Question(s) from Search: IIT

The total number of local maximum and minimum of the function
is

a) 0

b) 1

c) 2

d) 3

The area enclosed by the curve y = sinx + cosx and y = |cosx – sinx| over the interval $\left[\mathrm{0,}\frac{\pi }{2}\right]$

is

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

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

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

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

If  and b_n = 1 – a_n then find the least natural number n₀ such that b_n > a_n for all n ≥ n₀

If  are unit coplanar vectors then the scalar triple product  

a) 0

b) 1

c)

d)

One or more than one correct option

If the line x = α divides the area of the region R = {(x, y) ∈ ℝ² : x³ ≤ y ≤ x, 0 ≤ x ≤ 1 into two equal parts then

a) ${2\alpha }^4-{4\alpha }^2+1=0$

b) ${\alpha }^4+{4\alpha }^2-1=0$

c) $\frac{1}{2}<\alpha <1$

d) $0<\alpha <\frac{1}{2}$

The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the Arithmetic mean of
 
 

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{(k-1)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{\mathrm{k\pi }}{6}\right)}$

a) $3-\sqrt{3}$

b) $2\left(3-\sqrt{3}\right)$

c) $2\left(\sqrt{3}-1\right)$

d) $2\left(2+\sqrt{3}\right)$

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

A ray of light along $x+\sqrt{3}y=\sqrt{3}$

gets reflected upon reaching X- axis, the equation of the reflected ray is

a) $y=x+\sqrt{3}$

b) $\sqrt{3}y=x-\sqrt{3}$

c) $y=\sqrt{3}x-\sqrt{3}$

d) $\sqrt{3}y=x-1$

If  and  where 0 < x ≤1, then in this interval

a) Both f (x) and g (x) are increasing functions

b) Both f (x) and g (x) are decreasing functions

c) f (x) is an increasing function

d) g (x) is an increasing function

The number of common tangents to the circles x² + y² – 4x − 6y – 12 = 0 and x² + y² + 6x + 18y + 26 = 0 is

a) 1

b) 2

c) 3

d) 4

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.

The function  is not differentiable at

a) – 1

b) 0

c) 1

d) 2

One or more than one correct option

Let RS be a diameter of the circle x² + y² = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) $\left(\frac{1}{3},\frac{1}{\sqrt{3}}\right)$

b) $\left({\frac{1}{4},}^{}\frac{1}{2}\right)$

c) $\left(\frac{1}{3},-\frac{1}{3}\right)$

d) $\left(\frac{1}{4},-\frac{1}{2}\right)$