Question(s) from Search: IIT

Find the derivative with respect to x of the function

 at x =

a)

b)

c)

d)

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

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

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

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, ∞)

Given
 
 
Prove that
 

Using mathematical induction, prove that

 for n > 1

Let p ≥ 3 be an integer and α, β be the roots of x² – (p + 1) x + 1 = 0. Using mathematical induction show that αⁿ + βⁿ
i) is an integer
ii) and is not divisible by p.

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

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 ,

Find all solutions of

a)

b)

c)

d)

In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = ………….

a) a+b

b) b+c

c) c+a

d) a+b+c

Show that the value of  wherever defined

a) always lies between  and 3

b) never lies between  and 3

c) depends on the value of x

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

Let a > 0, b > 0, c > 0 then both the roots of the equation  

a) are real and positive

b) have negative real parts

c) have positive real parts

d) none of these

If sinA sinB sinC + cosA cosB = 1then the value of sinC is

A plane passes through (1, −2, 1) and is perpendicular to the two planes  and  The distance of the plane from the point (1, 2, 2) is.

The number of ordered pairs satisfying the equations
 is

a) 4

b) 2

c) 0

d) 1

Show that the sum of the first n terms of the series
1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .  .  .
is  when n is even, and  when n is odd.

Let U₁ = 1, U₂ = 1, U_{n + 2} = U_{n + 1} + U_n, n > 1. Use mathematical induction to show that
 
for all integers n > 1

Compute the area of the region bounded by the curves
y = exlnx and

a)

b)

c)

d)

What normal to the curve y = x² forms the shortest normal?

a)

b)

c)

d) y = x + 1

The circle x² + y² = 1 cuts the X–axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the X–axis and the line segment PQ at S. Find the maximum area of ΔQRS.

a)

b)

c)

d)

From a point A common tangents are drawn to the circle  and the parabola . Find the area of the quadrilateral formed by the common tangents drawn from A and the chords of contact of the circle and the parabola.

Let
where a is a positive constant. Find the interval in which  is increasing.

a)

b)

c)

d)