CBSE Class 12 Mathematics Application Of Derivatives MCQs Set E

Question. The equation of tangent to the curve (x/a)ⁿ+(y/b)ⁿ=2at(a ,b) is
(a) x a+ y/b=2 
(b) x/a + y/b=1/2
(c) x/b -y/b =2 
(d) ax by + = 2
Answer: a

Question. The least value of a such that the function f given by f (x)=x² + ax+1  is strictly increasing on (1, 2) is
(a) –2
(b) –1
(c) 0
(d) 2
Answer: a

Question. The position of a point in time ‘ t’ is given by x=a + bt-ct² , y= at+bt². Its acceleration at time ‘t ’ is
(a) b- c
(b) b+ c 
(c) 2b -2c
(d) 2√b²+c²
Answer: d

Question. The equation of normal to the curve y=(1+x)v + sin^-1(sin²x) at x = 0, is
(a) x+ y  = 1
(b) x- y = 1
(c) x+ y  = -1
(d) x- y  = -1
Answer: a

Question. Water is dripping out from a conical funnel of semi-vertical angle π/4 at the uniform rate of 2 cm²/s in the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is 4 cm, the rate of decrease of the slant height of water, is
(a) √2/4π cm/s
(b) 1/4π cm/s
(c) 1/π √2 cm/s
(d) None of these
Answer: a

Question. The total cost C(x) in rupees associated with the production of x units of an item is given by c(x) = 0.007x³ – 0.003x² + 15x+4000.  
The marginal cost when x = 17 units are produced is
(a) R.s.19.196
(b) R.s.20.225
(c) R.s.20.967
(d) R.s.21.297
Answer: c

Question. Let f be a function defined on [a, b] such that f (x) > 0, for all x ∈[a ,b ]. Then, f is an increasing function on
(a) (a, b)
(b) (a, b]
(c) [a, b]
(d) [ a ,b) 
Answer: a

Question. The function x^x is increasing, when
(a) x>1/e
(b) x<1/e
(c) x < 0
(d) for all real x 
Answer: a

Question. If the tangent at (x₁,y₁) to the curve x3 + y3 = a3  meets the curve again at (x₂,y₂), then
(a) x₂/x₁ + y₂/y₁ =-1
(b) x₂/y₁ + x₂/y₂ =-1
(c) x₁/x₂ + y₁/y₂ =-1
(d) x₂/x₁ + y₂/y₁ =1 
Answer: a

Question. If y=4x-5  is tangent to the curve y²= px³+q at(2,3), then (p ,q)  is
(a) (2,7)
(b) −2,7)
(c) (−2,−7)
(d) (2,−7)
Answer: d

Question. Moving along the x-axis there are two points with x=10 +6t,x=3+ t². The speed with which they are reaching from each other at the time of encounter is (x is in centimetre and t is in seconds)
(a) 16 cm/s
(b) 20 cm/s
(c) 8 cm/s
(d) 12 cm/s 
Answer: c

Question. y= log(1+x) – 2x/2+x, x>-1,  is an increasing function of x throughout in, 
(a) x > −1
(b) x > 1
(c) x < 0
(d) x > 0
Answer: a

Question. If the line joining the points (0, 3) and 5,-2) ) is a tangent to the curve y=c/x+1′, then the value of c is
(a) 1
(b) -2
(c) 4
(d) None of these 
Answer: c

Question. The interval in which the function f(x) =xe^2-x increases, is
(a) (-∞,1)
(b) (2,∞)
(c) (0,2)
(d) None of these
Answer: a

Question. If f (x)x³ + 4x² +λx+ 1 is a monotonically decreasing function of x in the largest possible interval (-2,-2/3),then
(a) λ = 4
(b)λ = 2
(c) λ = -1
(d) λ has no real value
Answer: a

Question. The triangle formed by the tangent to the curve f(x) =x² + bx-b at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is
(a) − 1
(b) 3
(c) − 3
(d) 1
Answer: c

Question. The condition for the curves x²/a² – y²/b²=1,xy= c² to intersect orthogonally, is 
(a) a² + b² = 0 
(b) a²-b² = 0 
(c) a= b 
(d) None of these
Answer: d

Question. An angle θ, 0<θ<π/2, which increases twice as fast as its sine, is
(a) π/2
(b) 3π/2
(c) π/4
(d) π/3
Answer: d

Question. A balloon, which always remains spherical, has a variable diameter 3/2(2x+1). The rate of change of its volume with respect to x, is
(a) 27π/8(2x+1)²
(b) 8π/27(2x+1)²
(c) 27π/8(x+2)²
(d) 8π/27(x+2)²
Answer: a

Question. Sand is pouring from a pipe at the rate of 12 cm 3/s.
The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the base. How fast height of the sand cone increasing when the height is 4/1 cm? 
(a) π/48 cm/s
(b) 1/48π cm/s
(c) 48/π cm/s
(d) 48π cm/s
Answer: b

Question. The sides of an equilateral triangle are increasing at the rate of 2 cm/s. The rate at which the area increases, when the side is 10 cm, is
(a) √3cm²/s
(b) 10cm²/s 
(c) 10 √3cm²/s 
(d) 10/√3 cm²/s 
Answer: c

Question. The point on the curve √x+ √y = 2a²,where the tangent is equally inclined to the axes, is
(a) (a⁴ , a⁴)
(b) (0,4a⁴)
(c) (4a⁴,0)
(d) None of these
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) Both (a) and (b)
(d) None of these
Answer: c

Question. If the subnormal at any point on y=(a-^1-nxⁿ is of constant length ,then the value of n is
(a) 1
(b) 0.5
(c) 2
(d) -2
Answer: b

 Question. From mean value theorem, f (b)- f(a)f’ (a) ;a≤x1<b,if f(x)=1/x’ , then x1 is equal to
(a) √ab
(b) a+ b/2
(c) 2ab/a +b 
(d) b- a/b+ a
Answer: a

Question. Gas is being pumped into a spherical balloon at the rate of 30 ft3/min. Then, the rate at which the radius increases when it reaches the value 15 ft, is
(a) 1/30π ft/min
(b) 1/15π ft/min
(c) 1/20 ft/min
(d) 1/15 ft/min
Answer: a

 Question. Let f (x) be differentiable ∀x. If f(1) = -2 and f’ (x)≥ 2 ∀ x ∈ [1,6] then
(a) f(6)< 8 
(b) f(6)≥8
(c) f(6) ≥ 5
(d) f(6) ≤ 5
Answer: b

 Question. If the area of the triangle, included between the axes and any tangent to the curve xyⁿ=aⁿ⁺¹ is constant, then the value of n is
(a) -1
(b) -2
(c) 1
(d) 2 
Answer: c

Question. An object is moving in the clockwise direction around the unit circle x² +y² = 1.  As it passes through the point (1/2,√3/2), its y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in unit per second)
(a) 2
(b) 3√3
(c) √3
(d) 2√3
Answer: b

Question. If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is
(a) a constant
(b) proportional to the radius
(c) inversely proportional to the radius
(d) inversely proportional to the surface area
Answer: d

Question. The rate of change of the surface area of a sphere of radius r, when the radius is increasing at the rate of 2 cm/s is proportional to
(a) 1/r
(b) 1/r²
(c) r
(d) r²
Answer: c

Question. If there is an error of k% in measuring the edge of a cube, then the per cent error in estimating its volume is
(a) k
(b) 3k
(c) k/3
(d) None of these
Answer: b

Question. A kite is moving horizontally at a height of 151.5 m.
If the speed of kite is 10 m/s, how fast is the string being let out, when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
(a) 8 m/s
(b) 12 m/s
(c) 16 m/s
(d) 19 m/s
Answer: a

Question. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then the approximate error in calculating its volume is 
(a) 3.12 π m³
(b) 3.92π m³
(c) 3.56 π m³
(d) 4.01π m³¹
Answer: b

Question. Two cyclists start from the junction of two perpendicular roads, their velocities being 3v m/min and 4v m/min. The rate at which the two cyclists are separating, is
(a) 7/2 v m/min
(b) 5 v m/min
(c) v m/min
(d) None of these
Answer: b

Question. At what point on the curve x³-8a² y=0, the slope of the normal is −2/3 ?
(a) ( a, a)
(b) (2a ,a) 
(c) (2a,a) 
(d) None of these
Answer: c

Question. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm/s. How fast is the area decreasing when the two equal sides are equal to the base? 
(a) 3 b cm²/s
(b) √2 b cm²/s
(c) b/√2 cm²/s
(d) √3 b cm²/s
Answer: d

Question. The total revenue in rupees received from the sale of x units of a product is given by R (x) =13x²+ 26x +15. 
 The marginal revenue when x = 7 is 
(a) R.s.197
(b) R.s.199
(c) R.s.205
(d) R.s.208
Answer: d 

Question. If the curves x²/a² + y²/b² =1 and x²/1²-y²/m²=1 cut each other orthogonally, then
(a) a² +b² =l²+ m² 
(b) a² – b²= l²– m²
(c) a² – b²= l²+ m²
(d) a² + b²=b2 = l²-m²
Answer: c

Question. The value of a for which the function (a+2 )x3-3×2+ 9ax-1decreases monotonically through out for all real x, are
(a) a < – 2
(b) a > – 2
(c) – 3 < a <0
(d) -∞< a ≤-3
Answer: d