Question(s) from Search: IIT

The coordinates of the in centre of the triangle that has the co ordinates of the mid points of its sides as (0, 1), (1, 1) and (1, 0) is

a) $2+\sqrt{2}$

b) $2-\sqrt{2}$

c) $1+\sqrt{2}$

d) $1-\sqrt{2}$

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

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

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

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)

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

The slope of the line touching both parabolas y² = 4x and x² = −32y is

a) $\frac{1}{2}$

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

c) $\frac{1}{8}$

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

Let a, r, s, t be non-zero real numbers. Let P(at², 2at), Q, R(ar², 2ar and S(as², 2as) be distinct points on the parabola y² = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)

The value of r is

a) $\frac{-1}{t}$

b) $\frac{t^2+1}{t}$

c) $\frac{1}{t}$

d) $\frac{t^2-1}{t}$

One or more than one correct option

If the normals of the parabola y² = 4x drawn at the end points of the latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² then the value of r² is

a) 4

b) 1

c) 2

d) 0

A curve passing through the point  has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the X-axis. Determine the equation of the curve.

Let a solution y = y (x) of the differential equation  satisfies

Statement 1 :

Statement 2 :

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

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

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

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

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

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

One or more than one correct option

Let y(x) be a solution of the differential equation $\left(1+e^x\right)y^{\prime }+y{e}^x=1$

. If y(0) = 2, then which of the following statements is/are true?

a) y(−4) = 0

b) y(−2) = 0

c) y(x) has a critical point in the interval (−1, 0)

d) y(x) has no critical point in the interval

Find  at x = , when

a) 0

b) 1

c) – 1

d) 2

Let f : (0, ∞) → ℝ and  If  then f(4) equals

a)

b) 7

c) 4

d) 2

There exists a function f(x) satisfying f (0) = 1,  and

f (x) > 0 for all x and

a)   for all x

b)  

c)   for all x

d)   for all x