(MATHEMATICS) FUNCTIONS

HOMEWORK
√−log0.3 ⁡(x−1)
1. The domain of the function f(x) = is
√x2 +2x+8

(A) (1,4) (B) (−2,4) (C) (2,4) (D) [2, ∞)

1
2. The domain of the function f(x) = log1/2⁡ (−log 2 ⁡ (1 + 4 ) − 1) is
√x

(A) 0 < x < 1 (B) 0 < x ≤ 1 (C) x ≥ 1 (D) null set

3. If q2 − 4pr = 0, p > 0, then the domain of the function,
f(x) = log⁡(px 3 + (p + q)x 2 + (q + r)x + r) is
q q
(A) R − {− } (B) R − [(−∞, −1] ∪ {− }]
2p 2p
q
(C) R − [(−∞, −1] ∩ {− }] (D) none of these
2p

4. The domain of the function √log1/3 ⁡ log 4 ⁡([x]2 − 5) is (where [x] denotes greatest integer

function)
(A) [−3, −2) ∪ [3,4) (B) [−3, −2) ∪ (2,3]
(C) R − [−2,3) (D) R − [−3,3]
5. If f(x) = 2sin2 ⁡ θ + 4cos⁡(x + θ)sin⁡ x ⋅ sin⁡ θ + cos⁡(2x + 2θ) then value of f 2 (x) +
π
f 2 ( − x) is
4

(A) 0 (B) 1 (C) -1 (D) x 2

6. Let P(x, y) be a moving point in the x − y plane such that [x]. [y] = 2, where [.] denotes
the greatest integer function, then area of the region containing the points P(x, y) is equal
to :
(A) 1 sq. units
(B) 2 sq. units
(C) 4 sq. units
(D) None of these
7. Total number of solution of 2cos⁡ x = |sin⁡ x| in [−2π, 5π] is equal to :
(A) 12 (B) 14 (C) 16 (D) 15
1 1 1 1 2 1 3 1 1999
8. The sum [ ] + [ + ]+[ + ]+[ + ]+ ⋯..+[ + ] is equal to (where [∗]
2 2 2000 2 2000 2 2000 2 2000

denotes the greatest integer function)

(A) 1000 (B) 999 (C) 1001 (D) None of these

APNI KAKSHA 1
(MATHEMATICS) FUNCTIONS
9. Total number of solutions of the equation x 2 − 4 − [x] = 0 are (where [.] denotes the
greatest integer function) :
(A) 1 (B) 2 (C) 3 (D) 4
10. y = 2[x] + 3&y = 3[x − 2] + 5 then [x + y] = ?
(A) 0 (B) 15 (C) 30 (D) 45
11. How many doots the following equation posses 3|x| (2 − |x|) = 1.
(A) 1 (B) 2 (C) 3 (D) 4
12. If f(x) = min{|x − 1|, |x|, |x + 1|}, then :
(A) f is odd (B) f is even
(C) f is periodic (D) None of these
−1 if x<0
13. Let g(x) = 1 + x − [x] and f(x) = {0 if x = 0, then ∀x, fog(x) equals (where [∗]
1 if x>0
represents greatest integer function).
(A) x (B) 1 (C) f(x) (D) g(x)
14. Domain of the function
3
f(x) = log e ⁡ {log |sin⁡ x| ⁡(x 2 − 8x + 23) − } is⁡given⁡by ∶
log 2 ⁡ |sin⁡ x|
(A) (3,5) (B) (3, π) ∪ (π, 5)
(C) (3, π) ∪ (3π/2,5) (D) None of these
15. Which of the following has range above y = 2
(A) f(x) = Sgn⁡(1 − |x|) (B) f(x) = Sgn⁡([x 2 − x])
(C) f(x) = Min⁡(|x|, x 2 , 2) (D) f(x) = Max⁡{|tan⁡ x|, cos⁡ |x|}xt[−π, π]
16. sin y = sin x has graph

(A) (B)

(C) (D) Not

APNI KAKSHA 2
(MATHEMATICS) FUNCTIONS
17. Find the domain of each of the following functions
x3 −5x+3 1
(i) f(x) = (ii) f(x) =
x2 −1 √x+|x|
1
(iii) f(x) = ex+sin⁡ x (iv) f(x) = + √x + 2
log10 ⁡(1−x)
1
(v) log x ⁡ log 2 ⁡ ( ) (vi) f(x) = √3 − 2x − 21−x
x−1/2

(vii) f(x) = √1 − √1 − x 2 (viii) f(x) = (x 2 + x + 1)−3/2

x−2 1−x
(ix) f(x) = √ +√ (x) f(x) = √tan⁡ x − tan2 ⁡ x
x+2 1+x

1 5x−x2
(xi) f(x) = (xii) f(x) = √log1/4 ⁡ ( )
√1−cos⁡ x 4

(xiii) f(x) = log10 ⁡(1 − log10⁡(x 2 − 5x + 16))

APNI KAKSHA 3
(MATHEMATICS) FUNCTIONS
ANSWER KEY
1. D 2. D 3. B 4. A 5. B 6. C 7. B
8. A 9. B 10. B 11. B 12. B 13. B 14. D
15. D 16. C
17. (i) R − {−1,1} (ii) (0, ∞) (iii) R
1 3
(iv) [−2,0) ∪ (0,1) (v) ( , 1) ∪ (1, ) (vi) [0,1]
2 2

(vii) [−1,1] (viii) R (ix) ϕ

π
(x) ⋃n∈1 [nπ, nπ + ] (xi) R − {2nπ}, n ∈ I (xii) (0,1] ∪ [4,5)
4

(xiii) (2,3)

APNI KAKSHA 4