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• 1. www.eenadupratibha.net QUARTERLY EXAMINATIONS X Class Mathematics (English Medium) MODEL PAPER 1Time: 2  Hours Parts: A & B Max. Marks: 50 2Time: 2 Hours Part - A Marks: 35 SECTION - INote: 1) Answer any 5 of the following questions choosing at least 2 from each group A & B. 2) Each question carries 2 Marks (5 × 2 = 10) et Group - A .)n1. Define conjunction and write truth table.2. If A, B are two sets. Prove that A - B = B - A3. h(a x+3 3x + 3 If f : R − {3} → R is defined by f(x) =  Show that f  = x (x ≠ 1) tib x-3 x-14. Find the value of k so that x3 - 3x2 + 4x + k is exactly divisible by (x - 2) pra Group - B  √35. du In an equilateral triangle with side a, prove that the area of the triangle is a2 4 na6. The mean of 10 observations is 16.3. By an error, one observation is registered ee) as 32 instead of 23. Find the correct mean.7. ( )( .) ( w 1 3 If 0 1 2 –1 = p -1 Find p.8. w w Write the merits of the Arithmetic mean. SECTION - IINote: 1) Answer any 4 of the following questions 2) Each question carries 1 Mark (4 × 1 = 4)9. n(A∪B) = 51, n(A) = 20, n(B) = 44 then find n(A∩B).10. Define constant function  11. Find the sum and product of the roots of the equation √ 3x2 + 9x + 6√ 3 = 012. Determine k so that k + 2, 4k - 6 and 3k - 2 are the three consecutive terms of an A.P. www.eenadupratibha.net
• 2. www.eenadupratibha.net13. The mean and median of a unimodal grouped data are 72.5 and 73.9 respec- tively. Find the mode of the data. 1 414. If A = 2 1 ( ) find A2 . SECTION - IIINote: 1) Answer any 4 of the following questions choosing at least two from each group A & B. Each question carries 4 Marks (4× 4 = 16) Group - A15. For any three sets A, B, C prove that A∪ (B∩C) = (A∪B) ∩ (A∪C).16. Let f, g, h be functions defined by f(x) = x + 2, g(x) = 3x - 1 and h(x) = 2x, show that ho (gof) = (hog) of.17. ( 5 9 Find the independent term of x in the expansion of 3x −  ) et a. 1 1 1 x2 n If (b+c) (c+a) and (a+b) are in H.P. show  ,  ,  will also be in H.P. h18. a2 b2 c2 tib Group - B ra19. Define Thales theorem and prove it.20. p Find the median of the marks scored by 50 students in a 50 Marks test u d Marks 1 - 10 11 - 20 21 - 30 31 - 40 41 - 50 na No. of students 3 12 16 14 5 ee21. Using matrix inversion method, solve the linear equations . 2x + 5y = 11, 4x - 3y = 9 w ) ( ) w( -2 1 2 022. Given A = 3 -1 , B = 5 -3 prove that (AB)−1 = B−1 A−1 w SECTION - IVNote: Answer any one of the following. Each question carries 5 marks. (1 \$ 5 = 5)23. Construct a triangle ABC in which BC = 4 Cm, A = 50° − and altitude through A = 3 Cm.24. Using graph of y = x2, solve the equation x2 − 4x + 3 = 0 www.eenadupratibha.net
• 3. www.eenadupratibha.net PART - BMax. Marks: 15 Time: 30 MinutesNote: Pick out the correct answer from the choice. Each question carries 1/2 mark.1. For some symbol of quantifier ( ) ⊃ A) B) ∈ C) ∨  D) ∃2. (A∪ B) = ( ) A) A ∪ B B) A ∩ B C) A∪ B D) A∩B3. For the following which one is bijection? ( ) A f B A f B A f B A f B a x a x a x a > x et > > > > > A) b y B) b > y C) b > y D) b > y > .n c z c > z c > z c z ha If f(x) = x2 - 3x + 2, f(-2) = tib4. ( ) A) 12 B) 4 C) -12 D) 0 ra5. Discriminent of 2x2 - 7x + 3 = 0 is ( ) A) 20 up B) 24 C) 25 D) 26 d6. n arithmetic means are there in between a and b, Then d = ( ) na a-b b−a a+b b-a A)  B)  C)  D)  n+1 n+1 n-1 n-17. .ee Angle in a semi circle is ( ) w A) 0° B) 90° C) 180° D) 360° w8. Mean of 12, 15, x, 19, 25, 44 is 25 then x = ( ) w  A) 20 B) 25 C) 30 D) 359.  If 2 −4 d 5 = 14 then d = ( ) A) −1 B) 1 C) 2 D) 410. ∆ ABC ∼ ∆PQR, A = 50°, B = 60° then R = ( ) A) 50° B) 60° C) 70° D) 80°II. Fill in the Blanks11. B and C are disjoint sets. (A - B) ∪ (A - C) = ------12. I is identity function, I-1 (4) = ----- www.eenadupratibha.net
• 4. www.eenadupratibha.net13. p Λ ∼p is example of ---14. K a, K b, K c (K ≠ 0) are in G.P. then a, b, c are in ----- x x x x15. Median of  , x, , ,  (x > 0) is 8 then x = ----- 5 4 2 316. Harmonic Mean of x & y is -----17. Value of x2 − x − 2 < 0 is in between -----, ----- ( )18. If P = 1 0 then P −1 = ----- 0 119. Angles in the same segment of a circle are -----20. If AT = A , A is called ----- Matrix. etIII. Match the following .n(i) Group A Group B ⊃ B then A ∩ B = a21. A ( ) A) B22. h In a G.P. a = 2, S∞ = 6 then r = ( ) B) A tib23. A∪A = ( ) C) ∅ ra24. If f(x) = x + 2, g(x) = x, fog(x) = ( ) D) µ p25. Quadrants of y = mx2 (m > 0) ( ) E) x + 2 du F) I, IV 2 na G)  3 H) I, II(ii) .ee Group A Group B w26. A.M. of a + 2, a, a − 2 ( ) I) 6028. w )27. 70, 60, 70, 80, 60, 100, 60 mode w )( (1 0 0 1 0 1 1 0 = ( ( ) ) J) 90° K) a29. In a ∆ ABC BC2 + AB2 = AC2 then B= ( ) L) ( ) 1 0 0 030. If A = ( ) ( ) 2 6 8 5 ,B= 2 x 6 5 A = B then x = ( ) M) I N) 70 O) 8 P) ( ) 0 1 1 0 www.eenadupratibha.net
• 5. www.eenadupratibha.net Answers SECTION - I Group - A1. Define Conjunction and write truth table.Sol: If two or more statements are combined with connective and then the Compound Statement is said to be conjunction. This is denoted by p Λ q.Ex: 2 is even and 2 prime number Definition of Truth table. If p & q both are true then p Λ q is true.Table: p q pΛq et T T T .n T F F F T F ha F F F tib2. If A, B are two sets. prove that A − B = B − A raSol: To Prove p B−A ⊃ (i) A - B du (ii) B − A ⊃ A - B na (i) x ∈ A - B ⇒ x ∈ A and x ∉ B ⇒ x ∉ A and x ∈ B ee ⇒x∈B−A . ∴ A - B ⊃ B − A ------ (1) ww (ii) x ∈ B − A ⇒ x ∈ B and x ∉ A ⇒ x ∉ B and x ∈ A w ⇒ x ∈ A - B ∴B−A ⊃ A - B ------- (2) from (1) & (2) A - B = B − A3. x+3 3x +3 ( ) If f : R-{3} ->R is defined by f(x) =  show that f  = x (x ≠ 1) x−3 x −1 3x+3 +3Sol: LHS= f  ( ) 3x + 3 x−1 = x−1 3x + 3  −3 x−1 www.eenadupratibha.net
• 6. www.eenadupratibha.net 3x + 3 + 3 (x − 1) / x − 1 =  3x + 3 − 3(x − 1) / x − 1 3x + 3 + 3x − 3 =  3x + 3 − 3x + 3 6x =  = x RHS 6 3x + 3 ( ) ∴f  = x x−14. Find the value of k so that x3 − 3x2 + 4x + k is exactly divisible by (x − 2).Sol: f(x) = x3 − 3x2 + 4x + k et f(x) is exactly divisible by x − 2 ∴ f(2) = 0 .n f(2) = 23 – 3(2)2 + 4(2) + k = 0 a h 8 – 12 + 8 + k = 0 tib 16 − 12 + k = 0 ra 4+k=0 p ∴ k = −4 du Group - B  √3 2 na5. In an equilateral triangle with side a, prove that the area of the triangle is a . 4 eeSol: side an equilateral triangle = a . A In ∆ ABC AB2 = AD2 + BD2 ww) ( a 2 a2 = x2 +  a x a w 2 a2 4a2−a2 3a2 x 2 = a2 −  =  =  4 4 4 B a/ D a/ C  2 2  ∴Height: x = √ 3a2 √ 3 a = 4 2 1 ∴ Area of a equilateral triangle  =2 × bh  1 √3 a =×a× 2 2 www.eenadupratibha.net
• 7. www.eenadupratibha.net  √ 3a2 =  Sq. units. 46. The mean of 10 observations is 16.3 By an error, one observation is registered as 32 instead of 23. Find the correct mean.Sol: The mean of 10 observations = 16.3 Sum of 10 observations = 16.3 × 10 = 163 Registered as 32 instead of 23 163 – 32 + 23 = 154 154 ∴ Correct men  15.4 10 = = et 1 3 2 p = .n7. 0 1 ( )( ) ( ) −1 −1 then p = ? a 1 3 2 p = hSol: 0 1 −1 −1 tib ( )( ) ( ) 1 × 2 + 3(−1) [ ][] 0 × 2 + 1(−1) = p −1 2−3 [ ][] 0−1 = pra p −1 −1 [ ][] du p na −1 = −1 ee ⇒ p = −1 .8. Write the merits of Arithmetic mean. wwSol: i) It is uniquely defined. i.e., it has one and only one value ii) It is based on all observations w iii) it is easily understood iv) it is easy to compute SECTION − II9. If n(A∪B) = 51, n(A) = 20, n(B) = 44 then find n(A∩B) = ?Sol: n(A∪B) = n(A) + n(B) - n(A∩B) 51 = 20 + 44 - n(A∩B) 51 = 64 - n(A∩B) ∴ n(A∩B) = 64 - 51 = 1310. Define Constant function. www.eenadupratibha.net
• 8. www.eenadupratibha.netSol: A function f : A → B is a constant function if there is an element c ∈ B such that f(x) = C for all x ∈ A Ex: f = {(x, 5) / x ∈ N}  11. Find the sum and product of the roots of the √ 3x2 + 9x + 6√ 3 = 0  Sol: a = √ 3, b = 9, c = 6√ 3   −b −9 √ 3 −9√ 3  The sum of roots  =  ×  =  = −3√ 3   a √3 √3 = 3  c 6√ 3 The product of roots = =6  a √3 =12. Determine k so that k + 2, 4k − 6 and 3k − 2 are the three consecutive terms of et an A.P. .nSol: 4k − 6 − (k + 2) = (3k − 2) − (4k − 6) a ⇒ 4k − 6 − k - 2 = 3k − 2 − 4k + 6 h tib ⇒ 3k − 8 = k + 4 ⇒ 3k + k = 4 + 8 ⇒ 4k = 12 pra u 12 ∴k==3 4 d na13. The mean and median of a unimodal grouped data are 72.5 and 73.9 respective- ee ly. Find the mode of the data. .Sol: Mode = 3 × median - 2 × mean ww = 3(73.9) -2 (72.5) = 221.7 -145 w( )14. If A = = 76.7 1 4 2 1 Find A2. 1 4 1 4Sol: A2 = A × A = 2 1 ( )( ) 2 1 1×1+4×2 1×4+4×1 = ( ) 2×1+1×2 2×4+1×1 1+8 4+4 = 2+2 8+1 ( ) www.eenadupratibha.net
• 9. www.eenadupratibha.net 9 8 = 4 9 ( ) SECTION - III Group - A15. For any 3 sets A, B, C prove that A∪(B∩C) = (A∪B)∩(A∪C) ⊃Sol: We have to prove that: (i) A∪(B∩C) (A∪B)∩(A∪C) ⊃ (ii) (A∪B)∩(A∪C) A∪(B∩C). (i) x ∈ A∪(B∩C) ⇒ x ∈ A or x ∈ (B∩C) ⇒ x ∈ A or (x ∈ B and x ∈ C) et ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C) .n ⇒ x ∈ (A∪B) ∩ (A∪C) ∴ A∪(B∩C) a (A∪B) ∩ (A∪C) ------ (1) ⊃ h tib (ii) x ∈ (A∪B) ∩ (A∪C) ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C) pra ⇒ x ∈ A or (x ∈ B and x ∈ C) ⇒ x ∈ A or x ∈ (B ∩ C) du ⇒ x ∈ A∪(B∩C) na ∴ (A∪B) ∩ (A∪C) A ∪ (B∩C) ------- (2) ⊃ ee From (1) & (2) A ∪ (B∩C) = (A∪B) ∩ (A∪C)16. w. Let f, g, h functions be defined by f(x) = x + 2, g(x) = 3x − 1 & h(x) = 2x. Then show that ho(gof) = (hog)of wwSol: (i) ho(gof)(x) gof(x) = g[f(x)] = g[x + 2] = 3(x + 2) − 1 = 3x + 6 − 1 = 3x + 5 ho(gof)(x) = h[gof(x)] = h[3x + 5] = 2(3x + 5) = 6x+ 10 -------- (1) www.eenadupratibha.net
• 10. www.eenadupratibha.net (ii) (hog)of(x) hog(x) = h[g(x)] = h[3x − 1] = 2(3x − 1) = 6x − 2 (hog)of(x) = hog[f(x)] = hog(x + 2) = 6(x + 2) − 2 = 6x + 12 − 2 et = 6x + 10 -------- (2) From (1) & (2) ho(gof) = (hog)of ( ) .n17. Find the Independent term in the expansion of 3x−  5 9 −5 ha x2 tibSol: a = 3x, b = , n = 9, r = r x2 ra − Tr+1 = nc xn−r yr r ( ) up = 9Cr (3x)9−r  −5 r d x2 na = 9Cr (3)9−r (x)9−r (−5)r. x−2r ee = 9Cr (3)9−r (−5)r. x9−r−2r . = 9Cr (3)9−r (−5)r. x9−3r w We have to find independent term ww∴ Power of x = 0 9 − 3r = 0 9 3r = 9 ⇒ r =  = 3 3 ∴T3+1 = 9C3 (3x)9−3  x2 ( ) −5 3 (−5)3 T4 = 9C3 (3)6. x6.  x6 T4 = 9C3 (3)6 (−5)3 www.eenadupratibha.net
• 11. www.eenadupratibha.net 1 1 118. If (b + c), (c + a) & (a + b) are in H.P. then show that , ,  will also be a2 b2 c2 in H.P.Sol: (b + c), (c + a), (a + b) --- are in H.P 1 1 1  ,  ,  --- are in A.P. b+c c+a a+b 1 1 1 1 ∴− =  −  c+a b+c a+b c+a b+c−c−a / / c+a−a−b / /  =  / (c + a) (b + c) / (a + b) (c + a) b−a c−b et  = b+a c+b  .n by cross multiplication ha (b − a) (b + a) = (c − b) (c + b) b2 − a2 = c2 − b2 tib ∴ a2, b2, c2 are in A.P. ra 1 1 1 ∴ , ,  are in H.P. a2 b2 c2 up Group − B19. d Define Thales theorem and prove it. naSol: In a triangle, a line drawn parallel to one side, will divide the other two sides in ee the same ratio. .Given: ∆ABC, in Which DE//BC and DE intersects AB in D and AC in E w AD AETo prove:  =  w DB EC A wConstruction: Join BE, CD and draw EF ⊥ BAProof: In ∆ADE, ∆BDE 1 ∆ADE  × AD.EF AD D F E 2  =  =  ------- (1) ∆BDE 1  × BD.EF DB 2 > ∆AED AE B C similarly:  =  ------- (2) ∆ECD EC But ∆BDE = ∆ECD (Triangles on the same base DE, and between the same parallel lines DE & BC) www.eenadupratibha.net
• 12. www.eenadupratibha.net AD AE From (1) & (2) We get  =  DB EC20. Find the median of the marks scored by 50 students in a 50 marks test. Marks 1 - 10 11 - 20 21 - 30 31 - 40 41 - 50 No. of 3 12 16 14 5 StudentsSol: CI f Cum.Frequency N 50 1 - 10 3 3  =  = 25 2 2 11 - 20 12 15F et 21 - 30 16f 31 L = 20.5 Median class 31 - 40 14 45 a.n 41 - 50 5 50 h N = 50 ( ) ra tib N -F 2 ×C Median = L+  f ( ) dup = 20.5 + 25 - 15  × 10 na 16 100 ee = 20.5 +  16 w. = 20.5 + 6.25 = 26.7521. ww Using Matrix inversion method, solve the linear equation 2x + 5y = 11, 4x-3y=9Sol: 4x - 3y = 9 2x + 5y = 11 A= ( ) () () 2 5 4 −3 X= x y B= 11 9 AX = B ⇒ X = A−1 B A = 2(−3) − 5 × 4 = −6 − 20 = −26 ≠ 0 ∴ A−1 exists. www.eenadupratibha.net
• 13. www.eenadupratibha.net d −b A−1 =  1 ad − bc ( ) −c a −3 −5 =  1 2(−3)−5 × 4 [ ] −4 2 [ ] 3 5   −3 −5 =  1 −6 −20 [ ] −4 2 = 26 4 26 26 −2  26 X = A−1 B ( )( ) 3 5   26 26 11 = 4 −2   9 et 26 26  + 45 33 78 () ( ) ( ) ()   .n x 26 26 26 3 X= = = = a y 44 − 18 26 1    h 26 26 26 tib ⇒ x = 3, y = 1 ra −2 1 2 0 ( ) ( )22. Given = 3 −1 , B = 5 −3 then Prove that (AB)−1 = B−1A−1. ( up) (−2)(2) + (1)(5) (−2)(0) + (1)(−3)Sol: AB = d na( ) (3)(2) + (−1)(5) (3)(0) + (−1)(−3) ( ee ) = −4 + 5 0 −3 = 1 3 w . ( ) (AB)−1 =  6−5 1 0+3 −1 3 3 3 ww (1)(3) − (−3)(1) −1 3 ( ) [ ] 3 3 1 3 3   =  = 6 6 3+3 −1 1 −1 1  6 6 ( )( ) 3  ( ) B= 2 5 −3 0 ⇒ B−1 =  1 −3 0 (2)(−3) − (0)(5) −5 2 = 6 5  6 0 2 − 6 ( ) A= −2 1 3 −1 ( )⇒ A−1 =  1 (−2)(−1) − (1)(3) −1 −3 −1 −2 www.eenadupratibha.net
• 14. www.eenadupratibha.net 1 =  2 −3 ( −1 −3 −1 −2 ) ( ) = 1 3 1 2 ( )( ) 3  0 1 1 B−1 A−1 = 6  − 5 2 3 2 6 6 ( )( ) 3 3 3 3  +0  +0   = 6 6 = 6 6 5 − 6 5 −  4 −   1 1   6 6 6 6 6 6 ∴ (AB)−1 = B−1 A−1.. et SECTION - IV23. Construct a triangle ABC in which BC = 4 cm, A = 50°. and altitude through − .n A = 3 cm. ha tib pra du na .ee ww wConstruction:(1) Draw a line segment BC = 4 cm and make an angle PBC = 50° with the help of a protractor.(2) Draw perpendicular bisector RQ of BC. Draw perpendicular EB to BP. Let RQ and EB intersect in a point say O. Let M be mid point of BC.(3) Taking O as centre and OB as radius draw a circle(4) Take a point L on RQ such that the line segment ML = 3 cm(5) Draw AA // BC through L intersecting the circle in two points say A and A. Join AB, AC and AB, AC. Either of the triangle ABC, ABC will be the required triangle. www.eenadupratibha.net
• 15. www.eenadupratibha.net24. Using graph of y = x2, solve the equation x2 − 4x + 3 = 0. etSol. x2 − 4x + 3 = 0; x2 = 4x − 3. y = x2 is parabola and y = 4x − 3 is straight line .n y = x2 x 0 ha 1 2 3 −1 −2 −3 ib y 0 1 4 9 1 4 9 t ra y = 4x − 3 p x 0 1 2 3 −1 du y −3 1 5 9 −7 a .een ww w www.eenadupratibha.net
• 16. www.eenadupratibha.net PART - B1) D .n 2) B et Answers 3) B 4) A5) C ha 6) B 7) B 8) D tib9) B 10) C 11) A 12) 4 ra 2xy13) Contradiction 14) Arithmetic progression 15) 24 16)  ( )dup x+y 1 017) -1, 2 18) 19) Equal 20) Symmetric matrix 0 1 na21) A 22) G 23) D 24) E .e25) H29) J e 26) K 30) O 27) I 28) P ww Writer: P.Venugopalw S.A.(Maths) Govt. TW AHS (Girls), Jaggannapet, Warangal District www.eenadupratibha.net