SOLUTIONS SET 1 MATHEMATICS CLASS X

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1 Tp Careers & Yu SOLUTIONS SET MTHEMTICS CLSS X Prime factrs f 84 are, and 7.. Sum f zeres Prduct f zeres Required plynmial x ( )x + ( 0) x + x 0. Given equatin is x + y 0 Fr x, y L.H.S becmes () + () R.H.S Hence, x, y is a slutin f the given equatin. 4. Here, first term Secnd term 8 Cmmn difference 8 Next term cs 5 sin 5 4 cs Nw, ct sin 4 9 ct Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

2 Tp Careers & Yu Prbability f lsing prbability f winning P cm C 4 cm y Pythagras therem C + C 5 cm Nw P C 5.5 cm 8. Let C tuches the circle at pint K Nw, P Q 0 cm and P K and CQ CK P P (i) (ii) P K C Q C Q QC and C K + KC Perimeter f C + C + C P P + K + KC + Q QC P + Q [using (i) and (ii)] cm Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

3 Tp Careers & Yu 9. Let r cm, r 4 cm and R be the radius f the required circle.t.q. πr πr + πr π(r + r ) R r + r R 5 cm 0. Median 0.5. Fr zers the quadratic plynmial is equated t O. i.e. x 9 x 0 r x x 9 0 x 6x + x 9 0 (x + ) (x ) 0 Nw, either x + 0 r x 0 x x x S, zers f the equatin are and Verificatin f relatin Here in the equatin by equating with general equatin we get a, b, c b ( ) (i) Nw sum f zers + and a i.e. sum f zers a b (ii) Prduct f zers x.x 9 ls a c 9 c cefficient cnstt i.e. prduct f zers a cefficient f x Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

4 Tp Careers & Yu 4 ase. Given: ct θ ct θ C perpendicular S in C, using Pythagras C + C (4k) + (k) (6 + 9)k 5k 4k C 5 k 5k (Neglecting Negative as length can t be less than zer) H k S k 4k and cs θ 5k 5k 5 cs θ 5 + cs θ OR tan 0 cs ec 70 + ct 0 sec 70 + tan 5. tan 45. tan 75 sin 0 cs 0 sin 70 + cs 0 sin 0 cs 70 + tan 5. tan (90 5 ) sin 0 cs 0 cs 0 + cs 0 sin 0 sin 0 + tan 5. ct 5 u sin g frmula sin (90 θ) cs θ cs (90 θ) tan (90 θ) ct θ (sin 0 + cs 0 ) +. tan 5 tan 5 + {using sin θ + cs θ } {using ct θ } tan θ. Given: (x, y ) (, k), (x y ) (4, ) C(x, y ) ( 9, 7) nd rea( C) 5 square units We knw rea f a triangle with crdinates (x, y ), (x y ) (x y ) is rea x (y y ) + x (y y ) + x (y y ) Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

5 Tp Careers & Yu 5 ( 7) + 4(7 k) 9 (k + ) k 9k 7 r k 9 0 either k 9 0 r ( k 9) 0 k 9 k k k k k k r 4. D E C P E C Given: DE C and (i) EC CP T prf: DC P Prfs: In C, DE C (Given) E D (ii) CE D line parallel t ne side f a triangle divide the ther tw sides in the same rati E C ls (Given) CE CP S we have Nw in P; DC P D C (iii) D CP D D C CD (just prved) ( line dividing tw sides f a triangle in the same rati is parallel t the third side) Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

6 Tp Careers & Yu 5. Ttal number f cards in the bx 65 (i) Number f cards f ne digit (6, 7, 8, 9) 4 4 Prbability f digit number 65 (ii) Ttal numbers divisible by 5 (0, 5, 0, 5, 0, 5, 40, 45, 50, 55, 60, 65 and 70) Prbability f number divisible by HCF 6 LCM HCF LCM Prduct f numbers HCF LCM Prduct f numbers. 7. x y 4x y 8 The pint f intersectin is (, 4). Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

7 Tp Careers & Yu 8. (a b)x + (a + b)y a ab b.() (a + b)x + (a + b)y a + b..() Subtracting () frm (), we get bx b + ab x a + b Substituting in () (a b) (a + b) + (a + b)y a ab b (a + b)y ab ab y a + b Fr real and distinct rts, D > 0 4( + m) 8m( + m) > m + 6m 4m 6m > 0 4 8m > 0 m < Fr equal rts, D 0 m OR 9. L.H.S. cs θ + cs θ + cs θ cs θ + + ct ct ct θ + cs ec θ θ + cs ec θ θ (cs ec θ ct ct θ + cs ec θ θ) + cs ec θ ct θ + cs ec (cs ec θ ct θ) (cs ec θ + ct ct θ + cs ec θ θ) (ct θ + cs ec θ ) [ cs ec θ + ct [ cs ec θ + ct θ] θ] Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

8 Tp Careers & Yu ct θ + csec θ R.H.S. OR R.H.S. tan θ + ct θ cs θ + cs θ sin θ + cs cs θ θ ls, R.H.S. (csec θ ) (sec θ cs θ) sin θ cs θ cs θ ( sin θ) ( cs cs θ θ) cs θ sin θ cs θ cs θ Hence, R.H.S. L.H.S. 0. The required.p. is 0,, 6,,97 It is an.p with a 0, d and n 0 S n [a + (n )d] 0 [ ] 605. P is the mid pint f the line segment jining (, ) and (5, 6) C-rdinates f P are P lies n x + y + k 0 7 7, k k 0 k 7 Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

9 Tp Careers & Yu. Given: triangle C with D as median Cnstructin : Draw M perpendicular t C. In M, M + M (D DM ) + (D + DM) D + D + D.DM In MC, C M + MC C (D DM) + (CD DM) C D + CD CD.DM dding (i) and(ii). (i) (ii) + C D + D + D.DM + D + CD CD.DM Since, D CD + C (D + D ) OR Given: triangle C with an acute angle and lines D and CE perpendicular n lines C and respectively. T prve: E C D In D and CE D EC 90 Hence, D CE S C E D C D E Hence prved [given] E D D M C C. Let (5, 6), (, 5), C(, ) and D(6, ) be the vertices f the square. T shw that it is a square we shuld use the prperty that all its sides shuld be equal and bth its diagnals shuld als be equal. Nw, ( 5 ) + (6 5) ( 4) + () 7 C CD D ( ) + (5 ) ( 6) + ( ) ( 6 5) + ( 6) Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

10 Tp Careers & Yu C ( 5 ) + (6 ) D ( 6) (5 ) Since, C CD D and C D, all the frm sides f the quadrilateral CD are equal and its diagnals C and D are als equal. Therefre CD is a square. 4. O P P and P are the required tangents. 5. Radius f circle is O R 4 cm. Radius f circle with diameter OD r 7 cm S area f this circle is πr cm. 7 rea f the semicircle (πr ) cm 7 rea f the shaded prtin (area f circle with radius r) + (area f semicircle with radius R) (area f the triangle) cm 6. Let the number f persns be n 6500 If Rs is divided equals than each willg et Rs. n If number f persn are increased by 5 i.e. n + 5 than each persn will get Rs. 0 less. S accrding t questin. Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

11 Tp Careers & Yu n + 5 n n 6500 n n n (n) (n + 5) (n) (n + 5) n + 5n 0 0 n + 65n 50n 50 0 n(n + 65) 50(n + 65) 0 n 50, 65 S number f persn are 50. OR Let the speed f train b v km/h 60 S time taken by train t cver 60 km hurs v Nw if the speed is increased by 5 km/h i.e. v + 5 km/h then time reduces by ne hur. S accrding t questin v v v 60 v 60 v (v) (v + 5) v + 5 (v) (v + 5) 800 v + 5v v + 45v 40v v(v + 45) 40(v + 45) 0 (v 40) (v + 45) 0 v 40, 45 Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

12 Tp Careers & Yu S riginal speed f train is 40 km/h 7. Let GH be the upper surface f the lake, C be the psitin f the clud, D be its reflectin in the lake and E be the eye f the bserver. Draw EF CD and EG GH. Then, GE 60, FEC 0 and FED 60. Clearly, FH GE h. Let height f the clud HC HD x. FC x 60 and FD x EF Nw, ct 60 FD E h G 0 60 C F H EF (x + 60) ct 60 (x + 60) D EF ls ct 0 FC EF (x 60) EF (x 60) (x + 60) x 80 x + 60 x 40 x 0 Height f the clud 0 m (x 60) (x + 60) 8. The length f tw tangents drawn frm an external pint t a circle are equal. Given: Tw tangents P and Q drawn frm a pint t a circle C(O, r). T prve: P Q. Cnstructin: Jin OP, OQ, and O. Prf: Since a tangent at any pint f a circle is perpendicular t the radius thrugh the pint f cntact, we have OP P and OQ Q. Nw, in right triangles OP and OQ, we have OP OQ (radii) and O O (cmmn) OP OQ Hence, P Q. P Q O Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

13 Tp Careers & Yu Part II In PQ P Q (Prved) (ngles ppsite t equal sides) (Sum f angles f s) (i) 4 5 (ngles ppsite t radii) dding (i) and (ii) ( + 4) (as OQ Q) (ii) P O and 6 are supplementary OR Q The rati f the areas f tw similar triangles is equal t the rati f the squares f the crrespnding sides. Given: C and DEF are tw similar triangles. T prve: ar ( C) C C (i) ar ( DEF) EF DE DF Cnstructin: Draw G C and DH EF. Prf: (i) C G ar ( C) ar ( DEF) EF DH (Qarea f base height) C EF G DH (i) G D C Nw, in s G and DEH, we have E (Q C DEF) G DHE (each equal t 90 ) G ~ DEH DE G DH ( Similarity) (QIf s are similar, the rati f their crrespnding sides is same) Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t E H F

14 Tp Careers & Yu C ut ( Q C DEF) DE EF G C (ii) DH EF ar ( C) C C C Nw, frm (i) and (ii), we have: ar ( DEF) EF EF EF ar ( C) C C Similarly we can prve ar ( DEF) EF DE DF ---- () If the area f tw similar triangle are equal i.e ar( G) ar( DEH) Using the abve relatin: ar ( C) ar ( DEF) C EF C C EF DE DF DE C DF We get C EF, DE and DF C Nw, in s G and DEH, we have C EF, DE and DF C G DEH (using SSS) 9. Radius f tp R 8 cm Radius f bttm r 7 cm Let height f the bucket h Capacity f the bucket 560 cm s capacity π h [R + r + Rr] π h 560 [R + r + Rr] (/7) (h/) ( x 7) h h 0 h r r R l Slant height f Frustum h + (R r) 0 + (8 7) Ttal surface area π[ l (R + r) + r ] 44 cm Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t

15 Tp Careers & Yu 0 Classes Frequency (f i ) Class mark f i x i Cumulative x frequency f i 50 f ixi 0 Mean fix f i i Calculatin f median Since n 50. n 50 S, 5 This bservatin lies in l 40, f, cf, h 0 n cf 50 We knw, median l + h f Calculatin f mde Here the maximum class frequency is and the class crrespnding t this frequency is S, mdal class is Lwer limit ( l ) f mdal class 40 Frequency (f ) f mdal class Frequency (f 0 ) f class preceding mdal class 0 Frequency (f ) f class succeeding mdal class 0 We knw, Mde l + f f0 f f0 f ( 0) h Fr the ther sets f CSE papers and free materials fr bard exam preparatin lg n t