Question(s) from Search: IIT

One or more than one correct options

The value(s) of $\underset{0}{\overset{1}{\int }}\frac{x^4{(1-x)}^4}{{1+x}^2}\mathrm{dx}$

is (are)

a) $\frac{22}{7}-\pi$

b) $\frac{2}{105}$

c) $0$

d) $\frac{71}{15}-\frac{3\pi }{2}$

 =

a) True

b) False

For non-zero vectors a, b, c,  holds if and only if

a) a . b = 0, b . c = 0

b) b . c = 0, c . a = 0

c) c . a = 0, a . b = 0

d) a . b = 0, b . c = 0, c . a = 0

$\underset{-2}{\overset{2}{\int }}\frac{{3x}^2}{1+e^x}\mathrm{dx}$

equals

a) 8

b) 2

c) 4

d) 0

The value of $\underset{0}{\overset{1}{\int }}{4x}^3\left[\frac{d^2}{{\mathrm{dx}}^2}{\left(1-x^2\right)}^5\right]\mathrm{dx}$

is

a) 4

b) 0

c) 2

d) 6

Let f be a non-negative function defined on the interval [0, 1]. If $\underset{0}{\overset{x}{\int }}\sqrt{1-{\left(f^{\prime }(t)\right)}^2}\mathrm{dt}=\underset{0}{\overset{x}{\int }}f\left(t\right)\mathrm{dt},0\le x\le 1$

and f(0) = 0, then

a) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

b) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)>\frac{1}{3}$

c) $f\left(\frac{1}{2}\right)<\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

d) $f\left(\frac{1}{2}\right)>\frac{1}{2}\wedge f\left(\frac{1}{3}\right)<\frac{1}{3}$

(One or more correct answers)
If E and F are independent events such that 0 < P (E) < 1 and 0 < P (F) < 1 then

a) E and F are mutually exclusive

b) E and  are independent

c)  are independent

d)

Match the following
Let  

Let p be the first of the n Arithmetic Means between two numbers and q be the first of n Harmonic Means between the same numbers. Then show that q does not lie between p and

a) – 1

b) 2

c) 1 + e⁻¹

d) None of these

One or more than one correct answer

Let P and Q be distinct points on the parabola y² = 2x such that the circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of triangle OPQ is $3\sqrt{2}$

then which of the following is/are the coordinates of P?

a) $\left(\mathrm{4,}2\sqrt{2}\right)$

b) $\left(\mathrm{9,}3\sqrt{2}\right)$

c) $\left(\frac{1}{4},\frac{1}{\sqrt{2}}\right)$

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

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