CBSE Class 12 Mathematics Applications Of Derivatives Assignment Set A

Question. The radius of the cylinder of maximum volume, which can be inscribed a sphere of radius R is:
a. 2/3 R
b. √2/3 R
c. 3/4 R
d. √3/4 R
Answer : B

Question. The abscissae of the points of the curve y3 = x in the interval [–2, 2], where the slope of the tangent can be obtained by mean value theorem for the interval [– 2, 2] are:
a. ± 2/√3 
b. ± √3/2
c. ± 3
d. 0
Answer : A

Question. If the line ax+by+c = 0 is a normal to the curve xy = 1, then:
a. a > 0,b > 0
b. a > 0,b < 0
c. a < 0,b > 0
d. a < 0,b < 0
Answer : B, C

Question. If f(x) = 

a. f (x) is increasing on[−1,2]
b. f (x) is continuous on[−1,3]
c. f '(2) does not exist
d. f (x) has the maximum value at x = 2
Answer : ALL

Question. Let 2 3 h(x) = f (x) − ( f (x)) + ( f (x)) for every real number x. Then:
a. h is increasing whenever f is increasing
b. h is increasing whenever f is decreasing
c. h is decreasing whenever f is decreasing
d. nothing can be said in general
Answer : A, C

Question. The function f(x) = ∫₋₁^x (e^t -1) (t - 1) (t - 2)3 (t - 3)5 dt has a local minimum at x equals to:
a. 0
b. 1
c. 2
d. 3
Answer : B, D

Question. In the mean-value theorem f (b) - f(a) / b - a = f(c), if a = 0, b = 1/2 and f (x) = x(x −1)(x − 2), the value of c is:
a. 1 - √15/6
b. 1 + √15
c. 1 - √21/6
d. 1 + √21
Answer : C

Question. If f (x) 

then:
a. g(x) has local maxima at 1 log 2 e x = + and local minima at x = e
b. f (x) has local maxima at x = 1 and local minima x = 2
c. g(x) has no local minima
d. g(x) has no local maxima
Answer : B, C

Question. Let f be a real-valued function defined on the interval (0,∞), by f (x) = Inx + ∫₀^x √1 + sintdt . Then which of the following statement(s) is(are) true?
a. f ′′(x) exists for all x∈(0,∞)
b. f ′(x) exists for all x∈(0,∞) and f ′ is continuous on (0,∞), but not differentiable on (0,∞)
c. there existsα > 1 such that╿f ′(x)╿< f (x)╿for all x∈(0,∞)
d. there exists β > 0 such that ╿f╿( x) + ╿f '( x)╿≤ β from all x∈(0,∞)
Answer : B, C

Question. If (x) = ∫₀^x e^t2 (t - 2) (t - 2) (t - 3) dt ∀x∈ ∞ then:
a. f has a local maximum at x = 2
b. f is decreasing on (2, 3)
c. there exists some c∈(0,∞) such that f ′′(c) = 0
d. f has a local minimum at x = 3
Answer : ALL

Question. A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
a. 24
b. 32
c. 45
d. 60
Answer : A, D

Question. The function f ( x) = 2│x│ + │x + 2│ −││x + 2│− 2│x ││has a local minimum or a local maximum at x is equal to:
a. –2
b. −2/3
c. 2
d. 2/3
Answer : A, B

Question. On the ellipse 4x² + 9y² = 1, the point at which the tangents are parallel to the line 8x = 9 y, are:
a. (2/5, 1/5)
b. (- 2/5, 1/5)
c. (- 2/5, - 1/5)
d. (2/5, - 1/5)
Answer : B, D

Question. If f (x) is cubic polynomial which has local maximum at x = −1. If f (2) = 18, f (1) = −1 and f ′(x) has local minimum at x = 0, then:
a. the distance between (−1,2) and (a, f (a)) where x = a is the point of local minima, is 2 5.
b. f ( x) is increasing for x∈ [1,2 √5]
c. f ( x) has local minima at x = 1
d. the value of f (0) = 5
Answer : B, C

Question. The function f (x) = x(x + 3)e^-1/2x satisfies all the condition of Rolle's theorem in [– 3, 0]. The value of c is:
a. 0
b. 1
c. – 2
d. – 3
Answer : C

Question. Let f : (0,∞) → R be given by

a. f (x) is monotonically increasing on [1,∞)
b. f (x) is monotonically decreasing on [0,1)
c. f (x) + f (1/x) = 0, for all x∈(0,∞)
d. (2 ) x f is an odd function of x on R
Answer : A, C, D

Assertion and Reason

Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
a. If both assertion and reason are true and the reason is the
correct explanation of the assertion.
b. If both assertion and reason are true but reason is not the
correct explanation of the assertion.
c. If assertion is true but reason is false.
d. If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Let f (x) = 2√x and g(x) = 3 - 1/x , x > 1?
Assertion: f (x) > g(x)(x > 1)
Reason: f (x) − g(x) increases on (1, ∞)
Answer : A

Question. Let f be a function defined by

Assertion: x = 0 is point of maxima of f
Reason: f ′(0) = 0
Answer : B

Question. Assertion:│cot x − cot│≤│x − y│for all x,y ∈ (- π/2, π/2)
Reason: If f is differentiable on an open interval and │f ′(x)│≤ M then │f (x) − f (y)│≤ M│x − y
Answer : D

Question. Let f (x) = x⁴ − 2x² +5be defined on [−2, 2] ?
Assertion: The rang of f (x) is [2, 13]
Reason: The greatest value of f is attained at x = 2
Answer : D

Question. Let y = x + a²/x (a, > 0) ?
Assertion: y_max  = −2a (local mix)
Reason: y_min  = 2a (local min)
Answer : B

Question. Assertion: The function f (x) = 2sin x + cos 2x(0 ≤ x ≤ 2π ) has minimum at x = π/3 and maximum at 5π/3
Reason: The function f (x) above decreases on (0, π/3) increases on (π/3, 5π/3) and decreases on (5π/3, 2π)
Answer : A

Question. Let f(x) tan^-1 1 - x / 1 + x ?
Assertion: The difference between the greatest and smallest value of f (x) on [0, 1] is π/4
Reason: If a function g decrease on [a, b] then the greatest value of g = g(a) and least value of g is g(b) .
Answer : A

Question. Let a,b ∈ R be such that the function f given by f (x) = log│x│+ bx² + ax, x ≠ 0 has extreme values at x = −1 and x = 2.
Assertion: f has local maximum at x = −1 and at x = 2.
Reason: a = 1/2 and b = 1/4
Answer : A

Question. Assertion e^π >π^e 
Reason: The function x^-1/x (x > 0) has local maximum at x = e.
Answer : D

Question. Let f(x) = x/logx ?
Assertion: The minimum value of f (x) is e.
Reason: log x >1 for x > e and < 1 for x < e
Answer : .A

Comprehension Based

Paragraph –I

Consider the function f : (−∞,∞)→(−∞,∞) defined by f (x) = x² - ax + 1 / x² + ax + 1, 0 < 2

Question. Which of the following is true?
a. (2 + a)² f "(1) + (2 − a)² f "(−1) = 0
b. (2 − a)² f "(1) − (2 + a)² f "(−1) = 0
c. f '(1) f '(−1) = (2 − a)²
d. f '(1) f '(−1) = −(2 + a)²
Answer : A

Question. Which of the following is true?
a. f (x) is decreasing on (–1, 1) and has a local minimum at x = 1.
b. f (x) is increasing on (–1, 1) and has a local maximum at x = 1.
c. f (x) is increasing on (–1, 1) but has neither a local maximum nor a local minimum at x = 1.
d. f (x) is decreasing on (–1, 1) but has neither a local maximum nor a local minimum at x = 1.
Answer : A

Question. Let f(x) = ∫₀^ax  f"(t) / 1 + t⁰ dt. Which of the following is true?
a. g '(x) is positive on (−∞,0) and negative on (0,∞)
b. g′(x) is negative on (−∞,0) and positive on (0,∞)
c. g′(x) changes sign on both (−∞,0) and (0,∞)
d. g′(x) does not change sign (−∞,∞)
Answer : B

Paragraph –II

Let f (x) = (1−x)² sin² x+x² for all x∈R and let

f (t) dt for all x ∈ (1,∞).

Question. Which of the following is true?
a. g is increasing on (1,∞)
b. g is decreasing on (1,∞)
c. g is increasing on (1, 2) and decreasing on (2,∞)
d. g is decreasing on (1, 2) and increasing on (2,∞)
Answer : B

Question. Consider the statements:
A.There exists some x ∈ R such that 2 f (x) + 2x = 2(1+ x )
B. There exists some x ∈ R such that 2 f (x) +1 = 2x(1+ x) Then,
a. Both A and B are true
b. A is true and B is false
c. A is false and B is true
d. Both A and B are false
Answer : C

Paragraph –III

Let f : [0,1]→ R (the set of all real numbers) be a function.
Suppose the function f is twice differentiable, f (0) = f (1) = 0 and satisfies "( ) 2 '( ) ( ) , x f x − f x + f x ≥ e x ∈ [0,1].

Question. Which of the following is true for 0 < x <1?
a. 0 < f (x) < ∞
b. - 1/2 < f (x) < 1/2
c. −1/ 4 < f (x) < 1
d. −∞ < f (x) < 0
Answer : D

Question. If the function e^−x f(x) assumes its minimum in the interval [0,1] a x =1/ 4, which of the following is true?
a. f (x) < f (x) 1/4 < x < 3/4
b. f (x) > f (x), 0 < x < 1/4
c. f (x) < f (x) 0 < x < 1/4
d. f (x) < f (x) 3/4 < x < 1
Answer : A

Integer

Question. The curve y = ax³ + bx² + cx + b touches the x-axis P(−2,0) and cuts the y-axis at a point Q where its gradient is 3, then the value of −10a −100b +1000c must be
Answer : 3080

Question. Tangent at a point P₁ (other than (0, 0)) on the curve y = x³ meets the curve again at 2P .The tangent at P₂ meets the curve again at P₃ and so on, then we get P₁, P₂, P₃, … Pn are is GP, then the ratio area (ΔP₁ P₂ P₃) / area (ΔP₂, P₃ P₄) λ/μ, then the value of 251/2 λ/μ⋅ must be
Answer : 2008

Question. If the value maximum slope of f (x) = x³ + 8x² − 13x − 18 is λ, then the value of 15λ must be:
Answer : 125

Question. Rolle’s theorem holds for the function f (x) = x³ + bx² + cx, 1 ≤ x ≤ 2 at the point 4/3 then the value of 100c − 500b must be:
Answer : 3300

Question. If the approximate value of 10 log (4.04) 0 abcdef, It is given that 4 log 4 = 0.6021 and 10 log e = 0.4343, then the value of abcd must be:
Answer : 1

Question. If the greatest and least values of the function f (x) = x³ − 6x² + 9x + 1 on [0, 2] are λ and μ , then the value of λ⁴ +μ8must be:
Answer : 626

Question. The three sides of a trapezium are equal each being 6 cm long. If area of trapezium when it is maximum is A, then the value of 4√3 Amust be:
Answer : 324

Question. Let f (x) = x³ + 6x² + ax + 2. If the largest possible interval in which f (x) is a decreasing function in (−3,−1), then the value of a must be:
Answer : 9

Question. The number of critical points of the function f (x) = │x│e^−x must be:
Answer : 2

Question. Let 

. If f (x) has greatest value at x = 1, then [b² ∈ (2, λ )]. Then,λ must be:
Answer : 130

Question. The indicated horse power I of an engines is calculated from the formula I = PLAN / 33000 where, A = π/4 d² Assuming that error of 10% may have been made in measuring P, L, N and d. If the greatest possible in I is λ%then λ must be:
Answer : 50

Question. A balloon is in the form of right circular cylinder of radius 1.5m and length 4m and is surmounted by hemispherical ends. If the radius is increased by 0.01 m and the length by 0.05 m, the percentage change in the volume of the balloon is a. bcd%, then the value of abcd must be:
Answer : 2389

Question. When travelling at x km/ h, a truck burns diesel at the rate of 1/300 (900/x + x) L/km If the diesel oil costs 40 paise/L and driver is paid Rs 1.50 per h, if the steady speed that will minimize the total cost of the trip of 500 km is λ km/ h, then the value of 50λ must be:
Answer : 2250

Question. If f (x) = tan^-1 (sin x + cos x)³ is an increasing function, then the value of x in (0, 2π ) is x ∈ (aπ/4, bπ/4) ∪ (cπ/4, dπ/4)
Then, the value of a +10b +100c +1000d must be:
Answer : 8

Question. If f (x) = (x −1)(x − 2)(x −3) and a = 0, b = 4, then c the using LMVT is 2 ±λ. Then value of 450 √2λ must be
Answer : 600

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