Question(s) from Search: IIT

Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c and d denote the lengths of the sides of the quadrilateral; prove that
2 ≤ a² + b² + c² + d² ≤ 4

The number of ordered pairs satisfying the equations
 is

a) 4

b) 2

c) 0

d) 1

Let O (0, 0), A(2, 0) and  be the vertices of a triangle. Let R be the region consisting of all those points P inside ΔOAB which satisfies d(P, OA) ≤ d(P, OB) . d(P, AB), where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area.

a)

b)

c)

d)

Let f(x) be a continuous function given by
 

Find the area of the region in the third quadrant bounded by the curve x = − 2y² and y = f(x) lying on the left of the line 8x + 1 = 0.

a) 192

b) 320

c) 761/192

d) 320/761

Let d be the perpendicular distance from the centre of the ellipse  to the tangent at a point P on the ellipse. Let F₁ and F₂ be the two focii of the ellipse, then show that

Find the area of the region bounded by the curves y = x², y = |2 – x²| and y = 2 which lies to the right of the line x = 1.

a)

b)

c)

d)

Prove that in an ellipse the perpendicular from a focus upon a tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

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 f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g(x) = |f(x)| for all x. Then g is

a) Onto if f is onto

b) One–one if f is one–one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

f(x) is a differentiable function and g(x) is a double differentiable function such that  
If  prove that there exists some c ε (−3, 3) such that .

If (x – r) is a factor of the polynomial f(x) = a_nxⁿ + .  .  . + a₀, repeated m times (1 < m ≤ n) then r is a root of  repeated m times.

a) True

b) False

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.

A hyperbola having the transverse axis of length 2sinθ is confocal with the ellipse . Then its equation is

a)

b)

c)

d)

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