Question(s) from Search: IIT

For a positive integer n, define
 then

a) a(100) ≤ 100

b) a(100) > 100

c) a(200) ≤ 100

d) a(200) > 100

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]Which of the following is true?

a) $f\left(x\right)<\infty$

b) $\frac{-1}{2}<f\left(x\right)<\frac{1}{2}$

c) $\frac{-1}{4}<f\left(x\right)<1$

d) $-\infty <f\left(x\right)<0$

If ω(≠1) is a cube root of unity and  then A and B are respectively

a) 0, 1

b) 1, 1

c) 1, 0

d) – 1, 1

If (1 + x)ⁿ = C₀ + C₁x + C₂x² + .  .  . + C_nxⁿ, then show that the sum of the products of the C_j’s is taken two at a time represented by
 is equal to

Let a, b, c and d be non-zero real numbers. If the point of intersection of lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lie in the fourth quadrants and is equidistant from the two axes, then

a) 2bc – 3ad = 0

b) 2bc + 3ad = 0

c) 2ad – 3bc = 0

d) 3bc + 2ad = 0

One or more than one correct option

Let α, λ, μ ∈ ℝ. Consider the system of linear equations αx + 2y = λ 3x – 2y = μWhich of the following statements is/are correct?

a) If α = −3, then the system has infinitely many solutions for all values of λ and μ

b) If α ≠ −3, then the system of equations has a unique solution for all values of λ and μ

c) If λ + μ = 0, then the system has infinitely many solutions for α = −3

d) If λ + μ ≠ 0, then the system has no solution for α = −3

Let  and f = R – [R] where [ ] denotes the greatest integer function. Prove that Rf = 4^{2n + 4}

One or more than one correct option

Circle(s) touching X – axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$

on Y – axis is/are

a) x² + y² – 6x + 8y + 9 = 0

b) x² + y² – 6x + 7y + 9 = 0

c) x² + y² – 6x − 8y + 9 = 0

d) x² + y² – 6x − 7y + 9 = 0

Using induction or otherwise, prove that for any non-negative integers m, n, r and k
 

Let V be the volume of the parallelepiped formed by the vectors  and . If a_r, b_r, c_r where r = 1, 2, 3 are non-negative real numbers and , show that V ≤ L³

One or more than one correct option

A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)² + y² = 16 and x² + y² = 1, then

a) Radius of S is 8

b) Radius of S is 7

c) Centre of S is (−7, 1)

d) Centre of S is (−8, 1)

The locus of the midpoint of a chord of the circle  which subtend a right angle at the origin is

a)

b)

c)

d)

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