Question(s) from Search: IIT

Let y(x) be the solution of the differential equation $\left(\mathrm{xlnx}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+y=2\mathrm{xlnx},\left(x\ge 1\right)$

. Given that y = 1 when x = 1, then y(e) is equal to

a) e

b) 0

c) 2

d) 2e

If C_r stands for  then the sum of the series
 
where n is a positive integer, is equal to

a) 0

b) (−)^n/2(n + 1)

c) (−)^n/2 (n + 2)

d) None of these

Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x  ℝ, f(x + T) = f(x). If  then the value of  is

a)

b)

c) 3I

d) 6I

One or more than one correct options

If y(x) satisfies the differential equation y′ − ytanx = 2xsecx and y(0) = 0, then

a) $y\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{8\sqrt{2}}$

b) $y^{\prime }\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{18}$

c) $y\left(\frac{\pi }{3}\right)=\frac{{\pi }^2}{9}$

d) $y^{\prime }\left(\frac{\pi }{3}\right)=\frac{4\pi }{3}+\frac{{2\pi }^2}{3\sqrt{3}}$

The sum if p > q is maximum when m is

a) 5

b) 10

c) 15

d) 20

At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by $\frac{\mathrm{dP}}{\mathrm{dx}}=100-12\sqrt{x}$

. If the firm employs 25 more workers then the new level of production of items is

a) 2500

b) 3000

c) 3500

d) 4500

If a, b, c; u, v, w are complex numbers representing the vertices of two triangles such that c = (1 − r)a + rb, w = (1 − r)u + rv where r is a complex number. The two triangles

a) have the same area

b) are similar

c) are congruent

d) none of these

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

If x is not an integral multiple of 2π use mathematical induction to prove that
 

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

Prove by induction that
P_n = Aαⁿ + Bβⁿ for all n ≥ 1
Where α and β are roots of the quadratic equation
x² – (1 – P) x – P (1 – P) = 0,
P₁ = 1, P₂ = 1 – P², .  .  .,
P_n = (1 – P) P_{n – 1} + P (1 – P) P_{n – 2}  n ≥ 3,
and ,

Let P be a point on the parabola y² = 8x which is at a minimum distance from the centre C of the circle x² + (y + 6)² = 1. Then the equation of the circle passing through C and having its centre at P is

a) x² + y² – 4x + 8y + 12 = 0

b) x² + y² –x + 4y − 12 = 0

c) x² + y² –x + 2y − 24 = 0

d) x² + y² – 4x + 9y + 18 = 0

Let  then points where f (x) is not differentiable is (are)

a) 0

b) 1

c) ± 1

d) 0, ± 1