SAMPLE QUESTION PAPER MATHEMATICS


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1 SAMPLE QUESTION PAPER 078 MATHEMATICS Time allowed : 3 hrs Maximum marks : 80 General Instructions : All questions are compulsory. The question paper consists of 30 questions divided into four sections A,B,C,D. Section A contains 6 questions of marks each 3. Section B contains 6 questions of marks each 4. Section C contains 0 questions of 3 marks each 5. Section D contains 8 questions of 4 marks each 6. There is no overall choice. However an internal choice has been provided in four questions of 3 marks each. You have to attempt only one of the alternatives in all such questions. 7. Use of calculator is not permitted SECTION A. Without performing actual division, state whether the rational number 3 expansion or a Nonterminating repeating decimal expansion.. Find the 0 th term from the last term of the A.P : 3,8,3,.,53 5 will have a terminating decimal 3. Find the value(s) of k, for which the quadratic equation kx(x ) + 6 = 0 has two equal roots. 4. Find the coordinates of the point on yaxis which is nearest to the point (5,). 5. In this given figure LM CB and LN CD prove that AM = AN AB AD 6. Express Sin 67 +Cos 75 in terms of trigonometric ratios of angles between 0 and If two positive integers p and q are written as p=a²b³ and y=a³b ; a,b are prime numbers, then verify LCM(p, q) X HCF(p, q) = pq 8. For which value of k will the following pair of linear equations have no solution? 3x + y = (k )x + (k )y = k + 9. Find the sum of first 4 terms of the list of numbers whose nth term is given by : an=3+n 0. If (, p ) is the midpoint of the line segment joining the prints (,0) and 3 (0, ), then show that the line 5x + 3y + 9 = 0 passes through the point (,3p). An Integer is chosen between 0 and 00. What is the probability that it s not divisible by 7.
2 . A bag contains 5 red balls and some blue balls. The probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag. SECTION C 3. Find all the zeroes of x 43x 33x²+6x, if you know that two of its zeroes are and  4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be row less. If 3 students are less in a row, there would be rays more. Find the number of students in the class. 5. In what ratio does the xaxis divides the line segment joining the points (4,6) and (,7)? Find the coordinates of the point of division? OR, The points A(4,), B(7,), C(0,9) and D(3,5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB. 6. Show that exactly one of the numbers n, n+, n+4 is divisible by ABC is right angled at C. If BC = a, CA = b and AB = c and P is the length of perpendicular drawn from C AB, then prove that: (I) PC = ab (II) p = a + b OR, In figure, the line segment XY is parallel to side AC of areas. Find the ratio AX AB ABC and it divides the triangle into two parts of equal 8. In the given figure, XY and X Y are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X Y at B. Prove that <AOB = Evaluate: cos 5 + cos 55 cosec 5 tan (tan3 tan3 tan30 tan67 tan77 ) 0. In the given figure, ABPC is a quadrant of a circle of radius 4cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.. In a rain water harvesting system, the rainwater from a roof of m x 0m drains into a cylindrical tank having diameter of base m and height 3.5m. If the tank is full, find the rainfall in cm. Write your views on the water conservation. OR, A cubical block of side 0cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs 5 per 00 sq.cm
3 . Find the mode of the following distribution of marks obtained by the students in an examination: Marks obtained Number of students Given the mean of the distribution is 53, using empirical relationship estimate the value of its median. SECTION D 3. A plane left 40 minutes late due to bad weather and in order to reach its destination 600km away in time, it had to increase its speed by 400km/h from its usual speed. Find the usual speed of the plane. OR, Check whether the equation 5x 6x = 0 has real roots and if it as, find them by the method of completing the square. Also, verify that roots obtained satisfy the given equation. 4. If the ratio of the sum of the first nterms of two A.P. is 7n+ : 4n+7, the find the ratio of their 9 th terms. 5. State and prove Pythagoras theorem. OR, State and prove the Basic Proportionality Theorem. 6. Draw a ABC with side BC = 7cm, <B = 75, < A = 05. Then construct a triangle whose sides are 4 times the 3 corresponding sides of ABC. Write the steps of construction and justify your answer. 7. Prove that Cos A Sin A+ Cos A+Sin A = Cosec A + Cot A using the identity Cosec A = + Cot A. 8. As observed from the top of a 75 m high light house from the sea level, the angles of depression of two ships are 30 ⁰ and 45⁰. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. 9. Two dairy owners A and B sell flavored milk filled to capacity in mugs of negligible thickness, which are cylindrical in shape with a raised hemispherical bottom. Or, The mugs are 4 cm high and have a diameter of 7 cm as shown in the given figure. Both A and B sell flavored milk at the rate of Rs 80 per litre. The dairy owner A uses the formula πr h to find the volume of milk in the mug and charges Rs 43. for it. The dairy owner B is of the view that the price of the actual quantity of milk should be charged. What according to him should be the price one mug of milk? Which value is exhibited by the dairy owner B? 30. In the retail market fruit vendors were selling apples kept in packing boxes. These boxes contained varying number of apples. If the median number of apples in a box was 4 and the total number of boxes was 550, then find the missing frequencies f and f in the following distribution table. Number of apples in a box Number of boxes 0 67 f f OR, Draw the move than type,give for the following data. Also, find the median from the graph. Class Interval Frequency
4 Marking Scheme ( class X) Sub : Mathematics Session : 078 SECTION A The number given = 3. 5 LCM of 5 = It is terminating. ( Any denominator no. that can be expressed in the form of m x 5 n is terminating) AP: 3, 8, 3,, 53 a = 3, d = 5, l = 53 From the last term: a = 53, d = 5, l = 53 a 0 (from last) = a + (n)d = (5) =5345 = 08 f(x) = kx(x) + 6 = 0 = kx x +6 = 0 f(x) have equal and real roots D = 0 b 4ac = 0 (k) 4.k.6 = 0 4k = 4k k = 6 The point is (5, ) The point on yaxis : ( 0, y) The point (0,) is nearest to te point (5, ) on yaxis. Given, LM CB, LN CD, ABCD is a quadrilateral. To prove: AM = AN AB AD Proof: In ABC, LM BC In AL LC = AM MB ADC, AL = AN MD ND ( BPT).(ii) Sin cos 75 o Sin (90 3) o + cos (90 5k) o Cos 3 o + sin 5 o ( BPT).(i) p = a b 3, q=a 3 b L.C.M => a 3 b 3 HCF => a b LCM X HCF = pq a 3 b 3 x a b = a b 3 x a 3 b a 3 b = a 3 b = (proved) When no solution: a = b c a b c 3x+y=0 (k)x+(k)y(k+)=0 a =3 ; b = ; c = a =(k) ; b =(k) ; c =(k+) 3 k = k / k+ 4
5 Case 3k3 = k k= 9) a n =3+n ; a =3+4 =>7 a = 3+ a =5 a=5 ; d= a =3 S n = 4 (0 + 3.) = (0+46) = 67 0) = 0+ (,0 ) p 3 = 9 X ; p 3 = 9 +0 (, p 3 (0, 9 ) P = 3 0(,3p) => (,) so, x =  ; y= 5x+3y+= = = 0 (Hence Proved) ) Numbers between 0 to 00 which are divisible by 7 is 4. So, numbers which are not divisible by 7 are 004=86 So, P = = ) A bag Contains 5 red balls and some blue balls. Let the number of blue balls be x So, Total no of balls = x x A T Q : 3( 5 5+x ) = x x+5 = x x+5 So, x =5 So, number of blue balls are a=bq+r, a r < b b=3 => r=0,, r=0, => n=3q, 3q+, 3q+4 r=, => n=3q+, 3q+3, 3q+5 r=, => n=3q+, 3q+4, 3q+6 Exactly one in the above 3 sets is divisible by The roots given and ...( x ) and ( x + )...( x ) ( x+ ) = (x) ( ) = x... The other zeros are x ) x 43x 33x + 6x ( x 3x + x 44x x 3 + x + 6x 3x 3 + 6x +  x x 5
6 x... x 3x+... The total zeros are = x x x + = x( x) (x) = (x ) (x) 5. Let the total rows be x Let the total columns be y. ATQ... xy= (x+3) (y) xy = xy  x + 3y  3 3= x + 3y.. (I)... xy =( x 3) (y +) xy =xy + x 3y = x 3y. (II) x + 3y = 3 x  3y = 6 x= y = 3 3y = y=4,,,  The total rows is 9 The total columns is (h,b) (x,0) (,7) 0 = 7k 6 6 FTk k= 6 (II) 7... Ratio is 6:7... x = x = 34 3 = Area ofabc= ½ cp =/ab Or, pc = ab Or, /p = c/ab Or, / p =c /a b = a + b / a b = / a + / b A 8.Given : XY ǁ X Y AB are tangent. To prove : ADB = 90⁰ Proof : In POA and OAC OPA = OCA = 90⁰ OR C B 6
7 8. PAC + QBA = 80 [Cointerior angles] POA COA [By ] 9.XY AC BXY = A& BYX = C ABC ~ XBY [By AA] AR. ABC AR. XBY = ( AB XB )² =  XB AB = AR. ABC AR. XBY = AB² XB² AR. ABC =. AR. XBY AB XB = XB AB = AB XB = AB AX AB = AX AB = OR 9. cos 35 + cos 55 cosec 5 tan (tan3 tan3 tan30 tan67 tan77 ) = cos 35 + sin 35 sec 75 tan (tan3 tan77 tan3 tan67 tan30 ) = + 3(cot77 tan77 cot67 tan67 tan30 ) = + 3 ( tan77 tan 77 tan67 tan30 ) = = + = 0. BC = AC + AB = BC = 3 BC = 4 r = 7 Given, ABPC is a quadrant, AC = AB= 4 cm = r θ = 90 Area of shaded region = AR. ABC + AR of semicircle AR. quadrant ABPC = AB AC + πr θ 360 πr = =554 = 98 cm. Given, =
8 length of the roof(l) = m breadth of the roof(b) = 0m height of the roof (h ) =? base diameter of tank = m base radius of tank (r) = m h = 3.5m Therefore, Vol. of tank = vol. of roof Or, πr h = l. b. h Or, 3.5 = 0 h 7 Or, h =.cm Views: a. Water is very necessary to carry out daily activities. b. We should not waste water as our lives depend on it. OR. TSA of the solid = CSA of the hemisphere + TSA of Cube Area of the Top = πr + 6a a = πr + 5a = x 3.4 x 5 x x 0 x 0 = 657cm Therefore, Rs5/00cm = = Rs Here modal class: l = 60, f = 9, f = 7, f 0 =, h = 0 f f 0 Mode = l + ( )n f f 0 f = 60 + ( 9 )n 58 7 = =68 Now, 3Median = Mode + Mean = 68 + (53) = =74 Median = 74 = 58 3 Median=58 3. For real roots, D 0 D = (6)  4x5x() = 76 0 So, the Eqn. has real roots. OR 3. Let, The usual speed be x km/hr. 600 x 600 x+400 = 3 Or, 800 ( x+400 x x(x+400) ) = 3 Or, x= 400± ( ) = 400± (00) = 800 km/hr. 5x 6x = 0 8
9 x 6 5 x 5 = 0 (x 6 0 ) ( 6 0 ) 5 = 0 x= 3± Given ratio of the n th term of A.P.s is 7n+:4n+7 So, n (a + (n )d ) n = 7n+ (a + (n )d ) 4n+7 Now, ratio of 9 th term = a + (9 )d a +(9 )d = a +8d a +8d Comparing both equation, So, Ratio of 9 th term = 7(7)+ = 0 = 4 4(7) Correct Statement Figure Given To prove Construction Proof Or Same. 6. Correct ABC Correct a A B C cos A+cosec A 7. L.H.S = 8. x 75 cot A cosec A+ cos A+cosec A (cosec A+cot A)(cosec A cot A) = cot A cosec A+ = cosec A + cot A = cot 45 x 6 0 = ± 9 5 (a + ( n )d) ((a + ( n ) d) = a + ( n ) d (a + ( n ) d ) = n = 8 n = 6 ; n=7 7n + 4n + 7 7n + 4n + 7 (dividing the numerator & denominator by sin A) Or x= 75 x + y 75 = cot30 Or, x+y = 75 3 Or, y=75( 3 ) 9. Height of the mug = 4cm =.4dm Diameter = 7cm = 0.7dm Radius = 3.5cm = 0.35dm Actual volume of the milk in the mug = volume of the cylinder volume of the raised hemispherical bottom 9
10 Now, Volume of the cylinder = πr h = = dm 3 Volume of the hemispherical bottom = 3 πr3 = = dm 3 So, Actual volume of the milk in the mug = =0.4497dm 3 Volume of the milk = The price of one litre = Rs. 80 The price of litres = = Rs According to the dairy B, the price of the milk should be Rs. 8 approximately. The dairy owner B is a humane person with morals. He believes in serving people at fair price and does not deceive people unlike dairy owner A. 30. Number of Apples Class mark Frequency Cumulative frequency f 87+f f 87+f+f f+f f+f f + f 550 Median = 4 Median of the data = l + ( Accordingly, 4 = 55 + ( 75 (47+f+f) 5 n +cf f ) h 4 = 75+8 f f 5 0 = ff f+f = = 93 () Now, The total number of boxes = 550 So, 7+f+f = Class F Cf Type Interval More than More than More than More than More than 40 ) 5 = 55 + ( f f ) 5 = 55 + ( 8 f f ) Cumulative Frequency 0
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