Class X Mathematics Sample Question Paper 08-9 Time allowed: Hours Max. Marks: 80 General Instructions:. All the questions are compulsory.. The questions paper consists of 0 questions divided into sections A, B, C and D.. Section A comprises of 6 questions of mark each. Section B comprises of 6 questions of marks each. Section C comprises of 0 questions of marks each. Section D comprises of 8 questions of marks each.. There is no overall choice. However, an internal choice has been provided in two questions of mark each, two questions of marks each, four questions of marks each and three questions of marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. Section-A. Find the value of a, for which point P (, ) is the mid-point of the line segment joining the points Q(-5,) and R(-,0).. Find the value of k, for which one root of the quadratic equation kx -x+8 = 0 is. Find the value(s) of k for which the equation x + 5kx + 6 = 0 has real and equal roots.. Write the value of cot θ If sinθ = cosθ, then find the value of tanθ + cos θ. If nth term of an A.P. is (n+), what is the sum of its first three terms? 5. In figure if AD= 6cm, DB=9cm, AE = 8cm and EC = cm and ADE = 8 0. Find ABC 6. After how many decimal places will the decimal expansion of terminate?
Section-B 7. The HCF and LCM of two numbers are 9 and 60 respectively. If one number is 5, find the other number. Show that 7 5 is irrational, give that 5 is irrational. 8. Find the 0 th term from the last term of the AP,8,,.,5 If 7 times the 7 th term of an A.P is equal to times its th term, then find its 8 th term. 9. Find the coordinates of the point P which divides the join of A(-,5) and B(,-5) in the ratio : 0. A card is drawn at random from a well shuffled deck of 5 cards. Find the probability of getting neither a red card nor a queen.. Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is a prime number. For what value of p will the following pair of linear equations have infinitely many solutions (p-)x+y = p px+py = Section-C. Use Euclid s Division Algorithm to find the HCF of 76 and 75.. Find the zeroes of the following polynomial: 5 5x +0x+8 5 5. Places A and B are 80 km apart from each other on a highway. A car starts from A and another from B at the same time. If they move in same direction they meet in 8 hours and if they move towards each other they meet in hour 0 minutes. Find the speed of cars. 6. The points A(,-), B(,), C (k,) and D(-,-) are the vertices of a parallelogram. Find the value of k. Find the value of k for which the points (k-,k-), (k,k-7) and (k-,-k-) are collinear. 7. Prove that cotθ tanθ = cos θ sinθcosθ Prove that sinθ( + tanθ) + cosθ( + cotθ) = secθ + cosecθ 8. The radii of two concentric circles are cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and intersecting the larger circle at P on producing. Find the length of AP.
9. In figure = and NSQ MTR, then prove that PTS~ PRQ. In ABC, if AD is the median, then show that AB +AC = (AD +BD ) 0. Find the area of the minor segment of a circle of radius cm, if length of the corresponding arc is cm.. Water is flowing at the rate of 5 km per hour through a pipe of diameter cm into a rectangular tank which is 50 m long and m wide. Find the time in which the level of water in the tank will rise by cm. A solid sphere of radius cm is melted and then recast into small spherical balls each of diameter 0.6cm. Find the number of balls.. The table shows the daily expenditure on grocery of 5 households in a locality. Find the modal daily expenditure on grocery by a suitable method. Daily Expenditure (in Rs.) No of households 00-50 50-00 00-50 50-00 00-50 5
Section-D. A train takes hours less for a journey of 00km if its speed is increased by 5 km/h from its usual speed. Find the usual speed of the train. Solve for x: = + + (abx) a b x, [ a 0, b 0, x 0, x (a + b)]. An AP consists of 50 terms of which rd term is and the last term is 06. Find the 9 th term. 5. Prove that in a right angled triangle square of the hypotenuse is equal to sum of the squares of other two sides. 6. Draw a ABC with sides 6cm, 8cm and 9 cm and then construct a triangle similar to ABC whose sides are of the corresponding sides of ABC. 7. A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes minutes for the angle of depression to change from 0 0 to 5 0, how long will the car take to reach the observation tower from this point? The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 0 0 and the angle of depression of its shadow from the same point in water of lake is 60 0. Find the height of the cloud from the surface of water. 8. The median of the following data is 55. Find the values of x and y if the total frequency is 00. Class Interval Frequency 0-00 00-00 5 00-00 x 00-00 00-500 7 500-600 0 600-700 Y 700-800 9 800-900 7 900-000
The following data indicates the marks of 5 students in Mathematics. Marks Number of students 0-0 5 0-0 0-0 0-0 0-50 50-60 60-70 7 70-80 9 80-90 7 90-00 8 Draw less than type ogive for the data above and hence find the median. 9. The radii of circular ends of a bucket of height cm are 5 cm and 5 cm. Find the area of its curved surface. 0. If secθ + tanθ = p, then find the value of cosecθ. 5
Class: X Mathematics Marking Scheme 08-9 Time allowed: hrs Maximum Marks: 80 Q No ( (), = ) = (, ) a = -9 SECTION A Marks K -8+8 = 0 K= 5 For roots to be real and equal,b ac = 0 (5k) 6 = 0 k = ± 8 5 cot θ. = cot θ cosec θ = - sinθ = cosθ θ = 5 tanθ + cos θ = + = a =, a = 7 s = ( + 7) = 5 5 AD = AE DB EC DE BC ADE = ABC = 8 0 6 places SECTION B 7 HCF LCM = Product of two numbers 9 60 = 5 nd number nd number = 7
Let us assume, to the contrary that 7 5 is irrational 7 5 =, Where p & q are co-prime and q 0 = 5 = is rational = 5 is rational which is a contradiction Hence 7 5 is irrational 8 0 th term from the end = l (n )d = 5-9 5 = 58 7a = a 9 X = = 0 Y = = a + 7d = 0 a = 0 7(a + 6d) = (a + 0d) 0 Probability of either a red card or a queen = 6 5 = 8 5 P(neither red car nor a queen) = = or Total number of outcomes = 6 Favourable outcomes are (,), (,), (,), (,), (,5), (5,) i.e. 6 Required probability = or For infinitely many solutions p -p = p or = p p -6p = 0 or p = ± 6 p = 0,6 p = 6 SECTION: C By Euclid s Division lemma 76 = 75 +76 75 = 76 + 99 76 = 99 +77 99 = 77 + 77= + = +0 HCF = 6 =
5 5x +0x+8 5 = 5 5x +0x+0x+8 5 = 5x( 5x + )+ 5( 5x + ) = ( 5x + ) (5x+ 5) Zeroes are = and 5 Let the speed of car at A be x km/h And the speed of car at B be y km/h Case 8x-8y = 80 x-y = 0 6 Case x + y = 80 x+y = 60 on solving x= 5 and y = 5 Hence, speed of cars at A and B are 5 km/h and 5 km/h respectively. Diagonals of parallelogram bisect each other midpoint of AC = midpoint of BD (, = k = - ) = (, ) For collinearity of the points, area of the triangle formed by given Points is zero. {(k )(k 7 + k + ) + k( k k + ) + (k )(k k + 7)} = 0 7 LHS = cotθ tanθ {(k )(k 5) k + 5k 5} = 0 k -k = 0 k = 0, = - = = = = RHS
LHS = sinθ( + tanθ) + cosθ( + cotθ) = sinθ + + cosθ + = sinθ + cosθ = (cosθ + sinθ) ( = ) = cosecθ + secθ = RHS SECTION: E 8 APB = 90 0 (angle in semi-circle) ODB = 90 0 (radius is perpendicular to tangent) ABP~ OBD = = AP = 6cm 9 = PT=PS....(i) From (i) and (ii) Also, NSQ MTR NQS=MRT PQR=PRQ PR=PQ...(ii) = TPS = RPQ (common) PTS~ PRQ
AD is median, So BD=DC. AB = AE +BE AC = AE +EC Adding both, AB +AC = AE +BE +CE = (AD -ED )+(BD-ED) +(DC+ED) = AD -ED +BD +ED -BD.ED+DC +ED +CD.ED = AD +BD +CD = (AD + BD ) 0 r = cm = θ = = 600 Area of minor segment = area of sector area of corresponding triangle = r = r [ ] = [ ] = [ ] = ( ) cm Volume of water flowing through pipe in hour = 5 000 = m Volume of rectangular tank = 50 = m Time taken to flow m of water = hours Time taken to flow m of water = Number of balls = =... = 000 = hours 5
00-50 is the modal class Mode = l + h = 00 + 50 = 00+0.59 = Rs. 0.59 Section D Let the usual speed of the train be x km/h = x +5x-750 = 0 (x+0)(x-5) = 0 x = -0,5 Usual Speed of the train = 5 km/h = + () = () -ab = x +(a+b)x x +ax+bx+ab = 0 (x+a)(x+b) = 0 x= -a,-b n=50, a = and a50 = 06 a+d = a+9d = 06 on solving, d= and a= 8 a9= a+8d = 8+8 = 6 5 Correct given, To prove, figure and construction Correct proof 6 Correct construction of ABC Correct construction of similar triangle = 6
7 Correct figure Let the speed of car be x m/ minutes In ABC, = tan 50 h = y In ABD, = tan 00 h = y+x y y = x y = = ( ) y = 6x( + ) Time taken from C to B = 6( + ) minutes Correct figure In ABE, = tan 00 x=h 7
In BDE, = tan 60 0 h+0 = x h+0 = h h = 0 h = 60 height of cloud from surface of water = (60 + 60)m = 0m 8 Class Interval Frequency cf 0-00 00-00 5 7 00-00 x 7+x 00-00 9+x 00-500 7 6+x 500-600 0 56+x 600-700 y 56+x+y 700-800 9 65+x+y 800-900 7 7+x+y 900-000 76+x+y N=00 76+x+y =00 x+y = (i) Median = 55 500 600 is median class 60-80 is the median class Median = l + h 500 + ) 00 = 55 ( x) 5 = 5 x = 9 from (), y = 5.96 8
Correct table Drawing correct Ogive Median=6 Marks Number of students cf 0-0 5 5 0-0 8 0-0 0-0 5 0-50 8 50-60 60-70 7 9 70-80 9 8 80-90 7 5 90-00 8 5 9 r = 5cm, r = 5cm h = cm l = h + (r r ) = + 0 = 6cm Curved surface area of bucket = π(r + r ) l = = = (5 + 5) 6 cm or 6.cm 0. Secθ + tanθ = p + = p +sinθ = pcosθ = p sin θ (+sinθ) = p ( sin θ) + sin θ + sinθ = p p sin θ ( + p ) sin θ + sinθ + ( p ) = 0 D = - (+p )(-p ) = - (-p ) =p Sinθ = ± ( ) Cosec θ =, ± = ( ) =, - 9