Mathematics_Today_-_October_2016

10. Let fx x

bbb

bb

x

xx

() ,

,

= − +

− + −

++

≤ <

−≤<

⎧

⎨

⎪

⎩⎪

3 32

2

1

32

01

231 3

If f(x) has least value at x = 1, then

(a) –2 < b < –1 (b) –1 < b < 0

(c) 0 < b < 1 (d) 1 ≤ b < ∞

11. Let the function f(x) = sinx + cos x, be defined in
    [0, 2π], then f(x)
    (a) increases in ππ
    42

⎛⎝⎜ , ⎞⎠⎟

(b) decreases in

ππ

4

5

4

⎛⎝⎜ , ⎞⎠⎟

(c) increases in 0

4

5

4

⎡,,ππ 2 π

⎣⎢

⎞

⎠⎟∪

⎛

⎝⎜

⎤

⎦⎥

(d) decreases in 0

42

⎡ ,,ππ 2 π

⎣⎢

⎞

⎠⎟

∪⎛

⎝⎜

⎤

⎦⎥

12. Let f(x) = 2x^2 – ln |x|, x ≠ 0, then f(x) is
    monotonically

(a) increasing in ⎝⎜⎛−^1 ⎠⎟⎞∪∞⎛⎝⎜ ⎞⎠⎟

2

0 1

2

,,

(b) decreasing in ⎝⎜⎛−^1 ⎠⎟⎞∪∞⎛⎝⎜ ⎞⎠⎟

2

0 1

2

,,

(c) increasing in ⎝⎜⎛−∞,,⎠⎟⎞∪⎛⎝⎜ ⎞⎠⎟

1

2

01

2

(d) decreasing in ⎝⎜⎛−∞ −,,^1 ⎟⎠⎞∪⎛⎝⎜ ⎞⎠⎟

2

0 1

2

13. Rolle’s theorem holds for the function

f(x) = x^3 + bx^2 + cx, 1 ≤ x ≤ 2 at the point^4

3

, then

(a) c = 8 (b) c = –5

(c) b = –5 (d) b = 8

Comprehension Type

If f(x) = |x – 1| + |x – 3| + |5 – x|, ∀ x ∈ R

14. If f(x) increases, then x ∈
    (a) (1, ∞) (b) (3, ∞) (c) (5, ∞) (d) (1, 3)
    15. If f(x) decreases, then x ∈
    (a) (– ∞, 1) (b) (– ∞, 3)
    (c) (– ∞, 5) (d) (3, 5)
    Matrix Match Type
    16. Match the columns:
    Column I Column II
    (P) f(x) = cosπx + 10x + 3x^2 + x^3 ,
    –2 ≤ x ≤ 3. The absolute minimum
    value of f (x) is

(1) 3/4

(Q) If x^ ∈ [–1, 1], then the minimum

value of f(x) = x^2 + x + 1, is

(2) 2

(R) Let f (x) = (4/3) x^3 – 4x, 0 ≤ x ≤ 2.

Then, the global minimum value

of the function is

(3) –15

(S) Let f (x) = 6 – 12x + 9x^2 – 2x^3 ,

1 ≤ x ≤ 4. Then the absolute

maximum value of f(x) in the

interval is

(4) – 8/3

P Q R S

(a) 2 1 3 4

(b) 3 1 4 2

(c) 3 2 1 4

(d) 1 2 3 4

Integer Answer Type

17. If the approximate value of log 10 (4.04) is 0.abcdef.
    It is given that log 10 4 = 0.6021 and log 10 e = 0.4343,
    then the value of a must be


18. The minimum value of the expression

34 4

3

3

4

bc

a

ca

b

ab

c

+ + + + + (a, b, c are +ve) is

19. The number of critical points of the function f ′(x)
    where fx

x

x

()=||− 22 is

20. The three sides of a trapezium are equal each being
    6 cm long. If area of trapezium when it is maximum
    is 27 A, then the value of A must be ””
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