Á Ã UËˇÊÊ - I,

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1 Á Ã UËˇÊÊ - I, 05-6 SUMMATIVE ASSESSMENT I, 05-6 ªÁáÊÃ / MATHEMATICS ˇÊÊ - X / Class X ÁŸœÊ Á UÃ ÿ: hours Áœ Ã 90 Time Allowed: hours Maximum Marks: 90 Ê Êãÿ ÁŸŒapple Ê. Ë Ÿ ÁŸflÊÿ Ò UBQOMN. ß Ÿ òê apple Ÿ Ò, Á ã apple øê U πá«uêapple,, ÃÕÊ Œ apple Ê UÊ ªÿÊ Ò πá«u- apple Ÿ Ò Á Ÿ apple àÿapple Ê Ò; πá«u- apple 6 Ÿ Ò Á Ÿ apple àÿapple apple Ò ; πá«u- apple 0 Ÿ Ò Á Ÿ apple àÿapple apple Ò ; ÃÕÊ πá«u-œ apple Ÿ Ò Á Ÿ apple àÿapple apple Ò. ß Ÿ òê apple Êappleß Áfl À Ÿ Ë Ò. Ò È apple U U Ê ÿêappleª flá Ã Ò General Instructions:. All questions are compulsory.. The question paper consists of questions divided into four sections A, B, C and D. Section-A comprises of questions of mark each; Section-B comprises of 6 questions of marks each; Section-C comprises of 0 questions of marks each and Section-D comprises of questions of marks each.. There is no overall choice in this question paper.. Use of calculator is not permitted. Ÿ ïÿê apple apple àÿapple Ê Ò Question numbers to carry one mark each πá«u- / SECTION-A DEW, apple ABEW Ò ÿáœ AD= cm., DE= cm. ÃÕÊ DW=cm Ò, ÃÊapple BD Ê ÊŸ ôêêã ËÁ In DEW, ABEW. If AD= cm, DE= cm and DW= cm, then find the value of DB. Page of 0

2 ÿáœ +cos θ= 5, ÃÊappleθ Ê ÊŸ ôêêã ËÁ If +cos θ= 5, find the value of θ. ÿ Êapple U M apple Á Áπ sec θ cosec θ Write the expression in simplest form : sec θ cosec θ. ÁŸïŸÁ ÁπÃ Ê U Ê UÃÊ UŸ apple Êäÿ flª ôêêã ËÁ ÁŸflÊ πø Íø Ê Ë ÊªÃ åãê Êapple Ë ïÿê From the following frequency distribution, find the median class : Cost of living index Number of weeks Ÿ ïÿê 5 apple0 apple àÿapple apple Ò πá«u- / SECTION-B Question numbers 5 to 0 carry two marks each. 5 Á h ËÁ Á + 5 Á U appleÿ ïÿê Ò Prove that + 5 is an irrational number. 6 fl apple «Ë ïÿê ôêêã ËÁ, Á apple 70 ÊÒ U 5 Êapple Êª ŒappleŸapple U Ê 5 ÊÒ U 8 Êapple U ÃÊ Ò Page of 0

3 Find the largest number which divides 70 and 5 learning remainder 5 and 8 respectively. 7 ÿáœ È Œ x 7x+k apple ÊÍãÿ α ÃÕÊ β Êapple ÃÕÊ α β= Êapple, ÃÊapple k Ê ÊŸ ôêêã ËÁ If zeroes α and β of a polynomial x 7x+k are such that α β=, then find the value of k. 8 ÁòÊ È ABC Ë È Ê Êapple AB ÃÕÊ AC Ê Á ãœèu X ÃÕÊ Y ß Ê U ÁSÕÃ Ò Á AX =, AY= cm AB ÃÕÊ YC=6 cm Ò ÃÊß XYBC Ò ÿê Ÿ Ë X and Y are points on the sides AB and AC respectively of a triangle ABC such that AX =, AB AY= cm and YC=6 cm. Find whether XYBC or not. 9 ÊŸ ÁŸ ÊÁ cosec cot 0 sec 77 tan 70 Evaluate : cosec cot 0 sec 77 tan 70 0 UÊ Ÿ Á apple U πapple ÃÊ Ò fl apple πapple apple ª 50 ÒøÊapple apple apple mê UÊ Á ª ÁflÁ UÊapple Ë ïÿê ŸËøapple ŒË ªß Ò apple mê UÊ Á ª ÁflÁ UÊapple Ê Êäÿ ôêêã ËÁ ÁflÁ UÊapple Ë ïÿê ÒøÊapple Ë ïÿê 5 0 Rajan plays cricket. His data regarding number of wickets taken in 50 matches he played in a year is given below. Calculate the mean of wickets taken by him. Number wickets Number of of Page of 0

4 matches πá«u- / SECTION-C Ÿ ïÿê apple 0 apple àÿapple s aò Question numbers to 0 carry three marks each. Œ ÊÊ ß Á Á Ë œÿêà ÍáÊÊZ Ê flª 6m+ ÿê 6m+5 apple M Ê Á Ë Ë ÍáÊÊZ m apple Á Ÿ Ë Êapple ÃÊ Ò Prove that the square of any positive integer cannot be of the form 6m+ or 6m+5 for any integer m. Áfl Êapple Ÿ ÁflÁœ apple ËÁ 5x y= x+y= Solve by elimination : 5x y= x+y= È Œ x 6x 0x Êapple È Œ x +x apple Êª ËÁ Êª» ÃÕÊ Êapple ôêêã ËÁ ÃÕÊ Áfl Ê Ÿ ªÊappleÁ UÕ Êapple àÿêá Ã ËÁ Divide the polynomial x 6x 0x by the polynomial x +x and verify the division algorithm. ÒUÁπ Ë UáÊ x=5y+ ŒûÊ Ò ãÿ ÒUÁπ Ë UáÊ ß Ê U Á Áπ Á ß ÿèç mê UÊ ÁŸM Á Ã appleuπê (i) (ii) (iii) ÁÃë UŒË Êapple ÊÃË Êapple Ê Ã U Êapple Page of 0

5 x=5y+ is given. Write another linear equation, so that the lines represented by the pair are : (i) (ii) (iii) intersecting coincident parallel 5 ABC apple, È Ê AC Ê äÿ-á ãœè X Ò ÿáœ XYAB Ò, ÃÊapple Á h ËÁ Á Y È Ê AB Ê äÿ-á ãœè Ò In ABC, X is middle point of AC. If XYAB, then prove that Y is middle point of AB. 6 ŒÊapple M ÁòÊ È Êapple ABC ÃÕÊ PQR apple Á U Ê Ê 80 cm ÃÕÊ 50 cm Ò ÿáœ QR=5 cm Ò, ÃÊapple BC ôêêã ËÁ The perimeters of two similar triangles ABC and PQR are 80 cm and 50 cm respectively. If QR=5 cm, then find BC. 7 5 sin 7 θ= ÁŒÿÊ ªÿÊ Ò, ÃÊapple sin θ cos θ Ê ÊŸ ôêêã ËÁ 5 Given sin θ=, find the value of : 7 sin θ cos θ 8 Á h ËÁ Á sin θ sin θ tan = θ cos θ cos θ Prove that : sin θ sin θ = tan θ cos θ cos θ Page 5 of 0

6 9 ÁŸïŸÁ ÁπÃ ÃÊÁ Ê ªÊ fl apple 00» Ê ÊappleZ apple ÒÄ U U apple à ÊÁŒÃ ªapple Í Êapple ÁŒπÊÃË Ò à ÊŒŸ (kg/ha) » Ê ÊappleZ Ë ïÿê apple Áœ apple Ê U Ê UŸ ŸÊß ÊÒ U ÃÊapple UáÊ πë Áø The following table gives the production yield per hectare of wheat of 00 farms of a village : Production yield (kg/ha) Number of farms Construct a more than type distribution and draw its ogive. 0 ÁŸïŸ UŸ Ê Êäÿ 8 Ò ÃÕÊ Ë Ê U Ê UÃÊ Êapple Ê ÿêappleª 50 Ò ÈåÃ Ê U Ê UÃÊ x ÃÕÊ y ôêêã ËÁ flª Ê U Ê UÃÊ 8 6 x y The mean of the following distribution is 8 and sum of all the frequencies is 50. Find the missing frequencies x and y. Class Frequency 8 6 x y Ÿ ïÿê apple apple àÿapple apple Ò πá«u-œ / SECTION-D Question numbers to carry four marks each. ÊÿÃÊ Ê U ÒŒÊŸ 8 m 7 cm Ê ÃÕÊ m 0 cm øêò«ê Ò ß apple Ë Ê Ë flªê Ê U UÊß ªÊŸË Ò ß Ê U Ë UÊß Êapple Ë ãÿíÿã ïÿê ôêêã ËÁ A rectangular courtyard is 8 m 7 cm long and m 0 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles. Page 6 of 0

7 È Á ÿêapple ÊÒ U apple Êapple Ê ÍÀÿ ` 00 Ò ÃÕÊ 5 È Á ÿêapple ÊÒ U apple Êapple Ê ÍÀÿ ` 750 Ò apple ÊÒ U È Ë Ê ª- ª ÍÀÿ ôêêã ËÁ chairs and tables cost ` 00 and 5 chairs and tables cost ` 750. Find the cost of one chair and one table separately. ÿáœ ÁmÉÊÊÃ È Œ f (x)=x 8kx+8x 9 Ê ÊÍãÿ ŒÍ appleu Ê áêêà Ò, ÃÊapple kx +kx+ apple ÊÍãÿ ôêêã ËÁ If one zero of the quadratic polynomial f (x)=x 8kx+8x 9 is negative of the other, then find the zeroes of kx +kx+.»ò Ä U UË Ê Ê Ê Ê 6x +8x +7x +x+7 apple M apple ŒÁ Ê Ã Á ÿê Ê ÃÊ Ò àÿapple ŒÍ U Êapple x +5 UÊ ÊË Á Ë ÊÒ U UŸapple apple ÊŒ x+ UÊ ÊË ø ªß ÒŸapple appleã U Ÿapple ß UÊÁ Ê apple «UÊÄ U UË ÊÿÃÊ apple Á» «U πêapple Ê Ÿ ŒÍ UÊapple Ë ïÿê ôêêã ËÁ, Á ã apple» «U Á Ê ß apple ÊÒŸ apple ÍÀÿ Œ ÊÊ ª Ò? A factory has a profit fund represented by 6x +8x +7x +x+7. The fund is equally divided between each worker of the factory. Each worker receives an amount of x +5, while after distribution, x+ amount is left. The management decides to use this amount to create a medical aid fund. Find the number of workers who received the fund. What values have been depicted here? 5 ÁøòÊ apple, ABC Ë È Ê AB U ŒÊapple Á ŒÈ D ÊÒ U E ß Ê ÁSÕÃU Ò Á AD=BE Ò ÿáœ DPBC ÊÒ U EQAC Ò, ÃÊapple Á h ËÁ Á PQAB Ò In the figure, there are two points D and E on side AB of ABC such that AD=BE. If DPBC and EQAC, then prove that PQAB. Page 7 of 0

8 6 ÊßÕÊªÊapple U appleÿ apple Áfl Êapple Ê ÕŸ Á Áπ ÊÒ U Á h ËÁ ß appleÿ apple Á áêê apple ÁŸïŸÁ ÁπÃ Êapple ËÁ ABC apple AB=6 cm, BC=6 cm ÊÒ U AC= cm Ò, ÃÊapple B ôêêã ËÁ State and prove converse of Pythagoras theorem. Using the above theorem, solve the following : In ABC, AB=6 AC= cm, find B. cm, BC=6 cm and 7 ÁòÊ ÊappleáÊÁ ÃËÿ ÊŸÊapple apple ÿêappleª Á ŸÊ ÊŸ ôêêã ËÁ cos 5 + cos 55 + tan.tan.tan0.tan67.tan77 cosec 5 tan 75 Without using trigonometric table, evaluate : ( ) cos 5 + cos 55 + tan.tan.tan0.tan67.tan77 cosec 5 tan 75 ( ) 8 ÿáœ x = seca + sina ÊÒ U y = seca sina Ò, ÃÊapple Á h ËÁ Á x y x y + = + If x = seca + sina and y = seca sina, prove that x y x y + = +. Page 8 of 0

9 9 fl Á Ê sin θ+ cos θ= Êapple Á h ËÁ ÊÒ U ß apple ÿêappleª apple Á h ËÁ Á sin θ cos θ = cos θ Ò Prove the identity sin θ+ cos θ= and use it to prove sin θ cos θ = cos θ 0 ÁŸïŸÁ ÁπÃ Ê U Ê UÃÊ UŸ Ê È ôêêã ËÁ flª Ã UÊ Ê U Ê UÃ Ê 7 7 Find the mode of the following frequency distribution Class interval f 7 7 ÁŸïŸ UŸ, ˇÊÊ apple 60 ÁfllÊÁÕ ÿêapple apple Ê U Œ ÊÊ ÃÊ Ò ßŸ ÁfllÊÁÕ ÿêapple apple Êäÿ ÃÕÊ È Ê U ôêêã ËÁ Ê U (kg apple ) Page 9 of 0

10 ÁfllÊÁÕ ÿêapple ïÿê Ë The following distribution gives the weights of 60 students of a class. Find the mean and mode weights of the students. Weight (in kg) Number of students o0o0o0o- Page 0 of 0

ªÁáÊÃ. (English+Hindi) MATHEMATICS. 1. Let f (x)=2 10 x+1 and g(x)=3 10 x 1. If (fog)(x)=x, then x is equal to :

ªÁáÊÃ. (English+Hindi) MATHEMATICS. 1. Let f (x)=2 10 x+1 and g(x)=3 10 x 1. If (fog)(x)=x, then x is equal to : MATHEMATICS ªÁáÊÃ. Let f (x)= 0 x and g(x)=3 0 x. If (fog)(x)=x, then x is equal to : 0 3 0 0 3 0 0 0 3 0 3 0 0 3 0 0 0 3. Let p(x) be a quadratic polynomial such that p(0)=. If p(x) leaves remainder when