Advanced Problems in Mathematics For Jee (Main & Advanced) (Black book maths)

393 Pages·2016·6.457 MB·English

by  Vikas Gupta and Pankaj Joshi (Author)

AAAdddvvvaaannnccceeeddd PPPrrrooobbbllleeemmmsss iiinnn
MMMAAATTTHHHEEEMMMAAATTTIIICCCSSS
fffooorrr
JJJEEEEEE
(((MMMAAAIIINNN &&& AAADDDVVVAAANNNCCCEEEDDD)))

by :
Vikas Gupta Pankaj Joshi
Director Director
Vibrant Academy India(P) Ltd. Vibrant Academy India(P) Ltd.
KOTA (Rajasthan) KOTA (Rajasthan)
[AN ISO 9001-2008 CERTIFIED ORGANIZATION]
CONTENTS
CALCULUS
1. Function 3 – 29
2. Limit 30 – 44
3. Continuity, Differentiability and Differentiation 45 – 74
4. Application of Derivatives 75 – 97
5. Indefinite and Definite Integration 98 – 127
6. Area Under Curves 128 – 134
7. Differential Equations 135 – 144
ALGEBRA
8. Quadratic Equations 147 – 176
9. Sequence and Series 177 – 197
10. Determinants 198 – 206
11. Complex Numbers 207 – 216
12. Matrices 217 – 224
13. Permutation and Combinations 225 – 233
14. Binomial Theorem 234 – 242
15. Probability 243 – 251
16. Logarithms 252 – 264
CO-ORDINATE GEOMETRY
17. Straight Lines 267 – 280
18. Circle 281 – 295
19. Parabola 296 – 302
20. Ellipse 303 – 307
21. Hyperbola 308 – 312
TRIGONOMETRY
22. Compound Angles 315 – 334
23. Trigonometric Equations 335 – 343
24. Solution of Triangles 344 – 359
25. Inverse Trigonometric Functions 360 – 370
VECTOR & 3DIMENSIONAL GEOMETRY
26. Vector & 3Dimensional Geometry 373 – 389

Function 3
1
F
UNCTION
Exercise-1 : Single Choice Problems
1. Range of the function f(x)(cid:61)log (2(cid:45)log (16sin2 x (cid:43)1)) is :
2 2
(a) [0,1] (b) ((cid:45)(cid:165),1] (c) [(cid:45)1,1] (d) ((cid:45)(cid:165),(cid:165))
2. The value of a and b for which |e|x(cid:45)b| (cid:45)a|(cid:61)2, has four distinct solutions, are :
(a) a(cid:206)((cid:45)3,(cid:165)), b(cid:61)0 (b) a(cid:206)(2,(cid:165)), b(cid:61)0 (c) a(cid:206)(3,(cid:165)), b(cid:206)R (d) a(cid:206)(2,(cid:165)), b(cid:61)a
3. The range of the function :
1
f(x)(cid:61)tan(cid:45)1 x (cid:43) sin(cid:45)1 x
2
(a) ((cid:45)(cid:112) 2,(cid:112) 2) (b) [(cid:45)(cid:112) 2,(cid:112) 2](cid:45){0} (c) [(cid:45)(cid:112) 2,(cid:112) 2] (d) ((cid:45)3(cid:112) 4,3(cid:112) 4)
4. Find the number of real ordered pair(s) (x,y) for which :
16x2(cid:43)y (cid:43)16x(cid:43)y2 (cid:61)1
(a) 0 (b) 1 (c) 2 (d) 3
|x|
(cid:230)1(cid:246)
5. The complete range of values of ‘a’ such that (cid:231) (cid:247) (cid:61)x2 (cid:45)a is satisfied for maximum number
(cid:232)2(cid:248)
of values of x is :
(a) ((cid:45)(cid:165),(cid:45)1) (b) ((cid:45)(cid:165),(cid:165)) (c) ((cid:45)1,1) (d) ((cid:45)1,(cid:165))
6. For a real number x, let [x] denotes the greatest integer less than or equal to x. Let f:R(cid:174) R be
defined by f(x)(cid:61)2x (cid:43)[x](cid:43)sinxcosx. Then f is :
(a) One-one but not onto (b) Onto but not one-one
(c) Both one-one and onto (d) Neither one-one nor onto
(cid:230)7(cid:45)5(x2 (cid:43)3)(cid:246)
7. The maximum value of sec(cid:45)1(cid:231) (cid:247) is :
(cid:232)(cid:231) 2(x2 (cid:43)2) (cid:248)(cid:247)
5(cid:112) 5(cid:112) 7(cid:112) 2(cid:112)
(a) (b) (c) (d)
6 12 12 3
4 Advanced Problems in Mathematics for JEE
8. Number of ordered pair (a,b) from the set A(cid:61) {1, 2, 3, 4, 5} so that the function
x3 a
f(x)(cid:61) (cid:43) x2 (cid:43)bx (cid:43)10 is an injective mapping (cid:34)x(cid:206)R :
3 2
(a) 13 (b) 14 (c) 15 (d) 16
9. Let A be the greatest value of the function f(x)(cid:61)log [x], (where [(cid:215)] denotes greatest integer
x
function) and B be the least value of the function g(x)(cid:61)|sinx|(cid:43)|cosx|, then :
(a) A(cid:62)B (b) A(cid:60)B (c) A(cid:61)B (d) 2A(cid:43)B (cid:61)4
10. Let A(cid:61)[a,(cid:165)) denotes domain, then f:[a,(cid:165))(cid:174) B, f(x)(cid:61)2x3 (cid:45)3x2 (cid:43)6 will have an inverse for
the smallest real value of a, if :
(a) a(cid:61)1,B (cid:61)[5,(cid:165)) (b) a(cid:61)2,B (cid:61)[10,(cid:165)) (c) a(cid:61)0,B (cid:61)[6,(cid:165)) (d) a(cid:61)(cid:45)1,B (cid:61)[1,(cid:165))
11. Solution of the inequation {x}({x}(cid:45)1)({x}(cid:43)2)(cid:179)0
(where {(cid:215)} denotes fractional part function) is :
(a) x(cid:206)((cid:45)2,1) (b) x(cid:206)I (I denote set of integers)
(c) x(cid:206)[0,1) (d) x(cid:206)[(cid:45)2,0)
12. Let f(x),g(x) be two real valued functions then the function h(x)(cid:61)2max{f(x)(cid:45)g(x),0} is
equal to :
(a) f(x)(cid:45)g(x)(cid:45)|g(x)(cid:45) f(x)| (b) f(x)(cid:43) g(x)(cid:45)|g(x)(cid:45) f(x)|
(c) f(x)(cid:45)g(x)(cid:43)|g(x)(cid:45) f(x)| (d) f(x)(cid:43) g(x)(cid:43)|g(x)(cid:45) f(x)|
13. Let R (cid:61) {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A(cid:61) {1, 2, 3, 4}. The relation
R is :
(a) a function (b) reflexive (c) not symmetric (d) transitive
(cid:230) 1 (cid:246) K(cid:112)
14. The true set of values of ‘K’ for which sin(cid:45)1(cid:231) (cid:247)(cid:61) may have a solution is :
(cid:232)1(cid:43)sin2 x(cid:248) 6
(cid:233)1 1(cid:249) (cid:233)1 1(cid:249)
(a) , (b) [1,3] (c) , (d) [2,4]
(cid:234) (cid:250) (cid:234) (cid:250)
(cid:235)4 2(cid:251) (cid:235)6 2(cid:251)
15. A real valued function f(x) satisfies the functional equation
f(x (cid:45) y)(cid:61) f(x)f(y)(cid:45) f(a(cid:45)x)f(a(cid:43) y) where ‘a’ is a given constant and f(0)(cid:61)1, f(2a(cid:45)x) is
equal to :
(a) (cid:45)f(x) (b) f(x) (c) f(a)(cid:43) f(a(cid:45)x) (d) f((cid:45)x)
16. Let g:R(cid:174) R be given by g(x)(cid:61)3(cid:43)4x if gn(x)(cid:61)gogogo......og(x)n times. Then inverse of
gn(x) is equal to :
(a) (x (cid:43)1(cid:45)4n)(cid:215)4(cid:45)n (b) (x (cid:45)1(cid:43)4n)4(cid:45)n (c) (x (cid:43)1(cid:43)4n)4(cid:45)n (d) None of these
x2 (cid:43)2x (cid:43)a
17. Let f:D(cid:174) R be defined as : f(x)(cid:61) where D and R denote the domain of f and
x2 (cid:43)4x (cid:43)3a
the set of all real numbers respectively. If f is surjective mapping, then the complete range of a
is :
(a) 0(cid:163)a(cid:163)1 (b) 0(cid:60)a(cid:163)1 (c) 0(cid:163)a(cid:60)1 (d) 0(cid:60)a(cid:60)1
Function 5
18. If f:((cid:45)(cid:165),2](cid:190)(cid:174)((cid:45)(cid:165),4], where f(x)(cid:61)x(4(cid:45)x), then f(cid:45)1(x) is given by :
(a) 2(cid:45) 4(cid:45)x (b) 2(cid:43) 4(cid:45)x (c) (cid:45)2(cid:43) 4(cid:45)x (d) (cid:45)2(cid:45) 4(cid:45)x
19. If [5sinx](cid:43)[cosx](cid:43)6(cid:61)0, then range of f(x)(cid:61) 3cosx (cid:43)sinx corresponding to solution set of
the given equation is : (where [(cid:215)] denotes greatest integer function)
(cid:230) 3 3 (cid:43)2 (cid:246) (cid:230) 3 3 (cid:43)4 (cid:246)
(a) [(cid:45)2,(cid:45)1) (b) (cid:231)(cid:45) ,(cid:45)1(cid:247) (c) [(cid:45)2,(cid:45) 3) (d) (cid:231)(cid:45) ,(cid:45)1(cid:247)
(cid:231) (cid:247) (cid:231) (cid:247)
5 5
(cid:232) (cid:248) (cid:232) (cid:248)
20. If f:R(cid:174) R, f(x)(cid:61)ax (cid:43)cosx is an invertible function, then complete set of values of a is :
(a) ((cid:45)2,(cid:45)1](cid:200)[1,2) (b) [(cid:45)1,1] (c) ((cid:45)(cid:165),(cid:45)1](cid:200)[1,(cid:165)) (d) ((cid:45)(cid:165),(cid:45)2](cid:200)[2,(cid:165))
(cid:233) x(cid:249) (cid:233) x(cid:249) (cid:233) x(cid:249)
21. The range of function f(x)(cid:61)[1(cid:43)sinx](cid:43) 2(cid:43)sin (cid:43) 3(cid:43)sin (cid:43)...(cid:43) n(cid:43)sin (cid:34)x(cid:206)
(cid:234) (cid:250) (cid:234) (cid:250) (cid:234) (cid:250)
(cid:235) 2(cid:251) (cid:235) 3(cid:251) (cid:235) n(cid:251)
[0,(cid:112)], n(cid:206)N ([(cid:215)] denotes greatest integer function) is :
(cid:236)(cid:239)n2 (cid:43)n(cid:45)2 n(n(cid:43)1)(cid:252)(cid:239) (cid:236)n(n(cid:43)1)(cid:252)
(a) (cid:237) , (cid:253) (b) (cid:237) (cid:253)
(cid:238)(cid:239) 2 2 (cid:254)(cid:239) (cid:238) 2 (cid:254)
(cid:236)(cid:239)n(n(cid:43)1) n2 (cid:43)n(cid:43)2 n2 (cid:43)n(cid:43)4(cid:252)(cid:239) (cid:236)(cid:239)n(n(cid:43)1) n2 (cid:43)n(cid:43)2(cid:252)(cid:239)
(c) (cid:237) , , (cid:253) (d) (cid:237) , (cid:253)
(cid:238)(cid:239) 2 2 2 (cid:254)(cid:239) (cid:238)(cid:239) 2 2 (cid:254)(cid:239)
x2 (cid:43)ax (cid:43)1
22. If f:R(cid:174) R, f(x)(cid:61) , then the complete set of values of ‘a’ such that f(x) is onto is :
x2 (cid:43) x (cid:43)1
(a) ((cid:45)(cid:165),(cid:165)) (b) ((cid:45)(cid:165),0) (c) (0,(cid:165)) (d) not possible
23. If f(x) and g(x) are two functions such that f(x)(cid:61)[x](cid:43)[(cid:45)x] and g(x)(cid:61){x}(cid:34)x(cid:206)R and
h(x)(cid:61) f(g(x)); then which of the following is incorrect ?
([(cid:215)] denotes greatest integer function and {·} denotes fractional part function)
(a) f(x) and h(x) are identical functions (b) f(x)(cid:61)g(x) has no solution
(c) f(x)(cid:43)h(x)(cid:62)0 has no solution (d) f(x)(cid:45)h(x) is a periodic function
(cid:233) x (cid:249)(cid:233) 15(cid:249)
24. Number of elements in the range set of f(x)(cid:61) (cid:45) (cid:34)x(cid:206)(0,90); (where [(cid:215)] denotes
(cid:234) (cid:250)(cid:234) (cid:250)
(cid:235)15(cid:251)(cid:235) x (cid:251)
greatest integer function) :
(a) 5 (b) 6 (c) 7 (d) Infinite
25. The graph of function f(x) is shown below :
O
1
Then the graph of g(x)(cid:61) is :
f(|x|)
6 Advanced Problems in Mathematics for JEE
(a) (b)
(c) (d)
26. Which of the following function is homogeneous ?
y x
(a) f(x)(cid:61)xsin y (cid:43) ysinx (b) g(x)(cid:61)xex (cid:43) yey
xy x (cid:45) ycosx
(c) h(x)(cid:61) (d) (cid:102)(x)(cid:61)
x (cid:43) y2 ysinx (cid:43) y
(cid:233)2x (cid:43)3 ; x(cid:163)1
27. Let f(x)(cid:61) . If the range of f(x)(cid:61)R (set of real numbers) then number of
(cid:235)(cid:234)a2x (cid:43)1 ; x(cid:62)1
integral value(s), which a may take :
(a) 2 (b) 3 (c) 4 (d) 5
28. The maximum integral value of x in the domain of f(x)(cid:61)log (log (log (x (cid:45)5)) is :
10 13 4
(a) 5 (b) 7 (c) 8 (d) 9
(cid:230) 4 (cid:246)
29. Range of the function f(x)(cid:61)log (cid:231) (cid:247) is :
2(cid:231) (cid:247)
(cid:232) x (cid:43)2 (cid:43) 2(cid:45)x(cid:248)
(cid:233)1 (cid:249) (cid:233)1 (cid:249)
(a) (0,(cid:165)) (b) ,1 (c) [1,2] (d) ,1
(cid:234) (cid:250) (cid:234) (cid:250)
(cid:235)2 (cid:251) (cid:235)4 (cid:251)
30. Number of integers statisfying the equation |x2 (cid:43)5x|(cid:43)|x (cid:45)x2|(cid:61)|6x| is :
(a) 3 (b) 5 (c) 7 (d) 9
31. Which of the following is not an odd function ?
(cid:230) x4 (cid:43) x2 (cid:43)1 (cid:246)
(a) ln(cid:231) (cid:247)
(cid:232)(cid:231)(x2 (cid:43) x (cid:43)1)2 (cid:248)(cid:247)
(b) sgn(sgn(x))
(c) sin(tanx)
(cid:230)1(cid:246) (cid:230)1(cid:246)
(d) f(x), where f(x)(cid:43) f(cid:231) (cid:247)(cid:61) f(x)(cid:215) f(cid:231) (cid:247) (cid:34)x(cid:206)R (cid:45){0} and f(2)(cid:61)33
(cid:232) x(cid:248) (cid:232) x(cid:248)