(Question paper - With Answers) STD. X - MATHEMATICS. [Time Allowed : 2½ Hrs.] [Maximum Marks : 100]

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1 GOVT SUPPLEMENTARY EXAM OCTOER - 06 (Question paper - With Answers) STD. X - MATHEMATICS [Time Allowed : ½ Hrs.] [Maimum Marks : 00] SECTION - I 8. The equation of a straight line passing through the point (, 7) and parallel to - ais is : Note: (i) Answer all the 5 questions. (a) (b) 7 (ii) Choose the correct answer from the (c) y 7 (d) y given four alternatives and write the 9. In the figure if ÐPA 0 then ÐPT option code and the corresponding answer. [5 5] C. For any two sets A and, {(A\)È(\A)}Ç (AÇ) is : (a) f (b) A È A (c) AÇ (d) A Ç. The 8 th term of the sequence,,,, 5, 8, P T... is : (a) 0 (b) 0 (a) 5 (b) 4 (c) (d) (c) 40 (d) 60. If the third term of a G.P is, then the product 0. The perimeters of two similar triangles of first 5 term is : are 4 cm and 8 cm respectively. If one side of (a) 5 (b) 5 (c) 0 (d) 5 the first triangle is 8 cm, then the corresponding 4. The sum of two zeros of the polynomial side of the other triangle is : f() + (p + ) + 5 is zero, then the value (a) 4 cm (b) cm of p is (c) 9 cm (d) 6 cm (a) (b) 4 (c) (d) 4. In the adjoining figure, AC 5. If a + b + c 0 has equal roots, then c is (a) 5 m C equal : (a) b (b) b (b) 5 m a 4a 5 (c) -b (d) -b (c) a 4a 60 (d) 5 m A 5m. ( cos q) ( + cot q) 6. If (5 ) (0), then the value of is : (a) sin θ (b) 0 (c) (d) tan θ 7. (a) 7 (b) 7. If the volume of a sphere is 9 p cu.cm, then (c) (d)0 its radius is : Area of the quadrilateral formed by the points (a) cm (b) 4 cm (,), (0,), (0,0), (, 0) is : (a) sq.units (b) sq.units (c) (c) 4 sq.units (d) sq.unit cm (d) cm [] 0

2 Sura s Mathematics - X Std. October 06 Question Paper with Answers 4. If t is the standard deviation of,y,z then the standard deviation of + 5, y + 5, z + 5 is : (a) t (b) t + 5 (c) t (d) yz 5. If p is the probability of the event A, then p satisfies: (a) 0 < p < (b) 0 p (c) 0 p < (d) 0 < p SECTION - II Note: (i) Answer 0 questions (ii) Question number 0 is compulsory. Select any 9 questions from the first 4 questions. [0 0] 6. For the given function F {(,), (, 5), (4, 7), (5, 9), (, )}, write the domain and range. 7. In a flower garden, there are rose plants in a first row, in the second row, 9 in the third row and so on. There are 5 rose plants in the last row. How many rows are there in the flower garden? 8. Find the quotient and the remainder when 7 + is divided by. 9. Form the quadratic equation whose roots are + 7, If A , then find the additive inverse of A.. Construct a matri A [a ij ] whose elements are given by a ij i j i+ j.. If the points (p, 0) and (0, q ) and (, ) are collinear, prove that +. p q. ACD is a quadrilateral such that all of its sides touch a circle. If A 6 cm, C 6.5 cm and CD 7 cm, then find the length of AD. 4. Prove the identity (sin 6 θ+cos 6 θ) sin θcos θ. 5. Find the angular elevation (angle of elevation from the ground level) of the Sun when the length of the shadow of a 0 m long pole is 0 m. 6. If the curved surface area of a solid hemisphere is 77 sq.cm, then find its total surface area. 7. Radius and slant height of a cone are 0 cm and 9 cm respectively. Find its volume. 8. Calculate the standard deviation of the first natural numbers. 9. If A and are two events P(A), P() 7, P(AÈ), find 0 (i) P(AÇ) (ii) P(A È ) 0. For A { - is a prime factor of 4}, { 5 <, Î N} and C {, 4, 5, 6}, find A È (ÈC). [OR] If the centroid of a triangle is at (, ) and two of its vertices are ( 7, 6) and (8, 5) then find the third verte of the triangle. SECTION - III Note: (i) Answer 9 questions. (ii) Question No. 45 is Compulsory. Select any 8 questions from the first 4 questions. [9 5 45]. Use Venn diagram to verify (AÇ) A È.. A function f : [ 7, 6) is defined as follows: + + ; 7 < 5 f( ) + 5 ; 5 ; < < 6 Find: (i) f ( 4) + f () (ii) f ( 7) f ( ) (iii) 4 f( ) + f( 4 ) f( 6) f(). The 0th and 8th terms of an A.P. are 4 and 7 respectively. Find the 7th term.

3 Sura s Mathematics - X Std. October 06 Question Paper with Answers 4. Find the total volume of 5 cubes whose edges are 6 cm, 7 cm, 8 cm,... 0 cm respectively. 5. Solve : Find the square root of If a and b are the roots of the equation form an equation whose roots are : (i) a b, b a a 8. If A c (ii) a+b, b+a b 0 d and I 0, then show that A (a + d)a (bc ad)i. 9. If the vertices of a DAC are A(, 4), (,) and C(,5). Find the equation of the straight line along the altitude from the verte. 40. Find the area of the quadrilateral whose vertices are ( 4, 5) (0, 7) (5, 5) and ( 4, ). 4. From the top and foot of 40 m high tower, the angles of elevation of the top of a lighthouse are found to be 0 and 60 respectively. Find the height of the lighthouse. Also find the distance of the top of the lighthouse from the foot of the tower. 4. The curved surface area and total surface area of solid circular cylinder are 880 sq.cm and 88 sq.cm respectively. Find its Volume. 4. Calculate the co-efficient of variation of the following data: 8, 0, 5,, In the class, 40% of the students participated in Mathematics - quiz, 0% in Science - quiz and 0% in both the quiz programmes. If a student is selected at random from the class, find the probability that the student participated in Mathematics or Science, or both quiz programmes. 45. (a) A circus tent is to be erected in the form of a cone surmounted on a cylinder. The total height of the tent is 49m. Diameter of the base is 4m and height of the cylinder is m. Find the cost of canvas needed to make te tent, if the cost of canvas is `.50/m (Take p 7 ) [OR] b) State and prove the converse of Angle isector theorem. SECTION - IV Note: Answer both the questions choosing either of the alternatives. [ 0 0] 46. (a) Draw a circle of diameter 0 cm. From a point P, cm away from its centre, draw the two tangents PA and P to the circle, and measure their lengths. [OR] (b) Construct a cyclic quadrilateral ACD with A 7 cm, ÐA 80, AD 4.5 cm and C 5 cm. 47. (a) Solve the equation 0 graphically. [OR] (b) Draw the Graph of y 0,, y >0. Use the graph to find y when 5, and to find when y 0.

4 4 Sura s Mathematics - X Std. October 06 Question Paper with Answers SECTION I. (a). (d). (b) 4. (c) 5. (b) 6. (b) 7. (d) 8. (c) 9. (d) 0. (d). (b). (c). (b) 4. (c) 5. (b) 6. Solution: SECTION II Domain {,, 4, 5, } Range {, 5, 7, 9, } 7. Solution: ANSWERS Let n be the number of rows in the flower garden. The number of rose plants in the st, nd, rd,..., n th rows are,, 9,..., 5 respectively. t, t, t 9 t n 5 Here d t t In an A.P. t n a+(n )d t n Þ + (n )( ) 5 Þ n n 5 n 5 5 n 0 n 0 So, there are 0 rows in the flower garden. 8. Solution: Let P() 7 + Given that the divisor is When 0 Þ So, 7 + ( 5+)+ Þ ( ) ( 5 + ) + Þ ( ) ( 5 + 7) + \ Quotient 5 + 7, Remainder 9. Solution : Sum of the roots ( + 7 )+( 7 ) 6 Product of the roots ( + 7 )( 7 ) \ Required quadratic equation (Sum of the roots) + (Product of the roots) Solution : A Additive inverse of A ( A) 6 6. Solution : a ij i j i+ j A a a a a a + 0, a +, a +, a Required Matri A 0. Solution: Given points are (p, 0) (0, q ), (, )

5 Sura s Mathematics - X Std. October 06 Question Paper with Answers 5 { p 0 0 q 0 p q q + p Þ p q q +p Dividing by p q we get, pq q p + pq pq pq + p q Þ +. Hence it is proved. p q. Solution: Let P, Q, R and S be the points where the circle touches the quadrilateral. We know that the lengths of the two tangents drawn from an eterior point to a circle are equal. 7 cm D C R \ A AS...() P Q...() S Q CR CQ...() P A DR DS...(4) 6 cm Adding the equations (), (), () and (4) we get, AP + P + CR + DR AS + Q + CQ + DS Þ A + CD AD + C. AD A + CD C Thus, AD 6.5 cm. 4. Solution: sin 6 θ + cos 6 θ (sin θ) + (cos θ) (sin θ +cos θ) sin θcos θ (sin θ + cos θ) ( a + b (a + b) ab(a + b)) sin θ cos θ. ( sin θ + cos θ ) 5. Solution: Let S be the position of the Sun and C be the pole. Let A denote the length of the shadow of the pole. p { Let the angular elevation of the Sun be q. 6.5 cm Given that A 0 In the Right D CA, tan q C A 0 0 m and C 0 m tan q A q 60 0 m Thus, the angular elevation of the Sun from the ground level is Solution: Curved surface area of a solid hemisphere pr sq.units. pr 77 pr 77 CSA 77sq.cm 86 Its total surface area pr 86 Total surface area of a hemisphere 458 sq.cm 7. Solution: Radius of a cone r 0 cm Slant height l 9 cm \ Height h l r cm Volume of the cone V pr h cu.units Volume of the solid cone 8800 cu.cm. 8. Solution: Standard deviation of the first n natural numbers n C 9 cm S 0 m 0 cm

6 6 Sura s Mathematics - X Std. October 06 Question Paper with Answers Thus, the standard deviation of the first natural numbers Solution: (i) P(A Ç ) P(A) + P() P(A È ) y addition theorem, (ii) P(A È ) P(A Ç) P(A Ç) Solution: A { / - is a prime factor of 4 } {,, 7} { / 5 <, Î N} {6, 7, 8, 9, 0,, } C {, 4, 5, 6} ÈC {6,7,8,9,0,,}È{,4, 5, 6} {, 4, 5, 6, 7, 8, 9, 0,, } \ AÈ(ÈC) {,,7} È {,4,5,6,7,8,9,0,,} {,,,4,5,6,7,8,9,0,,} (OR) Solution: Centroid of triangle G (, y) (, ), Vertices of triangles are A (, y ) ( 7, 6), (, y ) (8, 5) C (, y )? \ Centroid of triangle G (, y) y y y G , G(, ) G y, + + y G, Equating the -co-ordinates, we get +. Solution: Þ Equating the y-co-ordinates, we get + y Þ y 9 \ Third verte C(, y ) (, ) (AÇ) A È SECTION III (A ) (A ) U U

7 Sura s Mathematics - X Std. October 06 Question Paper with Answers 7 A A From () & (5) it is clear that (AÇ) A È. Solution: (i) f ( 4) ( 4) + 5 (Since 4 lies in the interval 7 < 5 use +5) f () U U U (Since lies in the interval 5 use + 5) f ( 4) + f () () + (7) + (ii) f ( 7) ( 7) + ( 7) ((Since 7 lies in the interval 7 < 5 use + + )) f ( ) ( ) + 5 (Since lies in the interval 5 use + 5) f ( 7) f( ) 6 4 (iii) f ( ) from (ii) f (4) 4 f ( 6) ( 6) + ( 6) (Since 6 lies in the interval 7 < 5 use ++) f () (Since lies in the interval 5 use + 5) \ 4 f( ) + f( 4 ) 4 ( ) + () f( 6) f() 5 6 ( ) Solution: Given, t 0 4 Þ a + (0 )d 4 Þ a + 9d 4... () and t 8 7 Þ a + (8 )d 7 Þ a + 7d 7... () Solving () and () a + 9d 4 a + 7d 7 ( ) ( ) ( ) 8d Substituting in () d 4 Þ a Þ a a 5 \ t 7 a + (7 )d a + 6d (4) \ The 7 th term Solution: The volume of the cubes from the series ( ) ( ) 0 5 0( 0 + ) 5( 5 + )

8 8 Sura s Mathematics - X Std. October 06 Question Paper with Answers (5 ) (5 8) ( ) (465 0) cu.cm The total volume of the 5 cubes 085 cu.cm 5. Solution: Given Þ Þ Þ ( + ) ( + ) Þ 0( + + ) 6( +) Þ Þ Þ Þ ( + 6) ( 5) 0 Þ 6, 5 6 \ Solution set is { 6, 5} 6. Solution: Given polynomial is already in descending powers of ( 5 + 6) 4 \ Solution: a and b are the roots of the equation a + b ( 6) ab (i) a b, b a Sum of the roots of required equation a b + b a ab(a + b) Product of the roots a b b a a b (ab) 7 \ Required equation (ii) Þ a+b, b+a Sum of the roots (a+b) + (b+a) a + b 6 Product of the roots (a+b) (b+a) 4ab+ a + b + ab 5ab+ (a + b ) 5ab+ ((a + b) ab) 5 + (() ) \ Required equation Þ

9 Sura s Mathematics - X Std. October 06 Question Paper with Answers 9 8. Solution: a A c Consider A A A A b d and I a c 0 0 b a d c a (a + d)a (a + d) c From equation () and () A (a + d)a b d a + bc ab+ bd ac + cd bc + d...() b d a + ad ab+ bd ac + cd ad + d...() a + bc ab+ bd ac + cd bc + d a + ad ab+ bd ac + cd ad + d bc ad 0 0 bc ad 0 (bc ad) 0 (bc ad) I \ A (a + d)a (bc ad)i Hence it is proved 9. Solution: Let (, y ) A (, 4) (, y ) (, 5). \ Slope of AC y y 5 ( 4 ) (,) A(, 4) D C(,5) m 9 Also D passes through to AC \ Slope of D is m \ m m ( D ^ AC) m m The equation of D at slope, and passes through (, ) is Þ y ( ) y Solution: \ The required equation is y The vertices of the quadrilateral are ( 4, 5) (0, 7), (5, 5) and ( 4, ) Area ( ) ( ) { } {( 58) ( 6) } { } 4. Solution: 60.5 Sq.units Required to find CD and AD. Let the distance of the top of the light house ED be m CD ( +40) m. From the rt.dle ED; tan 0 y Þ y ( 4, 5) A(0, 7) C ( 4, ) D(5, 5) Þ y...() y

10 0 Sura s Mathematics - X Std. October 06 Question Paper with Answers From the rt Dle ACD; \ tan y Þ 40 + y Þ y () Sub.() in () ( ) 40 + Þ 40 \ 40 or 0 \ Height of the light house CD From the rt D le ACD; sin AD Þ + 40 AD \ ( + 40 ) AD 0 60 m. 60 ( ) ( 60) 40 m Tower 40 m A 0 0 y m 60 0 y m D m E Light house 40 m Distance of the top of the lighthouse from the foot of the tower 40 m C 4. Solution: The curved surface area of Solid Cylinder (CSA) 880 sq.cm Þ prh () Total surface area (TSA) 88 sq.cm Þ pr(h+r) () Substituting () in () we get Þ prh+pr 88 Þ pr 88 Þ pr Þ pr 08 Þ pr 54 Þ r Þ r Þ r 7 cm Substituting r 7 cm in equation () we get 7/ h h cm 44 volume of the cylinder pr h 7 7/ cm 7 4. Solution: Average d d Sd 0 Sd 98

11 Sura s Mathematics - X Std. October 06 Question Paper with Answers s 44. Solution: Σd n Co-efficeient of variation (C.V) σ \ Co-efficeient of variation 4.6 n(s) 00 No. of students participated in Maths Quiz 40% No. of students participated in Science Quiz 0% No. of students participated in both quiz programmes 0% Let A denote the event of taking part in Math Quiz. \ n(a) 40 Þ P(A) n ( A) n( S ) Let denote the event of taking part in Science quiz. \ n() 0 Þ P() n ( ) n( S ) 0 00 and (AÇ) denotes the event of taking part in both Maths and Science Quiz \ n(aç) 0 Þ P(AÇ) n ( A ) 0 n( S) 00 Using addition theorem on probabilities 45. (a) Solution: P(AÈ) P(A) + P() P(AÇ) Cylindrical Part Conical Part Diameter, r 4 m Radius, r m Radius, r m Height, h 49 8 m Height, h m Slant height,l h + r Total area of the canvas needed CSA of the cylindrical part prh + prl pr (h + l) ( + 5) m + CSA of the conical part Therefore, area of the canvas 508 m Now, the cost of the canvas per sq.m `.50 Thus, the total cost of the canvas `655. (OR) (b) Theorem: If a straight line through one verte of a triangle divides the opposite side internally (eternally) in the ratio of the other two sides, then the line bisects the angle internally (eternally) at the verte. Case : (Internally)

12 Sura s Mathematics - X Std. October 06 Question Paper with Answers Given : In DAC, the line AD divides the opposite side C internally such that D DC A... () AC To Prove : AD is the internal bisector of ÐAC i.e., to prove ÐAD ÐDAC Construction : Through C draw CE AD meeting A produced at E. Proof : Since CE AD, by Thales theorem, we have D DC A AE... () Thus, from () and () we have, A A AE AC \ AE AC Now, in D ACE, we have Ð ACE Ð AEC ( AE AC )... () 46. Solution: (a) Given: Diameter of the circle 0 cm 0 \ radius 5 cm OP cm SECTION IV 5 cm O Rough Diagram A D C Since AC is a transversal of the parallel lines AD and CE, we get, ÐDAC ÐACE (alternate interior angles are equal)... (4) Also E is a transversal of the parallel lines AD and CE. we get ÐAD ÐAEC ( corresponding angles are equal)... (5) From (), (4) and (5), we get ÐAD ÐDAC \ AD is the angle bisector of ÐAC. Hence the theorem. cm P A E 5 cm

13 Sura s Mathematics - X Std. October 06 Question Paper with Answers Fair Diagram Steps of Construction: (i) (ii) O 5 cm Let the two circles intersect at A and With O as the centre draw a circle of radius 5 cm A cm (iii) Draw the perpendicular bisector of OP. Let it meet OP at M. (iv) (v) (vi) Verification: With M as centre and MO as radius, draw another circle. Let the two circles intersect at A and Join PA and P. They are the required tangents. Length of the tangent, PA cm In the right angled D OPA, cm cm PA OP OA cm \ PA P cm M P (OR)

14 4 Sura s Mathematics - X Std. October 06 Question Paper with Answers (b) Given: In the cyclic quadrilateral ACD, A 7 cm, A 80º, AD 4.5 cm and C 5 cm. Rough Diagram Fair Diagram Steps of Construction: 4.5 cm D 80 X O A 7 cm (i) Draw a rough diagram and mark the measurements. C Draw a line segment A 7 cm. (ii) Through A draw AX such that AX 80º. (iii) With A as centre and radius 4.5 cm, draw an arc intersecting AX at D and join AD. (iv) Draw the perpendicular bisectors of A and AD intersecting each other at O. (v) With O as centre and OA OD as radius, draw the circumcircle of DAD (vi) With as centre and radius 5 cm, draw an arc intersecting the circle at C. (vii) Join C and CD. Now, ACD is the required cyclic quadrilateral. 5 cm 4.5 cm D 80 A 7 cm C 5 cm

15 Sura s Mathematics - X Std. October 06 Question Paper with Answers (a) Solution: 0 Þ y Table y Plot the points (, ), (, 5), (, 0), (0, ), (, 4), (, ), (, 0), (4, 5) Draw the graph by joining the points by a smooth curve. y O Scale: ais cm unit y ais cm units y y The curve intersects the -ais at the points (, 0) and (, 0). The -coordinates of the above points are and. Thus the solution set is {, }.

16 6 Sura s Mathematics - X Std. October 06 Question Paper with Answers 47. (b) Solution: We form the table for Rectangular hyperbola y 0 Here y y Plot the points (, 0), (, 0), (4, 5) and (5, 4) and join them y Y (,0) (,0) (4,5) (5,4) y y 0 Scale: ais cm unit y ais cm units (0,) \ The relation y 0 is a rectangular hyperbola as ehibited in the graph. From the graph we find, (i) When 5, y 4 (ii) When y 0,.

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 -