Question(s) from Search: IIT

A hemispherical tank of radius 2 meters is initially full of water and has an outlet of 12cm² cross section area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law  where g(t) and h(t) are respectively the velocity of the flow through the outlet and the height of the water level above the outlet at the time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: Form a differential equation by relating the decrease of water level to the outflow).

a)

b)

c)

d)

Let P be a point on the ellipse . Let the line parallel to Y–axis passing through P meets the circle  at the point Q such that P and Q are on the same side of the X–axis. For two positive real numbers r and s find the locus of the point R on PQ such that PˆR : RˆQ = r : s and P varies over the ellipse.

Find the area bounded by the curves
x² = y, x² = − y and y² = 4x – 3

a) 1

b) 3

c) 1/3

d) 1/9

Let E = {1, 2, 3, 4} and F = {1, 2}, then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

For a twice differentiable function f(x), g(x) is defined as  If for a < b < c < d < e, f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2, f(e) = 0 then find the maximum number of zeros of g(x).

a) 1

b) 2

c) 3

d) 6

Find the equation of the normal to the curve

The larger of cos (lnθ) and ln (cosθ) if  is

a) cos(lnθ)

b) ln(cosθ)

The angle between the pair of tangents from a point P to the parabola y² = 4ax is 45°. Show that the locus of the point P is a hyperbola.

The integral $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}\mathrm{dx}$

is equal to

a) $\pi -4$

b) $\frac{2\pi }{3}-4-4\sqrt{3}$

c) $4\sqrt{3}-4$

d) $4\sqrt{3}-4-\frac{\pi }{3}$

A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw is

a)

b)

c)

d)

Let F : ℝ → ℝ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ε (1, 3). Let f(x) = x F(x) for all x ε ℝ.The correct statement(s) is/are

a) f′(1) < 0

b) f(2) < 0

c) f′(x) ≠ 0 for every x ε (1, 3)

d) f′(x) = 0 for some x ε (1, 3)

Let A, B , C be three mutually independent events. Consider the two statements S₁ and S₂

S₁ : A and B ∪ Care independent

S₂  : A and B ∩ C are independent. Then

a) Both S₁ and S₂ are true

b) Only S₁ is true

c) Only S₂ is true

d) Neither S₁ nor S₂ is true

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The point of contacts of C with its sides PQ, QR and RP are D, E, F respectively. The line PQ is given by  and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ. Equations of lines QR and RP are

a)

b)

c)

d)

Let f(x) = 7tan⁸x + 7tan⁶x – 3tan⁴x – 3tan²x for all $x\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$

Then the correct expression(s) is (are)

a) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\mathrm{xf}\left(x\right)\mathrm{dx}=\frac{1}{12}$

b) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}f\left(x\right)\mathrm{dx}=0$

c) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\mathrm{xf}\left(x\right)\mathrm{dx}=\frac{1}{8}$

d) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}f\left(x\right)\mathrm{dx}=1$

Consider the lines
L₁: x + 3y – 5 = 0, L₂: 3x – ky – 1 = 0, L₃: 5x + 2y – 12 = 0.
Match the statement/expressions in column 1 with the statement/expression in column 2.

The number of quadratic polynomials f(x) with non-negative integer coefficients ≤ 3 satisfying f(0) = 0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)\mathrm{dx}=1$

is

a) 8

b) 2

c) 4

d) 0

A function f : ℝ → ℝ, where ℝ is the set of real numbers, is defined by . Find the interval of values of α for which f is onto. Is the function one to one for α= 3? Justify your answer.

Let f : ℝ → ℝ be a function defined by $f\left(x\right)=\{\begin{array}{cc}[x]& x\le 2\\ 0& x>2\end{array}$

where [x] denotes the greatest integer less than or equal to x. If $I=\underset{-1}{\overset{2}{\int }}\frac{\mathrm{xf}\left(x^2\right)}{2+f(x+1)}\mathrm{dx}$ then the value of (4I – 1) is

a) 1

b) 3

c) 2

d) 0

Let f: [0, 2] → ℝ be a function which is continuous on [0, 2] and differentiable on (0, 2) with f(0) = 1. Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\left(\sqrt{t}\right)\mathrm{dt}\mathrm{for}x\in [\mathrm{0,}2]$

. If F′(x) = f′(x) Ɐ x ∈ [0, 2] then F(2) equals

a) e² – 1

b) e⁴ – 1

c) e – 1

d) e²

(Multiple correct answers)

Let M and N are two events, the probability that exactly one of them occurs is

a) P (M) + P (N) − 2P (M ∩ N)

b) P (M) + P (N) − P ()

c)

d)

The area (in square units) of the region y² > 2x and x² + y² ≤ 4x, x ≥ 0, y > 0 is

a) $\pi -\frac{4}{3}$

b) $\pi -\frac{8}{3}$

c) $\pi -\frac{4\sqrt{2}}{3}$

d) $\frac{\pi }{2}-\frac{2\sqrt{2}}{3}$

Let f and g be real valued functions on (−1, 1) such that g’(x) is continuous, g(0) ≠ 0, g’(0) = 0, g’’(0) ≠ 0 and f(x) = g(x)sinx
Statement 1 -
Statement 2 – f’(0) = g(0)

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1

b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

The area of the region $\left\{\left(x,y\right)\in R^2\mathrm{:}y>\sqrt{\left|x+3\right|},5y\le x+9\le 15\right\}$

is equal to

a) $\frac{1}{6}$

b) $\frac{4}{3}$

c) $\frac{3}{2}$

d) $\frac{5}{3}$

The area (in square units) bounded by the curves $y=\sqrt{x},2y-x+3=0$

, X – axis and lying in the first quadrant is

a) 9

b) 6

c) 18

d) $\frac{27}{4}$

One or more than one correct option

Let S be the area of the region enclosed by $y=e^{{-x}^2}$

, y = 0, x = 0 and x = 1, then

a) $S\ge \frac{1}{e}$

b) $S\ge 1-\frac{1}{e}$

c) $S\le \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$

d) $S\le \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$