M 68 MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maimum Marks : [ Q. to 6 carry one mark each ]. If sin sin sin y z, then the value of 9 y 9 z 9 9 y 9 z 9 A. B. C. D. is equal to. Let p, q, r be the sides opposite to the angles P, Q, R respectively in a triangle PQR. If r sin P sin Q pq, then the triangle is A. equilateral B. acute angled but not equilateral C. obtuse angled D. right angled. Let p, q, r be the sides opposite to the angles P, Q, R respectively in a triangle PQR. Then pr sin P Q R equals A. p +q + r B. p + r q C. q + r p D. p + q r. Let P (, ), Q (, ) be the vertices of the triangle PQR. If the centroid of PQR lies on the line + y =, then the locus of R is A. + y = 9 B. y = 9 C. + y = 5 D. y = 5 5. lim A. does not eist B. equals log e 6. If f is a real-valued differentiable function such that f f A. f () must be an increasing function B. f () must be a decreasing function C. f () must be an increasing function D. f () must be a decreasing function 7. Rolle's theorem is applicable in the interval [, ] for the function C. equals D. lies between and ' for all real, then A. f () = B. f () = C. f () = + D. f () = d y dy 8. The solution of 5 y, y, y e 5 d d is A. y = e 5 + e 5 B. y = ( + )e 5 C. y e 5 D. y e 5
9. Let P be the midpoint of a chord joining the verte of the parabola y = 8 to another point on it. Then the locus of P is A. y = B. y = C. y D. y.the line = y intersects the ellipse y at the points P and Q. The equation of the circle with PQ as diameter is A. y B. y C. y D. y 5. The eccentric angle in the first quadrant of a point on the ellipse y at a distance units from 8 the centre of the ellipse is A. 6 B. C.. The transverse ais of a hyperbola is along the -ais and its length is a. The verte of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is A. 6 y = a B. y = a C. 6y = a D. y = a. A point moves in such a way that the difference of its distance from two points (8,) and ( 8,) always remains. Then the locus of the point is A. a circle B. a parabola C. an ellipse D. a hyperbola. The number of integer values of m, for which the -coordinate of the point of intersection of the lines + y = 9 and y = m + is also an integer, is A. B. C. D. 5. If a straight line passes through the point () and the portion of the line intercepted between the aes is divided equally at that point, then y is A. B. C. D. D. 6. The maimum value of z when the comple number z satisfies the condition z z is 7. If A. B. C. D. 5 i 5 iy, where and y are real, then the ordered pair (,y) is A. (, ) B. (, ) C. (, ) D.,
8. If z z is purely imaginary, then A. z B. z C. z D. z 9. There are students in a class. In an eamination, 5 of them failed in Mathematics, 5 failed in Physics, failed in Biology and failed in eactly two of the three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects. A. is B. is C. is D. cannot be determined from the given information. A vehicle registration number consists of letters of English alphabet followed by digits, where the first digit is not zero. Then the total number of vehicles with distinct registration numbers is A. 6 B. 6 P P C. 6 P 9 P D. 6 9. The number of words that can be written using all the letters of the word 'IRRATIONAL' is A.! (!) B.! (!) C.!! D.!. Four speakers will address a meeting where speaker Q will always speak after speaker P. Then the number of ways in which the order of speakers can be prepared is A. 56 B. 8 C. D.. The number of diagonals in a regular polygon of sides is A. 95 B. 85 C. 75 D. 65. Let the coefficients of powers of in the nd, rd and th terms in the epansion of ( + ) n, where n is a positive integer, be in arithmetic progression. Then the sum of the coefficients of odd powers of in the epansion is A. B. 6 C. 8 D. 56 5. Let f( ) a b c, g( ) p q r such that f( ) g( ), f( ) g( ) and f( ) g( ). Then f( ) g( ) is A. B. 5 C. 6 D. 7 6. The sum! +! +... + 5 5! equals A. 5! B. 5! C. 5! + D. 5! 7. Si numbers are in A.P. such that their sum is. The first term is times the third term. Then the fifth term is A. 5 B. C. 9 D. 8. The sum of the infinite series 5. 57 6.. 6 9...... 6.. 9. is equal to A. B. C. D.
9. The equations + + a = and + a + = have a common real root A. for no value of a B. for eactly one value of a C. for eactly two values of a D. for eactly three values of a. If 6, 7, 6 are the P th, Q th and R th terms of a G.P., then P + Q is equal to A. R B. R C. R D. R. The equation y y k represents a parabola whose latus rectum is A. B. C. D.. If the circles y ky 6 and y ky k intersect orthogonally, then k is equal to A. or B. or C. or. If four distinct points (k, k), (, ), (, ), (, ) lie on a circle, then D. or A. k < B. < k < C. k = D. k >. The line joining A(b cos, b sin ) and B(a cos, a sin ), where a b, is produced to the point M(, y) so that AM : MB = b : a. Then cos ysin A. B. C. D. a + b 5. Let the foci of the ellipse y subtend a right angle at a point P. Then the locus of P is 9 A. + y = B. + y = C. + y = D. + y = 8 6. The general solution of the differential equation dy d y y is A. log y 6y c e B. log y 6y c e C. log y 6y c e D. log y 6y c e / sin cos 7. The value of the integral / 6 sin cos d is equal to A. 6 B. 8 C. D. 8. The value of the integral d is equal to (tan ) A. B. 6 C. 8 D.
l o g 9. The integrating factor of the differential equation log dy y log e e is given by d A. ( e ) B. log e (log e ) C. log e D. log e. Number of solutions of the equation tan sec cos, [, ] is A. B. C. `D.. The value of the integral sin cos sin d is equal to A. log e B. log e C.. Then at. Let y sin log e, log e D. dy, equals d A. B. C. D.. Maimum value of the function f A. B. 9 8. For, the value of A. B. d d on the interval 8, 6 is tan cos sin 5. The value of the integral sin e d C. is equal to is equal to A. B. e C. e 6. If D. C. D. D. and are the roots of the equation p q 6 are and B. and and D. the roots of the equation p q p p q A. C. 7 8 log e sin sin where p and q are real, then and
is 7. The number of solutions of the equation log A. B. C. D. 8. The sum of the series n n n C C... C n n is equal to A. n n B. C. n n n n D. n n 9. The value of... r r! is equal to r A. e B. e C. e D. e 5. If p Q PP T,, then the value of the determinant of Q is equal to A. B. C. D. 5. The remainder obtained when!!... 95! is divided by 5 is A. B. C. D. 5. If P, Q, R are angles of triangle PQR, then the value of cos R cos Q cos R cos P cos Q cos P is equal to A. B. C. 5. The number of real values of for which the system of equations D. y 5z 5 y z y 5y z z has infinite number of solutions is A. B. C. D. 6 5. The total number of injections (one-one into mappings) from a, a, a, a is to b, b, b, b, b, b, b A. B. C. 8 D. 8 5 6 7
c r r and 7 d r r. If P c r 55. Let r 7 r 5 r and Q d r, then P Q r A. B. 8 C. 6 D. is equal to 56. Two decks of playing cards are well shuffled and 6 cards are randomly distributed to a player. Then the probability that the player gets all distinct cards is A. 5C 6 / C 6 B. 5C 6 / C 6 C. 5 C 6 / C 6 D. 6 5 C 6 / C 6 57. An urn contains 8 red and 5 white balls. Three balls are drawn at random. Then the probability that balls of both colours are drawn is A. B. 7 58. Two coins are available, one fair and the other two-headed. Choose a coin and toss it once ; assume that C. the unbiased coin is chosen with probability. Given that the outcome is head, the probability that the two- headed coin was chosen is D. A. 5 B. 5 C. 5 D. 7 59. Let R be the set of real numbers and the functions f : R R and g : R R be defined f and g. Then the value of for which fg gf is A. B. C. D. 6. If a, b, c are in arithmetic progression, then the roots of the equation a b c are A. and c a B. a and c C. and c a [ Q. 6 to 8 carry two marks each ] D. and c a 6. Let y be the solution of the differential equation dy d y y log satisfying y() =. Then y satisfies A. y y B. y y C. y y 6. The area of the region, bounded by the curves y sin quadrant, is A. B. C. D. y y and y sin D. in the first
5 6. The value of the integral d is equal to A. B. 8 C. D. 6 6. If f() and g() are twice differentiable functions on (, ) satisfying f() = g(), f() =, g() = 6, f() =, g() = 9, then f() g() is A. B. C. D. 65. Let [] denote the greatest integer less than or equal to, then the value of the integral d is equal to A. B. C. D. 66. The points representing the comple number z for which arg z z lie on A. a circle B. a straight line C. an ellipse D. a parabola 67. Let a, b, c, p, q, r be positive real numbers such that a, b, c are in G.P. and a p = b q = c r. Then A. p, q, r are in G.P. B. p, q, r are in A.P. C. p, q, r are in H.P. D. p, q, r are in A.P. 68. Let S k be the sum of an infinite G.P. series whose first term is k and common ratio is the value of k Sk k is equal to k k A. log e B. log e C. log e D. log e 69. The quadratic equation a 8a a a (k > ). Then possesses roots of opposite sign. Then A. a B. < a < C. a < 8 D. a 8, then 7. If log e 6 log e A. < 5 B. < or > C. 5 D. < or > 5 7. The coefficient of in the epansion of... is A. 9 C 9 B. C C. C D. C 7. The system of linear equations + y + z = y z = 6 + y + z = has A. Infinite number of solutions for and all B. Infinite number of solutions for = and = C. No solution for D. Unique solution for = and =
7. Let A and B be two events with P(A C ) =., P(B) =. and P(A B C ) =.5. Then P(B\A B C ) is equal to A. B. 7. Let p, q, r be the altitudes of a triangle with area S and perimeter t. Then the value of is p q r s t s s A. B. C. D. t s t t 75. Let C and C denote the centres of the circles + y = and ( ) + y = respectively and let P and Q be their points of intersection. Then the areas of triangles C PQ and C PQ are in the ratio A. : B. 5 : C. 7 : D. 9 : 76. A straight line through the point of intersection of the lines + y = and + y = meets the coordinates aes at A and B. The locus of the midpoint of AB is A. ( + y) = y B. ( + y) = y C. ( + y) = y D. + y = y 77. Let P and Q be the points on the parabola y = so that the line segment PQ subtends right angle at the verte. If PQ intersects the ais of the parabola at R, then the distance of the verte from R is A. B. C. D. 6 78. The incentre of an equilateral triangle is (, ) and the equation of the one side is + y + =. Then the equation of the circumcircle of the triangle is A. y y B. y y C. y y D. y y 79. The value of lim n n! n n A. B. is e C. C. e D. D. e 8. The area of the region bounded by the curves y =, y A. log e B., = is log e C. log e D. 5 log e
M 68 SUBJECT : MATHEMATICS KEY ANSWERS WITH EXPLANATIONS. Given sin sin sin y z. sin sin y sin z y z 9 y9 z9 9. y9. z9 Ans. C.. r.sin P. sin Q pq p r q. R R pq r R r = R side = circumradius = diameter So, triangle is right angled triangle.. Given P + Q + R = 8 P R 8Q sin 8Q Now pr cos Q sin 9 Q cos Q pr p r q p r q pr. Let R = (h, k) Centroid = h k, h k. h k h k 9 Required locus y 9 Ans. A. h, k
5. lim lim lim. log log 6. Given f. f R and f() is differentiable f() is continuous function and f() and either f() > or f() < but It can not cut the -ais When f() > then f f() is decresing function then When f f f() is increasing function We can say that f is decreasing function. 7. Check the options individually. Take option (b) f f are opposite of sign Now, Rolle s theorem is applicable. 8. Let y e m dy d dy d me m me m 5mem mem em 5m m 5m
m. 5 5 5 General solution is y A B e y () = = A. A y e 5 e 5 A B e 5 = ( + B) B = y e 5 Ans. C. 9. y 8. y a a = 8 a = t h t or t = h t k k t or t k h k h y y y y y y y P,
Q, equation of circle PQ as diameter is y y y y 5. Let the pt on the first quadrant cos, 8 sin distance from the centre = cos 8sin 9 8 cos 9 cos cos [ pt lie on lst quadrant]. Let the hyperbola y a b Length of the transverse ais = a verte = (a, ) focus (ae, ) ae, a, ae centre = (, ) a e e b a b a Equation : y a a. The distance between (8, ) and ( 8, ) = 6 >. According to the defination of hyperbola the locus is a hyperbola.
. m I y 9 y m m 5 m 9 5 m I m 5,,, 5 when m I m can take only two values m =,. 5. Let the equation of the line a Its passes through, y b a b, is the mid points of (a, ) and (, b) a and b a, b Equation of the line Ans. C. 6. z z y y Now, z z z z z z z z z z z z z z Ans. C.
7. 8. 5 i 5 iy 5 i iy 5 i i5 iy i5. w5 iy iy i w iy w i iy z ik, k R, k z z ik ; by comp.-div. ik ik z ik z ik ik k k 9. n(m) = 5 = No. of failed in maths. n(p) = 5 n (B) = n M P n M B n P B n M P B We have to find n M P B Total no of student = 99 nm np nb nm P nm B np B nm P B 5 5 nm P B nm P B 99 5 nm P B 99 nm P B nm P B n M P B 99 Ans. C.
. 5 6 two alphabet we can choose 6 ways. and st number we can choose 9 ways. net numbers we can choose ways.. I - R - A - T - N - O - L - Number of words Ans. A.!!. Required number of ways is which the order of speakers can be prepared!! = [Taking speakers P & Q as identical]. No. of diagonals in a regular polygon C 99 = 5 99 = 95 = 85. n C, n C, n C are in A.P. n n n n n 6 n n or, n 6 n
or, 6n 6 = n n + + 6 or, n 9n + = n = 7, n = not acceptable. n 7 sum 6 6 5. f() = a + b + c, g() = p + q + r... () a b b q c r Now f() = g() a + b + c = p + q + r a b b q c r f() = g() a + b + c = p + q + r... () a b b q [using ()] f() g() = 9(a p) + (b q) + (c r) = 8(a p) + (b q) = [using ()] (a p) + (b q) = (a p) = (using ()) Now, f() g() = 6(a p) + (b q) + (c r) Ans. C. = 5 (a p) + (b q) (using ()) = 5.() + ( ) = 5 9 = 6 6.! +! +! +... + n n! = (n + )!! +! +... + 5 5! = 5! 7. Si numbers are in A.P. a 5d, a d, a d a + d, a + d, a + 5d 6a = a p b q a 5 d d
5d d d Fifth term = a + d 9 8 = 8. n n n n!... comparing, n n n 6 n n
9. Let be the common root a a a a a Either a = or, = Put = in the st equation + + a = a = Put a = + + = + + = They have no real common root. Put a = + = & + = or, = they have a one real common root.. Let A be the st term & r be the c.r. A. r p = 6 = 6 A. r q = 7 = A. r R = 6 =. Now, A 6 A r r p 6 q 6 R A r p q R 6 A r A r A r r p q R 6 6 6 r p q R 6 6 p + q = R Ans. C.
. y y k y y k y k k = Y AX L.R. = A = unit.. Apply, gg ff C C k k 6 k k 6 k or, k k 6 or, k k k 6 or, k k k or, k k k =, Ans. A.. Equation of circle is y y y y y y k 9k k 9k k k k =, k = Ans. C.
ab cos ab cos. b a = = ab b a cos cos ab b a ab sin ab sin y b a y = ab b a sin cos sin sin cos sin 5. cos ysin Ans. A. y 9 a =, b = e b a 9 8 9 e ae mps mps y y y 8 y 8 y 8
6. dy y d y Let + y = z dy d dy d dz d dz d dz d z z dz d z z = z z z = z z or, or, or, or, or, or, z dz d z z dz d c z z dz c z z d z z c z log z = + c 9 log y y c 9 or, 6 6y log y 9 c or, 6y log y c
7. sin cos cos sin cos sin cos sin 6 = cos sin cos sin d 6 = sin = 6 sin sin 6 = = 8. I sin cos d = = cos d cos sin sin d sin cos I d = a Apply f d f a d a I
9. dy y d loge d I e log e = e = e d loge loge log log e = e log e log e = log e. sin cos cos cos sin sin sin sin sin sin sin sin sin sin cos sin sin sin Ans. C. 5, 6 6
. sin cos sin d / sin cos d sin cos dzz z log z log log log, Putting sin cos = z cos sin d dz z sin log. y e sin loge e sin log sin sin log dy d cos sin loge cos dy d = Ans. A.
. f 8 f 8 For ma & min. f 8 8 = 6 = +, 6 f f at = f 8 f() is ma. Ans. A.. Ep. d d d d d d d d tan cos sin tan cos sin tan tan tan tan tan cos sin cos sin
Ep d. d 5. sin e d e d sin e d e d e = (e ) Ans. C. 6. p q roots are and p p p q p q q Now, 8 6 6 6 6 Sum of the roots = p q Product of the roots = Ans. A. p q
7. 8. C =, Ans. C. C C Cn n n d C d C d C d C d n C C C C n n n n n n 9. n C C Cn n n Ans. A. r r r! r r! r r! r e Ans. C. 5. P P T
Q = PP T 6 6 8 8 + 6 + + 9 + Q 66 6 Ans. A. 5.! +! +! +! = + + 6 + = 5 Required remainder = 5. Putting P = Q = R = cos R cos Q cos R cos P cos Q cos P 5. y 5z 5 y z 5y z 5 5 5
7 5 5 5 5 5 8 5 8 8 a b c c a b b c a It has one real solution and two imaginary solution. Ans. A. 5. n(a) = n(b) = 7 no. of mappings = 7 p 7!! /! 5 6 7! 55. C C... C / 6 7 = 8 C C C C6 C8 C 7 d d... d7 d d d5 d7 P 9 8 Q 6 6 9 a b c abc 56. Total no. of possible outcomes = C 6, which are equally likely. Number of casses that the player gets all distinct cards = 6 C 5C 6 6 5 C6 Required probability 6 5 C6 C 6
57. casei R, W caseii R, W 8R 5W 8C 5 5 C 8 C C C 8. 7 5. 5 8 6 8 6 / 58. Let X = the event that outcame is head Given that P A P B P B X / 8 59. f : R R g : R R / 5 8 5 / f 8 f g f g = g f g Ans. A. A B H/ T H/ H
6. b a c a b c a a c c a a c c a c a c c, a Ans. A. 6. dy d y y log d. log log t dy y y or, dt dy y t y d. dy dt dy or, dt dy t y y I. F. e y dy elog y y d t. y y dy k t. y log y k y.log log y k log k k = y.loge loge y y y 6. Solving, sin sin or,, Required area = d
Ans. C. 6 sq. unit. 6. Area = + (+6). Ans. C. = + 8 = + 8 = sq. unit 6. h f g h f g h f g h h c f g c f g c 6 = c c = f g f g c
f g. c 9 c 6 c c f g f g. = 65. d d d d d d = Ans. A. 66. arg z z z lies on a circle Ans. A. 67. a, b, c are in G.P. b = ac log b log a log c... () Now, a p b q c r
p log a q log b r log c log a q log c, log b p log b From () q r q q p r p r q p, q, r in H.P. Ans. C. 68. k sk k k k k k k k kk k k k k k 5 5 6 6... 5 6... =... 5 6... 5 6
=... 5 6... 5 6 = ln + = ln 69. O lies between the roots f() < a a a a a 7. loge 6 loge 6 5 5 5 5 5 6 6, Req. Answer 5 Ans. A.
. 7.... Required coefficient = C [From N r found coefficien of ] Ans. C. 7. To get infinite no of solution. u = Now + or, or, = 7. P A C. P B. P A P B C. 7. 6 PA B C. 5 PB / A B C P B A BC P A BC = P B A P A B C
P B A PA B C P A P A BC P A BC. 7. 5 P A P B C P A B C.. 7. 6. 5 Ans. A. 7. S ap bq cr, where a, b, c are the sides of the triangle a + b + c = t Now a b c t p q r s s s s 75. y y equation of common chord. y y y y y y t s y P ( 7 /, ) R (, ) C (, 6) C (, ) Q y
76. 7 7 Here R C Q 7, 9 RQ 6 PQ C PQ C PQ 5 5 7 5.. 7. 5 6.. C PQ CPQ Ans. C. 7 5 5 6 The point of intersectgion of the given lines is Any line through this point is y m Then coordinates of A & B are A,. m m, m and B, m m and k. Let P (h, k) be the mid-point of AB. Then h m Eliminating m from the above two repations, we get h k hk The required locus is (+y) = y..
77. Any chord of a parabola y = a, which subtends right angle at the verte, always pass through a fied point (a, ) on the ais of the parabola. In this case R is (, ) The distance of R from the verte is. Ans. C. 78. (,) The incentre, circumcentre, centroid of an equlateral triangle are same. Inradius = Circumradius = Circumcircle is y i.e., y y 79. Let L lim n n! n n n n log L lim log... n n n n n n lim log log... log n n n n a n lim log n n r log d log = L e e r n
8. Area d = = 6 log 5 log log log sq. unit.