Objective Mathematics
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1 . In BC, if angles, B, C are in geometric seq- uence with common ratio, then is : b c a (a) (c) 0 (d) 6. If the angles of a triangle are in the ratio 4 : :, then the ratio of the longest side to the perimeter is : (a) : ( ) : (c) : ( ) (d) :. Let BC and BC' be two non-congruent triangles with sides B = 4, C = C' = and angle B = 0º. The absolute value of the difference between the area of these triangles is : (a) 8 4 (c) 6 (d). In an isosceles triangle if one angle is 0º and radius of its incircle is square units is :, then area of the triangle in (a) 7 7 (c) 7 (d) 4 4. If a, b and c denote the length of the sides opposite to angles, B and C of a triangle BC, then the correct relation is given by : B C (a) ( b c)sin a cos B C ( b c)cos asin B C (c) ( b c)cos asin B C (d) ( b c)sin a cos 5. Three circular coins each of radii cm are kept in an equilateral triangle so that all the three coins touch each other and also the sides of the triangle. rea of the triangle is (a) (4 ) cm (c) (48 7 ) cm 4 ( 7 ) cm 4 (d) (6 4 ) cm 7. In a triangle BC, let C /. If r is the in-radius and is the circum-radius of the triangle then (r + ) is equal to : (a) a + b (c) c + a b + c (d) a + b + c 8. In a triangle BC, B / and C / 4. Let D divides BC internally in the ratio : then sin BD sin CD equal to : (a) / 6 / (c) / (d) / 9. If B, C a units, then 4 area (in sq. units) of traingle BC is : and (a) 6 4 (c) (d) 4 0. Let r, be respectively the radii of the inscribed and circumscribed circles of a regular polygon of n sides such that r 5, then n is equal to : (a) 5 6 (c) 0 (d) 8 r r. In a triangle BC, is equal to : bc ca ab (a) r r r (c) r (d) r [ ] Mathematics for JEE-0
2 Solution of Triangle a b c. If for a triangle BC, b c a 0 then c a b sin sin B sin C is equal to : (a) sin + sinb + sinc sin sinb sinc (c) sin + sin B + sin C (d) sin sin B sin C a b c. In a triangle BC if, then ratio of the radius of the circumcircle to that of the incircle is (a) 5/4 /5 (c) 6/7 (d) 6/ 4. In a triangle BC let D be the altitude form. If b > c, equal to o abc C and D b c (a) º º (c) 47º (d) 57º 5. In triangle BC, if cos cos B cosc a b, then a b c bc ca (a) = 90º B = 90º (c) C = 90º (d) C = 75º then B is 8. If D is the mid-point of side BC of a triangle BC and D is perpendicular to C, then (a) b = a c a = b c (c) b = a c (d) a + b = 5c 9. If two sides of a triangle are the roots of the equation 4x ( 6) x 0 and the included angle is 60º, then the third side is (a) / (c) / (d) / 0. In a triangle BC, if (a + b + c) (b + c a) = bc, then : (a) 0 6 (c) 0 4 (d) 4. Internal bisector of angle of triangle BC meets side BC at D. line drawn through D perpendicular to D intersects the side C at E and the side B at F. If a, b, c represent sides of (a) E is H.M. of b and c bc D cos b c 4bc (c) EF sin b c (d) the triangle EF is isosceles BC, then. If a triangle BC with side a = units is inscribed in a circle of radius 0 units, then in-radius of triangle BC can be : (a) 4 units (c) 5 units 8 units (d) units 6. In a triangle BC, if then C is equal to : (a) 0º 60º (c) 75º (d) 90º a c b c a b c 7. In a triangle with one angle, the lengths of the sides form an.p. If the length of the greatest side is 7 cm, the radius of the circumcircle of the triangle is (a) 7 cm (c) cm 5 cm (d) cm. Let the two adjacent sides of a cyclic quadrilateral be, 5 and the angle between them is. If the area of quadrilateral is 4 square units, then the remaining sides can be : (a) 4 (c) (d) 6 4. Which of the following expressions on solving reduce to the area of triangle BC? (all the notations are having their usual meaning). (a) r( r r r ) r r 4 ( r r ) r r (c) r r r cot r (sin ) (d) r r c b [ 4 ] Mathematics for JEE-0
3 5. For triangle BC, which of the following statements are true? (a) Product of all the side lengths of BC ( r s ). r r r r (c) If r r, then BC is right-angled. (d) If r, then BC is equilateral. 7. In triangle BC, let the side lengths be a = 6, b = 8 and c = 0. Statement : Distance between the circum-centre and in-centre of BC is equal to 5 units Statement : For any triangle, distance between the circum-centre and in-centre is equal to r, where, r represents the circum-radius and in-radius of the triangle. Following questions are assertion and reasoning type questions. Each of these questions contains two statements, Statement (ssertion) and Statement (eason). Each of these questions has four alternative answers, only one of them is the correct answer. Select the correct answer from the given options : (a) Both Statement and Statement are true and Statement is the correct explanation of Statement. Both Statement and Statement are true but Statement is not the correct explanation of Statement. (c) Statement is true but Statement is false. (d) Statement is false but Statement is true. 6. Let be the area of n-sided regular polygon inscribed in a circle 'C' of unit radius and be the area of n-sided regular polygon circumscribing the circle 'C'. Statement : If 4( ), then the number of sides ' n' of the regular polygon are Statement : 4 tan n. 8. Consider an acute-angled triangle BC in which the altitudes are P, BQ and C. Statement : Incentre of triangle PQ is the orthocentre of triangle BC Statement : orthocentre of triangle I I I is the in-centre of triangle BC, where I, I, I denote the centre of escribed circles for triangle BC. 9. Consider a triangle BC, having side lengths a, b, c and circum-radius (). If r, r, r denote the ex-radii of triangle BC, then ab ac bc Statement : 6 r r r a b b c c a Statement : 6 b a c b a c 00. Statement : In triangle BC, if the sides b, c and the angle BC is known, then a unique triangle can b only be formed if sin B and B is acute c b Statement : If sin B and B is obtuse, then c BC doesn't exist. [ 5 ] Mathematics for JEE-0
4 Solution of Triangle 6. If BC = 4 units and the area of BC is ' ' square units, then : Comprehension passage () ( Questions No. - ) (a) 4 tan sin Let circum-radius of BC be '' and the line joining the circum-centre 'O' and in-centre 'I' is parallel to side BC. If,, are the radii of circumcircles of triangles OBC, OC and OB respectively, then answer the following questions. a b c. Value of is equal to : (a) a b c (c) abc. Value of ( cosb + cosc ) is : a bc (d) (a) / (c) / (d) / a b c. For given BC the in-radius is given by : (a) cos B (c) cos C cos (d) none of these Comprehension passage () ( Questions No. 4-6 ) In triangle BC, let the altitude, internal angular bisector and the median from vertex meet the opposite side BC at D, E and F respectively. If BD, and DE EF CF, then answer the following questions. (c) tan tan cot tan (d) tan tan ( ) = Comprehension passage () ( Questions No. 7-9 ) Let triangle BC of area square units be inscribed in a circle of radius 4 units, where 0,. If p, p and p denote the length of altitudes of triangle BC from the vertices, B and C respectively, then answer the following questions. 7. The value of cos cos B cosc 4, is equal to : p p p (a) (c) (d) 4 8. If sides a, b, c are in.p., then maximum value of is equal to : p p p (a) 8 (c) 6 4 (d) 4. If { p } denotes the fractional part of p, where p = [ p ] + { p }, then : (a) {tan B} 0 {sin } = / (c) {cos B} (d) {tan B} {tan C} 9. Minimum value of the expression a p b p c p is equal to : b c a (a) 4 6 (c) 8 (d) 5. Value of tan cos (cos( C)) is equal to : B (a) tan (c) sin (B) tan 4 (d) tan B + tan C 0. Let a, b, c represent the sides of triangle BC, where (b a) = (c b) = and a, b, c N. If C, then value of (c b a) is equal to... [ 6 ] Mathematics for JEE-0
5 . If sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the largest side of triangle is.... Let a, b and c represent the sides of triangle BC opposite to the vertices, B and C respectively. If a b c b c a ( b c ) 0, then value of sec () is equal to.... Let three circles touch one-another externally and the tangents at their points of contact meet at a point whose distance from any point of contact is units. If ratio of the product of radii to the sum of radii of cricles is k :, then k is equal to If 0 is the area of formed by joining the points of contact of incircle with the sides of the given triangle whose area is, similarly, and are the corresponding area of the formed by joining the points of contact of excircles with the sides, then value of 0 is equal to In triangle BC, let the orthocentre (H) and circum-centre (C 0 ) be (, ) and (4, ) respectively. If side BC of the triangle lies on line y = 0 and internal angles are, B, C, then match the following columns (I) and (II). Column (I) Column (II) (a) ( C0) cos (p) sec HB (q) (c) H (r) 4 (d) HC (s) sec 6. In triangle BC, let CH and CM be the lengths of the altitude and median to base B. If side lengths (t) a 5, b 97 and c =, then match the following columns I and II. Column (I) Column (II) (a) Value of cos(tan ( MH )) is (p) Length of in-radius of triangle MHC is (q) (c) If BC is extended to P such that triangle PB is right angled at P, and area of PC is ' ' square units, (r) 5 then integer(s) less than can be (s) MH (d) If PH, then value of tan is more than (t) / [ 7 ] Mathematics for JEE-0
6 Solution of Triangle. (c).. (c) (d) Ex 6. (c) 7. (a) 8. (a) 9. (a) 0. (a). (d).. (c) 4. (a) 5. (a) (a) 8. (a) (c). (a, b, c, d). (a, d). (a, c) 4. (a, b, c, d) 5. (b, c, d) 6. (c) 7. (a) (a) 0. (d) E.. (a). 4. (d) 5. Ex (d) ( 9 ). ( 6 ). ( 4 ). ( 4 ) 4. ( ) 5. (a) t 6. (a) t p q (c) q (c) p, q, s (d) s (d) p, q, t [ 8 ] Mathematics for JEE-0
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