[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

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Eustacia Harvey
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1 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 of the function f (x) = ( x cos x 0)( x + cos x 0), is (A) Q. In the inequality below the value of the angle is expressed in radian measure. Which one of the inequalities below is true? (A) < < < < < < < < Q. The expression cot 9 + cot + cot + cot 8 (A) 80 none of these Q.5 In a triangle ABC, angle B < angle C and the values of B and C satisfy the equation tan x k ( + tan x) = 0 where (0 < k < ). Then the measure of angle A is : (A) / / / / Q. The value of expression 8 0 tan 0 equals (A) 5 9 Q. If M and m are maximum and minimum value of the function f (x) = then (M + m) equals (A) tan x tan x 9, tan x Q.8 The value of cos sec tan (A) 5 5 zero Q.9 The set of values of a for which the equation, cos x + a x = a possess a solution is (A) (, ) [, ] (, ) () Q.0 The value of 0 sec0 (A)

2 Q. For 0 < A <, the value of log cos A sec A (A) 0 Q. The sum (A) sec () cosec () cot () none Q. If tan, tan are the roots of the equation x + px + q = 0, then the value of ( + ) + p ( + ) cos( + ) + q cos ( + ) is (A) independent of p but dependent on q independent of q but dependent on p independent of both p and q dependent on both p and q Q. The range of k for which the inequality k cos x k cos x + 0 x (, ), is (A) k < k > k k 5 Q.5 The value of the expression x cosx x 9x equals cos9x cos x (A) (tan 9x tan x) (tan 9x tan x) (tan x tan x) (tan x tan x) cos Q. If f () = cos then value of f ( ) f ( ) equals (A) Q. The value of the product cos cos cos cos cos cos , is (A) Q.8 The variable 'x' stisfying the equation x cos x + tan x cot x =, belongs to the interval (A) 0,,, non existent Q.9 Let be a real number such that 0. If f (x) = cos x + cos(x + ) + cos(x + )

3 takes some constant number c for any x R, then the value of [c + ] (Note : [y] denotes greatest integer less than or equal to y.) (A) 0 Q.0 In a triangle ABC, if cosa cosb + A + B + C =, then triangle ABC is (A) right angle but not isosceles isosceles but not right angled right angle isosceles obtuse angled Q. If tanb = n cosa n cos A then tan(a + B) equals (A) ( n) cosa ( n ) cosa (n ) cosa (n ) cosa Q. If P = (tan ( n + ) tan ) and Q = n r 0 r ( ) r, then cos( ) (A) P = Q P = Q P = Q P = Q Q. 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 ) (A) cos n n 0 Q. If A, B, C are interior angles of triangle ABC such that (cos A + cos B + cos C) + ( + B + C) = 9 then number of possible triangles is (A) 0 infinite Q.5 If cosec + cosec + cosec 8 + cosec + cosec = cot k, then the value of k is (A) 9 8 Q. The value of (cos º + cos º + cos º + + cos 9º ) ( º + º + º + + 9º) equals (A) cos 0 Q. Suppose that 'a' is a non-zero real number for which x + y = a and cos x + cos y = a. The value of cos (x y), is (A) a a 9a 5a Q.8 Let P(x) = (cosx cosx cosx) ( x x x) then P(x) (A) + cos x + x cos x none

4 Q.9 If the maximum value of the expression 5sec tan cosec (where p and q are coprime), then the value of (p + q) is (A) 5 8 q p Q.0 Let f n () = 5... (n ). Then the value of f cos cos cos5... cos(n ) (A) cosx x Q. The minimum value of x cos x is cos x x (A) Q. If a = cos (0 ), b = sec (0 ) and c = tan (0 ) then (A) a < b < c b < c < a c < b < a a = b < c A B C Q. In triangle ABC, the minimum value of sec sec sec (A) 5 5 Q. Let S = cos( r ) and P = r cos r r, then 5 (A) log S P = P = S cosec S > cosec P tan P < tan S Q.5 If the expression, cos is simplified, then it simplifies to (A) cos 0 cos 0 cos 0 5 cos 0 cos9 9 Q. Find the smallest natural 'n' such that tan(0n) =. cos9 9 (A) n = n = n = n = 5 Q. The number of ordered pairs (x, y) of real numbers satisfying x x + = y and x + y, (A) 0 8 Q.8 In a triangle ABC, + cos B = and cos A + B = then C can be (A) Q.9 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 is : (A) 9(8 + ) 8(9 + ) 98 89

5 Q.0 If a cos + a cos = m and a + a cos = n. Then (m + n) / + (m n) / : (A) a a / a / a Q. As shown in the figure AD is 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 cosa + cosb + cosc a x b y c z cota + cotb + cotc coseca + cosecb + cosecc Q. If A = 0 0 then is identical to (A) Q. One side of a rectangular piece of paper is cm, the adjacent sides being longer than cms. One corner of the paper is folded so that it sets on the opposite longer side. If the length of the crease is l cms and it makes an angle with the long side as shown, then l is (A) cos cos Q. The average of the numbers n n for n =,,, cos (A) cot tan Q.5 A B In a triangle ABC, C =. If tan and tan are the roots of the equation ax + bx + c = 0 (a 0) then (A) a + b = c b + c = a a + c = b b = c Q. A circle is inscribed inside a regular pentagon and another circle is circumscribed about this pentagon. Similarly a circle is 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 is true? 5 (A) A = A 5 A = A 9 9 A = A 5 A = A Q. The value of x satisfying the equation, x = x is (A) cos 0 cos 0 cos 0 cos 80

6 Q.8 Exact value of cos 5 (A) Q.9 If x + y + z = 0 = cos x + cos y + cos z then the expression, cos( x) + cos( y) + cos( z), for R is (A) independent of but dependent on x, y, z dependent on but independent of x, y, z dependent on x, y, z and independent of x, y, z and Q.50 Let x, y, z, t be real numbers x + y = 9 ; z + t = and xt yz = then the greatest value of P = xz, is (A) [COMPREHENSION TYPE] Paragraph for question nos. 5 to 5 Let f () = cos, where R and m f () M. Q.5 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 Q.5 The value of (m + ) (A) 0 5 Q.5 Sum of all values of x satisfying the equation x = m... m m, is (A) Q.5 If A + B + C = then [REASONING TYPE] Statement-: cos A + cos B + cos C has its minimum value. Statement-: Maximum value of cos A cos B cos C = 8. (A) 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-. Statement- is true, statement- is false. Statement- is false, statement- is true.

7 Q.55 If P = Statement- : P 90 P is irrational Statement- : tan is irrational (A) 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-. Statement- is true, statement- is false. Statement- is false, statement- is true. Q.5 Statement-I : In any triangle ABC, cot A + cot B + cot C is always positive. Statement-II : Minimum value of cot A + cot B + cot C =. (A) 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-. Statement- is true, statement- is false. Statement- is false, statement- is true Q.5 Statement-: In ABC, + B + C is always positive. Statement-: In ABC, + B + C = 8 A B C. (A) 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-. Statement- is true, statement- is false. Statement- is false, statement- is true. Q.58 Statement-: 5 5 tan tan tan tan tan tan Statement-: If = +, then tan tan tan = tan tan tan. (A) 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-. Statement- is true, statement- is false. Statement- is false, statement- is true. [MULTIPLE OBJECTIVE TYPE] Q.59 If t + cos t = 5 then tan t (A) Q.0 If x (0, ) and cos x + x = then tan x (A) ( )

8 8 Q. The value of the expression tan + tan + tan + 8cot (A) cosec + cot tan cot cos cos cos Q. Let P = and Q = Which of the following relation(s) is(are) correct? (A) P + Q = 0 P Q = 0 P + Q = P Q = 0 n Q. For 0 < <, if x = cos, y =, z = cos, then n0 n0 n0 (A) xyz = xz + y xyz = xy + z xyz = x + y + z xyz = yz + x n n n Q. Let P(x) = cot x tan x tan x cot x cot x + cos x cosx x x ( x cos x) following is(are) correct? (A) The value of P(8 ) + P( ) is. The value of P(8 ) + P( ) is. The value of P P is. The value of P P is.. Then which of the ( ) cos( ) Q.5 It is known that = cos 5 and 0 < < then the value of is (A) independent of for all in (0, ) 5 for tan > 0 ( cot ) for tan < 0 zero for tan > 0. 5 Q. In cyclic quadrilateral ABCD, if cot A = and tan B = 5, then which of the following is (are) correct? (A) D = (A + B) = 5 5 cos D = (C + D) = 5 Q. If the equation cos x + cos x a = 0 has solutions then a can be (A) 8 5

9 Q.8 For a positive integer n, let f n () = n (cos )( cos )(cos ) cos( )... cos( ) Which one of the following hold(s) good? (A) f (/) = 0 f (/8) = f (/) = f 5 (/8) =. Q.9 Two parallel chords are drawn on the same side of the centre of a circle of radius R. It is found that they subtend an angle of and at the centre of the circle. The perpendicular distance between the chords is (A) R cos cos R cos cos R R Q.0 If x cos x then which of the following holds good? (A) tan x = tan x = 5 x cos x 9 5 x cos x Q. If x and y are complementary angles and tan (x + y) =, then which of the following is(are) correct? (A) (x + y) = tan(x y) = cot x + cot y = 5 tan x tan y = Q. If cos + = sec where (0, ) then which of the following can be correct? (A) cos = tan = = cot = Q. The value of x in (0, /) satisfying the equation, x cos x (A) 5 is Q. Which of the following statements are always correct? (where Q denotes the set of rationals) (A) cos Q and Q tan Q (if defined) tan Q, cos and tan Q (if defined) if Q and cos Q tan Q (if defined) if Q cos Q

10 [MATCH THE COLUMN] Q.5 Column-I Column-II (A) The expresion wherever defined simplifies to (P) cos xcot( x) sec(x ) cosec x tan (x )(x ) The expression, (Q) x cos x simplifies to (wherever it is defined) (R) x cos x tan x cot x If < x < and y = log0 (tan x + sec x). Then the expression (S) sec x + tan x y y 0 0 E = simplifies to one of the six trigonometric functions. Then the trigonometric function, is Q. Column-I Column-II (A) Suppose cos = (P) 0 then the value of cos is ( R) Minimum value of the function (Q) f (x) = ( + x)( + cos x) x R, is Given that the sum of the solutions of the equation (R) / x tan x x + tan x = 0 over [0, ] = k, where k Q then the value of k equals (S) / The expression tcos t when simplified reduces to tcos t [SUBJECTIVE] Q. Let x 0, and log x ( cos x) =, then find the value of cosec x. Q.8 If x and y are non zero real numbers satisfying xy(x y ) = x + y, find the minimum value of x + y. Q.9 Find the sum of maximum and minimum value of the sum of the squares of the roots of the equation x + ( )x + cos = 0. For what value of in (, ) these extreme values occur. Q.80 Ug the identity x = 8 cos x + 8 cos x or otherwise, if the value of 5 = b a where a and b are coprime, find the value of (a b). Q.8 In any triangle, if ( + B + C) ( + B C) = B, find the angle C (in degree).

11 Q.8 Find the exact value of the expression T = Q.8 Find the exact value of the expression 5 9 tan tan + tan tan + tan Q.8 Let x = n n cosn n find the greatest integer that does not exceed 00 x. Q.85 Find (in degree) satisfying the equation, tan 5 tan 5 tan 5 = tan, where (0, 5 ) Q.8 Find the exact value of cosec0 + cosec50 cosec0. Q.8 Determine the smallest positive 'x' (in degrees) which satisfy the equation tan x = tan(x + 0 ) tan(x + 0 ) tan(x + 0 ). Q.88 If cos 5 = cos 5, where 0,, then find the possible values of (sec + cosec + cot ). Q.89 Compute the value of the expression tan tan Q.90 Compute the square of the value of the expression tan... tan. sec 0. cosec 0 ANSWER KEY Q. B Q. A Q. D Q. C Q.5 C Q. A Q. C Q.8 D Q.9 B Q.0 B Q. D Q. B Q. A Q. C Q.5 C Q. A Q. B Q.8 D Q.9 D Q.0 C Q. A Q. A Q. B Q. D Q.5 A Q. B Q. D Q.8 D Q.9 D Q.0 B Q. A Q. B Q. B Q. D Q.5 A Q. B Q. B Q.8 A Q.9 A Q.0 C Q. A Q. D Q. A Q. B Q.5 A Q. D Q. C Q.8 D Q.9 D Q.50 B Q.5 C Q.5 B Q.5 D Q.5 A Q.55 B Q.5 A Q.5 C Q.58 A Q.59 B,C Q.0 C Q. A,C,D Q. B,D Q. B,C Q. B,C Q.5 B,C Q. A,B,D Q. B,C Q.8 A,B,C Q.9 B,D Q.0 B,C Q. B,C Q. A,B,C,D Q. A,D Q. A,B,C Q.5 (A) R; S; Q Q. (A) Q; P; S; R Q. 9 Q.8 Q.9 50 Q.80 5 Q.8 0 Q.8 Q.8 5 Q.8 Q.85 5 Q.8 Q.8 5 Q.88 5 Q.89 5 Q.90

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

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