Q.1 If a, b, c are distinct positive real in H.P., then the value of the expression, (A) 1 (B) 2 (C) 3 (D) 4. (A) 2 (B) 5/2 (C) 3 (D) none of these

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1 Q. If a, b, c are distinct positive real in H.P., then the value of the expression, b a b c + is equal to b a b c () (C) (D) 4 Q. In a triangle BC, (b + c) = a bc where is the circumradius of the triangle. Then the triangle is () Isosceles but not right right but not isosceles (C) right isosceles (D) equilateral Q. Consider the triangle pictured as shown. If 0 < < / then the number of integral values of c is () 5 (C) 4 (D) 5 Q.4 The sum to infinity of the series is equal to : () 5/ (C) (D) none of these Q.5 In an acute angled triangle BC, point D, E and F are the feet of the perpendiculars from, B and C onto BC, C and B respectively. H is the intersection of D and BE. If sin = /5 and BC = 9, the length of H is () (C) 5 (D) 54 Q.6 triangle has sides 6, 7, 8. The line through its incentre parallel to the shortest side is drawn to meet the other two sides at P and Q. The length of the segment PQ is () (C) 7 (D) 9 Q.7 The smallest integer greater than, is log log4 () (C) (D) 4 Q.8 long a road lies an odd number of stones placed at intervals of 0 m. These stones have to be assembled around the middle stone. person can carry only one stone at a time. man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is () 5 9 (C) (D) 5 Q.9 Triangle BC is right angled at. The points P and Q are on the hypotenuse BC such that BP = PQ = QC. If P = and Q = 4 then the length BC is equal to () 7 6 (C) 45 (D) 54 Q.0 If S = (99) then the value of the sum (00) is () S S (C) 4S (D) S

2 Q. In an isosceles triangle BC, B = C, BC = 08 and BD D trisects BC and BD > DC. The ratio is DC () (C) 5 (D) 5 Q. In an.p. with first term 'a' and the common difference d (a, d 0), the ratio ' ' of the sum of the first n terms to sum of n terms succeeding them does not depend on n. Then the ratio d a and the ratio ' ', respectively are (), 4, (C), (D), Q. In BC if a = 8, b = 9, c = 0, then the value of () tan C sin B (C) 4 is 8 (C) 5 Q.4 In a triangle BC, CD is the bisector of the angle C. If cos has the value and l (CD) = 6, then has the value equal to a b () 9 (C) 6 (D) none Q.5 If the angles subtended by the sides of a triangle at orthocentre and incentre are equal,then the triangle is () Scalene Isosceles but not equilateral (C) Equilateral (D) Obtuse angled Q.6 The arithmetic mean of the nine numbers in the given set {9, 99, 999, } is a 9 digit number N, all whose digits are distinct. The number N does not contain the digit () 0 (C) 5 (D) 9 Q.7 With usual notations, in a triangle BC, a cos(b C) + b cos(c ) + c cos( B) is equal to abc abc 4abc abc () (C) (D) 4 Q.8 If for an.p. a, a, a,..., a n,... a + a + a 5 = and a a a = 8 then the value of a + a 4 + a 6 equals () 6 (C) 8 (D) Q.9 With usual notations in a triangle BC, ( I I ) ( I I ) ( I I ) has the value equal to () r r (C) 4 r (D) 6 r

3 Q.0 sector OBO of central angle is constructed in a circle with centre O and of radius 6. The radius of the circle that is circumscribed about the triangle OB, is () 6 cos 6 sec (C) (cos + ) (D) sec Q. n H.M. is inserted between the number / and an unknown number. If we diminish the reciprocal of the inserted number by 6, it is the G.M. of the reciprocal of / and that of the unknown number. If all the terms of the respective H.P. are distinct then () the unknown number is 7 the unknown number is /7 (C) the H.M. is 5 (D) the G.M. is Q. Let a b c be the lengths of the sides of a triangle T. If a + b < c then which one of the following must be true? () ll angles of T are acute. Some angle of T is obtuse. (C) One angle of T is a right angle. (D) No such triangle can exist. Q. Let there exist a unique point P inside a BC such that PB PBC PC. If P = x, PB = y, PC = z, = area of BC and a, b, c, are the sides opposite to the angle,b,c respectively, then tan is equal to () a b c 4 a b c (C) a b c 4 (D) a b c Q.4 If x, the numbers (5 +x + 5 x ), a/, (5 x + 5 x ) form an.p. then 'a' must lie in the interval () [, 5] [, 5] (C) [5, ] (D) [, ) Q.5 Let triangle BC be an isosceles triangle with B = C. Suppose that the angle bisector of its angle B meets the side C at a point D and that BC = BD + D. Measure of the angle in degrees, is () (C) 0 (D) 0 Q.6 If the sum of the first terms of an arithmetical progression equals that of the first 9 terms, then the sum of its first 0 terms, is () equal to 0 equal to (C) equal to (D) non unique Q.7 In a BC, the value of a cos b cos B c cos C a b c is equal to : () r r (C) r (D) r Q.8 With usual notation in a BC, if = k r r r r r r r r r r r r where k has the value equal to () (C) /4 (D) 4 Q.9 Let s, s, s... and t, t, t... are two arithmetic sequences such that s = t 0; s = t and 0 5 s = i i i i t. Then the value of s t s t () 8/ / (C) 9/8 (D) is

4 Q.0 If the incircle of the BC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to : () r r (C) r (D) r Q. BC is an acute angled triangle with circumcentre 'O' orthocentre H. If O = H then the measure of the angle is 5 () (C) (D) 6 4 Q. If 5...upto n terms upto n terms 0 7log 0 x 4 and n = log x log x log x log x , then x is equal to () (C) 0 6 (D) 0 7 Q. Let a n, n N is an.p. with common difference 'd' and all whose terms are non-zero. If n approaches infinity, then the sum... will approach a a a a a a n n () a d a d (C) a d (D) a d Q.4 In a BC if b + c = a then cot B cot C has the value equal to : () 4 (C) (D) Q.5 Let f, g, h be the lengths of the perpendiculars from the circumcentre of the BC on the sides a, b and c respectively. If a b f c g h = a b c then the value of is : f g h () /4 / (C) (D) Q.6 The sum of the first three terms of an increasing G.P. is and the sum of their squares is 89. Then the sum of its first n terms is () ( n ) n (C) 6 n (D) 6 ( n ) Q.7 In a BC if b = a and C = 0 0 then the measure of the angle is () (C) 75 0 (D) 05 0 Q.8 5 p 4p In a BC, a = a =, b = a, c = a such that a p+ = a p a p p p 5 where p =, then () r = r r = r (C) r = r (D) r = r p

5 Q.9 If O is the circumcentre of the BC and, and are the radii of the circumcircles of triangles OBC, OC and OB respectively then () a b c a b c a b c has the value equal to: (C) 4 (D) 4 Q.40 The value of n () 5 n n ( ) n equals (C) 6 5 (D) 6 5 Q.4 If a and ln a + (ln a ) + (ln a ) +... = 4 l n a ( ln a) ( ln a) ( ln a)... then 'a' is equal to () e /5 e (C) e (D) 4 e Q.4 The medians of a BC are 9 cm, cm and 5 cm respectively. Then the area of the triangle is () 96 sq cm 84 sq cm (C) 7 sq cm (D) 60 sq cm Q.4 If r, r, r be the radii of excircles of the triangle BC, then () cot B cot cot (C) Q is equal to () 4 (C) r r r is equal to : tan (D) tan (D) Q.45 In a BC with usual notations, if r =, r = 7 and =, then the BC is () equilateral acute angled which is not equilateral. (C) obtuse angled. (D) right angled. Q.46 If x, y and z are the distances of incentre from the vertices of the triangle BC respectively then a b c x y z is equal to () tan cot (C) tan (D) sin Q.47 circle of radius r is inscribed in a square. The mid points of sides of the square have been connected by line segment and a new square resulted. The sides of the resulting square were also connected by segments so that a new square was obtained and so on, then the radius of the circle inscribed in the n th square is () n r n r (C) n r (D) 5n r

6 Q.48 If in a BC, cos cosb + sin sinb sinc = then, the statement which is incorrect, is () BC is isosceles but not right angled BC is acute angled (C) BC is right angled (D) least angle of the triangle is 4 Q.49 The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to : () (C) (D) 4 [ where is the area of the triangle BC ] Q.50 Given and are the roots of the quadratic equation x 4x + k = 0 (k 0). If, +, + are in geometric progression then the value of 'k' equals 6 () 4 7 (C) 7 (D) Q.5 If abcd = where a, b, c, d are positive reals then the minimum value of a + b + c + d + ab + ac + ad + bc + bd + cd is () 6 0 (C) (D) 0 Q.5 triangle has base 0 cm long and the base angles of 50 and 70. If the perimeter of the triangle is x + y cos z where z (0, 90) then the value of x + y + z equals () (C) 50 (D) 40 Q.5 sequence of equilateral triangles is drawn. The altitude of each is times the altitude of the preceding triangle, the difference between the area of the first triangle and the sixth triangle is 968 The perimeter of the first triangle is () 0 (C) 6 (D) 8 square unit. Q.54 Let BC be a triangle with BC = and B = x such that (B)(C) =. If x varies then the longest possible length of the angle bisector D equals () / / (C) / (D) / Q.55 If a, b and c are three consecutive positive terms of a G.P. then the graph of y = ax + bx + c is () a curve that intersects the x-axis at two distinct points. entirely below the x-axis. (C) entirely above the x-axis. (D) tangent to the x-axis. Q.56 Triangle BC has BC = and C =. The maximum possible value of the angle is () (C) (D) 6 4 Q.57 Let a, b, c be the three sides of a triangle then the quadratic equation b x + (b + c a )x + c = 0 has () both imaginary roots both positive roots (C) both negative roots (D) one positive and one negative roots.

7 Q is the ratio of two relative prime positive integers m and n. The value of k k k (k ) k (m + n) is equal to () 4 4 (C) 9 (D) 7 Q.59 Let L and M be the respective intersections of the internal and external angle bisectors of the triangle BC at C and the side B produced. If CL = CM, then the value of (a + b ) is (where a and b have their usual meanings) () (C) 4 (D) 4 Q.60 The sum n 4 is equal to n n 4 () /4 / (C) /8 (D) / 00 Q.6 The sum k 4950 () 00 k 4 k k is equal to (C) 00 (D) none Q.6 In a triangle BC, BC = 0, B = and BC = 4. If perpendicular constructed on the side B at and to the side BC at C meets at D then CD is equal to () 8 Q.6 For which positive integers n is the ratio, n k n k k k (C) 5 an integer? 0 (D) () odd n only even n only (C) n = + 6k only, where k 0 and k I (D) n = + k, integer k 0 Q.64 If x >, y >, z > are in G.P., then log ex e, log ey e, log ez e are in ().P. H.P. (C) G.P. (D).G.P.

8 Q.65 Let l, l, l be the lengths of the internal bisectors of angles, B, C respectively of a BC. Statement-: because cos l B cos l C cos l a b c Statement-: l a bc b c, l b ca c a, l c ab a b () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. 9 Q.66 Statement-: In any BC, maximum value of r + r + r =. because Statement-: In any BC, r. (ll symbols used have their usual meaning in a triangle) () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, Statement- is true and Statement- is NOT the correct explanation for Statement-. (C) Statement- is true, Statement- is false. (D) Statement- is false, Statement- is true. Q.67 Let BC be an acute triangle and 'O' be its circumcentre. D, E and F are the foot of the perpendiculars dropped from 'O' to BC, C and B respectively. Statement-: rea of BC is four times the area of DEF because Statement-: atio of the areas of two similiar triangles is the square of the ratio of proportional sides. () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.68 Statement-: If 7 abc (a + b + c) and a + 4b + 5c = then = 0 ; where a, a b c b, c are positive real numbers. Statement-: For positive real numbers.m. G.M. () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true.

9 Q.69 Let BC be an acute triangle whose orthocentre is at H. ltitude from is produced to meet the circumcircle of the triangle BC at D. Statement-: The distance HD = 4 cos B cos C where is the circumradius of the triangle BC. because Statement-: Image of orthocentre H in any side of an acute triangle lies on its circumcircle. () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.70 Statement-: The difference between the sum of the first 00 even natural numbers and the sum of the first 00 odd natural numbers is 00. because Statement-: The difference between the sum of the first n even natural numbers and sum of the first n odd natural numbers is n. () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.7 Let BC be an acute angle triangle and D, E, F are the feet of the perpendicular from, B, C to the sides BC, C and B respectively. Statement- : Orthocentre of triangle BC is the Incentre of triangle DEF. because Statement- : Triangle DEF is the excentral triangle of triangle BC. () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.7 Statement-: Circumradius and inradius of a triangle can not be and 8 respectively. because Statement-: Circumradius (inradius) () Statement- is true, statement- is true and statement- is correct explanation for statement-. Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true.

10 n altitude BD and a bisector BE are drawn in the triangle BC from the vertex B. It is known that the length of side C =, and the magnitudes of the angles BEC, BD, BE, BC form an arithmetic progression. Q.7 The area of circle circumscribing BC is () 8 4 (C) (D) Q.74 Let 'O' be the circumcentre of BC, the radius of circle inscribed in BOC is () (C) (D) 8 4 Q.75 Let B' be the image of point B with respect to side C of BC, then the length BB' is equal to () 4 4 (C) (D) Let a m (m =,,...,p) be the possible integral values of a for which the graphs of f (x) = ax + bx + b and g (x) = 5x bx a meets at some point for all real values of b. Let p t r = ( r a m ) and S n = t r, n N. m Q.76 The minimum possible value of a is () n r (C) 8 (D) 4 Q.77 The sum of values of n for which S n vanishes is () 8 9 (C) 0 (D) 5 Q.78 The value of is equal to r5 t r () 6 (C) 5 (D) 8 Consider a triangle BC with b =. ltitude from the vertex B meets the opposite side in D, which divides C internally in the ratio :. circle of radius passes through the point and D and touches the circumcircle of the triangle BCD at D. Q.79 If E is the centre of the circle with radius then angle ED equals () sin 5 4 sin 4 (C) sin 4 5 (D) sin 6

11 Q.80 If F is the circumcentre of the triangle BDC then which one of the following does not hold good? () FCD = sin 5 4 (C) triangle DFC is an isosceles triangle FDC = cos 4 (D) rea of DE = (/4) th of the area of DBC Q.8 If is the circumradius of the BC, then equal () 4 6 (C) 6 5 (D) Q.8 Let a, a, a... and b, b, b... be arithmetic progressions such that a = 5, b = 75 and a 00 + b 00 = 00. Then () the difference between successive terms in progression 'a' is opposite of the difference in progression 'b'. a n + b n = 00 for any n. (C) (a + b ), (a + b ), (a + b ),... are in.p. 00 (D) ( a r br ) = 0000 r Q.8 In a EX, T is the mid point of XE, and P is the mid point of ET. If the PE is equilateral of side length equal to unity then which of the following alternative(s) is/are correct? () X = ET = 90 (C) cos XE = (D) T = Q.84 If sin(x y), sin x and sin (x + y) are in H.P., then sin x. sec y = () (C) (D) Q.85 In a BC, following relations hold good. In which case(s) the triangle is a right angled triangle? (ssume all symbols have their usual meaning) () r + r = r r a + b + c = 8 (C) If the diameter of an excircle be equal to the perimeter of the triangle. (D) = r r a Q.86 Consider a sequence {a n } with a = and a n = a n n for all n, terms of the sequence being distinct. Given that a and a 5 are positive integers and a 5 6 then the possible value(s) of a 5 can be () (C) 64 (D) 6

12 Q.87 The sum of the first three terms of the G.P. in which the difference between the second and the first term is 6 and the difference between the fourth and the third term is 54, is () (C) 7 (D) 7 Q.88 a, b, c are the first three terms of geometric series. If the H.M. of a and b is and that of b and c is 6 then which of the following hold(s) good? () Sum of the first term and common ratio of the G.P. is. Sum of the first five terms of the G.P. is 948. (C) If the value of the the first term and common ratio of the given G.P. is taken as the first term and common difference of an.p. then its 8 th term is 9. (D) The number 648 is one of the term of the G.P. Q.89 If the roots of the equation, x + px + qx = 0 form an increasing G.P. where p and q are real, then () p+q = 0 p (, ) (C) one of the roots is unity one root is smaller than (D) and one root is greater than. Q.90 In a BC, a semicircle is inscribed, whose diameter lies on the side c. If x is the length of the angle bisector through angle C then the radius of the semicircle is () 4 abc (sin sin B) x C s(s a)(s b)(s c) (C) x sin (D) s Where is the area of the triangle BC and 's' is semiperimeter. Q.9 triangle BC has the feature, () triangle is right angled (C) a = b = c a cos b cos B ccos C = then the correct statement(s) is/are : a b c = r sin (D) = sin Q.9 If the triplets log a, log b, log c and (log a log b), (log b log c), (log c log a) are in arithmetic progression then () 8(a + b + c) = 8(a + b + c ) + ab a, b, c are in G.P. (C) a, b, c are in H.P. (D) a, b, c can be the lengths of the sides of a triangle (ssume all logarithmic terms to be defined) Q.9 x, x are the roots of the equation x x + = 0; x, x 4 are roots of the equation x x + B = 0, such that x, x, x, x 4 form an increasing G.P., then () = B = (C) x + x = 5 (D) x + x 4 = 0

13 Q.94 Let P be an interior point of BC. Match the correct entries for the ratios of the rea of PBC : rea of PC : rea of PB depending on the position of the point P w.r.t. BC. Marks will be given only if all the entries of column-i are correctly matched. Column-I Column-II () If P is centroid (G) (P) tan : tan B : tan C If P is incentre (I) (Q) sin : sin B : sin C (C) If P is orthocentre (H) () sin : sin B : sin C (D) If P is circumcentre (S) (S) : : (T) cos : cos B : cos C Q.95 Let the lengths of the altitudes drawn from the vertices of a BC to the opposite sides are, and. If the area of BC is then find the value of. Q.96 If the sum to first n terms of a series, the r th term of which is given by (r + ) r can be expressed as (n n ) + S n + T, then find the value of ( + S + T). Q.97 In a triangle BC, if the sides a, b, c are the roots of x x + 8x 40 = 0. If the value of cos cos B cosc m + + can be expressed as a rational number as in the lowest form then find the a b c n value of (m + n). Q.98 Let S be the set of integers which are divisible by 5, and let T be the set of integer which are divisible by 7. Find the number of positive integers less than 000 and not in (S T).

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Objective Mathematics

Objective Mathematics . 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