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1 CBSE Sample Question Paper (07-8 Time: Hours Maximum Marks: 80 General Instructions: (i All questions are compulsory. (ii The question paper consists of 0 questions divided into four sections A, B, C and D. (iii Section A contains 6 questions of mark each. Section B contains 6 questions of marks each. Section C contains 0 questions of marks each. Section D contains 8 questions of 4 marks each. (iv There is no overall choice. However, an internal choice has been provided in four questions of marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v Use of calculators is not permitted. SECTION A. Write whether the rational number 7 7 will have a terminating decimal expansion or a non-terminating repeating decimal expansion. Sol So, it is a non-terminating repeating decimal expansion.. Find the value(s of k, if the quadratic equation x k x has equal roots. Sol. Given equation x k x ( On comparing eqn. ( with ax + bx + c 0, we get a, b k, c 4, For equal roots, we have b 4ac 0 i.e. b 4ac \ On using b 4ac, we get ( k 4 4 k 4 4 k 6 k ± 4. Find the eleventh term from the last term of the AP: 7,, 9,..., 6. Sol. Given series: 7,, 9,..., 6 Here, a 7, d 7 4, l 6 nth term from the end (last l (n d \ th term from the end (last l ( d l 0d 6 0 ( Find the coordinates of the point on y-axis which is nearest to the point (,. 8

2 Sol. (, P A Y 4 y B X 0 X Let Co-ordinate of point A on the Y-axis is (0, y or (0, OA From the graph, it is clear that OA y \ Co-ordinate of point A is (0,. Given PS cm, SR 4 cm. From given figure, we have PR PS + SR cm + 4 cm 7 cm ar ( PST PS Q ar ( PRQ PR [using Area-ratio theorem] ar ( PST ( \ 9 ar ( PRQ ( If cos A, find the value of tan A Sol. Given: cos A, then sec A cos A tan A 4 ( + tan A 4 sec A 4 Y SECTION B c m If two positive integers p and q are written as p a b and q a b; a, b are prime numbers, then verify: LCM (p, q HCF (p, q pq Sol. LCM (p, q a b and HCF (p, q a b LCM (p, q HCF (p, q a b 4 (a b (a b pq 8. The sum of first n terms of an AP is given by S n n + n. Find the sixteenth term of the AP. Sol. S n n + n S a S a + a 4 a 9 d a a 4 a 6 a + d + ( Find the value(s of k for which the pair of linear equations kx + y k and x + ky have infinitely many solutions. Sol. For pair of equations kx + y k and x + ky CBSE Sample Question Paper (07-8 n 9 P T S R

3 a We have: a k b c, k, b k c a For infinitely many solutions, b c a b c k \ k k k,...(i k and k k k...(ii From (i and (ii, k 0. If, p d n is the mid-point of the line segment joining the points (, 0 and c0, m, then 9 show that the line x + y + 0 passes through the point (, p. Sol. We know that y y + y p A P B (, 0 (0, (, p x y 9 x,y x y p Now, co-ordinate of point (, p is equal to c, m i.e. (,. The given line x + y + 0 passes through the point (, as ( + ( A box contains cards numbered to. A card is drawn at random from the box. Find the probability that the number on the drawn card is (i a square number (ii a multiple of 7 Sol. Total number of cards in the box are. (i Cards having square number of them between and are 6,, 6, 49, 64, 8, 00, ; so there are total eight (8 square numbers. 8 \ P (square number (ii Cards having a number, which is multiple of 7 are 4,, 8,, 4, 49, 6, 6, 70, 77, 84, 9, 98, 0,, 9. So, there are total sixteen (6 number, which are multiple of 7. 6 \ P (multiple of 7. A box contains balls of which some are red in colour. If 6 more red balls are put in the box and a ball is drawn at random, the probability of drawing a red ball doubles than what it was before. Find the number of red balls in the bag. Sol. Let number of red balls be x x \ P(red ball If 6 more red balls are added: The number of red balls x + 6 P(red ball x x + 6 x Since, c m x 8 \ There are red balls in the bag. 0 n Mathematics X

4 SECTION C. Show that exactly one of the numbers n, n + or n + 4 is divisible by. Sol. Let n k, k + or k +. (i When n k: n is divisible by. n + k + n + is not divisible by. n + 4 k + 4 (k + + n + 4 is not divisible by. (ii When n k + : n is not divisible by. n + (k + + k + (k + n + is divisible by. n + 4 (k k + (k + + n + 4 is not divisible by. (iii When n k + : n is not divisible by. n + (k + + k + 4 (k + + n + is not divisible by. n + 4 (k k + 6 (k + n + 4 is divisible by. Hence, exactly one of the numbers n, n + or n + 4 is divisible by. 4. Find all the zeroes of the polynomial x 4 + 6x x 0x if two of its zeroes are and. Sol. Since and are the two zeroes therefore, x x + c mc m (x is a factor of given polynomial. We divide the given polynomial by x. For other zeroes, x + x + 0 (x + 0, x, \ Zeroes of the given polynomial are,, and.. Seven times a two-digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is, determine the number. Sol. Let the tens and the units digit be y and x respectively. So, the number is 0y + x. The number when digits are reversed is 0x + y. Now, 7(0y + x 4(0x + y y x (i Also x y (ii Solving ( and (, we get y and x 6. Hence, the number is 6. CBSE Sample Question Paper (07-8 n

5 6. In what ratio does the x-axis divide the line segment joining the points ( 4, 6 and (, 7? Find the coordinates of the point of division. The points A(4,, B(7,, C(0, 9 and D(, form a parallelogram. Find the length of the altitude of the parallelogram on the base AB. Sol. Let x-axis divides the line segment joining ( 4, 6 and (, 7 at the point P in the ratio : k. 4k 7 k Now, coordinates of point of division P c 6, m k + k + 7 6k Since P lies on x-axis, therefore k k 0 k 6 7 Hence the ratio is : 6 : 7 6 J 7 7 N J c c N m m 6 6 Now, the coordinates of P are K, ie.., O K + c 4, 0 m. L 6 6 O K P L 6 O P Let the height of parallelogram taking AB as base be h. Now AB ( ( units. 49 Area (DABC [4( 9 + 7( ( ] sq. units Now, AB h h 49 h 9.8 units. 7. In the given figure, and DMTR, then prove that DPTS ~ DPRQ. Sol. In an equilateral triangle ABC, D is a point on the side BC such that BD BC. Prove that 9AD 7AB SQN TRM (CPCT as DMTR Since, P + + P + PQR + PRQ (Angle sum property + PQR + PRQ n Mathematics X

6 Also And PQR (as and PQR PRQ PQR PRQ SPT QPR (Common \ DPTS ~ DPRQ (By AAA similarity criterion Construction: Draw AP ^ BC In DADP, AD AP + DP AD AP + (BP BD AD AP + BP + BD (BP (BD AD AB BC BC + c BCm c m c m AD 9 7 AB ( BC AB 9AD 7AB 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 90. Sol. Join OC In DOPA and DOCA OP OC (Radii of same circle PA CA (Length of two tangents AO AO (Common \ DOCA (By SSS congruency criterion Hence, (CPCT Similarly, 4 Now, PAB + QBA AOB 90 (Angle sum property CBSE Sample Question Paper (07-8 n

7 cosec 6 + tan 4 sin 6 + cos 6 sin 7 + sin 7 sec 6 9. Evaluate: + cot 66 + sec 7 ( cosec 6 tan If sin q + cos q, then evaluate: tan q + cot q cosec 6 + tan 4 sin 6 + cos 6 sin 7 + sin 7 sec 6 Sol. + cot 66 + sec 7 ( cosec 6 tan cosec 6 + tan 4 tan ( cosec ( 90 7 sin 6 + cos 6 cos( sin 7 cosec ( [ cosec 6 cot ( 90 ] cosec 6 + tan 4 tan 4 + cosec 6 sin 6 + cos sin 7 cosec ( cosec 6 cot 6 ( sin q + cos q (sin q + cos q ( sin q + cos q + sin q cos q + sin q cos q sin q cos q...(i We know, sin q + cos q...(ii Dividing (ii by (i, we get sin θ+ cos θ sinθcos θ / tan q + cot q 0. In the given figure, ABPC is a quadrant of a circle of radius 4 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region. Sol. We know, AC r In DACB, BC AC + AB BC AC ( AB AC BC r Required area DACB + semicircle on BC as diameter quadrant ABPC r r r + p d n π r 4 r + πr πr r 96 cm 98 cm n Mathematics X

8 . Water in a canal, 6 m wide and. m deep, is flowing with a speed of 0 km/h. How much area will it irrigate in 0 minutes, if 8 cm of standing water is needed? A cone of maximum size is carved out from a cube of edge 4 cm. Find the surface area of the remaining solid after the cone is carved out. Sol. Let the area that can be irrigated in 0 minutes be A m. Water flowing in canal in 0 minutes c 0, 000 # m m 000 m Volume of water flowing out in 0 minutes ( m 4000 m (i 8 Volume of water required to irrigate the field A m...(ii 00 Equating (i and (ii, we get 8 A 4000 A 600 m. 00 Given: Side of cube (a 4 cm l r + h side 7 and r 4 cm 7 cm Surface area of remaining solid 6a pr + prl, where r and l are the radius and slant height of the cone (0 + 4 cm. 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 above distribution is, using empirical relationship estimate the value of its median. Sol. f f0 Mode l + e o # h f f 0 f c m # So, the mode marks is 68. Empirical relationship between the three measures of central tendencies is: Median Mode + Mean Median 68 + Median 8 marks CBSE Sample Question Paper (07-8 n

9 6 n Mathematics X SECTION D. A train travelling at a uniform speed for 60 km would have taken 48 minutes less to travel the same distance if its speed were km/hour more. Find the original speed of the train. 4 Check whether the equation x 6x 0 has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained satisfy the given equation. Sol. Let original speed of the train be x km/h. 60 Time taken at original speed x hours 60 Time taken at increased speed x + hours Now, x x ; x x + E x + x 0 0 x 4 or 0 (as speed cannot be negative x 4 km/h Discriminant b 4ac 6 4 ( 76 > 0 So, the given equation has two distinct real roots x 6x 0 Multiplying both sides by. (x (x 0 (x (x (x 9 x! 9! 9 x Verification: d n 6 d n Similarly, d n 6 d n 0 4. An AP consists of 7 terms. The sum of the three middle most terms is and the sum of the 4 last three terms is 49. Find the AP. Sol. Let the three middle most terms of the AP be a d, a, a + d. We have, (a d + a + (a + d a a 7 Now, the AP is a 8d,, a d, a d, a, a + d, a + d,, a + 8d Sum of last three terms: (a + 8d + (a + 7d + (a + 6d 49 a + d 49 a + 7d d 4 d 4 Now, first term a 8d 7 8(4 \ The AP is, 7,,, 47.. Show that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 4

10 Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Sol. Given: A right triangle ABC right angled at B. To prove: AC AB + BC Construction: Draw BD ^ AC Proof: In DADB and DABC ADB ABC (Each 90 BAD CAB (Common \ DADB ~ DABC (By AA similarity criterion AD AB Now, (Corresponding sides are proportional AB AC AB AD AC (i Similarly, DBDC ~ DABC BC CD AC (ii Adding (i and (ii, AB + BC AD AC + CD AC AB + BC AC (AD + CD AB + BC AC, Hence Proved. Given: DABC ~ DPQR To prove: TABC AB BC CA d n d n c m TPQR PQ QR RP Construction: Draw AM ^ BC, PN ^ QR TABC # BC # AM BC AM #...(i TPQR QR PN # QR # PN In DABM and DPQN, B Q ( DABC ~ DPQR M N (Each 90 DABM ~ DPQN (AA similarity criterion AM AB Therefore, (ii PN PQ AB BC But CA ( DABC ~ DPQR (iii PQ QR RP Hence, TABC BC AM # TPQR QR PN [From (i] AB AB PQ PQ AB d n [From (ii and (iii] PQ CBSE Sample Question Paper (07-8 n 7

11 TABC AB BC CA d n d n c m TPQR PQ QR RP [Using (iii] 6. Draw a triangle ABC with side BC 7 cm, B 4, A 0. Then, construct a 4 triangle whose sides are times the corresponding sides of DABC. Sol. Draw DABC in which BC 7 cm, B 4, A 0 and hence C 80 ( Construction of similar triangle A BC as shown below: A A B 4 7 cm 0 C C B B B B 4 cosθ sin θ+ 7. Prove that cos θ+ sin θ cosec q + cot q 4 cosθ sin θ+ cosθ sin θ+ cosθ+ sin θ+ Sol. LHS cosθ+ sin θ cos θ+ sin θ cos θ+ sin θ+ ( cosθ+ sin θ cos θ+ + cos θ sin θ ( cosθ+ sin θ cos sin θ+ θ+ sin θ cos θ cos θ+ cos θ sinθcos θ cosθ( cos θ+ cosθ + sinθcos θ sin θ cosec q + cot q RHS 8. The angles of depression of the top and bottom of a building 0 metres high as observed from the top of a tower are 0 and 60, respectively. Find the height of the tower and also 4 the horizontal distance between the building and the tower. Sol. Correct Figure X TP In DBTP tan 0 BP 8 n Mathematics X TP BP

12 BP TP...(i TR TR TR In DGTR, tan 60 GR...(ii GR TR GR Now, TP (as BP GR TP TP + PR 0 TP BG TP m m Now, TR TP + PR ( + 0 m. Height of tower TR 7 m. TR Distance between building and tower GR 7 GR m m 9. Two dairy owners A and B sell flavoured milk filled to capacity in mugs of negligible thickness, which are cylindrical in shape with a raised hemispherical bottom. The mugs are 4 cm high and have diameter of 7 cm as shown in given figure. Both A and B sell flavoured milk at the rate of ` 80 per litre. The dairy owner A uses the formula pr h to find the volume of milk in the mug and charges ` 4. for it. The dairy owner B is of the view that the price of actual quantity of milk should be charged. What according to him should be the price of one mug of milk? Which 4 value is exhibited by the dairy owner B? cuse π m 7 Sol. Capacity of mug (actual quantity of milk pr h pr π r h c rm # # # c4 # m 7 69 cm 6 Amount dairy owner B should charge for one mug of milk # ` Value exhibited by dairy owner B: honesty (or any similar value 0. The following distribution shows the daily pocket allowance of children of a locality. The 4 mean pocket allowance is ` 8. Find the missing frequency k. Daily pocket allowance (in ` Number of children 6 9 k 4 The following frequency distribution shows the distance (in metres thrown by 68 students in a Javelin throw competition. Distance (in m Number of students Draw a less than type Ogive for the given data and find the median distance thrown using this curve. CBSE Sample Question Paper (07-8 n 9

13 Sol. Daily pocket Number of Mid-point (x i xi 8 allowance (in ` children (f i ui f i u i k 0 k Σf i 40 + k Σf i u i k 8 Σ fu i i Mean x a + h e o Σ fi k c 8 m 40 + k 8 k Less than Number of Students Median distance is value of x that corresponds to N Cumulative frequency 68 4 Therefore, Median distance 6 m 0 n Mathematics X

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