Rao IIT Academy/ ICSE - Board 2018_Std X_Maths_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS. X - ICSE Board MATHS - QP + SOLUTIONS

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1 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS X - ICSE Board Date: MATHS - QP + SOLUTIONS SECTION - A (40 Marks) Attempt all questions from this Section Question 1 (a) Find the value of x and y if : [] x y x y x y x y Using equality of matrix x 6 10 and y x 4 y 5 15 x y 0 y 10 Topic: Matrices_Subtopic:Algebra of Matrices_ Level: 1_Std. X ICSE Board / Mathematics (b) Sonia had a recurring deposit account in a bank and deposited Rs. 600 per month for 1/ years. If the rate of interest was 10% p.a., find the maturity value of this account. [] Recurring deposite = 600 per month Period = 1/ yrs = 0 months R.O.I = (r) = 10% Total principal per 1 month 1 1

2 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions n n Rs., 79, 000 PRT Interest = Rs Maturity vale = =Rs.05 Topic: Banking_Subtopic: Recurring Deposits_ Level: 1_Std. X ICSE Board / Mathematics (c) Cards bearing numbers, 4, 6, 8, 10, 1, 14, 16, 18 and 0 are kept in a bag A card is drawn at random from the bag. Find the probability of getting a card which is : [4] (i) a prime number (ii) a number divisible by 4 (iii) a number that is a multiple of 6 (iv) an odd number n S C 1 10 (i) A = a prime number = {} P A 1 10 (ii) B = Number divisible by 4 = {4, 8, 1, 16, 0} 5 1 P B 10 (iii) C = a number that is multiple of 6 = {6, 1, 18} P C 10 (iv) D = an odd number = { } 0 P D 0 10 Topic: Probability_Subtopic: Probability_ Level: 1_Std. X ICSE Board / Mathematics

3 Question Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions (a) The circumference of the base of a cylindrical vessel is 1 cm and its height is 5 cm. Find the [] (i) radius of the cylinder (ii) volume of cylinder Use 7 (i) Given circumference r 1 r 7 r 7 1cm Radius = 1 cm (ii) Volume of cylinder r h = 4,650 cm Topic: Mensuration_Subtopic: Volume of cylinder_ Level: 1_Std. X ICSE Board / Mathematics (b) If (k ), (k + 1) and (4k + ) are three consecutive terms of an A.P., find the value of k. [] k, k 1, 4k are consecutive numbers in AP. k k k 1 4 4k k 4k k Topic: Progression_Subtopic: A.P._ Level: 1_Std. X ICSE Board / Mathematics (c) PQRS is a cyclic quadrilateral. Given QPS 7, PQS 55 and PSR 8, calculate :[4] S 8 R P 7 55 Q (i) (ii) QRS RQS (iii) PRQ

4 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions From diagram (i) SPQ QRS 180 (Opposite angles are supplimentary) 7 QRS 180 QRS (ii) PSR PQR 180 (iii) 8 PQR 180 PQR PQR 98 But PQR PQS RQS RQS RQS 4 S 8 o 55 o R o o = 5 o P 7 o 55 o Q PRQ 5 o Topic: Circles_Subtopic:Circles_ Level: _Std. X ICSE Board / Mathematics Question (a) If (x + ) and (x + ) are factors x ax b, find the values of a and b. [] x is factor of x ax b a b 0 8 a b 0 a b 8...(i) x is factor of x ax b a b 0 7 a b 0 4 4

5 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions a b 7...(ii) (ii) (i) a b a b 7 8 a b a b 7 8 a 19 Put a = 19 in (i) ( 19) b = 8 8 b = = b = 0 Topic: Remainder & Factor_Subtopic:Factor theory Level: _Std. X ICSE Board / Mathematics (b) Prove that sec cosec tan cot [] L.H.S sec cosec tan 1 cot 1 tan cot tan tan cot cot tan cot 1 tan cot tan cot Topic: Trigonometric Functions_Subtopic: Identities Level:_Std. X ICSE Board / Mathematics (c) Using graph paper draw a histogram for the given distibution showing the number of runs scored by 50 batsman. Estimate the mode of the data : [4] Runs Scored No. of batsman

6 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions A B Scale : On x axis 1 cm = 1000 units On y axis 1 cm = units 1 K MODE approx 4550 or 4600 Topic:Graphical Representation_Subtopic: Histogram_ Level: _Std. X ICSE Board / Mathematics Question 4 (a) Solve the following inequation, write down the solution set and represent it on the real number line : 10x 1x x, x Z [] 10x 1x 10 and 1x x 1 x and x x and x x Z x = {-4,-,-,-1,0,1,,,4} Topic: Inequality_Subtopic: Inequality_ Level: _Std. X ICSE Board / Mathematics (b) If the straight lines x 5y 7 and 4x ay 9 0 are perpendicular to one another, find the value of a. [] 6 6

7 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Slope of x 5y 7 is m1 5 5 Slope of 4x ay 9 0 is m 4 a lines are m 1 m a 1 5 a a 1 5 Topic: Coordinate Geometry_Subtopic: Equation of line_ Level: 1_Std. X ICSE Board / Mathematics (c) Solve x 7x 7 and give your answer correct to two decimal places. [4] x 7x x 7x x x 4 x x x 77 7 x = 0.88 or x = 7.88 Topic: Algebra_Subtopic: Qudratic Equation_ Level: 1_Std. X ICSE Board / Mathematics 7 7

8 Question 5 (a) Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions SECTION - B (40 Marks) Attempt any four questions from this Section The 4 th term of a G.P. is 16 and the 7 th terms is 18. Find the first term and common ratio of the series. [] Let the first term of a G.P. a and common ratio r a a 4 ar 16...(i) 6 7 ar 18 ii...(ii) i ar ar r 8 r Put r in equation (ii) a 16 a First term (a) = Common ratio = Topic: Progression_Subtopic: G.P._ Level:1 _Std. X ICSE Board / Mathematics (b) A man inversts Rs.,500 in Rs.50 shares available at 10% discount. If the dividend paid by the company is 1%, calculate : [] (i) The number of shares purchased (ii) The annual dividend received (iii) The rate of return he gets on his investment. Give your answer correct to the nearest whole number. Actual price = 50 Rs./Share Price after discount = 50 10% of 50 = Rs.45 (i) Total shares bought (Purchased) (ii) Annual dividend received % 8 8

9 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Rs. 000 (iii) ROR = % 500 aprrox. 1% Topic: Shares_Subtopic:Shares_ Level: _Std. X ICSE Board / Mathematics (c) Use graph paper for this question (Take cm = 1 unit along both x and y axis). ABCD is a quadrilateral whose vertices are A(, ), B(, ), C(0, 1) and D(0, 1). (i) Reflect quadrilateral ABCD on the y-axis and name it as A' B' CD. (ii) Write down the coordinaes of A'and B '. (iii) Name two points which are invariant under the above reflection. (iv) Name the polygon A' B' CD. A'(, ) A(,) 1 D (0,1) C(0, 1) B'(, ) B(, ) (ii) A', B ', (iii) C and D (iv) A' B' CD is trapezium Topic: Coordinate Geometry_Subtopic: Reflection_ Level: _Std. X ICSE Board / Mathematics 9 9

10 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Question 6 (a) Using properties of proportion, solve for x. Given that x is positive : [] x 4x 1 4 x 4x 1 x 4x x 4x 1 Applying componendo and dividendo x 4x 1 x 4x 1 x 4x 1 x 4x x 5 4x 1 Squaring on both sides, we get 4x 5 4x 1 9 6x 100x 5 64x 5 5 x 64 5 x 8 Topic: Algebra_Subtopic: Ratio & Proportion_ Level: _Std. X ICSE Board / Mathematics (b) If 0 4 A, B and C 1 0, 1 4 find AC B 10 C. [] A, B and C AC B 10 C AC B 10C

11 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions AC B 10C AC B 10C 1 Topic:Algebra_Subtopic:Matrices_ Level:1 _Std. X ICSE Board / Mathematics (c) Prove that Taking LHS : 1 cot cosec 1tan sec [4] 1 cot cosec 1 tan sec cos 1 sin sin sin cos cos sin cos 1 sin cos 1 sin cos sin cos 1 sin cos sin cos sin cos 1 sin cos 1 sin cos 1 sin cos sin cos sin cos = Topic: Trigonometry_Subtopic:Trigonometrical Identities_ Level: _Std. X ICSE Board / Mathematics Question 7 (a) Find the value of k for which the following equation has equal roots. [] x kx k k Sol. 4 0 x 4kx k k 0 following equation having equal roots b 4ac 0 here a 1, b 4k, c k k 4k 4 1 k k

12 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions 16k 4k 4k 8 0 1k 4k 8 0 k k 0 k k k 0 k k 1 k 1 0 k 1k 0 k 1 or k Topic: Algebra_Subtopic: Quadratic Equation_ Level: _Std. X ICSE Board / Mathematics (b) Sol. On a map drawn to a scale of 1 : 50,000, a rectangular plot of land ABCD has the following dimensions. AB = 6cm, BC = 8 cm and all angles are right angles. Find : (i) the actual length of the diagonal distance AC of the plot in km. (ii) the actual area of the plot in sq km. [] D C 8cm A 6cm Using pythagoras theory AC AB BC AC 6 8 AC 100 AC 10 cm 1) Actual length B cm km ) Area of ABCD cm sq. km Topic:Mensuration_Subtopic: Area_ Level: _Std. X ICSE Board / Mathematics (c) A (, 5), B(-1, ) and C(5, 8) are the vertices of a triangle ABC, M is a point on AB such that AM : MB = 1 :. Find the co-ordinates of M. Hence find the equation of the line passing through the points C and M. [4] 1 1

13 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Sol. A,5, B 1, and C5,8 A (, 5) 1 M B (-1, ) C (5, 8) Let the co-ordinates of M is (x, y) x y 4 1 point M = (1, 4) Equation of line passing through C (5, 8) and M (1, 4). 4 8 y 8 x y 8 x 5 4 y 8 1 x 5 y 8 x 5 x y 0 Topic: Coordinate Geometry_Subtopic: Equation of line Level: _Std. X ICSE Board / Mathematics Question 8 (a) Rs were divided equally among a certain number of children. Had there been 0 less children, each woule have received Rs. 100 more. Find the original number of chlidren. [] Sol. Let the original number of person be x, then 7500 divided equally between x person, 7500 each one get x 7500 divided equally between x - 0 children 7500 each one get 75 x 0 1 1

14 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions According to the question x 0 x x x 0 x 7500x x x 75 x x 0 75 x 75x 75x x x x 0x x x 0 80 x x or x x 50 or x 0 (not possible) original number of children = 50 Topic: Algebra_Subtopic: Quadratic Equation_ Level:1 _Std. X ICSE Board / Mathematics (b) If the mean of the following distribution of 4, find the value of a [] Marks Number of students Sol. Mean = 4 7 a Class 0-10 Frequency (f i ) 7 Class mark (x) i 5 f i xi a a Total 0 + a a 14 14

15 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions f x f i i Mean a 4 0 a a 70 4a 90 9a i a 10 Topic: Statistics_Subtopic: Mean_ Level: 1_Std. X ICSE Board / Mathematics (c) Using ruler and compass only, construct a ABC such that BC = 5 cm and AB = 6.5 cm and ABC 10 o [4] (i) Construct a circm - circle of ABC (ii) Construct a cyclic quadrilateral ABCD, such that D is equidistant from AB and BC. Sol. Step of construction : (i) Draw BC = 5 cm (ii) At B, draw X A O XBC 10 o 6.5cm (iii) From BX, cut off AB = 6.5 cm 10 o 5cm (iv) Join AC to get ABC B C (v) Draw the perpendicular bisector of BC and AB. These bisectors meet at O. With O as centre and radius equal to OA, draw a circle, which passes through A, B and C. This is the required circumcircle of ABC (vi) Produce the perpendicular bisector of BC so that it meets the circle at D. Join CD and AD to get the required cyclic quadrilateral ABCD. Topic: Geometry_Subtopic: Construction_ Level:1 _Std. X ICSE Board / Mathematics Question 9 (a) Priyanka has a recurring deposit account of Rs per month at 10% per annum. If she gets Rs as interest at the time of maturity, find dthe total time for which the account was held. [] Sol. Amount of recurring deposit per month = Rs Rate of interest = 10 % p.a. let period = n months Amount of interest = (1) 1000 n n 1 Total principal for one month 1000n n Interest n n...() From (1) nd (), we get n n D

16 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions 5n 5n 00 5n 5n 00 0 n n 1 0 n n n n n n n n n 6 Topic: Commercial Arithmetic_Subtopic: Banking Level: _Std. X ICSE Board / Mathematics (b) In PQR, MN in parallel to QR and PM MQ O (i) Find MN QR Q R (ii) Prove that OMN and ORQ are similar. (iii) Find. Area of OMN : Area of ORQ [] Sol. In PQR, MN QR is such a way that PM : MQ : (i) In PQR, MN QR PM PN MQ MQ NR PM Adding 1 on both sides, MQ 1 1 PM M P N PM MQ PM PQ PM 5 PM PQ 5 Now in PMN and PQR, PMN PQR (corresponding angles) P P (Common) PMN PQR PM MN PN PQ QR NR (AA postulates) 16 16

17 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions But PM PQ 5 MN QR 5 (ii) In OMN and ORQ (a) MON QOR (Vertically opposite angles) Since MN QR, (b) MNO OQR (Alternate angles) (c) NMO ORQ (Alternate angles) By AAA postulates, OMN ORQ (iii) Ar OMN MN Ar ORQ QR Ar Ar OMN ORQ 4 5 Topic: Geometry_Subtopic: Similarity_ Level: _Std. X ICSE Board / Mathematics (c) The following figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. This common radius is 7 cm. The height of the cylinder and cone are each of 4 cm. Find the volume of the solid. [4] 4 cm 4 cm 7 cm Sol. Volume = Volume of cone + Volume of cylinder + Volume of hemishpere 1 1 r h H r r h r H r cm Topic: Mensuration_Subtopic: Cylinder_ Level: _Std. X ICSE Board / Mathematics 17 17

18 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Question 10 (a) Use Remainder theorem to factorize the following polynomial : [] x x 9x 10 Sol. px x x 9x 10 p x 1 is a factor of px Now, dividing px by x 1, we get x 1 x x 10 x x 9x 10 x x x 9x 10 x x 10x 10 10x 10 0 x x 9x 10 x 1x x 10 x 1 x 5x 4x 10 x 1 x x 5 1x 5 x 1 x x 5 Topic: Algebra_Subtopic:Remainder & Factor theorem_ Level:1 _Std. X ICSE Board / Mathematics (b) In the figure given below O is the center of the circle. If QR = OP and o ORP 0. Find the value of x giving reasons. [] T P x 0 O Q S 0 0 R 18 18

19 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Sol. T P o 60 x 0 40 O Q S 0 0 R QR OP QR OP OQ Hence OQR is isoscleles QRO o OQP 40 linear pair of OQR o OPQ 40 as OPQ is isosceles o o o o POQ o o o o POT x Topic: Geometry_Subtopic:Circle_ Level: _Std. X ICSE Board / Mathematics (c) The angle of elevation from a point P of the top of a tower QR, 50 m high is 60 o and that of the tower PT freom a point Q is 0 o. Find the height of the tower PT, correct to the nearest metre. [4] R T P o 60 o 0 Q Sol. R T 50 P o 60 x o 0 Q 19 19

20 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions Let the height of the tower PT is h. and PQ is x In PQT tan 0 o 1 PT PQ H x x h...(1) In PQR 50 tan 60 o x x 50...() h 50 h h h Topic: Trigonometry_Subtopic: Heights & Distances_ Level: _Std. X ICSE Board / Mathematics Question 11 (a) The 4 th term of an A. P. is and 15th term is 66. Find the first term and the common difference. Hence find the sum of the series to 8 terms. [4] Sol. Let the first term at a A.P. is a and common difference is d. a a d...(1) 4 a a 14d 66...() 15 11d 44 a = 4 put d = 4 in equation (1) a 4 a 1 a 10 n Sn a n 1 d 8 S S

21 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions S Topic: Algebra_Subtopic: A.P._ Level: 1_Std. X ICSE Board / Mathematics (b) Use graph paper for htis questin. A survey regarding height (in cm) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded : [6] Height in cm No. of boys Taking cm = height of 10 cm along one axis and cm = 10 boys along the other axis draw an ogive of the above distribution. Use the graph to estimate the following : (i) the medium (ii) lower Quarile (iii) if above 158 cm is considered as the tall boys of the class. Find the number of boys in the class who are tall. Topic: Statistics_Subtopic: Median & Quartiles_ Level:_Std. X ICSE Board / Mathematics Sol. Height in cm No.of boys C. f Median = 60 th 0 item 1 1

22 Rao IIT Academy/ ICSE - Board 018_Std X_Maths_QP + Solutions A C 5 0 B (i) Meadian = (ii) Lower quartile = 14.5 (iii) 10 boys are above 158 cm

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