Question(s) from Search: IIT

If AB is a diameter of a circle and C is any point on the circumference of the circle then

a) The area of ΔABC is maximum when it is isosceles

b) The area of ΔABC is minimum when it is isosceles

c) The perimeter of ΔABC is minimum when it is isosceles

d) None of these

Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g (x) = |f (x)| for all x. Then g is

a) Onto if f is onto

b) One to one if f is one to one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

If n is a positive integer and 0 ≤ v < π then show that

A tangent PT is drawn to the circle x² + y² = 4 at the point $P\left(\sqrt{3},1\right)$

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)² + y² = 1A possible equation of L is

a) $x-\sqrt{3}y=1$

b) $x+\sqrt{3}y=1$

c) $x-\sqrt{3}y=-1$

d) $x+\sqrt{3}y=5$

Let 0 < A_i < π for i = 1, 2, .  .  . n. Use mathematical induction to prove that
 
where n ≥ 1 is a natural number.

The centre of those circles which touch the circle x² + y² – 8x – 8y = 0, externally and also touch the X- axis, lie on

a) A circle

b) An ellipse which is not a circle

c) A hyperbola

d) A parabola

Solve

 for every 0 < α, β < 2.

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio of 1 : 3. Then the locus of P is

a) x² = y

b) y² = 2x

c) y² = x

d) x² = 2y

The value of  where x > 0 is

a) 0

b) – 1

c) 1

d) 2

The value of

a) 5050

b) 5051

c) 100

d) 101

Let the curve C be the mirror image of the parabola y² = 4x with respect to the line x + y + 4 = 0. If A and B are points of intersection of C with the line y = −5 then the distance between A and B is . . .?

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.