DISCUSSION CLASS OF DAX IS ON ND MARCH, TIME : 9- BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE] Q. Let y = cos x (cos x cos x). Then y is (A) 0 only when x 0 (B) 0 for all real x (C) 0 for all real x (D) 0 only when x 0 Q. If u, v, w are real distinct numbers such that u + v + w = uvw, then the quadratic equation ux + vx + w = 0 has (A) real roots (B) roots lying on either side of unity. (C) both roots are negative (D) non-real roots Q. In ABC, if a = 9, b = 8 and c = x satisfies cos C =, then the value of x is [Note: All symbols used have usual meaning in triangle ABC.] (A) x = (B) x = (C) x = (D) x = 7 Q. Total number of ordered pairs (x, y) of real numbers satisfying tan y + (sin x) = 0 and x + y is equal to (A) (B) (C) 9 (D) Q. If n 0 n cos =, then the most general values of are given by (A) n ± (B) n ± (C) n ± (D) n ± (where n I) Q. If ab > 0 and ac > 0 then the line ax + by + c = 0 cannot pass through which one of the following quadrants? (A) First (B) Second (C) Third (D) Fourth Q.7 In triangle ABC if AB =, BC = and AC =, then the value of sin A sin B sin C is equal to (A) (B) (C) Q.8 The largest integral value of b for which the roots of quadratic equation x + ( b)x + (b ) = 0 are opposite in sign, is (A) (B) (C) (D) Q.9 The sum of a certain infinite geometric series is 0. When all the terms in the series are squared, the sum of the resulting series is 80. If the first term of the original series is expressed in lowest terms as (D) p, (p, q N) then the value of (p + q) is q (A) (B) (C) (D) []
Q.0 Let ABCD be a square on the coordinate plane such that its sides have slopes,,, respectively. If m be the positive slope of one of the diagonal then m is equal to (A) (B) (C) 9 (D) 9 Q. Let f() = sin cos, for all { : f() > 0} then the minimum value of f(), is (A) 0 (B) (C) (D) Q. Given and are roots of the quadratic equation x x + c = 0. If, +, + (in that order) are in arithmetic progression then the value of c is (A) (B) (C) 7 (D) 8 Q. If cos B cos A = sin 80 and A + B = 0 where 0 < A < 80, then the largest possible value of A, is (A) 0 (B) 80 (C) 0 (D) 0 Q. The number of integral values of 'm' for which atleast one solution of the inequality x mx + m 0 satisfies the inequality x x + 8 0, is (A) (B) (C) (D) 7 Q. If the sum of first n terms of an arithmetic progression is denoted by S n and S n = n sec + n(n ) sin ( tan ) cos ( cot ), where 0,. then the minimum value of common difference of the arithmetic progression, is (A) 0 (B) (C) (D) Q. If, are the roots of equation x + x = 0, then the value of is equal to (A) (B) (C) (D) ( ) ( ) Q.7 The general values of x satisfying simultaneously the equations sin x + = 0 and tan x + = 0 is given by (A) n +, n I (B) n +, n I (C) n +, n I (D) n +, n I []
log Q.8 If x = x 0 is solution of the equation log (x) (x) 0, then the value of x 0 is equal to x0 7 (A) (B) log log log log (C) log (D) Q.9 The true solution set of inequality log (sin ) > log (cos ) is equal to (A) n, n ni (B) n, n ni (C) n, n ni 7 (D) n, n ni Q.0 If x x + + + a = cos y where x, y, a R, then the value of (a + x + y) can be (A) (B) (C) (D) Q. If cos ( ) = cos ( cos ( ), then ) (A) tan, tan, tan are in A.P. (C) tan, tan, tan are in H.P. (B) tan, tan, tan are in G.P. (D) tan, tan, tan are NOT in A.P./G.P./H.P. Q. The maximum value of expression cos x sin x (A) (B) (C) is equal to (D) Q. Number of ordered pairs (x, y) of real numbers satisfying the system of equations sin x = sin y and cos x = sin y where 0 x and 0 y, is (A) (B) (C) (D) y xy Q. The maximum value of the expression, is x y (A) (B) (C) (D) Q. The true solution set of the inequality sin x + sin x 0 in (0, ) is (A) 0,, (B) 0,, (C),, 0,, (D) []
Q. If x is real, then the maximum value of (A) x x 9x 7 9x 7 7 (B) (C) (D) 7 is Q.7 If, x, y are in harmonic progression (x, y 0), then the number of integral ordered pairs (x, y) is (A) (B) (C) (D) Q.8 If the value of [0, ] such that the inequality sin x sin( x) 0 is true for all real numbers x, is equal to p where p and q are relatively prime positive integers, then the value of q (p + q) is (A) 7 (B) (C) (D) Q.9 If the roots of the quadratic equation (sin + sin 0 + cos 0) x + x + k k + = 0, is such that < 0 <, then the true set of values of k is, (A) ( ) (, ) (B) (, ) (C) ( ) (, ) (D) (, ) Q.0 Number of integral values of m for which x mx + m < 0, x (, ], is (A) 8 (B) 9 (C) 0 (D) Q. If the equations x x = 0 and x bx c = 0 (b, c R) have exactly one root in common, then the value of (b + c) is (A) (B) (C) (D) Q. The line that divides the circle x + x + y + y 0 = 0 into two equal parts, is (A) y = mx + (B) y = mx + m (C) y = mx m + (D) y = mx + 0 [COMPREHENSION TYPE] Paragraph for question nos. & x (x ) Let a, b, c (a < b < c) be three integers satisfying the inequality 0. (x ) (x x ) Also f (x) = x (p + ) x +, x R. Q. The value of (a + b + c ) is equal to (A) (B) (C) (D) Q. The true set of values of p for which range of f(x) is [0, ) for all x R, is (A) [, ] (B) [, ) (C) {, } (D) R []
Paragraph for question nos. & Consider, P(t) = t t t 0, t R t and Q(x) = x mx + m, where x, m R. Q. The sum of all integral values in the range of P(t) is (A) 9 (B) 0 (C) (D) Q. If Q(x) + P(t) x R then true set of values of m is (A) [, ] (B) [, ] (C) [ 7, ] (D) [, 7] [MULTIPLE OBJECTIVE TYPE] Q.7 Let x + cx + = 0 and x + x + c = 0 be two quadratic equation. Then which of the following statement(s) is (are) correct? (A) They have common root if c =. (B) They have common root if c =. (C) They have atleast one common root for c = and c =. (D) They have non-real common roots if c =. Q.8 Number of solutions of the equation x + + x + + x + = a, where x [, ] and 'a' is a parameter can be (A) 0 (B) (C) (D) Q.9 In cyclic quadrilateral ABCD, if cot A = and tan B = correct?, then which of the following is (are) (A) sin D = (B) sin (A + B) = (C) cos D = (D) sin(c + D) = Q.0 If sin x + sin x = (a + ), then which of the following statement(s) is(are) correct? (A) Number of integral values of a for real solution to exist is. (B) There exists no solution for a < 9 or a > 0. (C) The minimum value of a for real solution is. (D) Number of prime values of a for real solution to exist is. []
[MATCH THE COLUMN] Q. Column-I Column-II (A) Number of solutions of the equation tan x = cot x (P) lying in the interval [0, ] is (B) Let f (x) = x + x, x R. (Q) If the equation f (x) = b has atleast one real solution then the number of integral values in the range of b, is (C) Let p and p be two real values of p for which the expression (R) f(x, y) = x + y + xy + x + py + can be factorised into two linear factors, then (p p ) equals (S) 9 (D) If one root of quadratic equation x x + k = 0 lies in (, ) (T) and other root in (, ) then the number of integral value(s) of k is Q. Let P(x) = x x + c x R where c is a real constant, then Column-I Column-II (A) If greatest value of p(x) for x [, ] is, then c equals (P) 8 (B) If smallest value of P(x) for x [, ] is, then c equals (Q) (C) If the greatest value of P(x) for x [, ] is, then c equals (R) (S) 7 [INTEGER TYPE / SUBJECTIVE] Q. Let a, b, c be sides of a triangle ABC, and denotes the area of triangle ABC. If a =, =, and a cos C + a sin C b c = 0, then find the value of (b + c). [Note: All symbols used have usual meaning in triangle ABC.] 9 Q. Let f (x) = x (k )x + and g (x) = x x +. If f (x) > minimum value of g (x) for every x R, then find the sum of all integral values of k. Q. An infinite geometric series has sum 0. A new series, obtained by squaring each term of the original series, has 0 times the sum of the original series. The common ratio of the original series is n m where m, n are relatively prime integers. Find the value of (m + n). Q. Find the number of solutions of the equation sin x + cos x + sin x = + cos x lying in interval [, ]. Q.7 If f () = (sec + cosec ) (sin + cos ) sec cosec lies completely between the roots of the quadratic equation (a ) x + ax + a + 8 = 0 for all permissible values of, then find the number of integral values of a. []
Q.8 Let a, a, a,..., a n are in arithmetic progression where n 0. If the sum of its first five even terms is equal to and the sum of the first three terms is equal to ( ) then find the seventh term of arithmetic progression. Q.9 If the expression (m + m + ) a (m + ) a + m m + b vanishes for every m R, then find the value of (a + b ). Q.0 Let x and y be two real numbers satisfying x + y + (x + y) =. If M and m be the maximum and minimum value respectively of the expression (y x), then find the value of (M m). 0 Q. If f(x) = log (x x ) takes real values then the true set of values of x is (, a] [b, ). Find the value of (b a). Q. Let P = cosec 8 cosec cosec 78 and Q = sec sec sec 8, then find the value of (PQ). ANSWER KEY Q. C Q. A Q. D Q. C Q. C Q. A Q.7 B Q.8 B Q.9 C Q.0 A Q. D Q. D Q. D Q. D Q. A Q. D Q.7 C Q.8 A Q.9 B Q.0 C Q. B Q. B Q. B Q. D Q. A Q. B Q.7 D Q.8 A Q.9 A Q.0 D Q. D Q. B Q. A Q. A Q. B Q. D Q.7 ABCD Q.8 ABC Q.9 ABD Q.0 AB Q. (A) R ; (B) T; (C) Q ; (D) P Q. (A) Q, (B) S, (C) R Q. 000 Q. 0009 Q. 000 Q. 000 Q.7 000 Q.8 000 Q.9 000 Q.0 000 Q. 000 Q. 00 [7]