Question(s) from Search: IIT

Find the point on the curve 4x² + a²y² = 4a², 4 < a² < 8 that is farthest from the point (0, −2).

a) (a, 0)

b)

c)

d) (0, - 2)

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

a) x = −a

b)

c)

d)

Complex numbers  are the vertices A, B, C respectively of an isosceles right angled triangle with right angle at B. Show that

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)

Let a line passing through the fixed point (h, k) cut the X–axis at P and Y–axis at Q. Then find the minimum area of ΔOPQ.

a) hk

b) h²/k

c) k²/h

d) 2hk

Match the following

Let A_n be the area bounded by the curve y = (tanx)ⁿ and the line
x = 0, y = 0 and . Prove that for  . Hence deduce that
 

Consider the circle x² + y² = 9 and the parabola y² = 8x. They intersect P and Q in the first and fourth quadrants respectively. Tangents to the circle at P and Q intersect the X–axis at R and tangents to the parabola at P and Q intersect the X- axis at S. The radius of the incircle of △PQR is

a) 4

b) 3

c)

d) 2

ABCD is a rhombus. The diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and (2 – i) respectively then find the complex number x + iy represented by A.

a)  

b)  

c)  

d)  

Find all possible values of b > 0, so that the area of the bounded region enclosed between the parabolas  and  is maximum.

a) b = 1

b) b ≥ 1

c) b ≤ 1

d) 0 < b < 1

Let f(x) = sinx and g(x) = ln|x|. If the ranges of the composition function fog and gof are R₁ and R₂ respectively then

a)

b) ,

c)

d)

Let C₁ and C₂ be the graph of the function y = x² and y = 2x respectively. Let C₃ be the graph of the function
y = f (x), 0 ≤ x ≤ 1, f (0) = 0. Consider a point P on C₁. Let the lines through P, parallel to the axes meet C₂ and C₃ at Q and R respectively (see figure). If for every position of P (on C₁) the area of the shaded regions OPQ and OPR are equal, determine the function f(x).

a) x² – 1

b) x³ – 1

c) x³ – x²

d) 1 + x² + x³

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 .