Question(s) from Search: IIT

Prove that

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.

One or more than one correct option

Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6) then L is given by

a) y – x + 3 = 0

b) y + 3x – 33 = 0

c) y + x – 15 = 0

d) y – 2x + 12 = 0

Let ABCD be a quadrilateral with area 18 with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. A circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

a) 3

b) 2

c)

d) 1

Multiple choices

The function f (x) = max  is

a) continuous at all points

b) differentiable at all points

c) differentiable at all points except x = 1 and x =

d) continuous at all points except at x=1 and x=-1 where it is discontinuous

Find the equation of the circle passing through ( 4, 3) and touching the lines x + y = 4 and .

A circle touches the line y = x at a point P such that  , where O is the origin. The circle contains the point  in its interior and the length of its chord on the line  is  . Determine its equation.