Question(s) from Search: IIT

If , i = 1, 2, 3 are polynomials in x such that  and

F(x) =  
then (x) at x = a is equal to

a) – 1

b) 0

c) 1

d) 2

If  then f (x) increases in

a) (−2, 2)

b) No value of x

c) (0, ∞)

d) (−∞, 0)

A curve passes through the point $\left(\mathrm{1,}\frac{\pi }{6}\right)$

. Let the slope of the curve at each point (x, y) is $\frac{y}{x}+\sec \left(\frac{y}{x}\right)$ , x > 0. Then the equation of the curve is

a) $\sin \left(\frac{y}{x}\right)=\mathrm{lnx}+\frac{1}{2}$

b) $\mathrm{cosec}\left(\frac{y}{x}\right)=lnx+2$

c) $\sec \left(\frac{2y}{x}\right)=\mathrm{tanx}+2$

d) $\cos \frac{2y}{x}=lnx+\frac{1}{2}$

The points  in the complex plane are the vertices of a parallelogram if and only if

a)

b)

c)

d) None of these

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}

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