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2 Sura s n Mathematics - X Std. n Quarterly Question Paper 58 0 th Std. QURTERLY EXMINTION Question Paper With nswers MTHEMTICS Time llowed : ½ Hrs.] Maximum Marks : 00 SECTION - I (Marks : 5) Note: nswer all the questions. 5 5]. If {p, q, r, s} {r, s, t, u} then \ is (a) {p, q} (b) {t, u} (c) {r, s} (d) {p,q,r,s} 7. The GCD of (x + ) and x is (a) x (b) x + (c) x + (d) x On dividing by the Quotient is 9 (a) (x 5) (x ) (b) (x 5) (x + ) (c) (x + 5) (x ) (d) (x + 5) (x + ). If f :, is a bijective function and if 9. Matrix aij] m n is a square matrix if n() 5, then n () is equal to (a) m < n (b) m > n (a) 0 (b) (c) 5 (d) 5 (c) m (d) m n. If a, b, c, l, m are in.p., then the value of 0. If is of order and is of order a b + 6c l + m is then the order of is (a) (b) (c) (d) 0 (a) (b). If a, b, c are in G.P, then D E (c) (d) not defined equal to E F. If the line segment joining the points (,) and (, ) meets the x axis at P, then the (a) D E D (b) (c) (d) F ratio in which P divides the segment is E D F D (a) : (b) : (c) : (d) : 5. If n k, then rea of the triangle formed by the points (0,0)... + n is equal to (,0) and (0,) is (a) k (b) k (a) Sq. Unit (b) Sq. Units (c) Sq. Units (d) 8 Sq. Units N ÉN (c) (d) (k + ) D. In figure, if, Ð 0 and C DC 6. The system of equations x y 8, ÐC 60 then D. x y (a) has infinitely many solutions (b) has no solution (c) has a unique solution 0 60 (d) may or may not have a solution D C (a) 0 58] (b) 50 (c) 80 (d) 0 Ph /

3 58 Sura s n Mathematics - X Std. n Quarterly Question Paper. The sides of two similar triangles are in the ratio :, then their areas are in the ratio (a) 9 : (b) : 9 (c) : (d) : to to get.. Construct a matrix a ij ] whose elements are given by a ij L M.. If µ µ additive inverse of. then find the 5. Find the centroid of the triangle whose vertices are (, 6), (, ), and (5,). 6. The side of a square CD is parallel to x 5. axis. Find the slope of CD. If tan R, then the value of D 7. Find the equation of the straight line which (a) cosq (b) sinq (c) cosecq (d) secq slope is ; and passing rhrough (5, ). SECTION - II (Marks : 0) Prove that cosec (90 q) cot (90 q). In DPQR, QR, If cm, P cm and Note: (i) nswer 0 questions PR 6 cm then find the length of QR. (ii) Question number 0 is compulsory. Select any 9 questions from the first 0. (a) {,,, } and f µ, : Ž ¼ questions. 0 0] ½ 6. If {,6,7,8,9}, {,,6} and C {,,,,5,6}, then find Ç(ÈC). Write down the range of f. Is f a function 7. If Ì, then find Ç and \ (use Venn diagram). from to? 8. Write the first 5 terms of the sequence if OR] F F land F n F n + F n, n,, Which term of the G.P.,,, 8, is sin Rsin R (b) Prove the identity 0? cos RcosR tan q 0. The Cost of Pencils and erasers is `.50 and the cost of 8 Pencils and erasers is `.8. SECTION - III Find the cost of a Pencil and a eraser?. Find the quotient and remainder when x + x 7x is dived by x. (Marks : 5) Note: (i) nswer 9 questions.. Which rational expression should be added (ii) Question No. 5 is Compulsory. Select any 8 questions from the first questions ]. Let {0, 5, 0, 5, 0,5, 0, 5, 50} {,5, 0, 5, 0, 0} and C {7, 8, 5, 0, 5, 5, 8}, Verify $ÇC) ($È(\C).. In a group of students 65 play football, 5 play hockey, play cricket, 0 play football and hockey, 5 play football and cricket, 5 play hockey and cricket and 8 play all the three Ph /

4 Sura s n Mathematics - X Std. n Quarterly Question Paper 585 games. Find the number of students sin the group. (ssume that each student in the group plays at least one game).. Let {, 6, 8, 0} and {,, 5, 6, 7} If f : is defined by f(x) x + then represent f by i) an arrow diagram ii) a set of ordered pairs and iii) a table. Find the sum of all natural numbers betwen 00 and 500 which are divisible by. 5. Find the sum of first n terms of the series Factorize : x x x + into linear factors. 7. Find the GCD of the Polynomials x + x x and x + x 5x +. C 5 µ.verify that ( + C) + C. µ 0. If then show that + 5I 0.. Find the area of the quadrilateral whose vertices are (, ), ( 5, 6) (, ) and (,).. Find the area of the triangle formed by the line 7x + 0y 70 0 and its co-ordinate axis.. State and prove asic Proportionality Theorem.. If tanq + sinq m, tanq sinq n and m ¹ n then show that m n PQ. 5. a) If a,b, c, d are in geometric sequence, then prove that (b c) + (c a) + (d b) (a d). OR] b) If P \, Q \ \,then find Q P Q P Q SECTION - IV (Marks : 0) Note: nswer both the question choosing either of the alternatives. 0 0] 6. (a) Take a Point which is 9 cm away from the centre of a circle of radius cm and draw two tangents to the circle from that 8. 8 men and boys can finish a peice of work point. in 0 days while 6 men and 8 boys can finish OR] the same work in days. Find the number of (b) Construct a DC such that 6cm, days taken by one man alone to complete the ÐC 0 and the altitude from C to is work and also one boy alone to complete the of length. cm. work. 9. µ 5µ If 6 7 and 7. (a) Draw the graph y x + x and hence find the roots of x x 6 0. OR] (b) bank gives 0% S.I. on deposits for senior citizens. Draw the graph for the relation between the sum deposited and the interest earned for one year. Hence find.. (i) the interest on the deposit of `.650. (ii) ÿthe amount to be deposited to earn an interst of `.5. $6:(56$6(5.(< SECTION I. (a). (c). (d). (a) 5. (a) 6. (a) 7. (c) 8. (a) 9. (d) 0.(b). (a).(b).(d).(b) 5.(a) Ph /

5 586 Sura s n Mathematics - X Std. n Quarterly Question Paper SECTION II 6. Solution : + \ () * C {,, 6} * {,,,, 5, 6} ( ) ( ) ( ) {,,,, 5, 6}? ( * C) {, 6,} 8 + \ 8... () 7. Solution : +LQW means common elements Now substitute x in () to find the value \ null set since of y. \ {/ and } Ÿ \ I We get, () + y 50 + y 50 Þ y 50 6 y 6 Þ y (i.e) y Therefore, x and y \ I Thus, the cost of a pencil is ` and that of 8. Solution: an eraser is `., Given that F F and F n F n + F n. Solution : for n,, 5,... Let P(x) x + x 7x. The zero of the Now F, F division is. So we consider. F F + F + F F + F + 7 F 5 F + F \ The first five terms of the sequence are 5 n Remainder,,,, 5 \ When P(x) is divided by x, the quotient 9. Solution : Given G.P. is,,, 8,... is x + x + 5 and the remainder is. Here a, r. Solution : Let the n th term be 0 \ t n ar n 0 Ÿ () n 0 0 \ n 0 Ÿ n. \ The th term is Solution : Let x denote the cost of a pencil in rupees and y denote the cost of an eraser in rupees. Then according to the given information we have S ( ) É 5 Ph /

6 Sura s n Mathematics - X Std. n Quarterly Question Paper 587. Solution : a () () a (, 6), (, ) and (5, ) is () a G(x, y) G 5 6 µ, G (, ). () a 6. Solution : \ D C () a 5 ª º () a 0 0 «\» 5 90 «0» O ¼. Solution : Slope of CD Slope of x axis ( Q x - axis, when two lines are parallel their slopes are equal) ¹ ¹ ¹ 7. Solution: 6 6¹ m, (x, y ) is (5, ) Required equation is y y m(x x ) \ dditive inverse of 6 6 (slope-point form) ¹ 5. Solution : i.e., y + (x 5) The centroid G(x, y) of a triangle whose vertices are (x, y ), (x, y ) and (x, y ) is Ÿ y + x 0 Ÿ x y 0 given by 8. Solution : (, 6) cosec (90 q) cot (90 q) F E G We have (x, y ) (, 6), (x, y ) (, ), (x, y ) (5, ) \ The centroid of the triangle whose vertices are R R cot (90 ) R R cot (90 ) Hint : (cosec + cot q) (, ) D C (5, ) G (x, y) G, \ \ \ µ Ph /

7 588 Sura s n Mathematics - X Std. n Quarterly Question Paper 9. Solution : Given is cm, P is cm PR is 6 cm and QR P +LQW : Range of a function is a subset of the co-domain (nd set) f ( ) f ( ) f () f () Q R In DP and DPQR ½? Range of f,,, ¾ ÐP ÐPQR (corresponding angles) and ÐP is common. f is not a function from to, since and do not have images in the co-domain. \DP ~ DPQR ( similarity criterion) OR] Since corresponding sides are proportional, 0. (b) Solution : QR P sin Rsin R PR Now, cos RcosR sin R Ésin R cosr Écos R PR QR t 6 P sin sin cos sin µ R µ R R R É Thus, QR 9 cm. cos R. cos R sin Rcos R 0. a) Solution : (sin q + cos q ) f {(x, ) : x } where cos Rsin Rµ {,,, } (tan q) tan q. cos Rsin R SECTION III. Solution : C {, 5, 0, 5, 0, 0} {7, 8, 5, 0, 5, 5, 8} {5, 0}? \ ( C) {0, 5, 0, 5, 0, 5, 0, 5, 50} \ {5, 0} {0, 5, 0, 5, 0, 5,50}...() \ {0,5,0,5,0,5,0,5, 50} \ {, 5, 0, 5, 0, 0} {5, 5, 0, 5, 50} \ C {0, 5, 0, 5, 0,5,0,5, 50} \ {7, 8, 5, 0, 5, 5, 8} {0, 5, 0, 0, 50} Ph /

8 Sura s n Mathematics - X Std. n Quarterly Question Paper C ?(\) * (\C) {5, 5, 0, 5, 50} * 8 5 {0, 5, 0, 0, 50} {0, 5, 0, 5, 0, 5, 50}...() () & () verifies \ ( C) ( \ ) *( \ C). Solution : a 08, l 95, d Let F, H and C represent the set of students Q S who play foot ball, hockey and cricket n {a + l} respectively. Then n(f) 65, n(h) 5, and O D n(c). but n G lso, n(f H) 0, n(f C) 5, S 8 8 ^ 08 ` 95 n(h C) 5 and n(f H C) 8. We want to find the number of students in the whole group; that is n(f * H *C). y the formula, we have 9{80} 77. n(f * H * C) n(f) + n(h)+ n(c) n(f H) 87 8 n(h *C) (F C)+ n(f H C) LQW Hence, the number of students in the group ) 00 ( Solution : f (x) x + n Remainder f () + 97 is exactly by ž ) 500 ( 5 f (6) f (8) n Remainder f (0) The sum of all natural numbers between 00 and 500 which are divisible by is 77. (i) I (ii) f {(, ), (6, ), (8, 5), (0, 6)} (iii) \ Solution : The natural numbers between 00 and 500 which are divisible by are; 08, 9, Ph /

9 590 Sura s n Mathematics - X Std. n Quarterly Question Paper 5. Solution : Note that the given series is not a geometric series we need to find S n ( to n terms) S n ( to n terms) 9 (Multiply and divide by 9) (0 ) + (00 ) + (000 ) to n terms] 9 ( n terms n] Ä + Remainder ( y 0) Thus S n Q 0(0 ) µ Q Therefore, GCD ( f (x), g(x)) x + x Solution : Let x denote the number of days 0 Q needed for one man to finish the work and y Q É0 8 9 denote the number of days needed for one boy to finish the work. Clearly, x ¹ 0 and y ¹ Solution : Let p(x) x x x + So, one man can complete part of the Now, p() ¹ 0 (note that sum of the coefficients is not zero) work in one day and one boy can complete?( x ) is not a factor of p(x). However, p ( ) ( ) ( ) ( ) + 0. \ part of the work in one day. So, x + is a factor of p(x).we shall use The amount of work done by 8 men and synthetic division to find the other factors. boys in one day is n Remainder Thus, we have 8 Thus, p(x) (x + )(x 5x + ) \ 0... () Now, x 5x + x x x + The amount of work done by 6 men and 8 (x ) (x ) boys in one day is Hence, x x x +. (x + ) (x ) (x ) 7. Solution : Thus, we have 6 8 \... () Let f (x) x +x x and g(x) x + x 5x + Here degree of f (x) > degree of g(x).?divisor Let a is x + x 5x +. and b \ Then () and () give, respectively, 0 Remainder Ph /

10 Sura s n Mathematics - X Std. n Quarterly Question Paper 59 8a + b 0 Þ a + 6b 0... () 0 5 µ µ µ µ a + 8b Þ a + b () Writing the coefficients of () and () for the cross multiplication, we have 6 D E Thus, we have D E D E i.e., 6 9 ¹ 8 7¹ 70 0 That is, a 0, b 80 ¹ 8 ¹ ; Thus, we have x D 0, y E 80. Hence, one man can finish the work individually in 0 days and one boy can finish the work individually in 80 days. 9. Solution: Now, + C Thus, ( +C) µ µ µ 6 8 µ µ µ 0 5 Now, + C... () 6 5 µ µ µ µ µ () From () and (), we have (+C) + C. 0. Solution :. ¹ 5I ¹ ¹ ¹ + 5I ¹ 8 ¹ 0 5¹ ¹ ¹ 0 + 5I 0. Solution : 5 6 Ph /

11 59 Sura s n Mathematics - X Std. n Quarterly Question Paper \ (, ) (, ). Solution : C (5, 6) D (, ) rea ^85860 ^ 86 ` sq.units. 7x + 0y 70 0 If x 0 then \ ` \ 70 0 If y 0 then 7x + 0 (0) x x 70 x 70 7 x 0 rea of b h 5 t7t0 5 sq.units. Solution : Converse of asic Proportionality Theorem ( Converse of Thales Theorem) If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. Given: line l intersects the sides and C of DC respectively at D and E such that D E... () D EC To prove : DE C Construction : If DE is not parallel to C, then draw a linedf C. Proof Since DF C, by Thales theorem we get, D F... () D FC From () and (), we get F E F + FC E + EC ž FC EC FC EC This is possible only when F and E coincide. Thus, DE C. O 0 7 (0)+ 0y Solution : 0y 70 0 Given that m tan q + sin q and y 7 n tan q sin q. Þ y 7 0 Now, m n (tanq + sinq) (tanq sin q) tan q + sin q + sinq tanq (tan q + sin q sinq tanq) sin q tan q.... () lso, PQ Étan RsinR(tan Rsin R) É tan Rsin R R µ sin R R sin cos cos µ sin R R sin RÉsec R sin Rtan R (Q sec q tan q) sinq tanq.... () From () and (), we get m n PQ. Ph /

12 Sura s n Mathematics - X Std. n Quarterly Question Paper (a) Solution : Given a, b, c, d are in a geometric sequence. Let r be the common ratio of the given sequence. Here, the first term is a. Thus, b ar, c ar, d ar Now, (b c) + (c a) + (d b) (ar ar ) + (ar a) + (ar ar) a (r r ) + (r ) + (r r) ] a r r + r + r r ++ r 6 r + r ] a r 6 r + ] a r ] ÉP Q ª ¹ P (ar a) (a ar ) (a d) Q ª ÉP+Q ¹ P+Q «º OR] x y + x+y 5. (b) Solution : x+y x+y x+y \ \ \ \ \ \ \ \ \ ( \) \ \ \ \ \ \ ( \) u \ ( \) ( \)( \) \ \ \ \ \ \ ( \) ( \) ( \) liter : Q Q P Q P Q P Q P+Q P Q É É Q q P+Q Q ª ¹ «º ª «¹ º P Q ª P+Q ¹ P Q P+Q Ph /

13 6. (a) Solution: Given: Radius of the circle cm. OP 9 cm. Fair DiagramRough Diagram 59 Sura s n Mathematics - X Std. n Quarterly Question Paper SECTION IV O T Tc cm 8.5 cm M 9 cm 8.5 cm P Steps of Construction: (i) With O as the centre draw a circle of radius cm. (ii) Mark a point P at a distance of 9 cm. from O and join OP. (iii) Draw the perpendicular bisector of OP. Let it meet OP at M. (iv) With M as centre and MO as radius, draw another circle. (v) Let the two circles intersect at T and Tc. (vi) Join PT and P Tc. They are the required tangents. Length of the tangent, PT 8.5 cm Verification: In the right angled triangle OPT, PT OP OT 9 T cm O cm cm (approximately) 7c 9 cm P? PT PTc 8.5 cm OR] Ph /

14 . cm Sura s n Mathematics - X Std. n Quarterly Question Paper (b) Solution: Given: In DC, 6 cm, ÐC 0 The length of the altitude from C to is. cm. Fair Diagram Rough Diagram C 0 H O 6 cm K 0 M Steps of Construction: (i) Draw a line segment 6 cm. (ii) Draw X such that ÐX 0 (iii) Draw Y ^ X. (iv) Draw the perpendicular bisector of intersecting Y at O and at M. Y (v) With O as centre and O as radius, draw the circle. (vi) The segment K contains the vertical angle 0. (vii) (viii) (ix) On the perpendicular bisector MO, mark a point H such that MH. cm. Draw CHC parallel to meeting the circle at C and at C. Complete the DC, which is one of the required triangles. Cb C 0 M Ph /

15 596 Sura s n Mathematics - X Std. n Quarterly Question Paper 7. a) Solution: First let us draw the graph of y x + x. Now let us assign integer values from to for x and form the following table. 8 \ ± Plot the points (, 5), (, 0), (, ), (, ), (0, ), (, 0), (, 5) and (, ) on the graph sheet and join the points by a smooth curve. \ \ b \ Scale: axis cm unit \axis cm units \b Ph /

16 Sura s n Mathematics - X Std. n Quarterly Question Paper 597 The curve thus obtained is the graph of parabola y x + x Now, by solving y x + x and x x 6 0 We get, \ 0 6 y x + y x + is a straight line. Now, for the straight line y x + form the following table. 0 \ 9 Draw the straight line by joining the above points. The points of intersection of the parabola and the straight line are (, ) and (,). The x-coordinates of the points are and. Hence the solution set for the equation x x 6 0 is {, } 7. b) Solution: Let us form the following table. OR] Deposit C S.I. earned C \ Ph /

17 598 Sura s n Mathematics - X Std. n Quarterly Question Paper,QWHUHVW 80 Scale - axis cm C00 \ - axis cm C (00, 0) \ 0 (00, 0) (00, 0) 'HSRVLWV (500, 50) (600, 60) (700, 70) Clearly y x and the graph is a straight line. 0 Draw the graph using the points given in the table. From the graph, we see that (i) The interest for the deposit of `650 is ` 65. (ii) The amount to be deposited to earn an interest of ` 5 is ` 50. Ph /

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -