BOOSTER - A
1) Find the equation of the normal to the curve y= (1+ x)ʸ + sin⁻¹(sin²x) at x= 0. x+ y=1
2) Find the equations of the tangents drawn to the curve y²- 2x³ - 4y + 8=0 from the point (1,2). 2√3x - y = 2(√3-1) or 2√3x + y = 2(√3+1)
3) Find the point of intersection of the tangents drawn to the curve x²y = 1- y at the points where it is intersected by the curve xy = 1- y. (0,1)
4) Find all the lines that pass through the point (1,1) and are tangent to the curve represented parametrically as x= 2t - t² and y= t + t². x=1 when t= 1, m--> ∞; 5x - 4y = 1 if t≠ 1, t= 1/3
5) The tangent to y= ax² + bx + 7/2 at (1,2) is parallel to the normal at the point (-2,2) on the curve y= x² + 6x + 10. Find the value of a and b. 1,-5/2
6) A straight line is drawn through the origin and parallel to the tangent to a curve {x + √(a² - y²)}/a = log{a +√(a² - y²)}/y at an arbitrary point M. Show that the locus of the point P of intersection of the straight line through the origin and the straight line parallel to the x-axis and passing through the point M is x² + y² = a².
7) Show that the segment of the tangent to the curve y= (a/2) log |{a+ √(a² - x²)}/{a - √(a² - x²)}| - √(a² - x²) contained between the y-axis and the point of tangency has a constant length.
8) A function is defined parametrically by the equations
f(t)= x = 2t+ t² sin(1/t) if t≠ 0
0 if t=0
And
g(t)= y= (1/t) sint² if t≠ 0
0 if t= 0
Find the equation of the tangent and normal at the point for t= 0 if exist. T: x - 2y= 0; N: 2x + y =0
9) Find all the tangents to the curve y= cos(x + y), - 2π≤ x ≤ 2π, that are parallel to the line x+ 2y=0. x + 2y =π/2 & x + 2y =- 3π/2
10) Show that the segment of the normal to the curve x= 2a sin t + a sin t cos²t; y= - a cos³t contained between the coordinate axes is equal to 2a.
11) Show that the normals to the curve x= a(cos t + t sin t); y= a(sin t - t cos t) are tangent lines to the circle x² + y² = a².
12) The chord of the Parabola y= - a²x² + 5ax - 4 touches the curve y= 1/(1- x) at the point x= 2 and is bisected by the point. Find a. 1
13) If the tangent at the point (x₁, y₁) to the curve x³ + y³ = a³ (a≠ 0) meets the curve again in (x₂, y₂) then show that x₂/x₁ + y₂/y₁ = -1.
14) Determine a differential function y= f(x) which satisfies f'(x)= [f(x)]² and f(0)= -1/2. Find also the equation of the tangent at the point where the curve crosses the y-axis. -1/(x +2); x - 4y =2
15) Tangent at a point P₁ (other than (0,0)] on the curve y= x³ meets the curve again at P₂. The tangent at P₂ meets the curve at P₃ and so on. Show that the abscissa of P₁, P₂, P₃, .....Pₙ form a GP.
Also find the ratio area(P₁P₂P₃))/Area P₂P₃P₄. 1/16
16) The curve y= ax³ + bx² + cx + 5, touches the x-axis at P (-2,0) cuts the y-axis at a point Q where its gradient is 3. Find a,b,c. -1/2, -3/4, 3
17) The tangent at a variable point P of the curve y= x²- x³ meets it again at Q. Show that the locus of the middle point of PQ is y= 1- 9x + 28x²- 28x³.
18) Show that the distance from the origin of the normal at any point of the curve x= aeᶿ(sin(θ/2)+ 2 cos(θ/2)) and y= aeᶿ(cos(θ/2) - 2 sin(θ/2)) is twice the distance of the tangent at the point from the origin.
19) Show that the condition that the curve x²⁾³ + y²⁾³ = c²⁾³ and (x²/a²) + (y²/b²) = 1 may touch if c= a+ b.
20) The graph of a certain function f contains the point (0,2) and has the property that for each number 'p' the line tangent to y= f(x) at (p, f(p)) intersect the x-axis at p+ 2. Find f(x). 2e⁻ˣ⁾²
21) A curve is given by the equations x= at² and y= ar³. A variable pair of Perpendicular lines through the origin 'O' meet the curve at P and Q. Show that the locus of the point of intersection of the tangents at P and Q is 4y² = 3ax - a².
22) A and B are points of the Parabola y= x². The tangents at A and B meet at C. The median of the triangle ABC from C has length 'm' units. Find the area of the triangle in terms of 'm'. m√m/√2
23) Find the value of n so that the subnormal at any point on the curve xyⁿ = aⁿ⁺¹ may be constant, n=-2
24) Show that in the curve y= a log(x² - a²), sum of the length of tangent and Subtangent varies as the product of the coordinates of the point of contact.
25) Show that the curve x²/(a² + K₁) + y²/(b² +K₁) = 1 and x²/(a² + K₂) + y²/(b² + K₂)= 1 intersect orthogonally.
26) If the two curves C₁ : x = y² and C₂ : xy= k cut at right angles, find the value of k. ±1/2√2
27) Show that the angle between the tangent at any point 'A' of the curve log(x² + y²) = C tan⁻¹(y/x) and the line joining A to the origin is independent of the position of A on the curve. tan⁻¹(2/C)
RATE MEASURE AND APPROXIMATION
BOOSTER - A
1) Water is being poured on to a cylindrical vessel at the rate of 1 m³/min. If the vessel has a circular base of radius 3m, find the rate at which the level of water is rising in the vessel. π/9 m/min
2) A man 1.5 tall works away from the lamp post 4.5 m high at the rate of 4kmph .
a) how fast is the farther end of the shadow moving in the pavement ? 6kmph
b) how fast is his shadow lengthening ? 2kmph
3) A particle moves along the curve 6y = x³+ 2. Find the points on the curve at which the y coordinate is changing 8 times as fast as the x coordinate. (4,11)&(-4,-31/3)
4) An inverted cone has a depth of 10cm and a base of radius 5cm. Water is poured into it at the rate of 1.5cm³/min. Find the rate at which level of water in the cone is rising, when the depth of water is 4cm. 3π/8 cm/min
5) A water tank has the shape of a right circular cone with its vertex down. Its altitude is 10cm and the radius of the base is 15cm. Water leaks out of the bottom at a constant rate of 1 cu cm/sec. Water is poured into the tank at a constant rate of C cu. cm/sec. Compute C so that the water level will be rising at the rate of 4 cm/sec at the instant when the water is 2 cm deep . 1/36π cu cm/sec
6) Sand is pouring from a pipe at the rate of 12 cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always 1/6th of the radius of the base . How fast is the height of the sand cane increasing when the height is 4cm. 1/48π cm/s
7) An open Can of oil is accidentally dropped into a lake; assume the oil spreads over the surface as a circular disc of uniform thickness whose radius increases steadily at the rate of 10cm/sec. At the moment when the radius is 1 m. the thickness of the oil slick is decreasing at the rate of 4mm/sec, how fast is it decreasing when the radius is 2 metres. 0.05 cm/sec
8) Water is dripping out from a conical funnel of semi vertical angle π/4, at the uniform rate of 2 cm³/sec through a tiny hole at the vertex at the bottom. When the slant height of the water is 4cm, find the rate of decrease of the slant height of the water . √2/4π cm/s
9) An Air force plane is ascending vertically at the rate of 100 km/h. if the radius of the earth is R km, how fast the area of the earth, visible from the plane increasing at 3 minute after it started ascending. Take visible area A= (2πR²h)/(R+ h) where h is the height of the plane in kms above the earth. 200πr³/(r +5)² km²/h
10) A variable ∆ ABC in the xy plane has its orthocentre at vertex B. A fixed vertex A at the origin and the third vertex C restricted to lie on the parabola y= 1+ 7x²/36. The point B starts at the point (0,1) at time t=0 and moves upward along with y axis at a constant velocity of 2 cm/sec. How fast is the area of the triangle increasing when t= 7/2 sec. 66/7
11) A circular ink blot grows at the rate of 2 cm² per second. Find the rate at which the radius is increasing after 28/11 seconds. Use π= 22/7. 1/4 cm/sec
12) Water is flowing out at the rate of 6 m²/min from a reservoir shaped like a hemispherical bowl of radius R= 13 m. The volume of water in the hemispherical bowl is given by V= πy²(3R - y)/3 when the water is y metre deep. Find
a) At what rate is the water level changing when the water is 8m deep. -1/24π m/min
b) At what rate is the radius of the water surface changing when the water is 8m deep. -5/288π m/min
13) if in a triangle ABC , the side c and the angle C remain constant, while the remaining elements are changed slightly, show that da/cosA + db /cosB = 0
14) At time t> 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t= 0, the radius of the sphere is 1 unit and at t= 15 the radius is 2units.
a) Find the radius of the sphere as a function of time t. ⁴√(1+t)
b) At what time t will the volume of the sphere be 27 times its volume at t= 0. 80
15) Yse differential to a approximate the values of :
a) √36.6. 6.05
b) ³√25. 80/27
16) if the radius of a sphere is measured as 9 cm with an error 0.03cm, then find the approximate error calculating its volume. 9.72π cm³
MIXED
BOOSTER - A
1) Find the equation of a straight line which is tangent at one and normal at another point of the curve x= 3t², y= 2t³.
2) If the normal to the curve, y= f(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis. Then f'(3)=
a) -1 b) -3/4 c) 4/3 d) 1
3) The point/s on the curve y³+ 3x²= 12y where the tangent is vertical is/are
a) (±4/√3,-2) b) (± √(11/3,1) c) 0,0) d) (±4/√3,2)
4) Tangent to the curve y= x²+ 6 at a point P(1,7) touch is the circle x²+ y²+ 16x + 12y + c = 0 at a Q. Then the coordinates of Q are
a) (-6,-11) b) (-9, -13) c) (-10,-15) d) (-6,-7)
5) The tangent to the curve y= eˣ drawn at the point (c, eᶜ) intersect the line joining the points (c -1), eᶜ⁻¹) and (c +1, eᶜ⁺¹)
a) on the left of x= c
b) on the right of x= c
c) at no point
d) at all points
6) The intercept on x-axis made by tangents to the curve y= ˣ₀ ∫ |t | dt, x belongs to R, which are parallel to line y= 2x, are equal to:
a) ±4 b) ± 1 c) ±2 d) ±3
7) The slope of the tangent at the curve (y - x⁵)² = x(1+ x²)² at the point (1,3) is
MONOTONICITY
BOOSTER - A
1) Find the intervals of monotonicity for the following functions and represent your solution set on the number line.
i) f(x)= 2 ₑx²- 4x
ii) f(x)=eˣ/x
iii) f(x)= x²e⁻ˣ
iv) f(x)= 2x²- log |x|
Also plot the graphs in each case and state their range.
2) Let f(x)= 1- x - x³. Find all the real values of x satisfying the inequality, 1- f(x) - f³(x) > f(1- 5x).
3) Find the interval of monotonocity of the function in [0,2π]
i) f(x)= sinx - cosx in x ∈ [0,2π]
ii) g(x)= 2 sinx + cos2x in (0≤ x ≤2π).
iii) f(x)= (4 sin x - 2x - x cos x)/(2+ cosx)
4) Let f(x) be a increasing function defined on (0, ∞). If f(2a²+ a+1)> f(3a²- 4a +1). Find the range of a.
5) Let f(x)= x³- x²+ x+ 1 and
g(x)= max {f(t) : 0≤ t≤ x}, 0≤ x ≤ 1
3 - x , 1< x ≤2
Discuss the continuous and differentiable of g(x) in the interval (0,2).
6) Find the set of all values of the parameter 'a' for which the function,
f(x)= sin2x - 8(a +1) sinx + (4a²+ 8a - 14)x increases for all x ∈R and has no critical points of for all x ∈ R.
7) Find the greatesr and the least values of the following functions in the given interval if they exist.
i) f(x)= sin⁻¹ {x/√(x²+1)} - log x in [1/√3, √3]
ii) f(x)= 12x⁴⁾³ - 6x¹⁾³, x [-1,1]
iii) y= x⁵ - 5x⁴ + 5x³ +1 in [-1,2]
8) Find the values of 'a' for which the function f(x)= sinx - a sin2x - (1/3) sin3x + 2ax increases throughout the number line.
9) If f(x)= {(a²- 1)/3} x³+ (a -1) x²+ 2x +1 is monotonic increasing for every x∈ R then find the range of values of 'a'.
10) Find the set of values of 'a' for which the function.
f(x)= {1- √(21- 4a - a²)/(a +1)}x³+ 5x +√7 is increasing at every point of its domain .
11) Let a+ b = 4, where a< 2 and let g(x) be a differentiable function, if dg/dx > 0 for all x, prove that ᵃ₀∫ g(x)= ᵇ₀∫ g(x)= dx increases as (b - a) increases.
12) Find the range of values of 'a' for which the function f(x)= x³+ (2a +3)x²+ 3(2a +1)x +5 is monotonic in R. Hence find the set of values of 'a' which f(x) in invertible.
13) Find the value of x > 1 for which the function
D(x)= ∫ (1/t) log{(t -1)/32} dt in (x²,x) is increasing and decreasing.
14) Find all the values of the parameter 'a' for which the function:
f(x)=8ax - a sin 6x - 7x - sin 5x increases and has no critical points for all x∈ R
15) If f(x)= 2eˣ - ae⁻ˣ + (2a +1)x -3 monotonically increases for every x∈ R then find the range of the values of 'a'.
16) Prove that x¹- 1> 2x log x > 4(x -1) - 2 log x for x > 1.
17) Prove that tan²x + 6 log secx + 2 cos x + 4 > 6 secx for x ∈ (3π/2, 2π).
18) If 0< x < 1 prove that y= x log x - (x²/2) + 1/2 is a function such that d²y/dx²> 0. Deduce that x log x > x²/2 - 1/2.
19) Find the set of values of x for which the inequality log(1+ x)> x/(1+ x) is valid.
20) If b> a, Find the minimum value of |(x - a)³|+ |(x - b)³|, x ∈R.
21) Discuss the monotonicity of function g(x)= 2f(x²/2) + f(6- x²) ∀ x ∈ R where f"(x)> 0 ∀ x ∈ R.
22) Find a for f(x)= sin³x - a sin²x should not have any critical point in [π/6,π/3].
23) Let f(x)= {b²+ (a -1)b + 2}x + ∫ (sin²x + cos⁴x) dx. f(x) be strictly increasing function ∀ x∈ R and for all real values of b, find a .
24) Let f(x)= 2/√3 tan⁻¹{(2x +1)/√3} - log(x²+ x +1)+ (b²- 5b +3)x + c is strictly decreasing for all real values of x, find b.
25) Find a for which f(x)=logₐ(4ax - x²) is strictly increasing ∀ x∈ [3/2,2].
26) If x> 0, let f(x)=5x²+ Ax⁻⁵, A> 0 (constant). Find the smallest A such that f(x)≥ 24 ∀ x> 0.
27) If ax²+ b/x ≥ c ∀ x> 0, where a,b > 0, then show that 27ab²≥ 4c³.
1) a) I in (2,∞) & D in (-∞,2) b) I in (1,∞) & D in (-∞,0) U (0,1) c) I in (0,2) and D in (-∞ , 0 U (2, ∞) d) I for x > 1/2 or -1/2< x < 0 & D for x < -1/2 or 0< x < 1/2
2) (-2,0) U (2, ∞)
3) a) I in [0, 3π/4) U (7π/4, 2π] & D in (3π/4, 7π/4)
b) I in [0,π/6) U (π/2, 5π/6) U (3π/2, 2π] & D in (π/6, π/2) U (5π/6,3π/2)]
c) I in [0,π/2) U (3π/2,2π] & D in (π/2,3π/2)
4) (0,1/3) U (1,5) 5) continuous but not diff. at x= 1. 6) a< -(2+√5) or a>√5
7) a) π/6+ (1/2) log 3, π/3 - (1/2) log 3
b) max at x= 1 and f(-1)= 18; min at x= 1/8 and f(1/8)= -9/4
c) 2 and -10
8) (t, ∞) 9) a ∈(-∞, -3) U [1, ∞). 10) [-7,-1) U [2,3]. 12) 0≤ a ≤ 3/2 13) max log+1,3) & min log(1,3) 14) (6, ∞). 15) a≥ 0. 19) (-1,0) U (0, ∞). 20) (b - a)³⁾⁴
21) D: (-∞,-2) U (0,2), increase: (-2,0) U (2, ∞)
22) (-∞, 3/4) U (3√3/4, ∞) 23) [1- √11, 1+ √11]
24) [(5-√5)/2, (5+√5)/2]
25) (1/2, 3/4] U (1, ∞) 26) 2(24/7)⁷⁾²
BOOSTER - B
1) Verify Rolle's Theorem for f(x)= (x - a)ᵐ(x - b)ⁿ on [a,b], m, n being positive integer.
2) Let f(x)= 4x³- 3x²- 2x +1 use Rolle's Theorem to prove that there is exist c, 0< c < 1 such that f(c)= 0.
3) Using LMVT prove that:
a) tanx > x in (0,π/2).
b) sin x < x for x> 0.
4) ledpt f be continuous on [a,b] and assume the second derivative f" exist on (a,b). Suppose that the graph of f and the line segment joining the point (a, f(a)) and (b, f(b)) intersect at a point (x₀, f(x₀)) where a< x₀< b. Show that there exist a point c ∈ (a, b) such that f"(c)= 0.
5) Prove that if f is differentiable on [a,b] and if f(a)= f(b)= 0 then for any real α there is an x ∈ (a,b) such that α f(x) + f'(x)=0.
6) For what value of a, m and b does the function
f(x)= 3 x= 0
-x²+3x +a, 0< x < 1
mx + b 1 ≤ x ≤ 2
satisfy the hypothesis of the mean value theorem for the interval [ 0,2].
7) Assume that f continuous on [a, b], a> 0 and differentiable on an open interval (a,b).
Show that f(a)/a = f(b)/b, then there exist x₀∈ (a,b) such that x₀ f'(x₀)= f(x₀).
8) Let f, g be differentiable on R and suppose that f(0)= g(0) and f'(x)≤ g'(x) for all x≥ 0. Show that f(x)≤ g(x) for all x ≥ 0.
9) Let f be continuous on [a,b] and differentiable on (a,b). If f(a)= a and f(b)= b, show that there exist distinct c₁, c₂ in (a,b) such that f'(c₁)+ f'(c₂)= 2.
10) Let f defined on [0,1] be a twice differentiable function such that |f'(x)|≤ 1 for all x∈ [9,1] if f(0) = f(1), then show that || f'(x)|< 1 for all x ∈ [0,1].
11) f(x) and g(x) are differentiable functions 0≤ x ≤ 2 such that f(0) = 5, g(0) = 0, f(2) = 8, g(2) = 1. Show that there exists a number c satisfying 0< c < 2 and f'(c) = 3 g'(c).
12) If f, φ, ψ are continuous in [a, b] and deriavable in ]a, b[ then show that there is a value of c lying between a and b such that,
| f(a) f(b) f'(c)
φ(a) φ(b) ψ'(c) = 0
ψ(a) ψ(b) ψ'(c)|
13) Assuming |f"(x)|≤ m for which x in interval [0,1] and assume f takea on its largest value at an interior point of this interval, show that |f'(0)|+ |f'(a)|≤ a m, Assume f"(x) is continuous in [0,a].
14) Let a> 0 and f be continuous in [-a, a]. Suppose that f'(x) exist and f'(x)≤ 1 for all x∈ (-a, a).
If f(a)= a and f(-a)= - a, show that f(0)= 9.
15) Prove that inequality eˣ > (1+ x) using LMVT for all x∈ R₀ and use it to determine which of the two numbers eˣ and πᵉ is greater.
16) If f"(x) exists ∀ x ∈ R such that f(x) = f(6- x), f'(0)= 0 = f'(2)= f'(5). Find the minimum number of roots of the equation (f"(x))²+ f'(x) f"'(x)= 0 in interval [0,6].
17) let f: [0,1] --> R be continuous with f(0) = f(1) = 0. Assume that f"(x) exist on 0< x < 1, with f"(x) + 2f'(x) + f'(x) ≥ 0. Show that f(x) ≤ 0 for all x ∈ [0,1].
1) c= (mb + na)/(m + n) which lies between a and b
6) a= 3, b= 4 and m= 1
16) 12
BOOSTER - C
1) For all x ∈ (0,1):
a) eˣ <1+ x b) logₑ(1+ x) < x c) sinx > x d) logₑ < x b
2) Consider the following statements S and R:
S: Both sin x and cosx are decreasing functions in the interval (π/2,π).
R: If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a,b).
Which of the following is true ?
a) both S and R are wrong
b) both S and R are correct, but R is not correct explanation for S.
c) S is correct and R is the correct explanation for S.
d) S is correct and R is wrong. d
3) Let f(x)= ∫ eˣ(x -1)(x -2) dx then f decreases in the interval :
a) (-∞, 2) b) (-2,-1) c) (1,2) d) (2,+∞). c
4) If f(x)= xeˣ⁽¹⁻ˣ⁾ then f(x) is
a) increasing on (-1/2,1)
b) decreasing on [-1/2,1]
c) increasing on R
d) decreasing on R
5) Let -1≤ p≤ 1. Show that the equation 4x³- 3x - p has a unique root in the interval [1/2,1] and identify it.
6) The length of a longest interval in which the function f(x)= 3 sinx - 4 sin³x is increasing, is
a) π/3 b) π/2 c) 3π/2 d) π
7) Using the relation 2(1- cosx)< x², x≠ 0 or otherwise, Prove the sin(tan x)≥ x, ∀x ∈ [0,π/4].
8) Let f: [0,4] --> R be a differentiable function.
i) Show that there exist a, b ∈ [0,4], (f(4))²= 8f'(a) f(b)
ii) Show that there exist α, β with 0< α< β< 2 such that
⁴₀∫ f(t) dt= 2(α f(α²) + β f(β)²)).
9) Let f(x) = xᵅ log x, x > 0
0, x= 0
Rolle's Theorem is applicable to f for x [0,1], if α =
a) -2 b) -1 c) 0 d) 1/2
10) If f is a strictly increasing function, then
lim ₓ→₀ {f(x⅖) - f(x)}/{f(x) - f(0)} is equal to
a) 0 b) 1 c) -1 d) 2
11) If p(x)= 51x¹⁰¹ - 232x¹⁰⁰ - 45x + 1035, using Rolle's Theorem, show that atleast one root of p(x) lies between (45¹⁾¹⁰⁰, 46).
12) If f(x) ∀ x is a twice differentiable function and given that f(1)= 1, f(2)= 4, f(3)= 9, then
a) f"(x)= 2, for ∀ x ∈ (1,3)
b) f"(x)= f'(x)= 2, for some x ∈ (2,3)
c) f"(x)= 3, for ∀ x ∈ (1,3)
d) f"(x)= 2, for som x ∈ (1,3)
13) Let f(x)= 2+ cosx for all real x.
Statement 1: For each real t, there exists a point 'c' in [t, t+π] such that f'(c)= 0.
Because
Statement -2: f(t)= f(t + 2π) for each real t.
a) Statement- 1 is true, Statement- 2 is true ; Statement- 2 is correct explanation for Statement- 1.
b) Statement- 1 is true, Statement- 2 is true; Statement- 2 is NOT a correct explanation for Statement- 1.
c) Statement- 1 is true, Statement- 2 is false .
d) Statement- 1 is false, Statement- 2 is true.
PARAGRAPH
14) If a continuous function f defined on the real line R, assume positive and negative values in R then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative then the equation f(x)= 0 has a root in R.
Consider f(x)= keˣ - x for all real x where k is a real constant.
i) The line y= x meets y= keˣ for k < 0 at
a) no point b) one point c) two points d) more than two points
ii) The positive value of k for which keˣ - x = 0 has only one root is
a) 1/e b) 1 c) e d) logₑ2
iii) For k> 0, the set of all values of k for which keˣ - x = 0 has two district roots is
a) (0,1/e) b) (1/e,1) c) (1/e, ∞) d) (0,1)
Match the Column
15) In the following [x] denote the greatest integer less than or equals to x.
Match the functions in Column I with the properties in Column B II
Column I
a) x|x|
b) √|x|
c) x + [x]
d) |x -1|+ |x +1|
Column II
p) continuous in (-1,1)
q) differentiable in (-1,1)
r) strictly increasing in (-1,1)
s) non differentiability atleast at point on (- 1,1)
BOOSTER - D
1) Let the function g: (- ∞, ∞) --> (-π/2, π/2) be given by g(u)= 2 tan⁻¹ (eᵘ) - π/2. Then, g is
a) even and is strictly increasing in (0, ∞)
b) odd and is strictly decreasing in (-∞, ∞)
c) odd and strictly increasing in (-∞, ∞)
d) neither even nor odd, but is strictly increasing in (-∞, ∞).
2) Let f(x) be a non constant twice differentiable function defined on (-∞, ∞) such that f(x)= f(1- x) and f'1/4)= 0. Then
a) f"(x) vanishes atleast twice on [0,1]
b) f'(1/2)=0
c) ¹⁾²₋₁/₂∫ f(x + 1/2) sinx dx= 0
d) ¹⁾²₀∫ f(t)ₑsinπt dt = ¹₁/₂∫ f(1- t) ₑsinπt dt.
3) For the function f(x)= x cos(1/x), x≥ 1.
a) for atleast one x in the interval [1, ∞), f(x +2) - f(x)< 2
b) lim ₓ→∞ f'(x)= 1
c) For all x in the interval [1, ∞), f(x +2) - f(x)> 2
d) f'(x) is strictly decreasing in the interval [1, ∞)
4) Let P(x) be a polynomial of degree 4 having extrimum at x= 1, 2 and
lim ₓ→₀{1+ p(x)/x²}= 2. Then the value of p(2) is
Paragraph
Consider the polynomial f(x)= 1+ 2x + 3x²+ 4x³. Let s be the sum of all distinct real roots of f(x) and let t = |s|
5) The real number lies in the interval
a) (-1/4,0) b) (-11,-3/4) c) (-3/4,-1/2) d) (0,1/4)
6) The area bounded by the curve y= f(x) and the lines x= 0, y= 0 and x = t, lies in the interval
a) (3/4,3) b) (21/64,11/16) c) (9,10) d) (0,21/64)
7) The function f'(x) is
a) increasing in (-t, -1/4) and decreasing in (-1/4, t)
b) decreasing in (-t, -1/4) and increasing in (-1/4, t)
c) increasing in (-t, t)
d) decreasing in (-t,t)
8) The number of distinct real roots of x⁴- 4x³+ 12x²+ x -1=0 is
Paragraph
Let f(x)= (1- x)² sin²x + x² for all x ∈R and let g(x)= ˣ₁∫ {2(t -1)/(t +1) - log t} f(t) dt for all x ∈ (1, ∞)
9) 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,∞)
10) Consider the statements:
P: there exists some x ∈ IR such that f(x)+ 2x = 2(1+ x²)
Q: There exists some x∈ IR such that 2f(x)+ 1= 2x(1+ x)
Then
a) both P and Q are true
b) P is true and Q is false
c) P is false and Q is true
d) both P and Q are false
BOOSTER - E
1) If f and g are differentiable functions in [0,1] satisfying f(0)= 2= g(1), g(0)= 0 and f(1)= 6, then for some c ∈ ]0,1[
a) f'(c)= 2g'(c)
b) 2f'(c)= g'(c)
c) 2f'(c)= 3g'(c)
d) f'(c)= g'(c)
2) Let f: (0, ∞)--> R be given by f(x)= ∫ₑ-(t + 1/t) dt/t at (∞, 1/x)
Then
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) f(2ˣ) is an odd function of x on R.
3) For every pair of continuous functions f, g: [0,1] --> such that max {f(x): x ∈ [0,1]= max{g(x): x∈ [0,1]}
the correct statement/s is/are
a) (f(c))²+ 3f(c)= (g(c))²+ 3g(c) for some c ∈ [0,1]
b) (f(c))²+ f(c)= (g(c))²+ 3g(c) for some c ∈ [0,1]
c) (f(c))²+ 3f(c)= (g(c))²+ g(c) for some c ∈ [0,1]
d) (f(c))²= (g(c))² for some c ∈ [0,1]
MAXIMUM AND MINIMUM
Booster - A
1) A cubic f(x) vanishes at x= -2 and has relative minimum/maximum at x= -1 and x= 1/3.
If ¹₋₁∫ f(x) dx = 14/3, find the cubic f(x). x⅔+ x²- x +2
2) Investigate for the maximum and minimum for the function,
f(x) = ˣ₁∫ [2(t -1)(t -2)³+ 3(t -1)²(t - 2)²] dt. max at x=1; f(1)=0, min at x= 7/5; f(7/5)= - 108/3125
3) Find the greatest and least value for the function,
a) y= x + sin2x, 0≤ x ≤ 2π. Max at x=2π; max value=2π, min at x= 0, min value=0
b) y= 2 cos2x - cos 4x, 0≤ x ≤π. max at x=π/6 and 5π/6 v: 3/2; min at x=π/2, v: -3
4) Suppose f(x) is real valued polynomial function of degree 6 satisfying the following conditions :
a) f has a minimum value at x= 0 and 2.
b) f has maximum value at x= 1
c) for all x, lim ₓ→₀(1/x) log | f(x)/x 1 0
0 1/x 1 = 2
1 0 1/x
Determine f(x). 2x⁶/3 - 12x⁵/5 + 2x⁴
5) Find the maximum perimeter of a triangle on a given bases 'a' and having the given vertical angle α. a(1+ cosec(α/2)(
6) The length of three sides of a trapeziums are equal, each being 10cms. Find the maximum area of such a trapezium. 75√3
7) The plan view of a swimming pool consists of a semicircle of radius R attached to a rectangle of length '2r' and width 's'. If the surface area A of the pool is fixed, for what value of 'r' and 's' the perimeter 'P' of the pool is minimum. r=√(2A/(π+4)), s=√{2A/(π+4)}
8) For a given curved surface of a right circular cone when the volume is maximum, prove that the semi vertical angle is sin ⁻¹(1/√3).
9) Of all the lines tangent to the graph of the curve y= 6/(x²+3), find the equation of the tangent lines minimum and maximum slope. 3x + 4y -9=0; 3x - 4y +9 =0
10) A statue 4 metres high sits on a column 5.8 metres high. How far from the column must a man, whose eye level is 1.6 metres from the ground, stand in order to have the most favourable view of statue. 4√2
11) By the post office regulations, the combined length and girth of a parcel must not exceed 3 metres. Find the volume of the biggest cylindrical(right circular) packet that can be sent by the parcel post. 1/π
12) A running track of 440ft is to be laid out enclosing a football field, the shape of which is a rectangle with semi circle at each end. If the area of the rectangular portion is to be the maximum, find the length of its sides. 110', 70'
13) A window of fixed perimeter (including the base of the arch ) is in the form of a rectangle surmounted by a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted with clean glass. The clear glass transmits three times as much light per square metre as the colored glass does. What is the ratio of the sides of the rectangle so that the window transmits the maximum light ? 6/(6+π)
14) A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost of the material per square foot for the bottom is 15 paise, for the top 25 paise and for the the sides 20 paise . The labour charges for making the box are Rs 3. Find the dimensions of the box when the cost is minimum . 10, 10
15) Find the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y= 12 - x². 32
16) A trapezium ABCD is inscribed into a semicircle of radius l so that the base AD of the trapezium is a diameter and the vertices B and C lie on the circumference. Find the base angle θ of the trapezium ABCD which has the greatest perimeter. 60°
17) if y= (ax + b)/{(x -1)(x -4)} has a turning value at (2,-1). Find a and b and show that the turning value is a maximum. 1,0
18) If r is a real number then find the smallest possible distance from the origin (0,0) to the vertex of the Parabola whose equation is y= x²+ rx +1. √3/2 when r=√2 or -√2
19) A sheet of poster has its area 18m². The margin at the top and bottom are 75cms and at the sides 50 cms. What are the dimension of the poster if the area of the printed space is maximum? 2√3,3√3
20) A paperpendicular is drawn from the centre to a tangent to an ellipse x²/a² + y²/b²= 1. Find the greatest value of the intercept between the point of contact and the foot of the perpendicular. |a - b|
21) Considered the function,
F(x)= ˣ₋₁∫ (t²- t) dt, x ∈ R.
a) Find the x and y intercept of F if the exist. (-1,0),(0,5/6)
b) deriavatives F'(x) and F"(x). (x²- x); 2x -1
c) The intervals on which F is an increasing and the intervals on which F is decreasing. (-∞,0) U (1, ∞); (0,1)
d) Relative maximum and minimum points. (0,5/6); (1,2/3)
e) Any inflexion point. 1/2
22) A beam of rectangular cross section must be sawn from a round log of diameter d. What should the width x and height y of the cross section be for the beam to offer the greatest resistance
a) to compression. x= y= d/√2
b) to bending. d/√3, √(2/3)d
Assume that the compressive strength of a beam is proportional to the area of the cross section and the bending strength is proportional to the product of the width of section by the square of its height.
23) What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle of base 36 ft and altitude 12ft ? Assume that one side of the rectangle lies in the base of the triangle. 6x 18
24) The flower bed is to be in the shape of a circular sector of radius R and Central angle θ. if the area is fixed and perimeter is minimum, find r and θ. √A, 2 radians
25) The circle x²+ y²= 1 cuts the x-axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangle QSR. 4/3√3
BOOSTER - B
1) The mass of a cell culture at time t is given by, M(t)= 3/(1+ 4e⁻ᵗ)
a) Find lim ₜ→∞ M(t) and lim ₜ→∞ M(t). 0,3
b) Show that dM/dt = (1/3) M(3- M).
c) Find the maximum rate of growth of M and also the value of t at which occurs. 3/4, t= log 4
2) Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length l of the median drawn to its lateral side. CosA= 0.8
3) From a fixed point A on the circumference of a circle of radius 'a', let the perpendicular AY fall on the tangent at a point P on the circle, prove that the greatest area which the ∆ APY can have is 3√3(a²/8) sq.units.
4) Given two points A(-2,0) and B(0,4) and a line y= x. Find the coordinates of a point M on this line so that the perimeter of the ∆ AMB is least. (0,0)
5) A given quantity of metal is to be casted into a half cylinder i,e., with rectangular base and semicircular ends. Show that in order that total surface area may be minimum, the ratio of the height of the cylinder to the diameter the semi circular ends is π/(π+2).
6) let α, β be real numbers with 0≤α≤β and f(x)= x²- (α+β)x + αβ such that
¹₋₁∫ f(x) dx = 1. Find the maximum value of ᵅ₀∫ f(x) dx. √6/108
7) Let f(x)= x³ - 3x + a, a ∈ (0,2) has 3 distinct real roots x₁, x₂, x₃, find {x₁} + {x₂} + {x₃}.
Where {.} denote fraction part function. 1
8) For a> 0, find the minimum value of the integral
¹⁾ᵃ₀∫ (a³ + 4x - a⁵x²)eᵃˣ dx. 4 when a=√2
9) Consider the function
f(x)=[√x log x when x > 0
0 for x= 0
a) Find whether f is continuous at x=0 or not. f is continuous at x=0
b) Find the minimum the maximum if they exist. -2/e
c) does f'(0) ? Find lim ₓ→₀ f'(x). Does not exist
d) Find the inflection points of the graph of y= f(x). Pt of information x=1
10) Consider the function y= f(x)= log(1+ sinx) with -2π≤ x ≤ 2π. Find
a) the zeros of f(x). x= -2π,-π, 0, π, 2π
b) inflection points if any on the graph. No inflexion point
c) local maximum and minimum of f(x). Max at x=π/2 & -3π/2 and no min
d) asymptoms of the graph. x=3π/2 and x= -π/2
e) Sketch the graph of f(x) and compute the value of the definite integral
₋∫ f(x) dx at (π/2, -π/2). -π log 2
11) The graph of the derivative f' of a continuous function f is shown with f(0)=0. If
i) f is monotonic increasing in the interval [(a,b) U (c,d) U(e,f)] and decreasing in (p,q) U (r,s).
ii) f has a local minimum at x= x₁ and x = x₂.
iii) f is concave up in (l,m) U (n,t].
iv) f has inflection point at x= k.
v) number of critical points of y= f(x) is 'w'.
Find the value of (a+ b+ c+ d+ e)+ (p+ q+ r+ s) + (l+ m+ n)+ (x₁ + x₂)+ (k + w). 74
12) The graph of the derivative f' of a continuous function f is shown with f(0)=0
i) On what intervals is increasing or decreasing ? l: (1,6)U(8,9) and D in (0,1) U(6,8)
ii) At what values of x does f have a local maximum or minimum? Min at x= 1 and x= 8 , max at x= 6
iii) On what intervals is f concave upward or downward? CU in (0,2) U(3,5) U(7,9) and CD in (2,3) U(5,7)
iv) State the x-cordinate/s of the point/s of inflexion. x= 2,3,5,7
v) Assume that f(0)=0, Sketch a graph of f.
13) Find the set of value of m for the cubic x³- 3x²/2 + 5/2= log₁/₄(m) has distinct solutions. m belongs to (1/32,1/16)
14) Find the positive value of k for the value of definite integral ∫| cosx - Kx| dx at (π/2,0) is minimised. k= (2√2/π) cos(π/2√2)
15) A cylinder is obtained by revolving a rectangle about the x-axis , the base of the rectangle lying on the x-axis and the entire rectangle lying in the region between the curve.
y= x/(x²+1) and the x-axis. Find the maximum possible value of the cylinder. π/4
16) The value of 'a' for which f(x)= x³+ 3(a -7)x²+ 3(a²-9)x -1 have a positive point of maximum lies in the interval (a₁, a₂) U (a₃, a₄). Find the value of a₂+ 11a₃ + 70a₄. 320
17) What is the radius of the smallest circular disc large enough to cover every acute isosceles triangle of a given perimeter L ? L/4
18) Find the magnitude of the vertex angle α of an isosceles triangle of the given area 'A' such that the radius 'r' of the circle inscribed into the triangle is the maximum. π/3
19) The function f(x) defined for all real numbers x has the following properties
i) f(0)=0, f(2)=2 and f'(x)= k(2x - x²)e⁻ˣ for some constant k> 0. Find
a) the intervals on which f is increasing and decreasing and any local maximum or minimum values. Inc in (0,2) , dec in (-∞,0) U(2, ∞), l.mv= 0 lmaxv= 2
b) the intervals on which the graph f is concave down and concave up. Concave up for (-∞, 2-√2) U (2+ √2, ∞) and concave down in (2- √2), (2+ √2)
c) the function f(x) and plot its graph. e²⁻ˣ.x²
20) Use calculus to show the inequality, sin x ≥ 2x/π in 0≤ x ≤π/2.
Use this inequality to show that, cos x ≤ 1- x²/π in 0 ≤ x ≤π/2.
BOOSTER - C
1) The function f(x)= ˣ₋₁ ∫ t(eᵗ -1)(t -2)³(t -3)⁵ dt has a local minimum at x=
a) 0 b) 1 c) 2 d) 3
2) Find the coordinates of all points P on the ellipse x²/a² + y²/b² = 1 which the area of the triangle PON is maximum, where O denotes the origin and N the foot of the perpendicular from O to the tangent at P.
3) Find the normals to the ellipse x¹/9 + y²/4 = 1 which are farthest from its Centre.
4) Let f(x)= [ |x| for 0<|x|≤2
1 for x= 0
Then at x= 0, 'f' has
a) local maximum b) no local maximum c) a local minimum d) no extrimum
5) Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle sides a and b, the right angle of the triangle coinciding with one of the angles of the rectangle .
6) a) Let f(x)= (1+ b²)x²+ 2bx +1 and let m(b) be the minimum value of f(x). As b varies , the range of m(b) is
a) [0,1](0,1/2][1/2,1](0,1]
b) The maximum value of (cosα₁) (cosα₂).....(Cosαₙ) , under the restrictions
0≤ α₁, α₂.....αₙ≤ π/2 and (cotα₁) (cotα₂).....(Cotαₙ) = 1 is
a) 1/2ⁿ⁾² b) 1/2ⁿ c) 1/2n d) 1
7) If a₁, a₂, .....,aₙ are positive real numbers whose product is a fixed number e, the minimum value of a₁ + a₂+ a₃ +.....+ aₙ₋₁ + 2aₙ is
a) n(2e)¹⁾ⁿ b) (n +1)e¹⁾ⁿ c) 2ne¹⁾ⁿ d) (n +1)(2e)¹⁾ⁿ
8) a) Find a point on the curve x²+ 2y²= 6 whose distance from the line x+ y = 7, is minimum .
b) For a circle x²+ y²= r², find the value of r for which the area enclosed by the tangents drawn from the point P(6,8) to the circle and the chord of contact is maximum .
9) a) Let f(x)= x²+ bx⅖+ CX + d, 0< b² < c. Then f
a) is bounded b) has a local Maxima c) has a local minimum d) is strictly increasing
b) prove that sinx + 2x ≥ {3x(x +1)}/π ∀ x∈ [0,π/2]. justify the inequality, if any used).
10) If P(x) be a polynomial of degree 3 satisfying P(-1)= 10, P(1)= -6 and P(x) has maximum at x= -1 and P'(x) has minima at x= 1. Find the distancebetween the local maximum and local minimum of the curve.
11) a) 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 Maxima at x=0, then
a) the distance between (-1,2) and (a, f(a)), where x= a is the point of local minimum is 2√5.
b) f(x) is increaseing for x ∈(1,2√5]
c) f(x) has local minimum at x= 1
d) the value of f(0)= 5
b) f(x)= eˣ 0≤ x ≤ 1
2 - eˣ⁻¹ 1< x≤2 and g(x)= ˣ₀∫ f(t) dt, x∈ [1,3] then g(x) has
x - e 2< x ≤3
a) local maximum at x= 1+ log 2 and local minimum at x= e
b) local Maxima at x= 1 and local minimum at x= 2
c) no local Maxima d) no local minimum
c) If f(x) is twice differentiable function such that f(a) = 0, f(b)=2, f(c)= -1, f(d)= 2, f(e)= 0,
Where a< b < c < d< e, then find the minimum number of zeros of g(a)= (f'(x))² + f(x). f"(z) in the interval [a,e].
12) a) The total number of local Maxima and local minimum of the function
f(x)= (2+ x)³, -3< x ≤-1
³√x², -1< x < 2 is
a) 0 b) 1 c) 2 d) 3
b) Comprehension:
Considere the function f: (-∞, ∞) --> (-∞,∞) defined by
f(x)= (x²- ax+1)/(x²+ ax +1), 0< a<2
i) Which of the following is true?
a) (2+ a)½ 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.
ii) 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 and nor a local minimum at x= 1.
d) f(x) is decreasing on (-1,1) but has neither a local maximum and nor a local minimum at x= 1.
iii) Let g(x) = ∫ f'(t)/(1+ t²) dt at (eˣ, 0)
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 on (-∞,0) and (0,∞)
13) The maximum value of the function f(x)= 2x³- 15x² + 36x - 48 on the set
A={x | x²+ 20≤ 9x} is
14) Let f g and h be real valued functions defined on the interval [0,1] by ₑx² + ₑ-x², g(x)=ₓₑx² + ₑ-x² and h(x)= x²ₑx² + ₑ-x², if a,b and c denote, respectively, the absolute maximum of f, g and h on [0,1], then
a) a= b and c≠ b
b) a= c and a≠ b
c) a≠ b and c≠ b
d) a= b= c
15) Let f be a function defined on R(the set of all real numbers ) such that f'(x)= 2010(x - 2009)(x - 2010)²(x - 2011)³(x - 2012)⁴, for all x ∈R. If g is a function defined on R with values in the interval (0,∞) such that f(x)= log (g(x)), for all x∈R, then the number of points in R at which g has a local maximum is
16) If f(x)= ˣ₀∫ (ₑt²) (t -2)(t -3) dt for all x ∈(0, ∞), 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 maximum at x= 3
17) Let f: IR ---> IR be defined as f(x)= |x|+ |x²- 1|. The total number of points at which f attains either a local maximum or a local minimum is
18) Let p(x) be a real polynomial of least degree which has a local maximum at x= 1 and a local minimum at x= 3. If p(1)= 6 and p(3)= 2, then P'(0) is
Comprehension
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 f"(x)- 2 f'(x)+ f(x)≥ eˣ, x ∈ [ 0 ,1].
19) if the function e⁻ˣ f(x) assumes its minimum in the interval [0,1] at 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
20 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
21) The function f(x)= 2|x|+ |x +2| - ||x +2| - 2 |x|| has a local minimum or a local maximum at x=
a) -2 b) -2/3 c) 2 d) 2/3
22) A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:15 is converted unto an innverted 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. Then the lengths of the sides of the rectangular sheets are :
a) 24 b) 32 c) 45 d) 60
23) If x= -1 and x= 2 are extreme points of f(x)= α log |x|+ βx²+ x then
a) α = 2, β= 1/2 b) α = -6, β= 1/2 c) α = -6, β= -1/2 d) α = 2, β= -1/2
1bd 2) ±a²/√(a²+ b²), ± b²/√(a²+ b²) 3) ±√3x ± √2 y= √5 4a
5) 2ab 6)a) d b) a
7a
8a) (2,1) b) 5
9) a) d 10) 4√64 11) a) b,c b) 6 solutions. 12a) c b) i) a ii) a iii) b
13) 7 14) d 15) 1 16) ABCD 17) 5 18) 9 19) c 20) d 21) ab 22) ac 23) d
∈ ∞ ⁻¹ α ∀
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