ _rq>a H$s bmjv < H$_ hmo, Vmo H$n S>o H$s bmjv _ H$moB n[adv Z Zht hmojm & H$n S>o H$m dmñv{dh$ à{v _rq>a _yë` kmv H$s{OE VWm H$n S>o H$s bå~mb ^r kmv H$s{OE & The total cost of a certain length of a piece of cloth is < 00. If the piece was 5 m longer and each metre of cloth costs < less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre? 4. {gõ H$s{OE {H$ d Îm Ho$ {H$gr {~ÝXþ na ItMr JB ñne -aoim Cg {~ÝXþ go JwµOaZo dmbr {ÌÁ`m na bå~ hmovr h & Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 30//1 9 P.T.O.

34 5. AmH ${V 4 _, O Ho$ÝÐ dmbo d Îm Ho$ ~mø {~ÝXþ T go TP EH$ ñne -aoim h & `{X PBT = 30 h, Vmo {gõ H$s{OE {H$ BA : AT = : 1. AmH ${V 4 In Figure 4, O is the centre of the circle and TP is the tangent to the circle from an external point T. If PBT = 30, prove that BA : AT = : 1. Figure go_r {ÌÁ`m H$m d Îm It{ME & Ho$ÝÐ go 7 go_r Xÿar na {~ÝXþ P go d Îm na Xmo ñne -aoime± It{ME & BZ XmoZm ñne -aoimam H$s bå~mb _m{ne & Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre draw two tangents to the circle. Measure the length of each tangent. 7. g_mz D±$MmB Ho$ Xmo Iå^o 80 _rq>a Mm S>r g S>H$ Ho$ XmoZm Amoa EH$-Xÿgao Ho$ gå_wi h & BZ XmoZm Iå^m Ho$ ~rm g S>H$ Ho$ {H$gr {~ÝXþ P na EH$ Iå^o Ho$ erf H$m CÞ`Z H$moU 60 h VWm Xÿgao Iå^o Ho$ erf go {~ÝXþ P H$m AdZ_Z H$moU 30 h & Iå^m H$s D±$MmB`m± VWm {~ÝXþ P H$s Iå^m go Xÿ[a`m± kmv H$s{OE & 30//1 10

35 Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of a pole is 60 and the angle of depression from the top of another pole at point P is 30. Find the heights of the poles and the distances of the point P from the poles. 8. EH$ ~m Šg _ g»`m 6 go 70 VH$ H$s {JZVr Ho$ H$mS> h & `{X EH$ H$mS> `mñàn>`m ~m Šg go ItMm OmE, Vmo àm{`h$vm kmv H$s{OE {H$ ItMo JE H$mS> na (i) (ii) EH$ A H$ H$s g»`m h & 5 go nyu {d^m{ov hmozo dmbr g»`m h & (iii) 30 go H$_ EH$ {df_ g»`m h & (iv) 50 go 70 Ho$ _Ü` H$s EH$ ^má` g»`m h & A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the box, find the probability that it bears (i) a one digit number. (ii) a number divisible by 5. (iii) an odd number less than 30. (iv) a composite number between 50 and EH$ g_~mhþ {Ì^wO ABC H$m AmYma BC, y-aj na pñwv h & {~ÝXþ C Ho$ {ZX}em H$ (0, 3) h & _yb {~ÝXþ AmYma H$m _Ü`-{~ÝXþ h & {~ÝXþAm A VWm B Ho$ {ZX}em H$ kmv H$s{OE & AV: EH$ AÝ` q~xþ D Ho$ {ZX}em H$ kmv H$s{OE {Oggo BACD EH$ g_mvw^w O hmo & The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, 3). The origin is the mid-point of the base. Find the coordinates of the points A and B. Also find the coordinates of another point D such that BACD is a rhombus. 30//1 11 P.T.O.

36 30. nmzr go ^am EH$ ~V Z CëQ>o e Hw$ Ho$ AmH$ma H$m h & Bg ~V Z H$s D±$MmB 8 go_r h & ~V Z D$na go Iwbm h {OgH$s {ÌÁ`m 5 go_r h & Bg_ 100 Jmobr` Jmo{b`m± S>mbr JBª {Oggo ~V Z H$m EH$-Mm WmB nmzr ~mha Am J`m & EH$ Jmobr H$s {ÌÁ`m kmv H$s{OE & A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball. 31. EH$ XÿY dmbo ~V Z, {OgH$s D±$MmB 30 go_r h, EH$ e Hw$ Ho$ {N>ÞH$ Ho$ AmH$ma H$m h, {OgHo$ {ZMbo VWm D$nar d Îmr` {gam o H$s {ÌÁ`mE± H«$_e: 0 go_r VWm 40 go_r h, _ ^am XÿY ~m T>> nr{ S>Vm Ho$ {be H $n _ {dv[av {H$`m OmZm h & `{X `h XÿY < 35 à{v brq>a Ho$ ^md go CnbãY h VWm EH$ H $n Ho$ {be H$_-go-H$_ 880 brq>a XÿY à{v {XZ Mm{hE, Vmo kmv H$s{OE {H$ Eogo {H$VZo ~V Zmo H$m XÿY à{v {XZ H $n Ho$ {be Mm{hE VWm XmVm EO gr H$mo à{v {XZ H $n Ho$ {be Š`m ì`` H$aZm n S>oJm & Cnamoº$ go XmVm EO gr Ûmam H$m Z-gm _yë` àx{e V {H$`m J`m h? Milk in a container, which is in the form of a frustum of a cone of height 30 cm and the radii of whose lower and upper circular ends are 0 cm and 40 cm respectively, is to be distributed in a camp for flood victims. If this milk is available at the rate of < 35 per litre and 880 litres of milk is needed daily for a camp, find how many such containers of milk are needed for a camp and what cost will it put on the donor agency for this. What value is indicated through this by the donor agency? 30//1 1

37 SUMMATIVE ASSESSMENT II Mathematics Class X Time allowed: 3:00 hours Maximum Marks: 90 General Instructions: a. This question paper contains four parts A, B, C and D. b. All questions are compulsory for all. Section-A comprises of 4 questions of 1 mark each, Section-B comprises of 6 questions of marks each, Section-C comprises of 10 questions of 3 marks each and Section-D comprises of 11 questions of 4 marks each. c. There is no overall choice d. Use of calculator is not permitted Section A Find sum of 10 terms of following A.P. :, 5,, A tower stands near an airport. The angle of elevation θ of the tower from a point on the ground is such that its tangent is 5, find the height of the lower, if the distance of the 1 observer from the tower is 10 meters. 3. A die is thrown once. Find the probability of getting at most. 4. A( 1, 1), B(6,1), C(8, 8), D(x, y) are the four vertices of a rhombus taken in order. Find the co-ordinates of point D. Section B 5. Ram Prasad saved `10 in the first week of a year and then increased his weekly savings by `.75. If in the nth week, his savings become `59.50, find n. 6. Find the roots of the quadratic equation 5x 10x + = Two tangents PA and PB are drawn to the circle with centre O such that that OA= 3 AP. 0 APB = 10 Prove 8. Draw a circle of radius 3.6 cm. Take a point P outside the circle and construct a pair of tangents to the circle from that point. 9. In two concentric circles, a chord of length 4 cm of larger circle becomes a tangent to the smaller circle whose radius is 5 cm. Find the radius of the larger circle. 10. In the given figure, OAPB is a sector of a circle of radius 3.5 cm with the centre at O. If 0 AOB = 10, then find the length of OAPBO. (use π = ) 7 Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

38 Section - C 11. Find the sum of n terms of the sequence < an > where an = 5 6n and n is a natural number. 1. The sum of a number and its reciprocal is 10 3, find the number. 13. Draw a circle of radius 3 cm. Construct two tangents at the extreme ties of a diameter of this circle. 14. A man observes the angle of elevation of a bird to be He then walks 100 m towards the birds which is stationary and finds that the angle of elevation is 60. Find the height at which the bird is sitting. 15. From a well shuffled pack of 5 cards, two black queens and two kings are removed. From the remaining cards, a card is drawn at random. What is the probability that drawn card is : a) a face card b) an ace 16. Show that the line-segments joining the points (4, ) and (-6,4) and (-10, 5) and (8, 1) bisect each other. 17. The coordinates of the vertices of ABC are A( 7, ), B (9,10) and C(1, 4). If E and F are the 1 mid points of AB and AC respectively, prove that EF = BC. 18. In a cylinder of base radius 10 cm, liquid is filled to the height of 9 cm. A metal cube of diagonal 8 3 cm is immersed completely in the liquid. Find the height by which the water will rise in the cylinder. 19. The wheel of a motor cycle is of radius 1 cm. How many revolutions per minute must the wheel make so as to keep a speed of 77 km/h? 0. A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. 1. If SECTION D th th th p,q and r term of an A. P. are a, b and c respectively, then show that: a(q r) + (r p) + c(p q) = 0. Solve: y y = ; y 3, 4 y 4 y If the equation ( ) 1+ m x + mcx + (c a ) = 0has equal roots, prove that 0 c = a (1 + m ) Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks

39 4. In the figure FG is a tangent to the circle with centre A. If GCE and BCE. 0 DCB = 15 and CE=DE, find 5. Draw ABC such that BC=5 cm, ABC` ABC with A`B ; AB = 3 :. 0 0 ABC = 60 and ACB = 30, now construct 6. Two pillars of equal heights stand on either side of a road, which is 00 m wide. The angles of elevation of the top of the pillars are and 30 at a point on the road between the pillars. Find the position of the point between the pillars and height of each pillar cards numbered 1,, 3.., 16, 17 are put a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is (a) odd (c) divisible by 3. (b) a prime (d) divisible by 3 and both. 8. Prove that the points A(0, 0), B( 0, ) C (, 0) are the vertices of an isosceles right triangle. Also, find its area. 9. If h, C and V respectively represent the height, curved surface area and volume of a cone, prove that C 3 3π Vh + 9V = h 30. The area of equilateral triangle is 196 3m. Three circles are drawn at the vertices of the triangle with radius equal to the half of side of triangle. Find the area of the triangle not included in the sectors. 31. A school thought to collect the rainwater from the roof of the building, whose dimensions are m 0mby draining into a cylindrical vessel having diameter 7 m and height 4. m. If the vessel is just full, find the rainfall recorded in cm. Why it is necessary to conserve water by doing, these type of activities? Material downloaded from and Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks