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KENDRIYA VIDYALAYA SANGATHAN QUESTION BANK SUBJECT: MATHEMATICS CLASS:X SA-II REGIONAL OFFICE JAIPUR YEAR 2016-17 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 1 PREFACE The responsibility of Review of Question Bank with key points of class X Mathematics for the session 2016-17 has been entrusted to KV Banar, Jodhpur. KVS Jaipur Region acknowledges the sincere efforts of Dr. M. M. A. USMANI, Principal KV Banar, Jodhpur. I am confident that the question bank of class X (Mathematics) will directly help the students to understand the concept well and meet quality expectation. Wish you all the best. (Dr. JAIDEEP DAS) Dy. Commissioner KVS (R.O) Jaipur Region QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 2 QUESTION BANK – MATHEMATICS FOR CLASS – X (2016-2017) PATRON Dr. JAIDEEP DAS Dy. Commissioner KVS (R.O.) Jaipur Region COORDINATOR Dr. SUKRITI RAIWANI Assistant Commissioner KVS (R.O.) Jaipur Region RESOURCE PERSON Dr. M.M.A.USMANI PRINCIPAL, KV BANAR JODHPUR SINCERE EFFORTS: Mr MAHESH SINGH SENGAR PGT (Math) Mr RAKESH NARAYAN TGT (Math) Mr MAHESH GOURA TGT (Math) QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 3 SUMMATIVE ASSESSMENT-II SECOND TERM (SA II) UNITS I ALGEBRA(contd) Quadratic equations, arithmetic progressions II GEOMETRY(contd) Circles, constructions III MENSURATION Areas related to Circles, Surface Area & Volumes IV TRIGONOMETRY(Contd) Heights and Distances. V COORDINATE GEOMETRY VI PROBABILITY TOTAL MARKS: 90 MARKS 23 17 23 08 11 08 90 Important: ∑ Slow achiever may revise the knowledge part first. ∑ Bright students may emphasize the application part of the question Bank. CLASS X Design of question paper S. No. Types of question Marks per question No. of questions Total marks 1 VSA 1 4 4 2 SA 1 2 6 12 3 SA 2 3 10 30 4 LA 4 11 44 31 90 TOTAL QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 4 DETAILS OF THE CONCEPTS TO BE MASTERED BY EVERY CHILD OF CLASS X WITH EXCERCISES AND EXAMPLES OF NCERT TEXT BOOK SUMMATIVE ASSESSMENT -II 01 Quadratic Equation Standard form of quadratic equation Solution of quadratic equation by factorization Solution of quadratic equation by completing the square Solution of quadratic equation by quadratic formula Nature of roots NCERT Text book Example 5 Examples 3,4,5, Ex. 4.2 Q. 1(iii), Q .5 Examples. 8,9 ,13 Ex. 4.3 Q. 1(iii) 3(ii),Q.8,9 Examples. 10,11,13,14,15 , Q2,3 (ii) Ex.4.3 Sum of first n terms of an A.P. Distance formula Section formula Examples 17 Q.1,2, Ex. 4.4 Exp-2, Ex. 5.1 Q.1(ii),(iv) Q.2(iv),(v) Q.4(v),(xi),(xii),(xiv) Exp. 4,6,8 Ex. 5.2 Q.3(iii), Q.4 Q.5(ii) Q.6,8,11,13,15,20 Examples 13,15, 16 Ex. Qs 1(iv), 2(i),3(iii),(v),(viii),Qs.4,8,9.10(ii),12,16,20 Examples 2,4 ,Ex7.1 Qs 3,5,7,8,10 Examples 7,8,9 Qs 2,3,6,8,10 Area of Triangle Examples 14,15 Qs.2,3,5 Heights and distances Examples- 2,3,5,6,7 Ex 9.1 3,5,5,8,10,13,15 and 16 05 Some application of Trigonometry Circles Tangents to a circle Number of tangents from a point to a circle Theorem 10.1 and 10.2 Examples 2 and 3 Ex. 10.2 Qs. 4,7,9,10,12 and 13 06 Constructions Division of line segment in the given ratio Construction of triangle similar to given triangle as per given scale Construction of tangents to a circle Construction 11.1 Circumference of a circle Area of a circle Length of an arc of a circle Area of sector of a circle Area of segment of a circle Combination of figures Example 1 Exercise 12.1 Q 2 and 4 Example 3 Exercise12.2 Q3 Example 2 Exercise 12.2Qs. 3,5,7,8,11,12 and 13 Ex 12.3 1,4,5,9,12,13 and 15 02 Arithmetic progression General form of an A.P. nth term of an A.P. 03 04 07 Coordinate geometry Area related to circles QUESTION BANK CLASS X (MATHEMATICS) Examples 1 and 2 Ex. 11.1 Qs.4,6 and 7 Construction 11.3 Ex. 11.2 Q 1,4,6 and 7 SA-II Page 5 08 Surface area and volumes Surface area of a combination of solids Volume of combination of a solid Conversion of solids from one shape to another Frustum of a cone 09 Probability Events Probability lies between 0 and1 Performing experiment QUESTION BANK CLASS X (MATHEMATICS) Examples 2 and 3 Exercise 13.1 Qs. 2,5,7 and 9 Examples 6 and 7 Ex. 13.2 Qs. 2,4,6 and 7 Examples 9,10 and 111 Exercise 13.3 Qs. 2,4,5,8 and 9 Example 12& 14 Exercise 13.4 Qs. 2,4 and 5 Examples 4,6,7,9, 11,12 and 13 Ex. 15.1 Qs. 5,6,7,10,11,15,17 24 and 25 SA-II Page 6 QUADRATIC EQUATIONS KEY POINTS 1. The general form of a quadratic equation is ax2+bx+c=0, a≠o. a, b and c are real numbers. 2. A real number ? is said to be a root of the quadratic equation ax2+bx+c=0, a ≠ o if a ? 2+b ? +c=0. The zeroes of the quadratic polynomial ax2+bx+c=0 and the roots of the corresponding quadratic equation ax2+bx+c=0 are the same. 3. Discriminant: - The expression b2 - 4ac is called discriminant of the equation ax2+bx+c=0 and is usually denoted by D. Thus discriminant D = b2-4ac. 4. Every quadratic equation has two roots which may be: distinct & real, equal & real or no real. 5. IF ? and ? are the roots of the equation ax2+bx+c=0 then ?? ???√?????? ?? ???√?????? ?? And ? = 6. Sum of the roots , ? + ? = - ? ? and product of the roots, ?? ? ? ? 7. Forming quadratic equation, when the roots ? and ? are given. x2-( ? + ?)x+ ?. ? =0 8. Nature of roots of ax2+bx+c=0 i. If D?0, then roots are real and not equal ii. D=0, then the equation has equal and real roots iii. D<0, then the equation has no real roots (distinct real roots) (coincident roots) LEVEL-1 1. If 2 is a root of the equation ?2+2?? +4 =0, then the value of ? is Ans -2 2. If D>0, then roots of a quadratic equation ax2+bx+c=0 are Ans ??? √? ?? 3. Discriminant of ?? ? ?? ? ? = 0 is Ans 5 4. The sum of roots of a quadratic equation ???+4 ? – 32 = 0 is Ans -4 5. The product of roots of a quadratic equation ? ??+ 7 ? – 4 = 0 is Ans -2 6. Values of K for which the equation ? ??+2kx - 1 = 0 has real roots are: Ans k≥3 or K≤-3 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 7 LEVEL-2 1. For what value of k, x=a is a solution of equation ??- (a+b)x+k =0 ? Ans. K=ab 2. Represent the situation in the form of quadratic equation:Rohan ‘s mother is 26 years older than him . The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. Ans .??+32x -273 = 0 where x(in years) is Rohan’s present age 3. Find the roots of ?? - 3x - 10 = 0 Ans . -2 ,5 4. Find two consecutive odd natural numbers, sum of whose squares is 130. Ans . 7,9 − 5. Find the roots of Quadratic equation??? 2√??+2 = 0 by using the quadratic formula. ? ? Ans . - ? ? , -? ? 6. Find the discriminant of the Quadratic equation ???-4x+3 = 0 and hence find the nature of its roots. Ans . D= -8<0 its no real roots. LEVEL - 3 1. If ? ? ? ??? ? ? ? are roots of the equation ??? − − 2. Solve the equation: ? ??? ?? ? ? ? ?? ?? ≠ ??? ≠ ??? 3. Solve the equation ??? − ? ??? ? ?? ? ? find the value of k and m. Ans. ? ? ?? ? ? ? ? by the method of completing square. ? ? ? 4. Using quadratic formula, solve the equation: ? ? ? ?? − − ?? ? ? ? ? ? ? ?? Ans.? ? ? ? Ans.? ? Ans.? ? 5. The sum of two numbers is 15, if the sum of their reciprocals is ?? ????? ?ℎ???? ????? ?? ? ?? ? ? ??? ? ? ? ?? ? ??? ? ? − ????? ? ?? ?? Ans. 10 and 5 LEVEL - 4 1. By reduction of Rs 2 per kg in the price of sugar. Anita can purchase 2kg sugar more for Rs 224. Find the original price of sugar per kg. What value of Anita is depicted in the question? Ans: Rs. 16, Economical survey, Leadership ? 2. Two water taps together can fill a tank in ? ? hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. 3. Find the roots of equation ? ??? ? ? + ??? = ??? , x≠ -1,-2,-4 4. Solve the following equation for ‘x’ ??? - 3(a +b)x + a b = 0 Ans. 15 hours,25 hours. ?? Ans . 2, ? 5. If the roots of the equation (a-b) ??+ (b-c) x + (c-a) = 0 are equal, prove that 2a = b+c. QUESTION BANK CLASS X (MATHEMATICS) SA-II ? ? Ans .? , ? Page 8 Self-Evaluation 1. For what value of k are the roots of the equation ??? ? ??? ? ?? ? ? real and equal. 2. The hypotenuse of right angled triangle is 20 m. If the difference between the lengths of the other Side is 4m find the other sides. 3. Find a and b such that x+1 and x+2 are factors of the polynomials ?? ? ??? 4. Find the quadratic equation whose roots are 2 + √? and 2 - √? − ?? ? ??. 5. Due to some technical Problems an aero plane started late by one hour from its starting point the pilot decided to increase the speed of the aero plane by 100 km / h from its usual speed, , to cover a journey of 12000km in time. Find the usual speed of the aero plane. What value of the pilot is represented in the question? Ans: 300km/h, punctuality, cleverness, Leadership − − 6. Divide 29 into two parts so that the sum of squares of the parts is 425. 7. Solve for x: ??? ??? ? ??? ???? ? ARITHMETIC PROGRESSION ∑ ∑ ∑ (Key Points) Arithmetic progression (A.P.):- An A.P. is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed number is called the common difference of the A.P. If a is first term and d is common difference of an A.P. , then the A.P is a , a+d , a+2d , 2+3d ….. The ??? term of an a.p is denoted by ?? and ?? = a+(n-1) d , where a = first term and d = common difference. ??? Term from the end = l – (n-1) d, where l = last term. Three terms a-d , a , a+d are in A.P with common difference d. Four terms a-3 d, a-d , a+d ,a+3d are in A.P with common diff. 2d . ∑ The sum of first n natural number is ∑ The sum of n terms of an A.P with first term a and common difference d is denoted by ∑ ∑ ∑ ∑ ∑ ∑ ? ? ?????? ? ?? = ? {2a+ (n-1) d} also, ?? = ?(a+l) where, l = last term. ?? = ?? -????. Where ?? =??? term of an A.P d = ?? -????. Where d = common difference of an A.P. LEVEL -1 1. Find ??? term of 5, 11, 17, 23.......... 2. Find the common diff. of A.P: Ans .-4n+1) -10 , -6 ,-2 ,2 ,……… QUESTION BANK CLASS X (MATHEMATICS) SA-II Ans . 4 Page 9 3. Find the A.P whose first term is 3 and common difference is – 2 Ans A.P. = 4 , 1 -2, -5, -8………… 4. In an AP, the sum of its first n terms is n2+2n. Find its 18th term. Ans . 37 5. If 5m, 7m+10, 3m+2 are in AP then find m. Ans . m=-10/3 6. If arithmetic mean between 3a and 2a - 7 is a+4, then find a. Ans . a= 5 7. Find sum of all odd numbers between 50 and 100. Ans: 1875 8. Which term of the AP: 21, 18, 15… is zero? Ans: 8th 9. For what value of n are the ??? term of two AP , 63 , 65 , 67 ,…… and 3 , 10 , 17 ,…….equal? Ans . n = 13. 10. If sum of n terms of an AP is ???+5n, then find its ??? term. Ans. 4n+3 1. Find ??? term of an AP is 5-3n, find its common difference. Ans. -4. 2. Which term of an AP 21, 18, 15… will be -81? Ans . 35th 3. Write the next term of an AP 5, ? ? √?, ? ? ?√?… Ans.√?. LEVEL - 2 4. Find the value of middle most term of AP -11, -7, -3, …… ,49 5. Find the sum of all three digit numbers which are divisible by 11 6. Which term of the AP 3, 8, 13, 18… will be 55 more than its ???? term? Ans: 17,21 Ans. 445500. Ans: 31th term 7. Check whether -150 is a term of the A. P. 11, 8, 5, 2, Ans No 8. How many three digit numbers are divisible by 8? Ans. ???? term 9. Find the sum of all multiples of 7 lying between 500 and 900. QUESTION BANK CLASS X (MATHEMATICS) SA-II Ans: 39900 Page 10 10. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the Ans: 10 rows Flower bed? What value you depict here? LEVEL- 3 1. Which term of AP : 38, 33, 28, 23,..... is the first negative term? Also find the sum of n terms. Ans. 9th term, ??? (81-5n) 2. How many terms are there in the sequence 3, 6, 9, 12 …111? Ans. 37 terms 3. The first term of an AP is -7 and the common difference 5, find its 18th term and the general term. Ans. a18 =78n & an = 5n – 12 4. Solve the equation: - 4 + (-1) + 2+ ……x = 437 Ans. = 50 5. If 5 times of 21st term of an AP is equal to its 5th term, show that 25th term is zero. 6. Find the sum of those integers from 1 to 500 which are multiples of 2 as well as of 5. Ans. 12750 7. Tarandeep saves Rs 2 on first day of the month, Rs 4 on the second day, Rs 6 on third day and so on. What will be her saving in the month of feb 2012? What value is depicted by Tarandeep. Ans: 870, saving habit Problems for self evaluation. 1. Show that the sequence defined by ?? = 4n+7 is an AP. 2. Find the number of terms for given AP: 7, 13, 19, 25,….., 205. 3. The 7th term of an AP is 32 and it 13th term is 62. Find AP. 4. Find the sum of all two digit odd positive nos. 5. If the sum of p th term and q th term of an AP be equal to rth term and s th term and d be the common difference of the AP, prove that p +q = r + s 6. Find the sum of 10 terms of AP: 2, 7, 12… 7. The sum of three numbers of AP is 3 and their product is -35. Find the numbers. 8. A man re-pays a loan of Rs 3250 by paying Rs 20 in the first month and then increases the payment by Rs 15 every month. How long will it take him to clear the loan ? 9. Divide 32 into four parts which are in AP such that ratio of product of extremes to product of means is 7: 15. 10. If the m th term of A.P. IS 1/n and nth term is 1/m , then show that the sum of mn terms is 1/2( mn + 1) 11. In an AP the sum of first n terms is ??? QUESTION BANK CLASS X (MATHEMATICS) ? ? ?? ? , Find it 25th term. SA-II Page 11 12. Correct the following statements / formulae: i) Sn = a + (n- 1 ) d ii) a n = n/2 [ 2a + (n-1) d ] iii) a n – a n-1 represents the last term of an AP iv) The sequence of perfect square numbers is an AP. CO-ORDINATE GEOMETRY KEYCONCEPTS 1. Distance Formula:The distance between two points A(x1,y1) and B (x2,y2) is given by the formula. AB=√(X2-X1)2+(Y2-Y1)2 COROLLARY:- The distance of the point P(x,y) from the origin 0(0,0) is given by = OP= √X2+Y2 OP= √(X-0)2 + (Y-0)2 2. Section Formula :The co-ordinates of the point P(x,y) which divides the line segment joining A(x1,y1) and B(x2,y2) internally in the ratio m:n are given by . ?? ? ????? ??? ?? ? ??? ? 3. Midpoint Formula:- ? ????? ??? ? ??? ? If R is the mid-point, then m1=m2 and the coordinates of R are ? = ????? ? , ? = ????? ? , 4. Co-ordinates of the centroid of triangle:The co-ordinates of the Centroid of a triangle whose vertices are P(x1,y1), Q(x2,y2) and R(x3,y3) are x1+x2+x3 3 , y1+y2+y3 3 5. Area of a Triangle:- The area of the triangle formed a by the points P(x1,y1), Q(x2,y2) and R(x3,y3) is the numerical value of the expression. ar (∆PQR) = ? [x (y -y )+x2(y3-y1)+x3(y1-y2)] ? 1 2 3 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 12 LEVEL- 1 1. Find the centroid of triangle whose vertices are (3, -7), (-8, 6) and ( 5, 10). Ans. (0, 3) ? 2. If P ( ?, 4) is the midpoint of the line segment joining the points Q (-6 , 5 ) and R (-2 , 3) , then find the value of β . Ans . -12 3. A line intersects y –axis and x-axis at the points P and Q respectively. If ( 2 ,-5) is the midpoint of PQ , Then find the coordinates of P and Q respectively. Ans. (0,-10) and (4,0) 4. Find the value of k if the point P (2,4 ) is equidistant from the point (5,k) and (k,7) Ans: k=3 6. If the point A(x,y), B(3,6) and C(-3,4) are collinear, Show that x -3y + 15 = 0 7. Find the coordinate of the point on x-axis which is equidistant from (5,-2)and (-3,2). Ans (1,0) 8. Find the coordinates of a point A, where AB is diameter of a circle whose centre is (2, -3) and B is (1, 4) Ans. (3, -10) 9. The length of a line segment is 13 units. If one end point is (5, 7) and abscissa of the other point is -7, Find the ordinate of the point. Ans. 12 or 2 LEVEL-2 1. Point P (5, -3) is one of the two points of trisection of the line segment joining the points A (7, -2) and B (1, -5) near to A. Find the coordinates of the other point of trisection. Ans. (3, -4) 2. Show that the point P (-4, 2) lies on the line segment joining the points A (-4 , 6) and B (-4, -6). 3. If A (-2, 4) , B (0, 0) , C (4, 2) are the vertices of a ∆ABC, then find the length of median through the vertex A. Ans. 5 units 4. In what ratio does the line x – y - 2 = 0 divides the line segment joining the points A(3,-1) and B (8, 9)? Ans: 2:3 5. If the points A (4,3) and B (x,5) are on the circle with centre O(2,3) then find the value of x. Ans. 2 6. What is the distance between the point A (c, 0) and B (0, -c)? Ans. √? c 7. For what value of k, are the points (-3, 9), (2, k) and (4, -5) collinear? Ans. ? ? QUESTION BANK CLASS X (MATHEMATICS) SA-II − ? Page 13 LEVEL-3 1. Show that the points A(3, 0) , B(4, 5) , C (-1,4) and D (-2, -1) taken in order are the vertices of a parallelogram. 2. Point P divides the line segment joining the points A(2,1) and B(5,-8) such that AP: AB=1:3.If P lies on the line 2x-y+k=0, then find the value of ?. Ans. k = -8 3. Points P, Q, R, and S in that order are dividing a line segment joining A (2, 6) and B (7, -4) in five equal parts. Find the coordinates of point P and R? Ans. P (3, 4) , R (5, 0) 4. Find a relation between x and y if the points (2, 1), (x, y) and (7, 5) are collinear.Ans. 4x - 5y + 3 = 0 5. find the area of the quadrilateral ABCD whose vertices are A (-3, -1) , B (-2, -4) , C (4, -1) and D (3, 4) . Ans. 28 sq. units 6. In a village, Sarpanch gave tender to dig a well at the centre of village passing through the houses A (6,-6) B (3,-7) and C (3,3). Find the centre of the village. What value is depicited by Sarpanch. Ans : (-3,2) 7. Find the point on y- axis which is equidistant from the points (5, -2) and (-3, 2) Ans. (0, -2) LEVEL-4 1. A (6, 1), B (8, 2), C (9, 4) are the three vertices of a parallelogram ABCD. If E is the midpoint of DC, ? then find the area of ∆ADE. Ans. ? ??????? 2. find the value of ‘k’ for which the points are collinear (7, -2) , (5, 1) , (3, k) Ans. ? ? ? 3. Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are (0, -1) , (2,1) and (0,3). Find the ratio of this area to the area of the given triangle. Ans. 1:4 4. If the mid- point of line joining A (5,-1) and B (k,2) is ( x, y ) and 2x +2y + 1 =0 . Find the value of k. Ans: k=-3 5. Find the area of the rhombus, if its vertices are (3,0), (4,5), (-1,4) and (-2,-1) taken in order. Ans. 24 sq. units QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 14 SELF EVALUATION 1. Two opposite vertices of a square are (-1,2) and (3, 2). Find the coordinates of the other two vertices. 2. Find the centre of a circle passing through the points (6,-6), (3, 7) and (3, 3). [Ans.3,-2] 3. If the points (p, q), (m, n) and (p-m, q-n) are collinear ,prove that p n =q m. 4. If A and B are (-2, -2) and (2, -4) respectively, Find the coordinates of Q such that AQ = on the line segment AB 3 AB and Q lies 7 Ans :(2/7, -20/7) 5. The area of triangle is 5 square units. Two of its vertices are A (1, 2) and B (6, 2). If the third vertex lies on the line x – 2y + 1 = 0, Find the third vertex. APPLICATIONS OF TRIGONOMETRY (HEIGHTS AND DISTANCE) KEY POINTS Line of sight Line segment joining the object to the eye of the observer is called the line of sight. Angle of elevation When an observer sees an object situated in upward direction, the angle formed by line of sight with horizontal line is called angle of elevation. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 15 Angle of depression When an observer sees an object situated in downward direction the angle formed by line of sight with horizontal line is called angle of depression. Trigonometric Ratios of some specific angles. q sin q cosq tan q cot q sec q cosec q 30o ½ ÷3/2 1/÷3 ÷3 2/÷3 2 45o 1/÷2 1/÷2 1 1 ÷2 ÷2 60o ÷3/2 1/2 ÷3 1/÷3 2 2/÷3 LEVEL- 1 1. A 20m long ladder rest against a wall. If the feet of the ladder is 10 m away from the wall, then find the angle of the elevation. Ans. 600 2. If √????? ? ???ℎ?? ???? ?ℎ???????? ????? − ????? Ans. -1/2 3. An observer 1.5m tall is 20.5 meters away from a tower 22m high. Determine the angle of elevation of the top of the tower from the eye of the observer. Ans. 45° 4. The ratio of length of rod and its shadow is 1: 3 . Find the angle of elevation of the sun.Ans: 300 5. In a rectangle ABCD, AB =20cm –BAC=600then find the length of the side AD. Ans. 20√?cm 6. In rectangle, the angle between a diagonal and a side is 300 and the length of its diagonal is 6 cm . Find Ans:9 3 cm2 the area of rectangle. O Q 7. In the adjacent figure, what are the angles of depression of the top and bottom from the top of a tower h m high: QUESTION BANK CLASS X (MATHEMATICS) SA-II A 450 300 of a pole M Page 16 Ans450, 600 LEVEL -2 1. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30 0 with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree. Ans: 8 3 m 2. A ladder 50m long just reaches the top of a vertical wall. If the ladder makes an angle of 600 with the wall, find the height of the wall. Ans. 25 m 3. A circus artist is climbing a 20m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 300. Ans. 10 m 4. A hoarding 1.46 metretall,stands on the top of a pole, which shows the need for abolition of child labour in the society.From a point on the ground, the angle of elevation of the hoarding is 600And from the same point the angle of elevation of the top of the pole is 450.Find the height of the pole. (i) What are the ill effects of child labour on a society? (ii) Write any two ways by which the child labour can be abolished? Ans: 72 m LEVEL - 3 1. The shadow of a tower standing on a level plane is found to be 50m longer when sun’s elevation is 300 then when it is 600. Find the height of the tower. Ans. ??√?? 2. The angle of depression of the top and bottom of a tower as seen from the top of a 60 3 m high cliff are 450 and 600 respectively. Find the height of the tower. Ans.43.92m 3. From a window (9m above ground) of a house in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are 300 and 600 respectively. Find the height of the opposite house and width of the street. QUESTION BANK CLASS X (MATHEMATICS) SA-II Ans.12m,3√?m Page 17 4. The angle of elevation of a jet fighter from a point A on the ground is 600.After a flight of 15 seconds, the angle of elevation changes to 30◦. If the jet is flying at a speed of 720km/ hr, find the constant height at which the jet is flying. Ans;1500m 5. A window in a building is at a height of 10m above the ground. The angle of depression of a point P on the ground from the window is 300. The angle of elevation of the top of the building from the point P is 600 . Find the height of the building. Ans; 30m 6. A boy , whose eye level is 1.3m from the ground , spots a balloon moving with the wind in a horizontal line at same height from the ground. The angle of elevation of the balloon from the eyes of the boy at any instant is 600. After 2 seconds, the angle of elevation reduces to 300 If the speed of the wind at that moment is 29√? m/s , then find the height of the ballon from the ground . Ans; 88.3m 7. A man on the deck on a ship 14m above water level observes that the angle of elevation of the top of a cliff is 600and the angle of depression of the base of the cliff is 300. Calculate the distance of the cliff from the ship and the height of the cliff. Ans ; h= 56m , distance 24.25m 8. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of tower with a uniform speed Six minutes later, the angle of depression of the car is found to be 60 . Find the time taken by the car to reach the foot of the tower. Ans.3 minutes 9. As observed from the top a light house, 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 300 to 600. Determine the distance travelled by the ship during the period of observation (use 3 = 1.732) Ans115.5m 10. Two hoardings on “Save energy” are put on two poles height 10 m and 15 m standing to opposite to each other on aground. if the distance between their feet is 5 3 m Find the distance between their tops . Why do we need to save energy? Ans10m SELF EVALUATION/HOTS QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 18 1. An aero plane when flying at a height of 3125m from the ground passes vertically below another Plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30°and 60°respectively. Find the distance between the two planes at that instant. Ans ; 6250m 2. From the top of a building 60m high, the angels of depression of the top and bottom of a vertical lamp post are observed to be 30° and 60°respectively. Find [i] horizontal distance between the building and the lamp post [ii] height of the lamp post. Ans. 34.64m, h=40m 3. If the angle of elevation of a cloud from a point h meter above a lake is a and angle of depression of its reflection in the lake be b . Prove that the distance of the cloud from the point of observation is 2h seca tan b - tan a 4. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30 0. A girl standing on the 20 m high building, Find the angle of elevation of the same bird to be 450. Both the boy and the girl are on opposite sides of the bird. Find the distance of the bird from the girl. Ans: 30 2 m 5. The angle of elevation of a cloud from a point 60m above a lake is 30◦ and the angle of depression of the reflection of the cloud in the lake is 60° . Find the height of the cloud from the surface of the lake. Ans 120m Circle Basic concepts QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 19 1. In figure (i) line P and the circle and no common point hence ,the line P known as non intersecting line with respect to the circle. 2. In the figure (ii) , a line P intersects the circle in two distinct points, it is called a secant of the circle . In the figure (iii) , the lint P intersects the circle in one and only one point a, and is set to be are tangent to the circle. The point A at which the tangent line means the circle is called point of contact. The word tangent to a circle has been derived from the latin word “ tang ere “,which means ‘to touch’ and was introduces by the Danish Mathematician Thomes Fineke in 1853. Key Words 1. Circumscribed circle of a polygon: It is a circle passing through all the vertices of the polygon. The centre of this circle is called Circumcentre and its radius is called circumradius. 2. Inscribed circle (Encircled of the polygon): It is the largest circle that can be contained in the polygon and it touches each side of the polygon at a point. The centre of this circle is called incentre and its radius is called inradius. THEOREM: 10.1 The tangent at any point of a circle is perpendicular to the radius through the point of contact. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 20 Given: A circle with cente O and a tangent AB to the circle at a point P. To prove: OP is perpendicular AB Construction: Take any point R, other than P on the tangent AB. Join OR. Suppose OR meets the circle at Q. Proof: OP=OQ (radius of the circle) OP < OQ +QR OP < OR Thus, OP is shorter than any other line segment joining O to a any Point of AB, other than P. We know that among all line segments joining the point O to a point AB, the shortest one is perpendicular to AB Hence OP is perpendicular to AB THEOREM 10.2 The length of tangents drawn from an External point P to a circle are equal Given: AP and AQ are two tangents from a point to a circle C (O,r) To Prove: AP = AQ Construction: Join OA, OP and OQ. Proof: AP is tangent at P and OP is the radius through P Therefore OP is perpendicular to AP Similarly AQ is a tangent at Q and OQ is the radius through Q Therefore OQ is perpendicular to AQ In right triangles OPA and OQA OP =OQ (equal radii of the same circle) AO = AO (common) –OPA = –OQA (each 900) ∆ ??? @ ∆ OQA (RHS congruence) AP =AQ (CPCT) AP = AQ LEVEL1 1. At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY and at a distance 8 cm from A. Ans. 8 cm QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 21 2. In a circle of a diameter 10 cm, length of each of the two equal and parallel chords is 8 cm. find the distance between these two chords. Ans: 6 cm 3. In the adjoining figure, ∆ ABC is circumscribing a circle, then find the length of BC. Ans. 9cm A 8cm M 3 cm N 4 cm B C V 4. Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25cm. Given that the radius of the circle is 7cm. Ans: 24cm L 5. From a point P, two tangents PA and PB are drawn to a circle C(O,r) . If OP =2r ,then what is the type of ∆ APB. Ans. Equilateral triangle 6. If the angle between two radii of a circle is 130?,then find the angle between the tangents at the end of the radii. Ans. 50?. 7. ABCD is a quadrilateral. A circle centered at O is inscribed in the quadrilateral. If AB = 7cm , BC = 4cm , CD = 5cm then find DA. Ans. 8 cm 8. In a ∆ ABC , AB = 8cm , –ABC = 90?. Then find the radius of the circle inscribed in the triangle. Ans. 2cm LEVEL2 1. From an external point P, two tangents PA and PB are drawn to the circle with centre O. Prove that OP is the perpendicular bisector of AB. 2. If PA and PB are two tangents drawn to a circle with centre O , from an external point P such that PA=5cm and –APB = 60?, then find the length of the chord AB. Ans. 5cm 3. CP and CQ are tangents from an external point C to a circle with centre O .AB is another tangent which touches the circle at R and intersects PC and QC at A and B respectively . If CP = 11cm and BR = 4cm, then find the length of BC. 4. Ans. 7cm If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 22 5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle. 6. In adjacent figure; AB & CD are common tangents to two circles of unequal radii. Prove that AB=CD. A C B D LEVEL3 1. Four roads have to be constructed by touching village Kanpur in circular shape of radius 1700m as shown in figure, Sarita got contract to construct the road AB and CD while Vijay got the contract to construct AD and BC Prove that AB + CD = AD +BC Which value is depicted by the contractor? 2. Prove that the angle between the two tangents to a circle drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact to the centre. 3. AB is a chord of length 9.6cm of a circle with centre O and radius 6cm.If the tangents at A and B intersect at point P then find the length PA. Ans. 8cm 4. The in circle of a ∆ABC touches the sides BC, CA & AB at D,E and F respectively. If AB=AC, prove that BD=CD. 5. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre of the circle. 6. PQ and PR are two tangents drawn to a circle with centre O from an external point P. Prove that –QPR=2–OQR. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 23 LEVEL4 1. Prove that the length of tangents drawn from an external point to a circle is equal. Hence, find BC, if a circle is inscribed in a ∆ABC touching AB,BC &CA at P,Q &R respectively, having AB=10cm, AR=7cm &RC=5cm. Ans. 8cm 2. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following: If O is the centre of two concentric circles, AB is a chord of the larger circle touching the smaller circle at C, then prove that AC=BC. 3. A circle touches the side BC of a ∆ABC at a point P and touches AB and AC when produced, at Q & R respectively. Show that AQ=1/2 (perimeter of ∆ABC). 4. From an external point P, a tangent PT and a line segment PAB is drawn to circle with centre O, ON is perpendicular to the chord AB. Prove that PA.PB=PN2-AN2. 5. If AB is a chord of a circle with centre O, AOC is diameter and AT is the tangent at the point A, then prove that –BAT=–ACB. 6. The tangent at a point C of a circle and diameter AB when extended intersect at P. If –PCA=1100 , find –CBA Ans. 700 7. If PA and PB are tangents such that–APB=600 and PB=7cm.Find the length of chord AB . Ans AB=7 Cm [Self Evaluation] 1. If PA and PB are tangents from an external point P to the circle with centre O, the find –AOP+–OPA. Ans. 900 2. ABC is an isosceles triangle with AB=AC, circumscribed about a circle. Prove that the base is bisected by the point of contact. 3. AB is diameter of a circle with centre O. If PA is tangent from an external point P to the circle with –POB=1150 Ans. 250 then find –OPA 4. PQ and PR are tangents from an external point P to a circle with centre . If –RPQ=1200, Prove that OP=2PQ. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 24 5. Show that tangent line at the end points of a diameter of a circle are parallel. 6. If a, b, c are the sides of a right triangle where c is the hypotenuse , then prove that radius r of the circle touches the sides of the triangle is given by r= (a + b - c)/2. 0 7. A circle is inscribed in a quadrilateral ABCD where –B=90 .If AD = 24 cm ,AB=30 cm and DS =8cm . Find the radius r of the circle . CONSTRUCTION KEY POINTS 1. Division of line segment in the given ratio. 2. Construction of triangles:a. When three sides are given. b. When two sides and included angle given. c. When two angles and one side given. d. Construction of right angled triangle. 3. Construction of triangle similar to given similar to given triangle as per given scale. 4. Construction of triangles to a circle. LEVEL - I 1. A farmer divides sugarcane of 7 feet length between his son and daughter equally. Divide it geometrically considering sugarcane a line of 7 cm using construction. Find the length of each part. Which value is depicted? 2 Draw a line segment of length 7 cm. Find a point P on it which divides it in the ratio 3:5. 3. Divide a line segment of 7cm internally in the ratio 3:4. 4. Draw a circle of radius 4 cm. Take a point P on it. Draw tangent to the given circle at P. 5. Construct an isosceles triangle whose base 7.5 cm and altitude is 4.2 cm. 6. Is it possible to divide a line segment in the ratio 3 : 1 QUESTION BANK CLASS X (MATHEMATICS) SA-II 3 ? Page 25 LEVEL –II 1. Construct a triangle similar to a given ∆ABC such that each of its sides is 2/3 of the corresponding sides of ∆ABC. It is given that AB=4cm BC=5cm and AC=6cm also write the steps of construction. 2. Draw a right triangle ABC in which –B=600 AB=4.5 cm, BC=6.5cm then construct another triangle ABC whose sides are 5/3 times the corresponding sides of ∆ABC. 3. Draw a pair of tangents to a circle of radius 5cm which are inclined to each other at an angle of 600. 4. Draw a circle of radius 5cm from a point 8cm away from its centre construct the pair of tangents to the circle and measure their length. 5. Construct a triangle ABC in which BC=6cm –C=300 and –A=1050. Construct another triangle whose sides are 2/3 times the corresponding sides of ∆ABC 6. Draw a circle of radius 4 cm .construct a pair of tangents to it, the angle between them 600. Also justify the construction. Measure the distance between the centre of circle and the point of intersection of tangents. AREAS RELATED TO CIRCLES KEY POINTS 1. Circle: The set of points which are at a constant distance of r units from a fixed point o is called a circle with centre o. ∑ Perimeters and areas of simple closed figures. ∑ Circumference and area of a circle. ∑ Area of a circular path (i.e., ring). ∑ Sector of a circle and its central angle ∑ Major and Minor sectors. ∑ Segment of a circle – Major and Minor segments. 2. Circumference: The perimeter of a circle is called its circumference. 3. Secant: A line which intersects a circle at two points is called secant of the circle. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 26 4. Arc: A continuous piece of circle is called and arc of the circle.. 5. Central angle:- An angle subtended by an arc at the center of a circle is called its central angle. 6. Semi Circle: - A diameter divides a circle into two equal arc. Each of these two arcs is called a semi circle. 7. Segment: - A segment of a circle is the region bounded by an arc and a chord, including the arc and the chord. 8. Sector of a circle: The region enclosed by and an arc of a circle and its two bounding radii is called a sector of the circle. 9. Quadrant: - One fourth of a circle disc is called a quadrant. The central angle of a quadrant is 900. Important Formula S.N NAME FIGURE PERIMETER AREA 1. Circle ???or ?? ??2 2. Semi- circle ??+ 2r ½ ??2 3. Ring (Shaded region) 2 ???+ R) ?(R2-r2) 4. Sector of a circle 5. ??? l+2r=???° ? Segment of a circle QUESTION BANK CLASS X (MATHEMATICS) ??? +2r ???° SA-II ?? ? ? Sin ???? ???° or ? ?? ? ???? ? ? - ? sinq ???° ? Page 27 a. Length of an arc AB= ? 2 ?? ??? 0 A B b. Area of major segment= Area of a circle – Area of minor segment c. Distance moved by a wheel in 1 rotation=circumference of the wheel d. Number of rotation in 1 minute =Distance moved in 1 minute / circumference LEVEL-1 1. If the perimeter of a circle and the area of the circle are numerically equal, then the diameter of the circle is [Ans-4] 2. The area of the square that can be inscribed in a circle of 8 cm is [Ans-128cm2] 3. Area of a sector to circle of radius 36 cm is 54 ?cm2 . Find the length arc of the corresponding arc of the circle is [Ans –3 πcm] 4. A wheel has diameter 84 cm. The number of complete revolution it will take to cover 792 m is. 5. The length of an arc of a circle with radius 12cm is 10 ? cm. The central angle of this arc is . 6. The area of a quadrant of a circle whose circumference is 22 cm is [Ans-300] [Ans-1500] [Ans-.625cm2] 7. If the perimeter of a circle is equal to that of square, then the ratio of their areas is LEVEL-2 1. If the diameter of a semicircular protractor is 14 cm, then find its perimeter . [Ans-14/11] [Ans-36 cm] 2. The radius of two circle are 3 cm and 4 cm . Find the radius of a circle whose area is equal to the sum of the areas of the two circles. [Ans: 5 cm] 3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. [Ans: 154/3 cm] 4. If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square? Ans: p : 2) QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 28 5. What is the angle subtended at the centre of a circle of radius 10 cm by an arc of length 5p cm. LEVEL-3 Ans:900 1. The inner circumference of a circular track is 440m. The track is 14m wide. Find the diameter of the outer circle [Ans-168m] of the track. [Take ? =22/7] 2. Find the area of the shaded region. [Ans: 4.71cm2] 3. A copper wire when bent in the form of a square encloses an area of 121 cm2 . If the same wire is bent into the [Ans 154 cm2] form of a circle, find the area of the circle (Use ?=22/7) 4. A wire is looped in the form of a circle of radius 28cm. It is rebent into a square form. Determine the side of the [Ans-44cm] square (use ? ? ????? LEVEL-4 1. In fig, find the area of the shaded region [use ? ? ????? 2. In fig find the shape of the top of a table in restaurant is that of a sector a circle with centre 0 and –BOD=900. If OB=OD=60cm find QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 29 i. ii. The area of the top of the table The perimeter of the table top (Take ? ? ?????[ [Ans 8478 cm2] Ans 402.60 cm] 3. An arc subtends an angle of 900 at the centre of the circle of radius 14 cm. Write the area of minor sector thus [Ans 49? cm2] form in terms of ?. 4. The length of a minor arc is 2/9 of the circumference of the circle. Write the measure of the angle subtended by the arc at the center of the circle. [Ans 800] 5. The area of an equilateral triangle is 49√3 cm2. Taking each angular point as center, circle is drawn with radius equal to half the length of the side of the triangle. Find the area of triangle not included in the circles. [Ans 777cm2] [Take ?√3=1.73] SELF EVALUATION 1. If the perimeter of a semi-circular protractor is 108 cm , find the diameter of the protractor (Take π =22/7). 2. Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 8a cm? Give reasons for your answer. 3. Find the area of a sector of a circle of radius 28 cm and central angle 45°. 4. The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/h? 5. A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20m × 16m. Find the area of the field in which the cow can graze. 6. In Fig. ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and distance between AB and DC = 14 cm. If arcs of equal radii 7 cm with centers A, B, C and D have been drawn, then find the area of the shaded region of the figure. 7. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5m long rope .Find (i) the area of that part of the field in which the horse can graze. (ii)The owner has not made any shed for the horse. What values are lacking in the owner? QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 30 Ans 19.63cm2, 58.87cm2 SURFACE AREAS AND VOLUMES (IMPORTANT FORMULA) SNo NAME 1 FIGURE LATERAL CURVED SURFACE AREA TOTAL SURFACE AREA VOLUME NOMENCLATURE Cuboid perimeter of rect.x height sq. unit = 2(l+b) x h sq. unit 2(lxb + bxh + hx l) sq. unit L=length, b=breadth, h=height 2 Cube = 2(l+l) x l sq. unit =2(2l)xl sq. unit = 4 l2 sq. unit 6 x (area of Square) sq.unit = 6 l2sq.unit 3 Right Circular Cylinder circumference of circle x height sq. unit = (2 p r)x h sq.unit 4 Right Circular Cone = semi perimeter of circle x slant heightsq.unit =p r l sq. unit 5 Sphere = 4X Area of circle sq. unit =4pr2sq. unit = curved surface area + 2x area of base(circle) sq. unit = (2 p r)x h + 2pr2sq.unit = curved surface area + area of circle sq. unit =p r l + pr2sq. unit = 4X Area of circle sq. unit =4pr2sq. unit = area of base(rec) x height cubic unit = lxbx h cu unit = area of base (square) x height) cubic unit = l2 x l cubic unit = area of base (circle) x height cubic unit = pr2 x h cubic unit 6 Hemisph ere = half x area of sphere sq. unit =2pr2sq. unit = 3x area of circle sq. unit =3pr2sq. unit 2 2 2 2 7 Spherical shell =2p(R + r ) sq. unit 3pR +pr sq. unit 8 Frustum of a cone =pl(R+r) sq. unit where l2=h2+(R-r)2 -p[R2 + r2 + l(R+r)] sq. unit QUESTION BANK CLASS X (MATHEMATICS) SA-II ? l=edge of cube r= radius h=height = volume of ? cylinder cubic unit ? ? pr2h cubic ? unit r=radius of base, h=height , l=slant height = √?? ?? = half of volume of sphere cubic unit ? = pr3cubic unit r=radius of hemisphere ? ? ? pr3cubic unit ? ? ? p(R3 - r3) ? cubic unit 2 2 =ph/3[R + r + Rr] cubic unit − r=radius of the sphere R=External radius, r=internal radius R and r = radii of the base, h=height, l=slant height. Page 31 9. Diagonal of cuboid = √?? ? ?? ? ℎ? 10. Diagonal of Cube = ÷3l LEVEL - 1 [1] Three cubes of volumes 64cm3 are joined end to end. Find the surface area of the resulting cuboid? Ans: 224cm2 [2] Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm? Ans:19.4 cm3. [3] A semicircular thin sheet of metal of diameter 28cm is bent and an open conical cup is made. Find depth. (7÷3) [4] Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.9 cm is?Ans:30.8cm3. [5] The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends are18cm and 6cm. Find the curved surface area of the frustum [use ? ? ?? ]. ? Ans: 48cm2. [6] A plumbline is a combination of which geometric shapes? ANS:A cone with hemisphere. [7] What is the ratio of the volume of a cube to that of sphere which will fit inside it? Ans : 6 : p [8] A cylinder and a cone are of same radius and of same height. Find the ratio of the volume of the cylinder to that of the cone. Ans : 3 : 1 LEVEL - 2 [1] The slant height of the frustum of a cone is 5 cm .If the difference between the radii of its two circular ends is 4cm . Write the height of the frustum. Ans:3cm [2] A cylinder, a cone and a hemisphere are of same base and of same height .Find the ratio of their volumes? Ans: [3:1:2]. [3] A cone of radius 4cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base, compare the volume of the two parts. Ans:1:7 [4] How many spherical lead shots each having diameter 3cm can be made from a cuboidal lead solid of dimensions 9cm X 11cm X 12cm. Ans:84 [5] Three metallic solid cubes whose edges are 3cm, 4cm, and 5cm are melted and converted into a single cube . Find the edge of the cube so formed? QUESTION BANK CLASS X (MATHEMATICS) Ans :- 6cm . SA-II Page 32 LEVEL-3 [1] 400 persons took dip a rectangular pool which is 40 m long and 30 m broad. What is rise in the level of water in thepool, if the average displacement of water by a pers0n is 0.3m 3? Ans : 10 cm [2] Find the number of metallic circular disk with 1.5cm base diameter and of height 0.2 cm to be melted to form a right circular cylinder of height 10cm and diameter 4.5cm ? Ans:-450 [3] From a solid cube of side 7cm,a conical cavity of height 7cm and radius 3cm is hollowed out . Find the volume Ans:-277cm3. of remaining solid? [4] A cubical block of side 7cm is surmounted by a hemisphere. what is the greatest diameter of the hemisphere Ans:- 7cm,332.5cm2. can have? Find the surface area of the solid? [5] A heap of rice is in the form of a cone of diameter 9m and height 3.5m .Find the volume of the rice .How much Ans:-74.25m3, 80.61 m2 . canvas cloth is required to just cover the heap? [6] A square field and an equilateral triangle park have equal perimeter .If the cost of ploughing the field at the rate of Rs 5/m2 is Rs 720. Find the cost of maintain the park at the rate of Rs10/m2? Ans:-Rs1108.48 [7] Student of IX class packed 500 packet of biscuits each of dimension 6 cm x 3 cm x2 cm in boxes each of Volume 1800 cm3 to be distributed to the children of the flood victims (i) Find the number of boxes required? (ii) Which mathematical concept is used here? (iii)What moral values are shown? Ans: 10 boxes, volume concept, sharing and caring, team work etc. [7} A Cone of maximum size is carved out from a cube of edge 14 cm. find the surface area of the cone and of Ans: 154(÷5 +1) cm2,(1022+154)cm2 the remaining solid left out after the cone is carved out. LEVEL -4 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 33 [1] A well in village of diameter 3cm and 14m deep in dug. The earth, taken out of it, has been evenly spread all around it in the shape of a circular ring of width 4m to form an embankment find the height of embankment? Grampradhan spend the money for the embankment, what value is shown by him? ? ? ANS:- m. [2] 21 glass spheres each of radius 2cm are packed in a cuboidal box of internal dimensions 16cmX8cmX8cmand then ANS:-320cm3. the box is filled with water. Find the volume of water filled in the box? [3] The slant height of the frustum of a cone is 4cm and the circumferences of its circular ends are 18cm and 6cm. Find . ANS:-48cm2, 76.63cm2. curved surface area and total surface area of the frustum [4] A farmer connects a pipe of internal diameter 25cm from a canal into a cylindrical tank in his field, which is 12m in diameter and 2.5m deep. If water flows through the pipe at the rate of 3.6km/hr, in how much time will the tank be filled? Also find the cost of water, if the canal department charges at the rate of Rs0.07/m3? Ans:-96min, Rs19.80 [5] A spherical glass vessel has a cylindrical neck 7cm long and 4cm in diameter. The diameter of the spherical part is ANS:-4939cm3. 21cm Find the quantity of water it can hold. [6] The surface area of a solid metallic sphere is 616cm2. It is melted and recast into a cone of height 28cm. Find the diameter of the base of the cone so formed. ANS:-14cm [7] “Water is precious , Save it” Rain water is collected at top of the building it flows from a pipe of internal Diameter 25cm into a cylindrical tank which is 12m in diameter and 2.5 m deep. If water flows through the pipe at the rate of 3.6 km /h in how much time will the tank be filled? What value is depicted by owner of the building? Ans: 96 minutes SELF EVALUTION [1] A spherical copper shell, of external diameter 18cm,is melted and recast into a solid cone of base radius 14cm And height 4cm. Find the inner diameter of the shell. Ans:-16cm. [2] A bucket is in the form of a frustum of a cone with a capacity of 12308.8cm3. The radii of the top and bottom Circular ends of the bucket are 20cm and 12cm respectively. Findthe height of the bucket and also the area of metal sheet used in making it [take ? 3.14]? Ans:-?? ???? ????? ? ????????? ?. [3] The volume of a solid metallic sphere is 616cm3.its is melted and recast into a cone of height 28cm. Find the Diameterof the base of the cone so formed? QUESTION BANK CLASS X (MATHEMATICS) Ans:-21cm. SA-II Page 34 [4] From a solid cylinder whose height is 8cm and radius 6cm , a conical cavity of height 8cm and of base radius 6cm , is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total Ans:-603.19cm3, 603.19cm2 Surface area of the remaining solid [take ?=3.14] ? [5] A cylindrical vessel, with internal diameter10cm and height 10.5 cm is full of water. A solid cone of base Diameter 7cm and height 6cm is completely immersed in water. Find the volume of:(i) Water displaced out of the cylindrical vessel. Ans:- (i): 77cm3 , (ii) 748cm3. (ii) Water left in the cylindrical vessel. [6] A wooden article was made by scooping out a hemisphere from each ends of a solid cylinder. If the height of The cylinder is 20cm, and radius of the base is 3.5cm , find the total surface area of the article. Ans:-544cm2. ?? 3 m of ?? [7] A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41 If the internal diameter of the building is equal to its total height above the floor, find the height of the building? air. Ans:-4m . [8] A shuttle cock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere. The external diameters of the frustum are 5cm and 2cm, the height of the entire shuttle cock is 7cm. Ans:-74.38cm2 Find the external surface area. [9] Five containers shaped like a right circular cylinder having diameter 12cm & height 15cm are full of ice cream this ice cream is to be filled into cans of height 12cm & diameter 6cm having a hemispherical shape on the top & is to be distributed to the children in an orphanage. i) Find the number of such cans which can be filled with ice-cream. ii)What values does a person doing such an act possess? PROBABILITY KEY CONCEPTS 1. Probability:- The theoretical probability of an event E, written as P(E) is defined as. P (E)= Number of outcomes Favorable to E Number of all possible outcomes of the experiment Where we assume that the outcomes of the experiment are equally likely. 2. The probability of a sure event (or certain event) is 1. 3. The probability of an impossible event is 0. 4. The probability of an Event E is number P (E) such that 0≤P(E)≤1. 5. Elementary events:- An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1. 6. For any event E,P(E)+P(??)=1, where?? stands for not E, E and ?? are called complementary event. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 35 7. Performing experiments:a. Tossing a coin. b. Throwing a die. c. Drawing a card from deck of 52 cards. 8. Sample space:-The set of all possible outcomes in an experiment is called sample space. LEVEL-1 1. The probability of getting bad egg in a lot of 400 is 0.035.Then find the no. of bad eggs in the lot. [ans.14] 2. Write the probability of a sure event. [ans.1] 3. What is the probability of an impossible event? [ans.0] 4. When a dice is thrown, and then find the probability of getting an odd number less than 3. [ans. ] ? ? 5. A girl calculates that the probability of her winning the third prize in a lottery is 0.08.If 6000 tickets are sold, how many ticket has she brought. [Ans.480] 6. What is probability that a non-leap year selected at random will contain 53 Sundays. [Ans. ] ? ? 7. A bag contains 40 balls out of which some are red, some are blue and remaining are black. If the probability of ?? ? drawing a red ball is ?? and that of black ball is?, then what is the no. of black ball. [Ans.10] 8. Two coins are tossed simultaneously. Find the probability of getting exactly one head. [Ans. ] 9. A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting an ace. [Ans. ] 10. In a lottery, there are 10 prizes and 25 blanks. Find the probability of getting a prize. [Ans. ] ? ? ? ?? ? ? LEVEL-2 1. Find the probability that a no. selected at random from the number 3, 4, 5, 6...25 is prime. QUESTION BANK CLASS X (MATHEMATICS) SA-II ? ?? [Ans. ] Page 36 2. A bag contains 5 red,4 blue and 3 green balls. A ball is taken out of the bag at random. Find the probability that ? ? [Ans.??,?] the selected ball is (a) of red color (b) not of green color. 3. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing ? ? , ] ?? ?? (a) A face card (b) card which is neither a king nor a red card [Ans. ? ? [Ans. ] 4. A dice is thrown once. What is the probability of getting a number greater than 4? 5. Two dice are thrown at the same time. Find the probability that the sum of two numbers appearing on the top of the ? ? dice is more than 9. [Ans. ] ? 6. Two dice are thrown at the same time. Find the probability of getting different numbers on both dice. [Ans.?] ? ? [Ans. ] 7. A coin is tossed two times. Find the probability of getting almost one head. 8. Cards with numbers 2 to 101 are placed in a box. A card selected at random from the box. Find the probability that ? the card which is selected has a number which is a perfect square. [Ans.???] 9. Find the probability of getting the letter M in the word “MATHEMATICS”. [Ans. ] ? ?? LEVEL-3 1.Cards bearing numbers 3,5,…………..,35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing (a)a prime number less than 15 (b)a number divisible by 3 and 5. ? ? ?? ?? [Ans. , ] 2. Two dice are thrown at the same time. Find the probability of getting (a)same no. on the both side (b)different no. on ?? ?? both sides. [Ans. , ] 3. A child game has 8 triangles of which three are blue and rest are red and ten squares of which six are blue and rest are red. One piece is lost at random. Find the probability of that is (a) A square (b) A triangle of red colour. ? ? ? ?? [Ans. , ] 4.Two dice are thrown simultaneously. What is the probability that: (a)5 will not come up either of them? (b)5 will come up on at least one? (C)5 will come at both dice? QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 37 ?? ?? ? ?? ?? ?? Ans. , , 5. The king, queen and jack of clubs are removed from a deck of 52 playing cards and remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (a)heart ?? ? ?? Ans.??,??,?? (b)Queen(c) clubs 6. A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses Give the same result, i.e., 3 heads or three tails and looses otherwise. Calculate the probability that Hanif will ? ? Lose the game. [Ans. ] 7. Cards bearing numbers 1,3,5,…………..,37 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing ? ? ?? (a)a prime number less than Ans. (b)a number divisible by 3 and 5. [Ans. ? ? ?? 8. A dice has its six faces marked 0, 1, 1, 1, 6, 6.Two such dice are thrown together and total score is recorded. (a) How many different scores are possible? (b) What is the probability of getting a total of seven? [Ans. {a} 5 scores (0, 1, 2, 6, 7, 12) ? ? {b} ] 9) In a school, 27 children out of 45 worked for making their class room clean. What is the Probability that a child selected at random worked for the school? (i) What social values are being reflected here? Ans27/45 10) Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish . What is the probability that the fish taken Out is a male fish? (i) Gopi does not take care of feeding the fish & it dies after 6 days ? What values he has violated? Ans5/13 Self Evaluation 1. Three unbiased coins are tossed together. find the probability of getting (i) all heads (ii) two heads (iii) one heads (iv) at least two heads ? Ans. ? ? Ans. ? ? Ans. ? 2. Two dice are thrown simultaneously .Find the probability of getting ? Ans. ? ? An even number as the sum. Ans.? 3. Cards marked with the number 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability that the number on the card is: (i) An even number QUESTION BANK CLASS X (MATHEMATICS) Ans. SA-II ? ? Page 38 (ii) A number less than 14 Ans. (iii) A number is perfect square Ans. (iv) A prime number less than 20 Ans. ? ?? ? ??? ? ?? 4. Out of the families having three children, a family is chosen random. Find the probability that the family has (i) Exactly one girl Ans. (ii) At least one girl Ans. (iii) At most one girl Ans. ? ? ? ? ? ? 5. Five card the ten, jack, queen, king, and ace of diamonds are well shuffled with their face downward . One card is picked up at random What is the probability that the card is the queen? (ii) If the queen is drawn and put aside what is the probability that the second card picked up is (a) An ace (b) A queen 6. In a cooperative society, 60 people go to the same office they all use their conveyance, 10 people use Ans. ? (i) ? ? Ans. ?? ? Theirscooters, 10 go by their cars and the rest use their motorcycles. (i) What is the probability of people going by motorcycles? (ii) One day they all decided to go by cars but a car can accommodated only 5 people. What is the probability of people going by a car now? (iii) Which value is shown in (ii) part? SAMPLE PAPER-I FOR SA-II CLASS – X QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 39 MATHEMATICS Time : 3 hours 45 minutes Maximum Marks : 90 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 31 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of 1 mark each, Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 11 questions of 4 marks each. 3. Use of calculator is not permitted. 4. An additional 15 minutes time has been allotted to read this question paper only. SECTION A 1. If the quadratic equation px – 2 √5 px + 15 = 0 has two equal roots, then find the value of p. 2. In given Figure, a tower AB is 20 m high and BC, its shadow on the ground, is 20 √3 m long. Find the Sun’s altitude. 2 3. Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is 6. 4. In given Figure, PQ is a chord of a circle with centre O and PT is a tangent. If –QPT = 600, find PRQ. SECTION B QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 40 1. In given Figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If –PRQ = 1200, then prove that OR = PR + RQ. 2. In given Figure, 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 triangle ABC is 54 cm2, then find the lengths of sides AB and AC. 3. Solve the following quadratic equation for x : 4x2 + 4bx – (a2 – b2) = 0 4. In an AP, if S5 + S7 = 167 and S10 = 235, then find the AP, where Sn denotes the sum of its first n terms. 9. The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right triangle, right-angled at B. Find the value of p. 10. Find the relation between x and y if the points A(x, y), B(– 5, 7) and C(– 4, 5) are collinear. SECTION C th 11.The 14 term of an AP is twice its 8th term. If its 6th term is – 8, then find the sum of its first 20 terms.. 12. Solve for x : √3 x2 – 2 √2 x – 2√3 = 0 13. seconds, 0 the angle of elevation changes to 30 . If the airplane is flying at a constant height of 1500 √3 m, find the speed of the plane in km/hr. 14. If the coordinates of points A and B are (– 2, – 2) and (2, – 4) respectively, find the coordinates of ? P such that AP = ?AB, where P lies on the line segment AB. ? 15. The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is ? . ? The probability of selecting a blue ball at random from the same jar is . If the jar contains 10 orange balls, ? find the total number of balls in the jar. 16. Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 600. Also find the ?? area of the corresponding major segment. [Use π = ? ] 17. 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 4.2 m and height 4 m with the conical upper part of same diameter but of height 2.8 m, and the canvas to be used costs Rs.100 per sq. m, find the amount, the associations will ?? have to pay. What values are shown by these associations ? [Use π= ? ] 18. A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer. 19. A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 41 can have ? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs 5 per 100 sq. cm. [ Use π = 3.14 ] 20. 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 π= ? ] SECTION D 21. 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. 22. Find the 60th term of the AP 8, 10, 12, ..., if it has a total of 60 terms and hence find the sum of its last 10 terms. 23. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed ? 24. Prove that the lengths of the tangents drawn from an external point to a circle are equal. 25. 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. 26. Construct a triangle PQR with PQ= 6cm, QR=7cm and –Q=600 and then another triangle similar to it whose sides are 5/4 of the corresponding sides of triangle PQR. 27. At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 300. The angle of depression of the reflection of the cloud in the lake, at A is 600. Find the distance of the cloud from A. 28. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is (i) a card of spade or an ace. (ii) a black king. (iii) neither a jack nor a king. (iv) either a king or a queen. 29. Find the values of k so that the area of the triangle with vertices (1, – 1), (– 4, 2k) and (– k, – 5) is 24 sq. units. 30. In given Figure, PQRS is a square lawn with side PQ = 42 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). 31. 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 4.2 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 π= ? ] EXPECTED ANSWERS/VALUE POINTS SECTION - A QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 42 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 43 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 44 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 45 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 46 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 47 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 48 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 49 QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 50 Time : 3 hours 45 minutes SAMPLE PAPER-II FOR SA-II CLASS – X MATHEMATICS Maximum Marks : 90 General Instructions: 5. All questions are compulsory. 6. The question paper consists of 31 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of 1 mark each, Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 11 questions of 4 marks each. 7. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks each and 3 questions of four marks each. You have to attempt only one of the alternatives in all such questions. 8. Use of calculator is not permitted. 9. An additional 15 minutes time has been allotted to read this question paper only. SECTION A 1. In the Arithmetic progression :6,9,12,15,……… the common difference is 2. If a sphere and a cone of height ‘h’ have same radius and same volume then the ratio r:h is 3. The sum of the roots of y2+6y+5=0 is 4. If the radius of sphere is 8 cm , then its surface area ( in cm2) is SECTION B 5. First term of an A.P is -5 and its last term is 45. If the sum of these terms is 120, find the number of terms. 6. Find the value of p for which the points (-1,-1),(2,p) and (8,11) are collinear . OR The co-ordinates of the mid point of the line segment joining (3p,4) and (-2,2q) are (5,3).Find the value p and q. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 51 7. In the figure ,AR=4cm,BR=3cm andAC=11cm.Find the length of BC . 8. Find the volume of a frustum of a cone whose height is 4cm and radii of the ends are 7m and 4m. 9. Determine the value of p for which 9y2-24y+p=0 has equal roots. 10. Two cubes each of side 10 cm are joined end to end.Find the surface area of the resulting cuboid. SECTION C 11. Volume of two hemispheres are in the ratio 64:27 . Find the ratio of their curved surface areas. 12. A card is drawn at random from awell shuffied deck of 52 playing cards. Find the probability that the card drawn is (a) ablack king (b)ace. 13. A tent is cylindrical upto a height of 3m and conical above it .If the diameter of the base is 105 m and slant height of the conical part is 53 cm, find the cost of canvas used to make the tent at Rs 10 per square metre. 14. Acircle is inscribed in a quadrilateral ABCDwhere –B=90 0.If AD = 24 cm ,AB=30 cm and DS =8cm . Find the radius r of the circle . 15. The sum of 4th and 8th term of an A.P.is 37 and the sum of 6th and 12th terms is 46. Find the firstterm of the A.P. 16. Construct a pair of tangents to a circle of radius 3 cm from a point P at a distance of 5 cm from the centre. OR Construct a triangle PQR with PQ= 6cm, QR=7cm and –Q=600 and then another triangle similar to it whose sides are 5/4 of the corresponding sides of triangle PQR. 17. An observer 1.7m tall is 30.3m away from the foot of a tower. The angle of elevation of top of the tower from her eyes is 450. Find the height of the tower. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 52 3a2x2+8abx+4b2=0 OR 1 2 6 + = (x π 0,1,2) x - 2 x -1 x 18. Solve for x: 19. Students of Class X packed 500 packets of biscuits each of dimension 6cmx3cmx2cm in boxes each of volume 18000cm3 to be distributed to the children of flood victims. i) Find the number of boxes required. ii) Which mathematical concept is used here ? iii) What moral values are represented by the class X students? 20. Determine the sum of all multiples of 9 lying between 100 aand 200. OR th Find the sum of first 20 terms of an A.P whose n term is 4n-1. SECTION D 21. Prove that the lengths of the tangents drawn from an external point are equal. 22. A plane covers a distance of 1200 km at an uniform speed. Had the speed been 100 km/ hr more, it takes 1 hour less for the journey. Find its original speed. OR A motor boat whose speed is 18 km/hr in still water, takes 1hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. 23. If P (2,-1), Q (3,4), R (-2,3)and S (-3,-2) are four points in a plane, show that PQRS is a rhombus but not square. 24. Cards marked with numbers 1,2,3,……..,40 are placed in a box and mixed thoroughly. One card is Drawn at random from the box. Find the probability that the number on the drawn card is a) Divisible by 3 and 5 b) a prime number c) a perfect square 25. The curved surface area of a 16 m deep cylinder is plastered with concrete at the rate of Rs 15 per m2 If the total cost of plastering the curved surface is Rs 5280, find the capacity of the cylinder. 26. Find the shaded area if ABCD is a square, region I is a semicircle with a diameter 14 cm, region II and III are quadrants with centres at A and B respectively. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 53 OR In D ABC, –A=900, AB=6 cm, BC=10 cm and AC=8 cm. Find the radius of the circle and the shaded area if O is the centre of the incircle of D ABC. 27. A man saves Rs 320 during the first month, Rs 360 in the second month,Rs 400 in the third month And so on. If he continues his savings in this squence, in how many months will he save a total of Rs 20000? 28. The sum of the radius of the base and the height of a solid cylinder is 37 cm. If total surface area of the cylinder is 1628 cm2. Find the radius and volume of the cylinder.(use p=22/7) OR The height of a cone is 32 cm .A small cone is cut off at the top by a plane parallel to its base. If its volume is 1/64 of the volume of the given cone, at what height about the base is the cone cut? (use p=22/7) 29. Find the ratio in which Y- axis divides the line segments joining the points (5,-6) and (-1,-4) also find the Coordinates of the point of division. 30. Two poles are erected on either bank of a river just opposite to each other. One pole is 40 m high freom the top and foot of this pole. The angles of elevation of the top of the other pole are 300 and 600 respectively. Find the height of the other pole and width of the river. 31. The sum of first 15 terms of an A.P is 105 and the sum of the next 15 terms is 780. Find the first three terms of the arithmetic progression. QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 54 ANSWER KEY: (Marking scheme) Section A 1. 3 2. 1:4 Section B 5. a=-5, an=45, Sn=120 (n/2) (a+ an) = 120 (n/2) (-5+45) = 120 n/2 =3 , n = 6 6. ½[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]=area ½[-1(p-11)+2(11+1)+8(-1-p)]=0 -p+11+24-8-8p=0 -9p=-27 p=3 Or (3p-2)/2=5, p=4 (4+2q)/2=3, q-=1 7. BR=BP=3 cm, AR= AQ=4 cm CQ=11-4=7 CP=7 cm BC=7+3 =10 1 8. Volume= ph[r2+R2+rR] 3 1 22 = x x4 x[42+72+4x7] 3 7 1 22 = x x4 x[16+49+28] 3 7 =389.7 cubic cm 9. b2-4ac=0 (-242-4x9p)=0 -36p=-576 p=16 10. l=20 cm, b=10 cm, h=10 cm Surface area=2x(lb+bh+lh) sq. unit =2x(200+100+200) =1000 cm2 11. 2/3pR32∏/3pr3=64/27 R3∏r3=64/27 R ∏r =4/3 2pR2∏2pr2=(4/3)2 = 16/9 3. -3 4. 256p ½ ½ ½ ½ ½ ½ ½ ½ 1 1 1 ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ SECTION B 1 ½ ½ 1 12. Total Cards=52 1 a) P(black king)=2/52 or 1/26 1 b) P(an ace)=4/52 or 1/3 QUESTION BANK CLASS X (MATHEMATICS) 1 SA-II Page 55 13. Canvas area=2pr h+pr l s q. unit =pr(2h+l) =(22/7)x(105/2)x (6+53) = 9735 sq. m cost= 9735x10=Rs 97350 1 ½ ½ 1 14. DR=DS, AR=AQ, BQ=BP (tangents from an external point) DS=8 cm \ DR=8 cm AR=24-16 = 16cm, AQ=16 cm BQ=14cm , BP=14 cm –B=900–OQB=–OPB=900 –POQ=900 Also BQ=BP \ OPBQ is square So, r=BP=14 cm ½ ½ ½ ½ ½ ½ 15. a4+a6=37 a+3d+a+7d=37, 2a+10d=37 a6+a12=46 a+5d+a+11d=46, 2a+16d=46 ½ ½ ½ ½ 16. Construct a circle Construction of tangents Or To draw a Triangle PQR Construction of tangents 1 2 1 2 17. x . 30.3 x 1= x=30.3 ., 30.3 Height of the tower=30.3+1.7=32 Tan450= ½ ½ 1 18. D=b2-4ac = (8ab)2-4(3a2)(4b2) QUESTION BANK CLASS X (MATHEMATICS) SA-II Page 56 =64a2b2-48a2b2 =16a2b2 X=(-b+√D) ∕2a =(-8ab+√16a2b2) ∕2(3a2) = (-8ab+4ab) ∕ 6a2 = -2b/a and -2b/3a 1 1 1 OR ???????? ? = 1 ??????? ? ???? = ? ½ ??????? ? 3x2-13x+12=0 X=3 , 4/3 ½ 1 19. i) volume of 1packet=l x b x h cubic unit =6x3x2 = 36 cm2 No. of packets in 1 box=1800/36=50 So, no of boxes needed =500/50=10 ii) Volume of a cuboid ½ iii)sharing and caring, empathy, teamwork etc. ½ ½ ½ 1 20. 108, 117 ,126,…………., 198 form an A.P a=108 , d= 9, an=198 1 108+ (n-1)9=198 (n-1 )9=90 n-1 =10 n = 11 1 ? Sn= ?=(a+an) 1 ?? S11= ? =(108+198)=1683 a1=4(1)-1=3, 1 Or a2=4(20-1)=7 1 d= 4 n=20 Sn=n/2[2a+(n-1)d] QUESTION BANK CLASS X (MATHEMATICS) ½ SA-II Page 57 = 10x (6+19x4) = 820 1 21. Figure, given, to prove ,construction 2 Correct proof 2 22. Let the original speed be x km/hr. Given, ½ ???? ???? ? -?????=1 1 1200 x + 120000 - 1200 x =1 x( x + 100 ) ½ X2+100x-120000=0 ½ D= b2-4ac = (100)2 – 4(1)(-120000) = 490000 X= ???√?????? ?? = - 400 , 300 1 Speed cannot be negative. So original speed = 300km/hr ½ Or Let the original speed be x km/hr. Given, ?? ½ ?? =1 ???? ???? 1 Solving x2+48x -324 = 0 ½ D= b2-4ac = (48)2 – 4(1)(-324) = 3600 X= ½ ???√?????? ?? = - 54, 6 QUESTION BANK CLASS X (MATHEMATICS) 1 SA-II Page 58 Speed cannot be negative. So original speed = 6 km/hr ½ 23. P (2,-1), Q (3,4), R (-2,3) and S (-3,-2) PQ2= (x2-x1)2+(y2-y1)2 PQ2 =(3-2)2+(4+1)2 = 26 1 QR2 = (3+2)2+(4-3)2 = 26 ½ RS2 = (-2+3)2+(3+2)2 = 26 ½ SP2 =(2+3)2+(-1+2)2 = 26 ½ PR2= 32 , QS2= 72 1 Four equal sides, but diagonals are not equal. So, PQRS is Rhombus ½ 24. Total number of card =40 1 (a) Numbers divisible by 3 and 5 =15,30 P(a no. divisible by 3and 5)=2/40 or1/20 1 (b) Prime numbers = 2,3,5,7,11,13,17,19,23,29,31 and 37 . P{a prime no.)= 12/40 or 3/10 1 (c) Perfect squares 1,4,9,16,25,36 P(a perfect square) =6/40 0r 3/20 1 25. Curved surface area of cylinder =5280/15 = 352 m2 1 2pr h=352 ½ 2 x (22/7) x r x 16 = 352 ½ r=7/2 m ½ capacity of cylinder = pr2h = 22 x 3.5x3.5x16 7 = 616 m3 QUESTION BANK CLASS X (MATHEMATICS) 1½ SA-II Page 59 26. Area of square=14 x14 =196 cm2 1 Area of semicircle= 1/2pr2 sq. unit ? ?? = ? ?? ?x7x7 = 77 cm2 1 Area of quadrant II =1/4 2pr2 sq. unit ? ?? = ? ?? ?x7x7 = 77/2 cm2 1 Area of quadrant II =1/4 2pr2 sq. unit ? ?? = ? ?? ?x7x7 = 77/2 cm2 Area of shaded region=196-(77+77) cm2 = 42 cm2 1 Or Area of D ABC= ½ x basex height sq. unit =1/2 x 6 x8 cm2=24 cm2 1 Join O to A, B and C (1/2x10xr)+(1/2x6xr)+(1/2x8xr)=24 1 1/2xrx(10+6+8)=24 Radius= 2 cm 1 Area of circle= pr2 sq. unit = 22/7x2 x2=88/7 cm2 1 27. 320, 360, 400,………. Form an A.P with a= 320, d= 40 1 ? 20000=?[2a+(n-1)d] ½ n(600+40n)=40000 QUESTION BANK CLASS X (MATHEMATICS) ½ SA-II Page 60 40n2+600n-40000=0 n2+15n-1000=0 1 n=25, -40 (rejected) So, his saving will be Rs20000 in 25 months ½ 28. r + h = 37 cm ½ Total surface area= 2pr(r + h)=1628 cm2 2x22/7 x r x 37 =1628 1 r=7cm 1 h=37-7=30cm ½ Volume =pr2h cubic unit = (22/7) x7 x 7 x 30 cm3 = 4620 cm3 1 Or Let radii of the cones be r and R ,their heights be h and H 1 2 1 2 pr h+ pR H=1/64 3 3 1 R2H+r2h=1/64 but r/R=h/H 1 So, h3+H3=1/64 , h3+(32)3=1/64 1 Solving, h= 8 cm ½ The cut is made 32-8=24cm above the base. ½ 29. Let the ratio be k:1, Point be (0,y) ½ X= (m1x2+m2x1)/m1+m2 0=k(-1)+1(5)/ k+1, k=5 1½ Ratio = 5:1 1/2 y= (m1y2+m2y1)/m1+m2 y=5(-4)+1(-6)/ 5+1, y=-13/2 QUESTION BANK CLASS X (MATHEMATICS) 1 SA-II Page 61 Point of division is (0, -13/2) ½ 30. Correct figure ½ Let the first pole be AB =40 m and second pole be CD. Let CP= X m PD=AB=40 m ?? ? ? In rt. D ACP, tan300 = ?? √?=?? AP = √3x………….(equation 1) ?? √? ???? In rt. D BCD, tan600 = ?? AP = √3x = 20√3 m = 1 ?? 1 3 x = x+4 x=20 1 Height of the other pole = 20+40 = 60 m ½ Width of the river = 20√3 m. ½ 31. Sum of first 15 terms = 105 Sum of next 15 terms = 780 \ sum of first 30 terms = 780+105 = 885 15 [2a+(15-1)d] =105 2 30 [2a+(30-1)d] =885 2 1 1 2a+ 14d = 14 and 2a + 29d =59 \ a = -14 and =3 The first three terms area, a+d, a+2d…….i.e -14,-11,-8 QUESTION BANK CLASS X (MATHEMATICS) 1 SA-II Page 62