V15PCAF I, 01 SUMMATIVE ASSESSMENT I, 01 / MATHEMATICS IX / Class IX 90 Time Allowed : hours Maximum Marks : 90 General Instructions: All questions are compulsory. 1 1 6 10 11 The question paper consists of 1 questions divided into four sections A, B, C and D. Section- A comprises of questions of 1 mark each; Section-B comprises of 6 questions of marks each; Section-C comprises of 10 questions of marks each and Section-D comprises of 11 questions of marks each. There is no overall choice in this question paper Use of calculator is not permitted. SECTION A 1 1 Question numbers 1 to carry 1 mark each. 1 1 7 Page 1 of 8
Find the decimal expansion of. 7 p(x)x x1 p( ) 1 If p(x)x x1, then find the value of p ( ). ABCD x 1 In the figure ABCD. Find the value of x. 1 The co-ordinate axes divide the plane into how many parts? / SECTION B 5 10 Question numbers 5 to 10 carry marks each. 5 : 51 Evaluate : 51 6 q(t)t t t1 t1 Check if the polynomial q(t)t t t1 is exactly divisible by t1. 7 Prove that Two distinct lines cannot have more than one point in common. 8 P Q PQQRPR Page of 8
P and Q are the centres of two intersecting circles. Prove that PQQRPR. 9 0 m m Find the area of a rhombus whose one side is 0 m and one diagonal is m. 10 5 cm, 5 cm 61 cm Find the area of the triangle with sides 5 cm, 5 cm and 61 cm. / SECTION C 11 0 Question numbers 11 to 0 carry marks each. 11 a b ab Let a and b be rational and irrational numbers respectively. Is ab an irrational number? Justify your answer. 1. 0 5 p q p q q 0 Express 0 5. in the form of p q where p and q are integers and q 0. 1 xyz0 x y z xyz If xyz0, show that x y z xyz. Page of 8
1 ab7 a b 85 a b If ab7 and a b 85, find a b. 15 ABC AC D ADCDBD ABC In ABC, if D is a point on AC such that ADCD BD, then prove that ABC is a right angles triangle. 16 PR > PQ PS QPR PSR >PSQ In the given figure PR > PQ and PS bisects QPR. Prove that PSR >PSQ 17 If the bisectors of a pair of alternate angles formed by a transversal with two given lines are parallel, prove that the given lines are parallel. 18 ABCD ABAD A AC ABC ADC BCDC In the figure, ABCD is a quadrilateral such that ABAD and AC is the bisector of the angle A. Show that ABC ADC and BCDC. Page of 8
19 cm 60 cm The base of an isosceles triangle measures cm and its area is 60 cm. Find its perimeter. 0 ABC 10 cm DBC D90 BD6 cm ( 1.7 ) In the given figure ABC is equilateral triangle with side 10 cm and DBC is right angled at D90. If BD 6 cm, find the area of the shaded portion ( 1.7 ) / SECTION D 1 1 Question numbers 1 to 1 carry marks each. 1 5 8 9 8 1 7 Page 5 of 8
Two classmates Salma and Anil simplified two different expressions during the revision hour and explained to each other their simplifications. Salma explains simplification of 5 and Anil explains simplifications of 8 9 8 1 7. Write both the simplifications. What value does it depict? 1 If x 1 x x x 7x5, find the value of x x 7x5. x 5x 7x60 x Show that x is a factor of x 5x 7x60. Also, find the other factors. (pq) 0 (pq)15 ---- 1 Factorise : (pq) 0 (pq)15 5 (xp) x 5 p x xp p x px Find the value of p if (xp) is a factor of x 5 p x xp. Hence factorise x px. 6 (x) x 1 ax 5xb ab If (x) and x 1 are both factors of ax 5xb, then show that ab. 7 ABAD, 1 APAQ Page 6 of 8
In figure ABAD, 1 and. Prove that APAQ. 8 ABC B C BD CD 180yx In ABC, BD and CD are internal bisector of B and 180yx. C respectively. Prove that 9 Show that the perimeter of a is greater than the sum of its three medians. 0 ABAD, ACAE BADCAE BCDE In the figure, ABAD, ACAE and BADCAE. Prove that BCDE. Page 7 of 8
1 P PS PTQR TPS 1 (QR) In figure, PS is the bisector of P and PTQR. Show that TPS 1 (QR). ***** Page 8 of 8
MARKING SCHEME V15PCAF SUMMATIVE ASSESSMENT I, MATHEMATICS Class IX SECTION A Question numbers 1 to carry 1 mark each. 1. 8571 1 1 1 (B). 1 1 1 SECTION B Question numbers 5 to 10 carry marks each. 5 6 51 8 Put t 1 8 8 1 6 1 1 1 1 q 1 1 1 1 1 0 8 Yes, t1 exactly divides the polynomial q(t) 7 Proving using Euclid s axiom.. 8 In a circle, having centre at P We have PRPQradii.½ In a circle, having centre at Q QRQPradii.½ Euclid s first axiom :- things which are equal to the same things are equal to one another.½ PRPQQR.½ 9 AOB90 OB 0 1 16 m BD16 m Area of ABCD 1 d 1d 1 8 m 10 5 5 61 150 s 75 Area of the triangle 75( 75 5)( 75 5)( 75 61) 75 0 1 1 0 5 sq. cm Page 1 of 6
SECTION C Question numbers 11 to 0 carry marks each. 11 Yes, ab is an irrational number Let a is an irrational number b 1.15 is a rational number so, ab 1.7 1.15.88.. is an irrational number as decimal expansion.88 is non terminating and non recurring. Similarly, taking aand b5, we have ab.15 58.115 which is irrational. 1 Let x. 05.055 10x0.55 1000x05.5 990x995 995 799 x 990 198 1 xyz0 xyz (xy) (z) (cubing both sides) x y xy (xy)z x y xy (z)z x y xyzz x y z xyz 1 ab7, a b (ab) ab 857 ab 859 ab ab18 a b (ab) ab (ab) 7 (18) (7) 5771 15 ABDA CBDC ABDCBDAC BAC ABC180 B90 16 In PQR PR>PQ PQR > PRQ But 1 Page of 6 --------------(i)
17 Adding (1), () --------------(ii) PQR Ð1 > PRQ Ð PSR > PSQ Given, to prove, figure.1½ Proof : GM HL ---------(1).½ Also 1 and --------().½ (1) and () 1 1 AGH DHG ABCD.½ 18 In ABC and ADC, ABAD (given) BACDAC (AC is angle bisector) AC common ABC ADC (SAS) BCDC (CPCT) 19 Let each equal side of the le be x cm x x s x 1 cm Area ss as bs c 0 60 x 1 x 1 xx 1 xx 1 60 1 x 1 5 x 1 5x 1 x 169, x1 Perimeter50 cm Area of equilateral triangle a 10 5 cm.. cm In BDC, DC 10 6 8 cm Area of BDC 1 68 cm Area of shaded region. 19. cm SECTION D Question numbers 1 to 1 carry marks each. 1 5 5 5 5 Page of 6
5 10 6 7 98 17 7 9 9 7 7 7 Value : cooperative learning among classmates without any gender and religious bias. 1 x x (x) x xx x10 Given expression (x x1) (x) (0) (x) Let p(x)x 5x 7x60 p 5 7 60 5 5 8 111 60 7 5 0 0 x is a factor of x 5x 7x60 Dividing p(x) by x x x0(x) (x5) The factors of p(x) are (x) (x5) (x) On putting (pq) a in (1) we get (pq) 0(pq)15a 0a15 a 5a5a (5) (5) (a 5a)[5a(5) (5)] a(a5)5(a5) (a5)(a5) () Replacing a by (pq) on both sides of (), we get Page of 6
(pq) 0(pq)15(pq5) (pq5) 5 Let Q(x) x 5 p x xp Q(p) p 5 p 5 pp 0 p 0 p1 x (1)x x x (x)(x1) 6 Let p(x)ax 5xb p()0 [ Q (x) is a factor] (i.e.) 9a15b0 or 9a15b (1) or 9ab15 p 1 0 1 Q x is a factor of p( x ) 1 (i.e.) a 5 1 b0 a 5 b0 9 a159b 0 9 a159b0 a9b15 () from (1) and () 9aba9b 9aa9bb 8a8b a b 7 1 1 BACDAC.1 In DAC and BAC ADAB (given) ACAC (common) DACBAC (proved above) DAC BAC (SAS).1 ADCABC (cpct).½ In ADQ &ABP ADAB (given) 1 (given) ADQABC (proved above) ADQ ABP (ASA).1 AQAP (cpct).½ 8 In BDC DBCDCBx180.1 (Angle sum property of ) DBC DCBx60.1 BCx60 Adding y on both sides Page 5 of 6
9 ybcx60y 180x60y x180y.1.1 Fig By Median theorem ABAC > AD, BC BA > BE and CA CB > CF (ABBCCA ) > (ADBECF) Sum of sides of ABC > Sum of three medians of ABC 0 BADDACCAECAD BACDAE Proving BACDAE BCDE 1 QPSSPR In PQT, QQPT90180 Q90QPT In PTR, RRPT90180 R90RPT QR 90 QPT 90 RPT RPTQPT TPS SPR QPS TPS TPS (or) TPS 1 (QR) Page 6 of 6