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10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
10thmaths online(e)
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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

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