[STRAIGHT OBJECTIVE TYPE] log 4 2 x 4 log. x x log 2 x 1

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Cleopatra Robertson
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1 [STRAIGHT OBJECTIVE TYPE] Q. The equation, log (x ) + log x. log x x log x + log x log + log / x (A) exactly one real solution (B) two real solutions (C) real solutions (D) no solution. = has : Q. The minimum value of the function f (x) = (sin x cos x 0)( sin x + cos x 0), (A) (B) (C) 8 (D) 8 Q. The expression cot 9 + cot 7 + cot 6 + cot 8 equal to (A) 6 (B) 6 (C) 80 (D) none of these Q. In a triangle ABC, angle B < angle C and the values of B and C satfy the equation tan x k ( + tan x) = 0 where (0 < k < ). Then the measure of angle A : (A) / (B) / (C) / (D) / (log n) Q.5 If log 0 sinx + log 0 cosx = and log 0 (sin x + cos x) = 0 then the value of 'n' (A) (B) 6 (C) 0 (D) Q.6 If M and m are maximum and minimum value of the function f (x) = then (M + m) equals (A) 0 (B) (C) 0 (D) 8 tan x tan x 9, tan x Q.7 The set of values of a for which the equation, cos x + a sin x = a 7 possess a solution (A) (, ) (B) [, 6] (C) (6, ) (D) () Q.8 The sum... sin 5sin 6 sin 7sin 8 sin 9sin 50 sin sin equal to... dk eku gksxk& sin 5sin 6 sin 7sin 8 sin 9sin 50 sin sin (A) sec () (B) cosec () (C) cot () (D) none Q.9 Number of zeros after decimal before a significant figure in (75) 0 : (use log 0 = 0.0 & log 0 = 0.77) (A) 0 (B) 9 (C) 8 (D) none Q.0 The range of k for which the inequality k cos x k cos x + 0 x (, ), (A) k < (B) k > sin cos Q. If f () = cos then value of f ( ) f ( ) equals (C) k (D) k 5 (A) (B) (C) (D)

2 Q. The value of the product sin cos cos cos cos cos cos , (A) 007 (B) 008 (C) 009 Q. The set of values of x satfying simultaneously the inequalities log (D) 00 (x 8) ( x) 0 and 0 log 5 7 x > 0 (A) a unit set (B) an empty set (C) an infinite set (D) a set consting of exactly two elements. Q. In a triangle ABC, if cosa cosb + sina + sinb + sinc =, then triangle ABC (A) right angle but not osceles (B) osceles but not right angled (C) right angle osceles (D) obtuse angled Q.5 If P = (tan ( n + ) tan ) and Q = n r 0 r sin( ) r, then cos( ) (A) P = Q (B) P = Q (C) P = Q (D) P = Q Q.6 For,,..., n 0,, if ln (sec tan ) + ln (sec tan ) ln(sec n tan n ) + ln = 0, then the value of cos (sec tan ) (sec tan )... (sec tan ) equal to (A) cos n n (B) (C) (D) 0 Q.7 Let S be the set of ordered triples (x, y, z) of real numbers for which log 0 (x + y) = z and log 0 (x + y ) = z +. Suppose there are real numbers a and b such that for all ordered triples (x, y, z) in S we have x + y = a 0 z + b 0 z. The value of (a + b) equal to 5 (A) (B) 9 0. (C) 5 (D) Q.8 Let P(x) = (cosx cosx cosx) (sin x sin x sin x) then P(x) equal to (A) + cos x (B) + sin x (C) cos x (D) none Q.9 If the maximum value of the expression 5sec tan cosec (where p and q are coprime), then the value of (p + q) (A) (B) 5 (C) 6 (D) 8 Q.0 The minimum value of (A) (B) cosx sin x sin x cos x cos x sin x (C) (D) equal to q p

3 log x log y Q. If (x 0, y 0 ) satfies the simultaneous equations log x + log y = and then value of (x 0 + y 0 ) equal to (A) 8 (B) (C) (D) 8 A B C Q. In triangle ABC, the minimum value of sec sec sec equal to (A) (B) (C) 5 (D) 6 5 Q. Let S = cos( r ) and P = r cos r r, then 5 (A) log S P = (B) P = S (C) cosec S > cosec P (D) tan P < tan S cos 96 sin 96 Q. Find the smallest natural 'n' such that tan(07n) =. cos 96 sin 96 (A) n = (B) n = (C) n = (D) n = 5 Q.5 Let 'a' denotes the logarithm of 0. to the base 0., 'b' denotes the logarithm of to the base 8 and 'c' denotes the number whose logarithm to the base 0.6 minus. Then the value of ab c, (A) lies between and 5. (B) odd composite. (C) odd prime. (D) lies in (, ) Q.6 The number of ordered pairs (x, y) of real numbers satfying x x + = sin y and x + y, equal to (A) 0 (B) (C) (D) 8 Q.7 In a triangle ABC, sin A + cos B = 6 and cos A + sin B = then C can be (A) 0 (B) 60 (C) 90 (D) 50 Q.8 An equilateral triangle has side length 8. The area of the region containing all points outside the triangle but not more than units from a point on the triangle : (A) 9(8 + ) (B) 8(9 + ) (C) 98 (D) 89 Q.9 A line x = k intersects the graph of y = log 5 x and the graph of y = log 5 (x + ). The dtance between the points of intersection 0.5. Given k = a b, where a and b are integers, the value of (a + b) (A) 5 (B) 6 (C) 7 (D) 0 Q.0 If a cos + a cos sin = m and a sin + a cos sin = n. Then (m + n) / + (m n) / equal to : (A) a (B) a / (C) a / (D) a

4 Q. As shown in the figure AD the altitude on BC and AD produced meets the circumcircle of ABC at P where DP = x. Similarly EQ = y and FR = z. If a, b, c respectively denotes the sides BC, CA and AB then has the value equal to (A) tana + tanb + tanc (C) cosa + cosb + cosc a x b y c z (B) cota + cotb + cotc (D) coseca + cosecb + cosecc Q. One side of a rectangular piece of paper 6 cm, the adjacent sides being longer than 6 cms. One corner of the paper folded so that it sets on the opposite longer side. If the length of the crease l cms and it makes an angle with the long side as shown, then l (A) (C) sin cos sin cos (B) (D) sin sin 6 cos Q. If P the number of natural numbers whose logarithms to the base 0 have the charactertic p and Q the number of natural numbers logarithms of whose reciprocals to the base 0 have the charactertic q then log 0 P log 0 Q has the value equal to (A) p q + (B) p q (C) p + q (D) p q Q. A circle inscribed inside a regular pentagon and another circle circumscribed about th pentagon. Similarly a circle inscribed in a regular heptagon and another circumscribed about the heptagon. The area of the regions between the two circles in two cases are A and A respectively. If each polygon has a side length of units then which one of the following true? 5 (A) A = A 7 5 (B) A = A 9 9 (C) A = A 5 (D) A = A Q.5 The value of x satfying the equation, x = x (A) cos 0 (B) cos 0 (C) cos 0 (D) cos 80 Q.6 If sin x + sin y + sin z = 0 = cos x + cos y + cos z then the expression, cos( x) + cos( y) + cos( z), for R (A) independent of but dependent on x, y, z (B) dependent on but independent of x, y, z (C) dependent on x, y, z and (D) independent of x, y, z and Q.7 Let x, y, z, t be real numbers x + y = 9 ; z + t = and xt yz = 6 then the greatest value of P = xz, (A) (B) (C) (D) 6

5 [COMPREHENSION TYPE] Paragraph for question nos. 8 to 0 log M N = +, where an integer & [0, ) Q.8 If M & are prime & + M = 7 then the greatest integral value of N (A) 6 (B) 6 (C) 5 (D) Q.9 If M & are twin prime & + M = 8 then the greatest integral value of N (A) 6 (B) 65 (C) 78 (D) 79 Q.0 If M & are relative prime & + M = 7 then minimum integral value of N (A) 5 (B) (C) 6 (D) 8 Paragraph for question nos. to Let f () = sin cos, where R and m f () M. Q. Let N denotes the number of solutions of the equation f () = 0 in [0, ] then the value of log (A) (N) log m m N equal to (B) (C) (D) Q. The value of (m + ) equal to (A) 0 (B) (C) 5 (D) 6 Q. Sum of all values of x satfying the equation (A) x = m..., m m (B) (C) (D) [REASONING TYPE] Q. Statement-I : In any triangle ABC, cot A + cot B + cot C always positive. Statement-II : Minimum value of cot A + cot B + cot C =. (A) Statement- true, statement- true and statement- correct explanation for statement-. (B) Statement- true, statement- true and statement- NOT the correct explanation for statement-. (C) Statement- true, statement- false. (D) Statement- false, statement- true Q.5 Statement-: In ABC, sin A + sin B + sin C always positive. Statement-: In ABC, sin A + sin B + sinc = 8 sina sinb sinc. (A) Statement- true, statement- true and statement- correct explanation for statement-. (B) Statement- true, statement- true and statement- NOT the correct explanation for statement-. (C) Statement- true, statement- false. (D) Statement- false, statement- true.

6 [MULTIPLE OBJECTIVE TYPE] Q.6 The value of x satfying the equation xlog x log (A) a prime number (C) an even number log =, x (B) a composite number (D) an odd number 8 Q.7 The value of the expression tan + tan + tan + 8cot equal to (A) cosec + cot 7 7 (B) tan cot (C) sin 7 cos 7 cos cos 7 7 (D) sin sin 7 7 n Q.8 For 0 < <, if x = cos, y = sin, z = cos sin, then n0 n0 n0 (A) xyz = xz + y (B) xyz = xy + z (C) xyz = x + y + z (D) xyz = yz + x sin( ) cos( ) Q.9 It known that sin = cos 5 and 0 < < then the value of 6 sin n n n (A) independent of for all in (0, ) (B) 5 for tan > 0 (C) ( 7 cot ) for tan < 0 (D) zero for tan > 0. 5 Q.50 If y = log 7 a (x + x + a + ) defined x R, then possible integral value(s) of a /are (A) (B) (C) (D) 5 Q.5 If the equation cos x + cos x a = 0 has solutions then a can be (A) (B) 8 (C) (D) 5 Q.5 For a positive integer n, let 007] f n () = n (cos )( cos )(cos ) cos( )... cos( ) Which one of the following hold(s) good? (A) f (/6) = 0 (B) f (/8) = (C) f (/) = (D) f 5 (/8) =.

7 Q.5 Two parallel chords are drawn on the same side of the centre of a circle of radius R. It found that they subtend an angle of and at the centre of the circle. The perpendicular dtance between the chords (A) R sin sin (B) cos cos R (C) cos cos R (D) R sin sin Q.5 The value of x in (0, /) satfying the equation, sin x cos x (A) 5 (B) 7 (C) (D) 6 [MATCH THE COLUMN] Q.55 Column-I Column-II (A) a If log b = and log 7, b (P) then the value of (a b ) equal to (Q) (B) If number of digits in 'd', and number of cyphers after (R) 6 decimal before a significant figure starts in (0.) 9 'c', then (d c) equal to (C) If N = antilog antilog (log 96) log, (S) 6 5 then the charactertic of log N to the base, equal to 5 Q.56 Column-I Column-II (A) Suppose sin cos = (P) 0 then the value of sin cos ( R) (B) Minimum value of the function (Q) f (x) = ( + sin x)( + cos x) x R, (C) Given that the sum of the solutions of the equation (R) / sin x tan x sin x + tan x = 0 over [0, ] = k, where k Q then the value of k equals (S) / (D) The expression sin tcos t 6 6 sin tcos t when simplified reduces to [SUBJECTIVE] Q.57 Let N be the number of integers whose logarithms to the base 0 have the charactertic 5, and M the number of integers the logarithms to the base 0 of whose reciprocals have the charactertic. Find (log 0 N log 0 M). Q.58 Let x 0, and log sin x ( cos x) =, then find the value of cosec x.

8 Q.59 If x and y are non zero real numbers satfying xy(x y ) = x + y, find the minimum value of x + y. Q.60 Find the sum of maximum and minimum value of the sum of the squares of the roots of the equation x + ( sin )x + cos = 0. For what value of in (, ) these extreme values occur. Q.6 If log 5 75 = a, log = b then find the value of ab. a b Q.6 Using the identity sin x = 8 cos x + 8 cos x or otherwe, if the value of sin sin 7 sin = b a where a and b are coprime, find the value of (a b). Q.6 In any triangle, if (sin A + sin B + sin C) (sin A + sin B sin C) = sin A sin B, find the angle C (in degree). Q.6 Find the exact value of the expression T = sin 0 sin 80 sin 0. sin 80 sin 0 sin 0 Q.65 If sum of the integral values of x satfying the equation x log xlog x x N, then find charactertic of logarithm of N to the base 5. Q.66 Let x = n n cosn sin n find the greatest integer that does not exceed 00 x. Q.67 Find the exact value of cosec0 + cosec50 cosec70. Q.68 If cos 5 = cos 5, where 0,, then find the possible values of (sec + cosec + cot ). Q.69 Let k be the unique positive value satfying the equation log log ( k) (9k) = 0, then find the value of (7 k). [th, , UT-(J)] sec 0 Q.70 Compute the square of the value of the expression. cosec 0

9 ANSWER KEY Q. D Q. A Q. C Q. C Q.5 D Q.6 C Q.7 B Q.8 B Q.9 C Q.0 C Q. A Q. B Q. A Q. C Q.5 A Q.6 B Q.7 B Q.8 D Q.9 D Q.0 A Q. D Q. B Q. D Q. B Q.5 D Q.6 B Q.7 A Q.8 A Q.9 B Q.0 C Q. A Q. A Q. A Q. D Q.5 C Q.6 D Q.7 B Q.8 D Q.9 C Q.0 C Q. C Q. B Q. D Q. A Q.5 C Q.6 AC Q.7 ACD Q.8 BC Q.9 BC Q.50 BCD Q.5 BC Q.5 ABC Q.5 BD Q.5 AD Q.55 (A) S, (B) R, (C) Q Q.56 (A) Q; (B) P; (C) S; (D) R Q.57 Q.58 9 Q.59 Q Q.6 5 Q.6 5 Q.6 60 Q.6 Q.65 Q.66 Q.67 6 Q.68 5 Q.69 Q.70

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these Q. Given a + a + cosec [STRAIGHT OBJECTIVE TYPE] F HG ( a x) I K J = 0 then, which of the following holds good? (A) a = ; x I a = ; x I a R ; x a, x are finite but not possible to find Q. The minimum value