SET THEORY
BOOSTER - A
1) The number of proper subset in a set consisting of four distinct elements is
a) 4 b) 8 c) 16 d) 64
2) The number of proper subsets in a set consisting of five distinct elements is
a) 5 b) 10 c) 32 d) 31
3) If x ∈A=> x ∈ B then
a) A= B b) A ⊂B C) A ⊆B d) B ⊆A
4) If A ⊆ B and B ⊆ A then
a) A= ∅ b) A ∩B = ∅ c) A= B d) none
5) For two sets if A ∪B = A ∩B then
a) A ⊆B b) B ⊆ A c) A= B d) none
6) A - B = ∅ iff
a) A≠ B b) A ⊂B c) B ⊂ A d) A ∩B = ∅
7) If A ∩ B = B then
a) A ⊆B b) B ⊆ A c) A= B d) A= ∅
8) If A and B are two disjoint sets then n(A ∪B)=
a) n(A)+ n(B) b) n(A) - n(B) c) 0 d) none
9) For any two set A and B, n(A)+ n(B) - n(A ∩B)=
a) n(A ∪B) b) n(A) - n(B) c) ∅ d) none
10) The dual of A ∪ U= U is
a) A ∪ U= U b) A ∪∅= ∅ c) A ∪∅ = A d) A ∩∅= ∅
11) The dual of A ∪(B ∩C) = (A ∪B) ∩ (A ∪ C) is
a) (A ∩ B) ∪ (A∩ C)
b) (A ∪B) ∪(A ∪C)
c) (A∩B) ∩ (A ∩ C)
d) (A∪B)∪ (A ∪C)
12) State which of the following statements is true?
a) Subset of an infinite set is so an infinite set
b) The set of even integers greater that 889 is an infinite set.
c) The set of odd negative integers greater than (-150) is an infinite set.
d) A={x : x is real and 0< x ≤1) is a singleton set.
13) State which of the following statements is not true?
a) If a ∈ A and a ∈ B then A ⊆B.
b) If A⊆B and B ⊆C then A ⊆ C.
c) If A ⊆ B and B ⊆A, then A= B.
d) For any set A, if A ∪∅ = ∅(∅ being the null set) then A= ∅.
14) State which of the following is the set of factors of the number 12
a) {2,3,4,6} b) {2,3,4,6,12} c) {2,3,4,8,6} d) {1,2,3,4,6,12}
15) State which of the following is a null set?
a) {0} b) {∅}
c) {x: x is an integer and 1< x <2}
d) {x: x is a real number and 1< x <2}
16) If B be power set of A, state which of the following is true?
a) A ⊃B b) B ⊃A c) A ∈B d) A= B
17) If x ∈ A ∪B, State which of the following is true?
a) x ∈A b) x ∈B c) x ∈ A∀ x ∈B d) x ∈A ∧ x ∈B
18) If x ∈ A ∩B, state which of the following is true?
a) x ∈ A ∧ x ∈B b) x ∈B c) x ∈A ∨ x ∈B d) x ∉ A
19) If A= {2,4,6,8}, state which of the following is true?
a) {2,4} ∈ A b) {2,4} ⊆A c) {2,4} ⊂ A d) {2,4} ∈ Aᶜ
20) State which of the following statements is true?
a) {a} ∈ {a, b,c}
b) a ∉ {a,b,c}
c) a ⊂ {a,b,c}
d) {a} ⊂ {a,b,c}
21) State which of the following four sets are equal?
a) A={0} b) B={∅}
c) C={x : x is a perfect square and 2≤ x ≤6}
d) D={x : x is an integer and -1< x < 1}
22) Some well defined sets are given below. Identify the null set:
a) A==x: x is the cube of an integer and 2≤ x≤7}
b) B={0} c) C={∅} d) D={x: x is an integer and 2< x ≤3}
23) State which of the following sets is an infinite set?
a) A={x : x is an integer and -1≤ x < 1}
b) B= set of negative even integers greater than (-100)
c) C= set of positive integers less than 100
d) D= {x: x is real and -1≤ x <1.
1c 2d 3c 4c 5c 6b 7b 8a 9a 10d 11a 12b 13a 14c 15c 16c 17c 18a 19c 20d 21a c 22a 23d
BOOSTER - B
1) If A be a set, then
a) A ∩ ∅= ∅ b) A ∩ ∅= A c) A ∪ ∅= A d) A ∪ ∅= ∅
2) If A={a,b,c,d} and B={b,c,d,e} be two sets then
a) A - B = {a}
b) B - A= {e}
c) A - B = {b,c,d}
d) B - A={b,c,d}
3) If A, B and C are three finite sets; n(A)= 10, n(B)= 15, n(C)= 20, n(A ∩B)= 8 and (B ∪C)= 9, then the value of n(A∪B ∪C) will be
a) 26 b) 27 c) 28 d) none
4) Given A, B, C are three sets, state which of the followings are true?
a) A ∪B= B ∪A
b) A∩B = B ∩A
c) A ∪(B ∩ C)= (A∪B) ∩ (A ∪ C)
d) (A∩ B) ∩ C = A ∩ (B ∩ C)
5) State which of the following are null set?
a) {x ∈R: x²+1=0}
b) {x ∈ C: x > x}
c) {x ∈ R: x²+ x =0}
d) {x ∈ R: x²+2=0}
1ac 2ab 3abc 4abc 5ab
BOOSTER - C
Column I
A) Number of subsets of set {12,3,4} excepting null set are---
B) If n(A)= 192, n(B)= 76 and B ⊂A then the value of n(A ∪B) will be
C) If n(S)= 20, n(A)= 12, n(B)= 9 and n(A ∩B)= 4, then the value of n(A ∪B)' will be
D) If number of elements of set A be 7 then the value of number of elements of P(A) is
E) If A={a,b,c,d} be a set then the number of subsets of A will be
Column II
p) 16
q) 3
r) 15
s) 192
t) 128
Ar Bs Cq Dt Ep
BOOSTER - D
Column I
A) A∩ (B - A)=
B) (A ∩B) - C =
C) (A∪B) ∩(A ∪C')=
D) (A∪B) - C=
E) A - (B ∪C)=
Column II
p) (A - C) ∩ (B - C)
q) A
r) (A - B) ∩ (A - C)
s) ∅
t) (A - C) ∪(B - C)
As Bp Cq Dt Er
BOOSTER - E
1) In a class there are 150 students of which 65 like cricket, 44 like football and 42 like hockey , 20 like both football and cricket, 25 like both cricket and hockey and 15 like both hockey and football. Further 8 of the students like all the three games.
i) Number of students who like at least one of these three games.
a) 98 b) 99 c) 100 d) 101
ii) Number of students who like exactly one game
a) 50 b) 55 c) 54 d) 56
iii) Number of students who like exactly two games
a) 34 b) 36 c) 35 d) 37
2) Let A={x: x ∈N), B={x: x ∈2n, n ∈N}; C={x: x= 2n -1, n ∈ N}; D={x : x is a prime number} then
i) A ∩C is
a) A b) C c) D d) {2}
ii) B ∩C is
a) ∅ b) {2} c) B d) D
iii) C ∩D is
a) B b) C c) D - {2} d) ∅
1) ic iid iiib
2) ib iia iiic
BOOSTER - F
A) Statement - I is true, Statement -II is true and Statement -II is a correct explanation for Statement- I
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false.
D) Statement- I is false, Statement- II is true
1) Let A= {1,2,3} and B= {3,8}
Statement 1: (A ∪B) x (A ∩B) = {(1,3),(2,3),(3,3),(8,3)}
Statement- II: (AxB) ∩ (B x A)= {(3,3)}
2) Let X and Y before two sets
Statement- I: X ∩(Y ∪X)'= ∅
Statement- II: if n(X ∪Y)= P and n(X ∩Y)= ∅ then n(X ∆ Y)= P - Q {where X ∆ Y = (A - B) ∪ (B - A)}
1b 2b
RELATION
BOOSTER - A
1) If (a+ b, 3a - 2b)= (-9, -2), then a and b are
a) 2 and 1 respectively
b) -1 and 2 respectively
c) 1 and 2 respectively
d) -4 and -5 respectively
2) If A={1,2,4}, B={2,4,5}; C={2,5}, then (A x B) x (B - C) is
a) {(1,4)} b) {(1,2),(1,5),(2,5)} c) (1,4) d) none
3) If n(A)=3, n(B)= 4 then n(A x A x B)=
a) 12 b) 48 c) 36 d) 10
4) If A{a,b} and B°{1,2,3} then (Ax B) ∩ (B x A)=
a) {(a,1),(a,2),(b,3)}
b) {(b,1),(b,2),(b,3)}
c) {(a,1),(b,1),(a,3)} d) ∅
1d 2a 3c 4d
BOOSTER - B
1) let A and B be two sets containing respectively m and n distinct elements. Then number of different relations can be defined from set A to set B is
a) 2ᵐ⁺ⁿ b) ₂nᵐ c) ₂mⁿ d) 2ᵐⁿ
2) If R={3,9},(3,12),(4,8),(4,12),(5,10),(6,12)} be a given relation, then domain of R=
a) {3,4,5,6} b) {8,9,10, 12} c) {3,5} d) none
3) If R={3,9,(3, 12),(4,8),(4, 12),(5,10),(6,12)}
a) {3,4,5,6} b) {8,9,10,12} c) {3,10,12} d) none
4) If R is a relation on the set A={1,2,3,4,5,6,7,8,9} given by xRy <=> y= 3x, then R =
a) {(3,1),(6,2),( 8,2),(9,3)}
b) {(3,1),(6,2),( 9,3)}
c) {(3,1),(2,6),( 3,9)} d) none
5) Let R be a relation from set A do a set B, then
a) R= A ∪B b) R= A ∩B c) R⊆ A x B d) R ⊆ B x A
6) Total number of relations that can be defined on set A= {a,b,c} is
a) 2⁹ b) 2⁶ c) 2⁸ d) 2³
7) State which of the following is the total number of relations from set A={1,2,3} to set B={4,5}?
8) Let the relation R on set A={1,2,3,4} be defined as follows:
R={(1,2),(2,1),( 2,2),(3,3),(4,1),( 2,4),(4,2)}
Then state which one is true in each of the following two cases viz., (i) and (ii)
i) A) 3R2 B) 4nnot R 1 C) 1R3 D) 2R4
ii) A) 2 not R 1 B) 3R2 C) 1 not R 4 D) 4R3
1d 2a 3b 4d 5c 6a 7d 8id iic
MAPPING
BOOSTER - A
1) Let R be the set of real numbers and the mapping f: R---> R be defined by f(x)= sin x (for all x ∈R); then the range of f is
a) {f(x) ∈ R: - ∞ ≤ f(x)≤ ∞}
b) {f(x) ∈ R: - ∞ ≤ f(x)≤ 1}
c) {f(x) ∈ R: - 1 < f(x)<1∞}
d) {f(x) ∈ R: - 1 ≤ f(x)≤ 1}
2) State which of the following statement is true ?
a) Suppose the rule f(x)= 2x²- 9 associates the elements of N to itself. Then f(x) defines a mapping from N to itself (N being the set of natural numbersl.
b) If f: A---> B defines a mapping of A into B, then one element of A cannot be associated with two distinct elements of B.
c) If A={2,3,4}; B={1,2,5} and
R₁={(2,1),(4,5)} be a relation from A to B, then the relation R₁ defines a mapping of A into B.
d) If {2,3,4}; B={1,2,4,5} and R₂={2,1}, (3,4),(4,5),(3,2)} be a relation from A to B, then R₂ defines a mapping of A into B.
3) let A={0,1,2, 3,4} and Z be the set of integers . If the maping f: A---> Z be defined by f(x)= x²- 5x +2, state which of the following is the pre-image of 2?
a) 5 b) there is no pre-image of 2 c) 1 and 4 d) 0
4) If A={-2,1,0,-1,2}; B={-6,-5,-3,0,3} and the mapping f: A---> B is defined by f(x)= 2x²+ x -6; State which of the following is the image of (-2)?
a) 0 b) 3 c) - 3 d) -5
5) Let Z be the set of integera and the mapping f: Z --> Z be given by f(x)= 2x -1; state which of the following sets is equal to the set {x: f(x)= 3}?
a) {3} b) {2} c) {0} d) {-1}
1d 2b 3d 4a 5b
FUNCTION
BOOSTER - A
1) If f(x +2)= 2x²- 3x +5 then f(1)=
a) 2 b) 5 c) 10 d) none
2) If f(x)= 4ˣ then f(log₄x)=
a) 4 b) x c) 4ˣ d) x⁴
3) State which of the following statements is true?
a) if y² = x then y may be regarded as a function of x.
b) The function f(x)= x²/x and ∅(x)= x are identical .
c) The equation y³ - 3y² - 2x +11=0 represents x as a function of y.
d) If f(x)= √(x² + 4x -1) then f(-2) exist.
4) State for which of the following, the two functions f(x)= x and ∅(x)= + √x² are identical
a) 0< x <∞ b) - ∞<x <∞ c) 0≤ x <∞ d) -∞< x≤ 0
5) If f(x)= 3x -9, state which of the following is the value of f(x²-1):
a) 3x² -9 b) 3x² -12 c) x² -10 d) 3x² -10
6) If f(x -1)= 7x -5, state which of the following is the value of f(x):
a) 7x+2 b) 7x -12 c) 8x -4 d) 7(x +1)
7) If 2f(x) + 3f(-x)= 15 - 4x, which of the following is the value of [f(1)+ f(-1)].
a) 5 b) 7 c) -6 d) 6
8) If 3f(x)+ 2f(-x)= 5(x -2), state which of the following is the value of f(0):
a) 0 b) -2 c) 2 d) 1
9) If f(x)= kog₃x and ∅(x)= x², state which of the following is the value of f{∅(3)}:
a) 0 b) 1 c) 2 d) 3
10) The domain of definition of the function f(x)= √(x +3) is:
a) (-∞,3) b) (-∞,3] c) (3, ∞) d) [-3, ∞)
1c 2b 3c 4c 5b 6a 7d 8b 9c 10d
MIXED- A
1) If f(x)= x²- 3x +4, then the values of x which satisfy the relation f(x)= f(2x +1) are
a) 2 b) -1 c) 2/3 d) 0
2) If y= f(x)= (3x -5)/(x²-1) (x≠ -1) then the range of function will be
a) y≤ 1/2 b) y ≤ 2 c) y ≥ 9/2 d) y ≥ 2/3
3) If f(a)= (a²+ a -6)/(2a²- a -6) then for what value/s of a, f(a) will be undefined?
a) -3 b) 2 c) -3/2 d) -2/3
4) If R={(x,y): x ∈N, y ∈ N and 2x + y =10}, then R⁻¹=?
a) (8,1) b) (6,2) c) (4,3) d) (2,4)
5) If R⁻¹= {(x,y): x ∈N, y ∈N and x+ y =8} then R=?
a) (7,1) b) (6,2) c) (5,3) d) (1,4)
1bc 2ac 3bc 4abcd 5abc
Mixed- B
Column I
1) Two sets A={1,2,3} and B={2,4}
A) Ax B
B) B x A
C) A x A
D) B x B
E) (Ax B) ∩ (B x A)
Column II
p) {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}
q) {(2,2}
r) {(2,1),(2,2),(2,3),(4,1),(4,2),(4,3)}
s) {(1,2),(2,4),(4,2),(4,4)}
t) {(1,2),(1,4),(2,2),(2,4),(3,2),(3,4)}
At Br Cp Dq Es
2) Column I
A) Domain of R={(1,2),(2,4),(5,7),(9,10)} will be
B) Range of R={(3,9),(3,12),(4,8),(4,12),5,10)} Will be
C) is R={(1,7),(2,6),(3,5)} then the domain of R⁻¹ will be
D) The total number of relations of set A={a,b,c} will be
E) The total number of relation R from the set A to the set B will be
Column II
p) {7,6,5}
q) {1,2,5,9}
r) {8,9,10,12}
s) 2⁶
t) 2⁹
Aq Br Cp Dt Es
Mixed - C
** Let f: R-->R be defined by f(x)= -x³+ x, g: [-1,1] --> R and g:[-1,1] --> R is defined by g(x)= min (f(x), 0) h(x)= max(f(x),0)
1) f: R--->R will be
a) decreasing b) odd c) increasing d) even
2) Range of g(x) will be
a) [-1,1] b) [-2/3√3, 2/3√3] c) [-2/3√3,0] d) none
3) Number of roots of the g(x)= -1/2 is
a) 0 b) 1 c) 2 d) infinite
4) Which one will be the odd function?
a) g(x) b) h(x) c) g(x)+ h(x) d) g(x) - h(x)
5) Which one will be both the odd and even function?
a) h(x)+ g(x) b) h(x). g(x) c) h(x)- g(x) d) |h(x)|+ |g(x)|
1b 2c 3a 4c 5b
Mixed - D
** x², when x < 0
f(x)= x, when 0≤ x<1
1/x, when x≥ 1
1) Value of f(1/2) is
a) 1 b) 1/2 c) 2 d) 0
2) Value of f(√3) is
a) √3 b) 1 c) 0 d) 1/√3
3) Value of f(-2) is
a) 0 b) 1 c) 4 d) 2
4) Value of f(1) is
a) 1 b) 1 c) 3 d) none
5) Value of f-√3) is
a) 3 b) 1 c) √3 d) none
1b 2c 3a 4c 5b
Mixed- E
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I
B) Statement- I is true, Statement- II is true but Statement- II is not correct explanation of Statement- I
C) Statement- I is true, Statement- II is false.
D) Statement -I is false, Statement- II is true.
1) Statement- I: The value of f(x)= (ax + b)/(cx + d) (ad - bc ≠ 0) will never be a/c
Statement- II: Domain of g(y)= (b - dy)/(cy - a) consists of all real values excepting a/c.
2) Statement - I: the range of the function f(x)= sin²x + p sin x + q, where |p|> 2, will be numbers between q - p²/4 and p+ q +1.
Statement- II: The function g(t)= t²+ pt+1 where t ∈ [-1,1] and |p|> 2 will attain the minimum and the maximum values -1 and 1.
1a 2d
TRIGONOMETRICAL ANGLES
BOOSTER - A
1) A radian is a
a) fundamental unit of angle
b) sexagesimal c) right angle d) compound angle
2) The angle between two hands of a clock at 3p.m is
a) πᶜ b) πᶜ/2 c) πᶜ/4 d) πᶜ/8
3) If the angle of a right angled triangle at in AP, then the smallest angle is
a) πᶜ b) πᶜ/2 c) πᶜ/3 d) πᶜ/6
4) The degree measure of radian measure of angle (-3)ᶜ is
a) -170°49'5" b) -171°49'5" c) -171°50' d) -171°51'
5) The angular diameter of the moon 30'. How far from the eye a coin of diameter 2.2cm be kept to hide the moon?
a) 250cm b) 251cm c) 252cm d) 253cm
6) If the arc of a circle of radius 60cm subtends and angle 2/3 radius at its centre, then the length of the arc will be
a) 40cm b) 40.1cm c) 41cm d) 4.5cm
1a 2b 3d 4b 5c 6a
BOOSTER - B
1) The angle subtended at the centre of a circle, by a chord whose length is equal to the radius of the circle will be
a) πᶜ/3 b) πᶜ/4 c) 200ᵍ/3 d) 50ᵍ
2) If π/6 and 150ᵍ are two angles of a triangle, then third angle will be
a) πᶜ/3 b) 50ᵍ/2 c) 200ᵍ/3 d) πᶜ/12
3) The angle between the minute hand of a clock and the hour hand when the time is 7:30 AM, will be
a) 5π/9 b) 50ᵍ+ 5π/18 c) 50ᵍ + 11π/36 d) 1000ᵍ/9
4) The angles of a quadrilateral are in AP and the greatest is twice the smallest angle. Find the value of the smallest angle
a) π/4 b) π.3 c) 200ᵍ/3 d) 50ᵍ
5) A circular ring of radius 3cm is cut and bent so as to lie along the circumference of a hoop whose radius is 48cm. The angle which is subtended by the arc at the centre of the hoop will be
a) 25ᵍ b) π.8 c) 50ᵍ d) π/4
BOOSTER - C
COLUMN I
A) πᶜ
B) 5πᶜ/6
C) 2πᶜ/3
D) 7πᶜ/6
E) 3πᶜ/4
Column II
p) 200ᵍ/3
q) 700ᵍ/3
r) 150ᵍ
s) 200ᵍ
t) 200ᵍ/3
As Bp Ct DqEr
BOOSTER - D
Column I
A) 350ᵍ
B) 309ᵍ
C) 400ᵍ
D) 450ᵍ
E) 500ᵍ
Column II
p) 5πᶜ/2
r) 7πᶜ/4
s) 3πᶜ/2
t) 9πᶜ/4
Ar Bs Cp Dt Eq
BOOSTER - E
** let D be the number of degrees. R be the number of radian and G be the number of grades of any angle of the triangle ABC.
Then the required relation among three systems of the measurement of an angle is D/90 = G/100 = 2R/π.
1) If angle A= 30° then which one is true:
a) π/3 b) (100/3)ᵍ c) (200/3)ᵍ d) none
2) If angle B= 100ᵍ then which one is true?
a) π/3 b) π/6 c) π/4 d) none
3) Angle C will be
a) π/3 b) 75ᵍ c) π/6 d) 50ᵍ
1b 2d 3a
BOOSTER - F
The sense of an angle is said to be positive or negative according to the initial side rotates in anticlockwise or clockwise direction to get to the terminal side. if a man moves along a circular path after completing 2 rotations he makes a positive angle of 150ᵍ. Then
1) Total angle along clockwise direction is
a) +855° b) -855° c) 585° d) -585°
2) In which quadrant he is ?
a) first quadrant
b) second quadrant
c) third quadrant
d) 4th quadrant
3) at this position how much angle he makes along clockwise direction ?
a) +225° b) +495° c) -495° d) -225°
1a 2b 3d
BOOSTER - G
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I.
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false.
D) Statement- I is false, Statement- II is true.
1) Statement- I: the radian measurement of the angle of a regular octagon in radian is (3π/4)ᶜ.
Statement- II: Angle of an n sided regular polygon = (2n -4)90°/n.
2) Statement- I: the moon's distance from the Earth is 36 x 10⁴ km and its diameter subtends an angle of 31ʳ at the eye of the observer.
The diameter of moon is 3247.62 km.
Statement- II: Angle subtends at any point, θᶜ= arc length/radius
1a 2a
TRIGONOMETRIC RATIOS (OR FUNCTIONS) OF POSITIVE ACUTE ANGLES
BOOSTER - A
1) sec²A - tan²A=
a) 0 b) 1 c) 2 d) none
2) tanA. cosA=
a) 0 b) 1 c) cotA d) sinA
3) If θ is a positive acute angle then the value of secθ cannot be
a) greater than 1 b) less than 1 c) equals to 1 d) 0
4) If θ is a positive acute angle, then the value of θ satisfies the equation √3 sinθ - cosθ= 0 is
a) π/2 b) π/3 c) π/6 d) π/8
5) The minimum value of sec²α+ cos²α is
a) 0 b) 1 c) 2 d) - ∞
6) The maximum value of sinθ. cosθ is
a) 1/2 b) 1 c) 2 d) ∞
7) If 0< θ< 90°, then minimum value of 9 tan²θ+ 4 cot²θ is
a) 11 b) 12 c) 13 d) 14
8) If sinα = 4/5 where α is a positive acute angle then cosα =
a) 3/5 b) -3/5 c) ±3/5 d) none
9) If x= sin²α + cosec²α, states which of the following is true?
a) 0<x <1 b) 1≤ x <2 c) x ≥ 2 d) x = 1.5
10) If tanθ = a/b, then which of the following is the value of
(a sinθ + b cosθ)/(a sinθ - b cosθ) ?
a) (a²+ b²)/(a²- b²)
b) a/√(a²- b²)
c) b/(a²- b²)
d) √(a²+ b²)
11) If 0°≤ A≤ 90° and sinA = cosA, state which of the following is the value of A?
a) 0° b) 30° c) 45° d) 60°
12) State which of the following relation is true?
a) cosθ= 7/5 b) sinθ= (a²+ b²)/(a²- b²) (a≠ ± b) c) tanθ= 45° d) secθ= 4/5
1b 2d 3b 4c 5c 6a 7b 8c 9c 10a 11c 12c
BOOSTER - B
1) If x= r sinθ cosφ, y= r sinθ sinφ and z= r cosθ then the value of √(x²+ y²+ z²) will be
a) cosθ b) - r c) r d) cosφ
2) If secθ = x + 1/x , then the value of secθ + tanθ will be
a) 2x b) x c) 1/2x d) 1/x
3) The maximum and minimum value of cos(cosx) are
a) 1 b) cos 1 c) 0 d) none
4) If 1+ sin²A= 3 sinA cosA, then the value of tanA will be
a) 1 b) -1 c) 1/2 d) -1/2
5) If tanα = (sinA - cosA)/(sinA + cosA) , then the value of sinA + cosA will be
a) √2 cosθ b) √2 sinθ c) -√2 cosθ d) -√2 sinθ
1bc 2ac 3ab 4ac 5ac
BOOSTER - C
Column I
A) If a cosθ + b sinθ = x and a sinθ - b cosθ = y, then
B) If (x/a) cosθ + (y/b) sinθ = 1 and (x/a) sinθ - (y/b) cosθ = -1, then
C) minimum value of sinθ cosθ is
D) maximum value of 12 sinθ - 9 sin²θ is
E) maximum value of sinx + cosx is
Column II
p) x²/a² + y²/b²= 2
q) a²+ b²= x²+ y²
r) √2
s) -1/2
t) 4
Aq Bp Cs Dt Er
BOOSTER - D
Column I
A) If sinθ = p and tanθ = q, then
B) If sinθ + cosθ = p and tanθ + cotθ = q, then
C) If sinθ - cosθ = p and secθ - cosecθ = q, then
D) If sinθ + cosθ = p and sinθ - cosθ = q, then
E) If p= 2θ and Q= θ¹, then
Column II
p) q(1- p²)= 2p
q) 1/p² - 1/q²= 1
r) q(p²-1)= 2
s) p²= 4q
t) 2(p²+ q²)= 1
Aq Br Cp Dt Es
BOOSTER - E
** ABC is an acute angled triangle in which cosec(B+ C - A)= 1 and cot(C + A - B) = 1/√3, then
1) sinA=
a) 0 b) 1/2 c) 1/√2 d) √3/2
2) tanC=
a) 0 b) 2-√3 c) 2+√3 d) √3
3) SecB=
a) 0 b) 1 c) 2 d) none
1c 2c 3c
BOOSTER - F
** If sinα = 12/13; cosβ= 4/5 and α, β are two acute angles, then
1) Value of sinα cosβ + cosα cosβ is
a) 61/65 b) 63/65 c) 1 d) 67/65
2) Value of (tanα - tanβ)/(1+ tanα tanβ) is
a) 33/56 b) 31/56 c) 29/56 d) none
3) Value of secα cotβ + tanα cosecβ is
a) 32/5 b) 33/5 c) 34/5 d) 7
1b 2a 3c
BOOSTER - G
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I.
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false.
D) Statement- I is false, Statement- II is true.
1) Statement- I: cos 1< cos7
Statement- II: In 1st quadrant the value of cosine is decreasing but the value of sine is increasing.
2) Statement- I: If sec²θ= 4xy/(x + y)⅖ be true, then x= y and x≠ 0
Statement- II: If angle θ be lying in the 3rd and the 4th quadrant, then the value of secθ is decreasing.
1b 2b
TRIGONOMETRIC RATIOS OF ASSOCIATED ANGLES
BOOSTER - A
1) If sinθ = -1/2, then θ=
a) 30° b) 120° c) 150° d) 210°
2) sin(θ - 540°)=
a) sinα b) - sinα c) cosα d) - cosα
3) If tan35°= 0.7 then tan(-665°)=
a) 0.7 b) 0.007 c) 10/7 d) 100/7
4) State which of the following is the value of cot(-870°)?
a) √3 b) 1/√3 c) -1/√3 d) -√3
5) Which of the following is the value of cos(-1170°)?
a) 1 b) -1 c) 0 d) -1/2
6) Which of the following is the value of sec(-945°)?
a) √2 b) -√2 c) 2 d) -2
7) Which of the following is the value of cos(5π/2 - 19π/3)?
a) √3/2 b) -√3/2 c) 1/2 d) -1/2
8) sec²θ= 4xy/(x + y)² is true if and only if
a) x+ y ≠0 b) x = y, x≠ 0 c) x= y d) x≠ 0, y≠ 0
9) If tanθ + secθ = eˣ, then cosθ equals
a) (eˣ + e⁻ˣ)/2 b) 2/(eˣ + e⁻ˣ) c) (eˣ - e⁻ˣ)/2 d) (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ).
1d 2b 3d 4a 5c 6b 7a 8b 9b
BOOSTER - B
1) If cos²α - sinα = 1/4, then the value of α(0≤α≤360°) will be
a) 30° b) 120° c) 150° d) 105°
2) If n be an odd number then the value of cos(nπ+ θ) will be
a) sinθ b) cosθ c) - sinθ d) - cosθ
3) If cosθ= 1/2, then the value of θ will be
a) 420° b) 60° c) 300° d) 330°
4) If 0<θ<π, then the value of √{(1- sinθ)/(1+ sinθ)} + √{(1+ sinθ)/(1- sinθ)} will be
a) 2 secθ b) -2 secθ c) secθ d) - secθ
5) If tanθ = -1/√5 , then the value of cosθ will be
a) √(5.6) b) 1/√6 c) √(5/6) d) 1/2
1ac 2bd 3abc 4ab 5ac
BOOSTER - C
Column I
A) cos(-1125°)=
B) sin(8π/3) cos(22π/6) + cos(13π/3) sin(35π/6) =
C) cos510° cos330° + sin390° cos120°=
D) If 0<θ<π/2, then the minimum value of sinθ + cosθ will be
E) If n be an even integer, then the value of sin[nπ+ (-1)ⁿπ/3] will be
Column II
q) -1
r) 1/√2
s) √3/2
t) 1/2
Ar Bt Cq Dp Es
BOOSTER - D
Column I
A) sec(-1680°) sin330°=
B) cosec(-1410°)=
C) cos306°+ cos234° + cos162°+ cos18°=
D) sin30° sin210° sin330°=
E) tan(19π/3 - 5π/2)=
Column II
p) -1/√3
q) 2
r) 1
s) 0
t) 1/8
Aq Br Cs Dt Ep
BOOSTER - E
** If A, B, C are three angles of ∆ ABC, then
1) tan{(A- B)/2}=
a) cot(B + C/2) b) cot(C + B/2) c) tan(B + C/2) d) none
2) cos(A+ B) + sinC=
a) sin(B + C) - cosA
b) sin(A+ B) - cosC
c) sun(A + C) - cosB. d) none
3) sin(B + C) + sin(C + A)+ sin(A + B)=
a) cosA + cosB + cosC
b) sinA + sinB - sinC
c) sinA + sinB + sinC
d) -(sinA + sinB + sinC)
** If A, B, C, D are the successive angles of a cyclic quadrilateral, then
4) cos{(A+ C)/2} + cos{(B + D)/2}
a) 1 b) -1 c) 0 d) 2
5) cosA + cosB + cosC + cosD=
a) 0 b) 1 c) -1 d) none
6) cotA + cotB + cotC+ cotD=
a) 1 b) 0 c) -1 d) 2
1a 2b 3c 4c 5a 6b
BOOSTER - F
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I.
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I: is true, Statement- II is false.
D) Statement- I is false, Statement- II is true.
1) Statement- I: if secθ + tanθ = m (≠0) then sinθ = (m²-1)/(m²+1)
Statement- II: For any θ for which cosθ ≠ 0, sec²θ - tan²θ= 1.
2) Statement- I: If x ∈ R, x ≠ 0 then x²+ 1/x² cannot be equal to cosθ for any θ.
Statement- II: Sum of a positive number and its reciprocal cannot be less than 2.
1a 2a
TRIGONOMETRIC RATIOS OF COMPOUND ANGLES
BOOSTER - A
1) sin(A+ B) sin(A - B)=
a) sin²A - sin²B
b) cos²A - sin²B
c) cos²B - sin²B
d) cos²A - cos²B
2) cos(A+ B) cos(A - B)=
a) sin²A - sin²B
b) cos²A - sin²B
c) cos²B - sin²B
d) cos²A - cos²B
3) sin(45° - θ)=
a) (1/√3) (sinθ - cosθ)
b) (1/√2) (sinθ - cosθ)
c) (1/√3) (cosθ - sinθ)
d) (1/√2) (cosθ - sinθ)
4) tan(π/4+ θ) tan(π/4 - θ)=
a) 0 b) 1/√3 c) 1 d) √3
5) cot2θ + tanθ =
a) sin²2θ b) cot²2θ c) cosec²2θ d) tan²2θ
6) 2 cos(π/3 + A)=
a) cosA - √3 sinA
b) cosA - √2 sinA
c) sinA - √3 cosA
d) sinA - √2 cosA
7) If sinA= 3/5, cosB= -12/13, where A and B both lie in second quadrant, then the value of sin(A+ B) is
a) 56/65 b) -56/65 c) 65/56 d) -65/56
8) (cos9° + sin9°)/(cos9° - sin9°)=
a) sin54° b) cos54° c) tan54° d) cot54°
9) If sinA + sinB = 2, then which of the following is the value of sin(A + B)?
a) 2 b) 0 c) 1 d) -1
10) If sinθ + sinφ = 2, then which of the following is the value of cos(θ+φ)?
a) 0 b) 1 c) -1 d) 2
11) If cosA + cosB = 2, then which of the following is the value of cos(A+ B)?
a) 1 b) 0 c) -1 d) 2
12) If tanA = 3/4 and tanB= 4/5, then which of the following is the value of (A+ B)?
a) π/4 b) 3π/4 c) π d) π/2
1a 2b 3d 4c 5c 6a 7b 8c 9b 10c 11a 12d
BOOSTER - B
1) If tan(π cosθ)= cot(π sinθ), then the value of cos(θ-π/4) will be
a) 1/2 b) 1/2√3 c) -1/2 d) -1/2√2
2) If √2 cosA = cosB + cos³B and √2 sinA = sinB - sin³B, then the value of sin(A - B) will be
a) 1/2 b) 1/3 c) -1/2 d) -1/3
3) If tanα = x +1 and tanβ = x -1, then the value of x will be
a) √(2cot(α-β))
b) √(2 tan(α-β)
c) - √(2 cot(α-β)
d) - √(2 tan(α-β)
4) If tanA =3/4 and tanB= -5/11, then the value of sin(A + B) is
a) -16/65 b) 16/64 c) -56/64 d) 56/65
5) If tanA= 1/2 and tanB= 1/3, then the value of A+ B will be
a) 225° b) 405° c) 585° d) 945°
1bd 2bd 3ac 4ab 5abcd
BOOSTER - C
Column I
A) tan315° cot(-405°)+ cot495° tan(-585°)=
B) sin600° tan(-690°)+ sec840° cot(-945°)=
C) If sin(α+β)= 1, sin(α-β)= 1/2 and 0<α, β< π.2 then the value of
tan(α+2β) - tan(2α+β) is equal to
D) cos(A- B)= 3/5 and tanA + tanB= 2 then cot(A + B)=
E) cos67°24' cos7°24' + cos82°36' cos22°36'=
Column II
p) 1
q) 1/2
r) -1/5
s) 3/2
t) 2
At Bs Cp Dr Eq
BOOSTER - D
Column I
A) tan87° - tan42° - tan87° tan42°=
B) cosA + cos(120+A) + cos(120- A)=
C) tanα = 2 tanβ, then (sin(α+β))/(sin(α-β))=
D) sinα sinβ= cosα cosβ -1 then cotα cotβ =
E) If 3 sinθ + 4 cosθ = 5, Then the value of 4 sinθ - 3 cosθ=
Column II
p) 0
q) 1
r) 4
s) 3
t) -1
Aq Bp Cs Dt Er
BOOSTER - E
** If sinα + sinβ = a and cosα + cosβ = b then
1) sin(α+β)=
a) 2ab/(a²+ b²)
b) 2ab/(a²- b²)
c) (a²+ b²)/2ab
d) (a²- b²)/2ab
2) cos(α+β)=
a) (a²- b²)/(a²+ b²)
b) - (a²- b²)/(a²+ b²)
c) (a²+ b²)/(a²- b²)
d) - (a² + b²)/(a² - b²)
3) (tanα + tanβ)/(1- tanα tanβ)=
a) (a²+ b²)/2ab
b) (b² - a²)/2ab
c) 2ab/(b²+ a²)
d) 2ab/(b² - a²)
** ABC is an obtuse angled triangle whose angle=135°
4) value of (1+ tanA)(1+ tanB) is
a) 0 b) 1 c) 2 d) 4
5) Value of (cotA -1)(cotB -1) is
a) 1 b) 2 c) 3 d) undefined
6) value of tanA + cotB is
a) 2 b) 2√2 c) 0 d) 1
1a 2b 3d 4c 5b 6b
BOOSTER - F
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false
D) Statement- I is false, Statement- II is true
1) Statement- I: x+ y + z= xyz, then sign of any is negative
Statement- II: For any obtuse angled triangle tanA + tanB + tanC = tanA tanB tanC.
2) Statement- I: if A, B, C be the three angles of a triangle where A is an obtuse angle, then tanB tanC> 1
Statement- II: For any triangle tanA= (tanB + tanC)/(tanB tanC -1).
1d 2d
TRANSFORMATION OF SUMS AND PRODUCTS
BOOSTER - A
1) sinC + sinD=
a) 2 sin{C+ D)/2} cos{(C - D)/2}
b) 2 sin{C+ D)/2} sin{(C - D)/2}
c) 2 cos{C+ D)/2} cos{(C - D)/2}
d) 2 sin{C+ D)/2} sin{(D - C)/2}
2) cosC + cosD=
a) 2 sin{C+ D)/2} cos{(C - D)/2}
b) 2 cos{C+ D)/2} sin{(C - D)/2}
c) 2 cos{C+ D)/2} cos{(C - D)/2}
d) 2 sin{C+ D)/2} sin{(D - C)/2}
3) sinC - sinD=
a) 2 sin{C+ D)/2} cos{(C - D)/2}
b) 2 sin{C+ D)/2} sin{(C - D)/2}
c) 2 cos{C+ D)/2} cos{(C - D)/2}
d) 2 sin{C+ D)/2} sin{(D - C)/2}
4) cosC - cosD=
a) 2 sin{C+ D)/2} cos{(C - D)/2}
b) 2 cos{C+ D)/2} sin{(C - D)/2}
c) 2 cos{C+ D)/2} cos{(C - D)/2}
d) 2 sin{C+ D)/2} sin{(D - C)/2}
5) 2 sin40° sin10°=
a) cos30° + cos50°
b) cos30° - cos50°
c) cos50° - cos30° d) none
6) 2 sin24° cos15°=
a) cos40° + sin10°
b) siné0° - sin10°
c) sin10° - sin40° d) none
7) State which of the following is true?
a) cos10° + cos25° can be expressed as one cosine only
b) cos20° - cos40° can be expressed as one sine only
c) cosA cosB can not be expressed as the sum of two cosines
d) sinA cosB can be expressed as the difference of two sines
8) State which of the following is equal to sin(5θ/2) sin(3θ/2)?
a) (1/2) (sin4θ - sinθ)
b) (1/2) (cos4θ - cosθ)
c) (1/2) (cosθ - cos4θ)
d) (1/2) (cosθ + cos4θ)
9) State which of the following is equal to √3 sin10°?
a) sin40° + sin20°
b) cos30° - cos70°
c) cos50° + cos70°
d) sin70° + sin50°
1a 2b 3c 4d 5b 6a 7b 8c 9b
BOOSTER - B
1) The value of sin47° + sin61° - sin11° - sin25° is
a) sin83° b) cos367° c) cos7° d) sin97°
2) The value of {cosα+ cosβ)/(sinα - sinβ)ⁿ + {sinα+ sinβ)/(cosα - cosβ)ⁿ (Where n is a whole number) is equal to
a) 0 2tanⁿ{A- B)/2} c) 2 cotⁿ{A- B)/2} d) 2 cotⁿ{A+ B)/2}
3) (cos29° + sin29°)/(cos29° - sin29°)=
a) tan74° b) cot16° c) cot196° d) tan254°
4) If sinα + sinβ = p and cosα + cosβ = q, then the value of {tan(α-β)/2} will be
a) √{(4- p²- q²)/(p²+ q²)}
b) √{(p² + q² -4)/(p²+ q²)}
c) - √{(4- p²- q²)/(p²+ q²)}
d) - √{(p²+ q²-4)/(p²+ q²)}
5) sin(π/12) sin(5π/12)=
a) 1/3 b) 353/706 c) 1253/2506 d) 1/2
1abcd 2ac 3abcd 4ac 5bcd
BOOSTER - C
Column I
A) sin47°+ cos47°=
B) sin23°+ sin37°=
C) sin105°+ cos105°=
D) sin50°+ sin10°=
E) cos(2π/3) + cos(4π/7) + cos(6π/7)=
Column II
p) - cos187°
q) sin(3π/4)
r) cos(2π/3)
s) cos343°
t) cos340'
As Bp Cq Dt Er
BOOSTER - D
Column I
A) (sin5A - sin3A)/(cos5A + cos3A)=
B) (sinA + sin3A)/(cosA + cos3A)=
C) (cosA + cos5A)/(sin7A - sin5A)=
D) (sin3A - sinA)/(cosA - cos3A)=
E) (sin5A + sin3A)/(cos5A + cos3A)=
Column II
p) tan2A
q) cot2A
r) tan4A
s) tanA
t) cotA
As Bp Ct Dq Er
BOOSTER - E
** If x cosα + y sinα = x cosβ + y sinβ then
1) tan{(α+β)/2} =
a) y/x b) x/y c) (x + y)/y d) (x - y)/y
2) 2(1- cos(α-β))/(cosβ - cosα)=
a) (x²- y²)/y²
b) (x²+ y²)/y²
c) (x + y)/y
d) (x - y)/y
3) (sinα - cosα - sinβ + cosβ)/(sinα + cosα - sinβ - cosβ)=
a) (y+ x)/(y - x)
b) (y- x)/(y + x)
c) (x - y)/(x + y)
d) (x -p+ y)/(x - y)
** A, B, C are three angles of a triangle and sin(A + C/2)= n sin(C/2)
4) tan(A/2) tan(B/2)=
a) (n +1)/(n -1)
b) (1- n)/(1+ n)
c) (n -1)/(n +1)
b) (1+ n)/(1- n)
5) sin(C/2)/cos{(A - B)/2}=
a) n b) -n c) 1/n d) -1/n
6) sin(C/2)/(cos(A/2)(cos(B/2)) =
a) 2/(n +1) b) 1/(n +1) c) 2/(n +1) d) -1/(n +1)
1a 2b 3d 4c 5d 6a
BOOSTER- F
A) Statement- I is true, Statement- II is true and Statement- II is a correct for Statement- I
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false
D) Statement- I is false, Statement- II is true
1) Statement- I: If cos(β-γ)+ cos(γ-α) + cos(α-β) = -3/2, then sinα + sinβ + sinγ = cosα + cosβ + cosγ=0
Statement- II: a²+ b²= 0 => a= 0 and b =0
2) Statement- I: In ∆ ABC, tanA + tanB + tanC
Statement- II: In ∆ ABC, A+ B + C =π
1a 2d
TRIGONOMETRIC RATIOS OF MULTIPLE ANGLES
BOOSTER - A
1) 1- Cos2θ =
a) 2 sin²θ b) 2cos²θ c) sin2θ d) 2 cos2θ
2) (1+ cos2θ)/(1- cos2θ)=
a) sin²θ b) cos²θ c) tan²θ d) cot²θ
3) θθθθθθθθθθθθθθθθθθθθθθθθθαααφθθφθφθθθθθθθθθθθθθθθθθθθθθθθθαββγβαβααβαβββββββββαβαβαβαβαβαβαβαβαβββααβαααβααβαβαβαβαβᶜᵒˢˣˢᶦⁿˣθθθθθθθ
°°°
ᶜ
∈ ∈ ⊂ ∩ ∪ ⊆ ⊆⊆⊆ ∅∩∅∪∩⊆⊆ ∅ ≠ ⊂ ⊂ ∩ ∉ ⊇ ⊃ ∀ ∧ ∨ ∞ ⁻¹
LAWS OF INDICES
LOGARITHM
MATHEMATICAL INDUCTION
COMPLEX NUMBERS
QUADRATIC EQUATIONS
BOOSTER - A
One root of equation is zero when the roots of the equation are acropol to 1 another one with the science of a n c are opposite to that of the then both ratio of the equation zero positive negative fraction the roots of the equation then both roads of the equation zero positive negative imagine the maximum number of distinct roots in a quadratic equation 1 2 3 infinite is a real and rational then one out there to listen real wrestling in the other rupees zero real and rajdhani imaginary notify then both roots of the equation zero real and personal imaginary not define the roots of the equation are equal numbers is positive but not a perfect square then both roots of decoration minimum value of 0123 the maximum value of 0123 A4 is the root of the equation then which of the following is another route 4233 state which of the following is the sum of the roots of the equation state which of the following is the product of the rules of the equation state which of the following equation has the routes 2 and 3 if the roots of the equation are ratio local of any other then which of the following is the value of k to the sum of the roots of the equation then which of the following is the value of the product of the roots of the equation which of the following
The list value of which makes the roots of the equation imaginary the question of the smallest degree with real copies in having as one of the road says if the dimension of a quadratic equation is length and zero then the roots of this equations are both real both imaginary one real friend another images be a root of the quadratic equation then the equation will be the a root of the equation then its another it will be let the quadratic equation has two purely Complex roots than is purely imaginary is purely real the roots of the values of X satisfying the equation if the equation had no real time the roots of equations the roots of are the roots of are the roots are the roots of the roots of are the roots of are the roots of the roots of the roots of power when the real roots are when the real roots minimum value when the value minimum minimum value of roots of consider unknown polynomial as remainder respectively be the reminder on the polynomial is divided by distinct real rules and value of it has no distinct real roots the least value of range
LINEAR INEQUATIONS
BOOSTER - A
1) If x ∈N and -5< 2x -7 ≤ 1, then the values of x is
a) 2≤ x ≤ 4 b) 2≤ x <4 c) 2<x ≤4 d) 2,3 and 4
2) If x is an integer which is a perfect square and 7≤ 2x - 3< 17, then x is
a) 9 b) 4 c) 16 d) 25
3) If x ∈ N and 0≤ (2x -5)/2 ≤ 7, then the maximum and minimum value of x are
a) 9, 3 respectively
b) 9,4 respectively
c) 9,3 respectively d) none
4) If x is an integer then the solution set of the equation - x²+ 7x - 6> 0 is
a) (2,4) b) (3,5) c) {2,3,4,5} d) {4,5}
5) Solution sets of the inequation (3x +5)/7 > (x +3)/4 (where x < 5 is an integer) is
a) {2,3,4} b) {1,3,4} c) {1,2,3} d) {1,2,3,4}
6) Solution sets of the inequation -2≤ (3x -1)/2 ≤ 1 (where x ∈ Z) is
a) {1,2,-1} b) {1,0,-1} c) {-1,0,1} d) {1,-1,0}
7) If x- y =3 and x + y≥ 9, then the minimum value of x is
a) 2 b) 4 c) 5 d) 6
8) If x and y are positive integers , then the solution sets of the inequation x≤ 3, y≤ 2 and 5x + 6y≤ 21 are
a) x: 1 1 2 3 b) x: 1 1 2 3
y: 1 2 4 1 y: 1 2 1 1
c) x: 1 2 3 4 d) x: 1 1 1 1
y: 2 1 1 1 1 2 3 4
1d 2a 3a 4c 5a 6c 7d 8b
BOOSTER - B
1) The set of values of x which satisfy the inequation (5x +8)/(4- x) < 2, are
a) (-∞,0) b) (0, -∞) c) (4, ∞) d) (-∞,4)
2) The region bounded by the inequation 2x + 3y≥ 3, 3x + 4y≤ 18, -7x + 4y= 14, x - 6y≤ 3, x≥0, and y≥ 0 are lying in the quadrant
a) 1st b) 2nd c) 3rd d) 4th
3) The region bounded by the inequation |y - x|≤ 3 are lying in the quadrant
a) 1st b) 2nd c) 3rd d) 4th
4) Two consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40, then the members are
a) 11,13 b) 15,13 c) 17,19 d) 17,15
5) The solution sets of the inequation (|x| -4)/(|x| -5)> 0 where x∈ R and x ≠ ±5, are
a) [-4,] b) (-∞,-5) d) (5, ∞) d) none
1ac 2abc 3abcd 4ac 5abc
BOOSTER- C
Column I
A) The solution of the inquisition (2x +4)/(x -1) ≥ 5 is
B) The solution of the inequation (x +3)/(x -2)≤ 2 is
C) The solution of the inequation 3/(x -2) < 2 is
D) The solution of the inequation 1/(x -1) ≤ 2 is
E) The solution of the inequation (2x -3)/(3x -7)> 0 is
Column II
p) (-∞,1) ∪[3/2, ∞)
q) (-∞,3/2) ∪ (7/3, ∞)
r) (1,3)
s) (-∞,2) ∪ [7, ∞]
t) (-∞,2) ∪ (5, ∞)
Ar Bs Ct Dp Eq
BOOSTER - D
Column I
A) Solution of the inequation x +3> 0, 2x < 14, x ∈R is
B) Solution of the inequation 2x -7> 5 - x, 11- 5x ≤ 1, x ∈R is
C) Solution of the inequation 4x -1< 0, 3 - 4x < 0, x ∈ R is
D) Solution of the inequation (2x +1)/(7x -1)> 5, (x +7)/(x - 8) > 2, x ∈ R is
E) Solution of inequation 0< -x/2 < 3, x ∈R is
Column II
p) (4, ∞)
q) (-6,0)
r) (2,6)
s) (-3,-7)
t) no solutions
As Bp Cr Dt Eq
BOOSTER - E
** An inequation of variable x can have infinite number of solutions. Thus if x ∈ R, then the solution of the inequation x ≤ 4 represents all real values of x less than or equal to 4. Clearly, x ≤ 4 has infinite number of solutions . However, an inequation can have finite number of solutions if some condition is imposed. Thus the solution of the inequation x ≤ 4, x ∈ N, where x is a positive integer is {1,2, 3,4}. The set of all values of the variable (or variates) which satisfy a given inequation is called its solution set. Thus the solution set of the inequation x ≤ 4 where x ∈Z is (-∞, ..., -3,-2,-1,0,1,2,3,4}; further if x is a positive integer then the solution set of the inequation x ≤ 4 is {1,2,3,4} and if x ∈R, then the solution set of the inequation x ≤ 4 will be {x ∈ R; x ≤ 4}
1) If (2x +3)/5 < (4x -1)/2, then the range of x will be
a) (-∞, 11/16) b) (11/16, ∞) c) [11/16, ∞] d) [11/16, ∞)
2) If 7x -2< 4 - 3x and 3x -1< 2 + 5x, then the range of x will be
a) (3/5,3/2) b) (-3/2,3/5) c) (3/5,3/2) d) [3/5,3/2)
3) If for some values of a the inequation x²+ |x + a| - 9 < 0 has atleast one negative solution, then the range of qpa will be
a) (9,37/4) b) [9,37/4] c) (-9,37/4) d) [-9, 37/4]
4) If (x -1)/(4x +5) < (x -3)/(4x -3), then the range of x will be
a) 3/4< x <5/4 b) -5/4< x ≤ 3/4 c) -3/4< x ≤ 3/4 d) -5/4< x <3/4
1b 2d 3c 4d
Booster - F
** Consider the inequality 9ˣ - a.3ˣ - a +3≤ 0, where a is a real parameter.
1) The given inequality has atleast one negative solution for a lying in
a) (-∞,2) b) (3,∞) c) (-2, ∞) d) (2,3)
2) The given inequality has atleast one positive solution for a lying in
a) (-∞,-2) b) (2,∞) c) (3,∞) d) (-2,∞)
3) The given inequality has atleast one real solution for each a lying in
a) (-∞,2) b) [3,∞) c) [2,∞) d) [-2,∞)
BOOSTER - G
A) Statement- I is true, Statement- II is true and Statement- II is a correct explanation for Statement- I.
B) Statement- I is true, Statement- II is true but Statement- II is not a correct explanation of Statement- I
C) Statement- I is true, Statement- II is false
D) Statement- I is false, Statement- II is true
1) Statement- I: if |x -2| + |x -7|= |2x -9|, then either x ≤ 2 or x ≥ 7.
Statement- II: |a|+ |b|= |a + b| if a, b > 0
2) Statement- I: if x²+ ax + 4 > 0 for all x ∈R, then -4< a<4.
Statement- II: The sign of a quadratic expression ax²+ bx + c (a≠ 0) is same as that of a for all real values of x when b⅖- 4ac < 0
1a 2a
∈ ∈ ⊂ ∩ ∪ ⊆ ⊆⊆⊆ ∅∩∅∪∩⊆⊆ ∅ ≠ ⊂ ⊂ ∩ ∉ ⊇ ⊃ ∀ ∧ ∨ ∞ ⁻¹
