Question(s) from Search: IIT

The question contains Statement – 1(assertion) and Statement – 2 (reason). 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  because

Statement 2: f (t) = f[t, t + 2π] for each real t

a) Statement 1 and 2 are true. Statement 2 is a correct explanation of Statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation of Statement 1.

c) Statement 1 is true and Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

Let f(x) = (1 – x)² sin²x + x² and $g\left(x\right)=\underset{1}{\overset{x}{\int }}\left(\frac{2(t-1)}{t+1}-\mathrm{lnt}\right)f\left(t\right)\mathrm{dt}$

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, ∞)

Use mathematical induction to prove: If n is an odd positive integer
then  is divisible by 24.

Let PS is the median of the triangle with vertices P(2, 2), Q(6, −1) and R(7, 3), then the equation of the line passing through (1, −1) and parallel to PS is

a) 4x – 7y – 11 = 0

b) 2x + 9y + 7 = 0

c) 4x + 7y + 3 = 0

d) 2x – 9y – 11 = 0

One or more than one correct option

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $2\sqrt{2}$

, then

a) a + b – c > 0

b) a − b + c < 0

c) a − b + c > 0

d) a + b – c < 0

Using mathematical induction, prove that
 
m, n, k are positive integers and  for p < q

If one of the diameters of the circle, given by the equation x² + y² – 4x + 6y – 12 = 0 is a chord of a circle S whose centre is at (−3, 2), then the radius of S is

a) $5\sqrt{2}$

b) $5\sqrt{3}$

c) $5$

d) $10$

If  for all k ≥ n then show that

The function  (where [y] is the greatest integer less than or equal to y) is discontinuous at

a) All integers

b) All integers except 0 and 1

c) All integers except 0

d) All integers except 1

If  are three non-coplanar unit vectors and α, β, γ are the angles between  , v and w, w and u respectively and x, y and z are unit vectors along the bisector of the angles α, β, γ respectively. Prove that
  

For how many values of p, the circlex² + y² + 2x + 4y – p = 0 and the coordinate axis have exactly three common points

a) 0

b) 1

c) 2

d) 3

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 common tangent to the circles is

a) x = 4

b) y = 2

c) $x+\sqrt{3}y=4$

d) $x+2\sqrt{2}y=6$

The integer n, for which  is a finite

non–zero number is

a) 1

b) 2

c) 3

d) 4

The locus of the middle points of the chord of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x² + y² = 9 is

a) 20(x² + y²) – 36x + 45y = 0

b) 20(x² + y²) + 36x − 45y = 0

c) 36(x² + y²) – 20x + 45y = 0

d) 36(x² + y²) + 20x − 45y = 0

Let  be a regular hexagon in a circle of unit radius. Then the product of the length of the segments  ,  and  is

a)

b)

c) 3

d)

f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then

a) there exists at least one x  (1, 2) such that

b) there exists at least one x  (2, 3) such that

c)

d) there exists at least one x  (1, 3) such that

The radius of a circle having minimum area which touches the curve y = 4 – x² and the line y = |x| is

a) $2\sqrt{2}$

b) $2\left(\sqrt{2}-1\right)$

c) $4\left(\sqrt{2}-1\right)$

d) $4\left(\sqrt{2}+1\right)$

Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is

a) A parabola

b) A circle

c) An ellipse

d) A pairing straight line

Given a circle 2x² + 2y² = 5 and a parabola $y^2=4\sqrt{5}x$

Statement 1: An equation of a common tangent to the curves is $y=x+\sqrt{5}$ Statement 2: If the line $y=\mathrm{mx}+\frac{\sqrt{5}}{m}(m\ne 0)$ is the common tangent then m satisfies m⁴ – 3m² + 2 = 0

a) Statement 1 is correct. Statement 2 is correct. Statement 2 is a correct explanation for statement 1

b) Statement 1 is correct. Statement 2 is correct. Statement 2 is not a correct explanation for statement 1

c) Statement 1 is correct. Statement 2 is incorrect.

d) Statement 1 is incorrect. Statement 2 is correct.

Find all solutions of

a)

b)

c)

d)

Multiple choices

Which of the following functions are continuous on (0, π)

a) tanx

b)

c)

d)

One or more than one correct option

If the normals of the parabola y² = 4x drawn at the end points of the latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² then the value of r² is

a) 4

b) 1

c) 2

d) 0

Multiple choices

Let  for every real number x then

a) h (x) is continuous for all x

b) h is differentiable for all x

c)  for all x > 1

d) h is not differentiable for two values of x

Number of divisors of the form 4n + 2(n ≥ 0) of integer 240 is

a) 4

b) 8

c) 10

d) 3

The smallest positive root of the equation tan x – x = 0 lies in

a)

b)

c)

d)

e) None of these