Question(s) from Search: IIT

Let S be the focus of the parabola y² = 8x and PQ be the common chord of the circle x² + y² – 2x – 4y = 0 and the given parabola. The area of △QPS is

a) 2 sq. units

b) 4 sq. units

c) 6 sq. units

d) 8 sq. units

Consider the parabola y² = 8x. Let △₁ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$

on the parabola and △₂ be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then $\frac{{△}_1}{{△}_2}$ is

Multiple choices

Let g (x) = x f (x), where   at x = 0

a) g is  but  is not continuous

b) g is  while f is not

c) f and g are both differentiable

d) g is  and  is continuous

A five digit number divisible by 3 is formed using the numerals 0, 1, 2, 3, 4, and 5 without repetition. Total number of ways this can be done is

a) At least 30

b) At most 20

c) Exactly 25

d) None of these

A rectangle with sides (2m – 1) and (2n – 1) is divided into squares of unit length by drawing parallel lines. Then the number of rectangles possible with odd side lengths is

a) mn (m + 1)(n + 1)

b)

c)

d)

If the normal to the curve y = f(x) at the point (3, 4) makes an angle  with the positive X–axis then

a) – 1

b)

c)

d) 1

A circle passes through points A, B and C with the line segment AC as its diameter. A line passing through A intersects the chord BC at D inside the circle. If ∠DAB and ∠CAB are α and β respectively and the distance between the point A and the midpoint of the line segment DC is d, prove that the area of the circle is
 

Domain of definition of the function f (x) =  for real valued x is

a)

b)

c)

d)

Find the values of a and b, so that the functions

Is continuous for 0 ≤ x ≤ π

a)

b)

c)

d)

C₁ and C₂ are two concentric circles, the radius of C₂ being twice of C₁ . From a point on C₂ tangents PA and PB are drawn to C₁. Prove that the centroid of ΔPAB lies on C₁.

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

a)

b)

c)

d)

Let α ε ℝ, then a function f : ℝ → ℝ is differentiable at α if and only if there is a function g : ℝ → ℝ which is continuous at α and satisfies f(x) – f(α) = g(x) (x – α) for all x ε ℝ.

a) True

b) False

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

If two functions f and g satisfy the given conditions  x, y ε ℝ, f(x – y) = f(x)g(y) – f(y)g(x) and g(x – y) = g(x) . g(y) + f(x) . f(y).

If the RHD at x = 0 exists for f(x) then find the derivative of g(x) at x = 0.

Let

be a real valued function. The set of points where f(x) is not differentiable are

a) {0}

b) {1}

c) {0, 1}

d) {∅}

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

Match the following

Let the function defined in column 1 have domain  and range ()

Let f(x) = [x] where [.] denotes the greatest integer function. Then the domain of f is .  .  .  ., points of discontinuity of f are .  .  .  .

a) ∀ x ε I

b) ∀ x ε I − {0}

c) ∀ x ε I – {0, 1}

d) ∀ x ε I – {0, 1, 2}

PQ and PR are two infinite rays, QAR is an arc.

		U

Points lying in the shaded region excluding the boundary satisfies

a)   |z + 1| > 2; |arg(z + 1)| <

b)   |z + 1| < 2; |arg(z + 1)| <

c)  

d)  

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

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

The positive value of k for which  has only one root is

a)

b) 1

c) e

d) ln2

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)