Question(s) from Search: IIT

Find all maximum and minimum of the curve y = x(x – 1)², 0 ≤ x ≤ 2. Also find the area bounded by the curve y = x(x – 2)², the Y–axis and the line y = 2.

a) Local minimum at x = 1, Local maximum at x = , Area =

b) Local minimum at x = , Local maximum at x =1, Area =

c) Local minimum at x = 2, Local maximum at x = , Area =

d) Local minimum at x = , Local maximum at x =2, Area =

A line is perpendicular to  and passes through (0, 1, 0). Then the perpendicular distance of this line from the origin is  . . .

Prove that for complex numbers z and ω,   iff z = ω or .

The curve y = ax³ + bx² + cx + 5 touches the X – axis at (− 2, 0) and cuts the Y–axis at a point Q where the gradient is 3. Find a, b, c.

a)

b)

c)

d)

Points A, B, C lie on the parabola . The tangents to the parabola at A, B, C taken in pair intersect at the points P, Q, R. Determine the ratios of the areas of ΔABC and ΔPQR.

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

 is a circle inscribed in a square whose one vertex is . Find the remaining vertices.

a)

b)

c)

d)

True/False
For the complex numbers  and  we write  and  then for all complex numbers z with  we have  . 

a) True

b) False

Let
where a is a positive constant. Find the interval in which  is increasing.

a)

b)

c)

d)

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)