Question(s) from Search: IIT

In [0, 1], Lagrange’s Mean Value theorem is not applicable to

a)

b)

c)

d)

The area bounded by the angle bisectors of the lines

x² – y² + 2y = 1 and the line x + y = 3 is

a) 2

b) 3

c) 4

d) 6

Multiple choice

Let  and

Then g(x) has

a) Local maximum at x = 1 + ln2 and local minima at x = e

b) Local maximum at x = 1 and local minima at x = 2

c) No local maximas

d) No local minimas

For all x in [0, 1], let the second derivative  of a function f(x) exists and satisfies . If f(0) = f(1) then for all x ε [0, 1]

a)  

b)  

c) None of these

If  for all positive x where a > 0 and b > 0 then

a) 9ab² ≥ 4c³

b) 27ab² ≥ 4c³

c) 9ab² ≤ 4c³

d) 27ab² ≤ 4c³

Let ABCD be a square with side of length 2 units. C₂ is the circle through the vertices A, B, C, D and C₁ is the circle touching all the sides of the square ABCD. L is a line through A.

If P is a point on C₁ and Q is another point on C₂, then  is equal to

a) 0.75

b) 1.25

c) 1

d) 0.5

Let . Find the intervals in which λ should lie in order that f(x) has exactly one minimum and exactly one maximum.

a)

b)

c)

d)

Consider a circle with centre lying on the focus of the parabola  such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

a) or

b)

c)

d)

A two metre long object is fired vertically upwards from the mid-point of two locations A and B, 8 metres apart. The speed of the object after t seconds is given by  metres per second. Let α and β be the angles subtended by the objects A and B respectively after one and two seconds. Find the value of cos(α − β).

a)

b)

c)

d)

Investigate for maxima and minima the function
 

a) Local maximum at x = 1, 7/5, 2

b) Local minimum at x = 1, 7/5, 2

c) Local maximum at x = 1, 2. Local minimum at x =  7/5

d) Local maximum at x = 1. Local minimum at x =  7/5

A window of perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square meter as the coloured glass. What is the ratio for the sides of the rectangle so that the window transmits the maximum light?

a)

b)

c)

d)

Let

Find all possible values of b such that f(x) has the smallest value at x = 1.

a) (−2, ∞)

b) (−2, −1)

c) (1, ∞)

d) (−2, −1) ∪ (1, ∞)

Use mathematical induction for
 
to prove that
I_m = mπ, m = 0, 1, 2 .  .  .  .

Determine the points of maxima and minima of the function
  where b ≥ 0 is a constant.

a) Minima at x = x₁, maxima at x = x₂

b) Minima at x = x₂, maxima at x = x₁

c) Minima at x = x₁, x₂, no maxima

d) Maxima at x =x₁, x₂, no minima

where x_{1 =}   and x₂ =   

Consider the circle x² + y² = 9 and the parabola y² = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The radius of the circum circle of △PRS is

a) 5

b)

c) 3

d)

One or more than one correct option

Consider the family of circles whose centre lies on the straight line y = x. If the family of circles is represented by the differential equation Py′′ + Qy′ + 1 = 0 where P, Q are functions of x, y and y′ $\left(\mathrm{where}{y}^{\prime }=\frac{\mathrm{dy}}{\mathrm{dx}},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}\right)$

, then which of the following statements is/are true?

a) P = y + x

b) P = y – x

c) P + Q = 1 – x + y + y′ + (y′)²

d) P − Q = x + y − y′ − (y′)²

Let $f\mathrm{:}\left[\frac{1}{2},1\right]\to R$

(the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\prime }\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$ . Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)\mathrm{dx}$ lies in the interval

a) $\left(\mathrm{2e-1,}2e\right)$

b) $\left(e-\mathrm{1,}2e-1\right)$

c) $\left(\frac{e-1}{2},e-1\right)$

d) $\left(\mathrm{0,}\frac{e-1}{2}\right)$

Let the population of rabbits arriving at time t be governed by the differential equation $\frac{dp(t)}{\mathrm{dt}}=\frac{1}{2}p\left(t\right)-200$

. If p(0) = 100, then p(t) is equal to

a) 400 – 300e^t/2

b) 300 – 200e^−t/2

c) 600 – 500e^t/2

d) 400 – 300e^−t/2

Let f:[0, 1] → ℝ (the set all real numbers)be a function. Suppose the function is twice differentiable, f(0) = f(1) = 0 and satisfiesf′′(x) – 2f′(x) + f(x) ≥ e^x, x ∈ [0, 1]If the function e^−x f(x) assumes its minimum in the interval [0, 1] at $x=\frac{1}{4}$

then which of the following is true?

a) $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$

b) $f^{\prime }\left(x\right)>f\left(x\right),0<x<\frac{1}{4}$

c) $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$

d) $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 square units. Then the orthocentre of the triangle is at the point

a) $\left(2,-\frac{1}{2}\right)$

b) $\left(\mathrm{1,}\frac{3}{4}\right)$

c) $\left(1,-\frac{3}{4}\right)$

d) $\left(\mathrm{2,}\frac{1}{2}\right)$

A straight line L through the point (3, −2) is inclined at an angle of 60° to the line $\sqrt{3}x+y=1$

. If the line L also intersects the X- axis then the equation of L is

a) $y+\sqrt{3}x+2-3\sqrt{3}=0$

b) $y-\sqrt{3}x+2+3\sqrt{3}=0$

c) $\sqrt{3}y-x+3+2\sqrt{3}=0$

d) $\sqrt{3}y+x-3+2\sqrt{3}=0$

The sides of a rhombus are along the lines x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at (−1, −2) then which one of the following is a vertex of the rhombus?

a) $(-\mathrm{3,}-9)$

b) $(-\mathrm{3,}-8)$

c) $\left(\frac{1}{3},-\frac{8}{3}\right)$

d) $\left(\frac{-10}{3},\frac{-7}{3}\right)$

The C be a circle with the centre at (1, 1) and radius 1. If T is the circle centred at (0, k) passing through origin and touches the circle C externally, then the radius of T is equal to

a) $\frac{\sqrt{3}}{\sqrt{2}}$

b) $\frac{\sqrt{3}}{2}$

c) $\frac{1}{2}$

d) $\frac{1}{4}$

One or more than one correct option

The circle C₁ : x² + y² = 3 with centre at O intersect the parabola x² = 2y at the point P in the first quadrant. Let the tangent to the circle C₁ at P touches other two circles C₂ and C₃ at R₂ and R₃ respectively. Suppose C₂ and C₃ have equal radii $2\sqrt{3}$

and centres Q₂ and Q₃ respectively. If Q₂ and Q₃ lie on the Y- axis, then

a) $Q_2{Q}_3=12$

b) $R_2{R}_3=4\sqrt{6}$

c) $\mathrm{area}\mathrm{of}△{}_2{R}_3\mathrm{is}R\sqrt{2}$

d) $\mathrm{area}\mathrm{of}△{\mathrm{PQ}}_2{Q}_3\mathrm{is}4\sqrt{2}$

The circle passing through the point (−1, 0) and touching the Y – axis at (0, 2) also passes through the point

a) $\left(\frac{-3}{2},0\right)$

b) $\left(\frac{-5}{2},0\right)$

c) $\left(\frac{-3}{2},\frac{5}{2}\right)$

d) $(-\mathrm{4,}0)$