Question(s) from Search: IIT

The number of ordered pairs satisfying the equations
 is

a) 4

b) 2

c) 0

d) 1

Find the coordinates of the point at which the circles x² + y² – 4x – 2y = – 4 and  x² + y² – 12x – 8y = – 36  touch each other. Also find the equation of the common tangents touching the circles at distinct points.

The area bounded by the curves

  and   is

a) 1

b) 2

c)

d) 4

Let ABC be an equilateral triangle inscribed in the circle x² + y² = a². Suppose perpendiculars from A, B, C to the major axis of the ellipse  (a > b) meet the ellipse respectively at P, Q, R so that P, Q, R are on the same side of the major axis. Prove that the normals drawn at the points P, Q and R are concurrent.

Tangents are drawn from P (6, 8) to the circle  . Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.

Multiple choice

If

a) f(x) is increasing on [– 1, 2]

b) f(x) is continuous on [– 1, 3]

c)  does not exist

d) f(x) has maximum value at x = 2

Multiple choice

f(x) is a cubic polynomial with f(2) = 18 and f(1) = − 1. Also f(x) has a local maxima at x = − 1 and  has a local minima at x = 0 then

a) The distance between (− 1, 2) and (a, f(a)), where x = a is the point of local minimum, is

b) f(x) is increasing for

c) f(x) has a local minima at x = 1

d) The value of f(0) = 15

From the point A (0, 3) on the circle , a chord AB is drawn and extended to a point M such that AˆM = 2AˆB. The equation of locus of M is . . . . .

A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is . . . . .

A swimmer S is in the sea at a distance d km. from the closest point A on a straight shore. The house of the swimmer is on the shore at a distance L km. from A. He can swim at a speed of
u km/hour and walk at a speed of v km/hr (v > u). At what point on the shore should he land so that he reaches his house in the shortest possible time.

a)

b)

c)

d)

Sketch the region bounded by the curves
 and y = |x – 1|
and find its area.

a)

b)

c)

d) 5π + 2

Tangents are drawn from the point (17, 7) to the circle .
Statement 1 – The tangents are mutually perpendicular, because

Statement 2 – The locus of points from which mutually perpendicular tangents are drawn to the given circle is .

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

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

d) Statement 1 is false. Statement 2 is true

Let  be the vertices of the triangle. A parallelogram AFDE is drawn with the vertices D, E and F on the line segments BC, CA and AB respectively. Using calculus find the area of the parallelogram.

a)  

b)  

c)  

d)  

Find the area of the region bounded by the curve C: y = tanx, tangent drawn to C at  and the X–axis.

a) ln2 – 1

b)

c)

d)

Sketch the curves and identify the region bounded by
 

Tangent at a point P₁ (other than (10, 0)) on the curve y = x³ meets the curve again at P₂. The tangent at P₂ meets the curve at P₃ and so on. Show that the abscissae of P₁, P₂, P₃, .  .  . , P_n form a Geometric Progression. Also find the ratio .

a) 32

b) 16

c)

d)

In what ratio does the X–axis divide the area of the region bounded by the parabolas y = 4x – x² and y = x² – x

a) 1:4

b) 21:1

c) 21:4

d) 3:4

Let C₁ and C_2, be respectively, the parabolas  and  . Let P be any point on C₁ and Q be any point on C₂. Let P₁ and Q₁ be the reflections of P and Q respectively with respect to y = x . Prove that P₁ lies on C₂ and Q₁ lies on C₁ and  . Hence or otherwise determine points P₂ and Q₂ on the parabolas C₁ and C₂ respectively such that  for all points (P, Q) with P on C₁ and Q on C₂ .

Show that the sum of the first n terms of the series
1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .  .  .
is  when n is even, and  when n is odd.

A curve y = f(x) passes through the point P:(1, 1). The equation to the normal at (1, 1) to the curve y = f(x) is (x – 1) + a(y – 1)  = 0 and the slope of the tangent at any point on the curve is proportional to the ordinate of the point. Determine the equation of the curve. Also obtain the area bounded by the Y–axis, the curve and the normal at P.

a)

b) y = ;

c)  ;

d)

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 ratio of areas of the triangle PQS and PQR is

a)

b) 1:2

c)

d) 1:8

The function y = f(x) is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{xy}}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$

in (−1, 1) satisfying f(0) = 0, then $\underset{\frac{-\sqrt{3}}{2}}{\overset{\frac{\sqrt{3}}{2}}{\int }}f\left(x\right)\mathrm{dx}$ is

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

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

c) $\frac{\pi }{6}-\frac{\sqrt{3}}{4}$

d) $\frac{\pi }{6}-\frac{\sqrt{3}}{2}$

Let y′(x) + y(x) g′(x) = g(x) g′(x), y(0) = 0, x ∈ ℝ where f′(x) denotes $\frac{d}{\mathrm{dx}}f(x)$

and g(x) is a given non constant differentiable function on ℝ with g(0) = g(2) = 0. Then the value of y(2) is

a) 1

b) 0

c) 2

d) 4

One or more than one correct option

A solution curve of the differential equation $\left(x^2+\mathrm{xy}+4x+2y+4\right)\frac{\mathrm{dy}}{\mathrm{dx}}-y^2=\mathrm{0,}x>0$

passes through the point (1, 3), then the solution curve

a) Intersects y = x + 2 exactly at one point

b) Intersects y = x + 2 exactly at two points

c) Intersects y = (x + 2)²

d) Does not intersect y = (x + 3)²

Let f(x) = (1 – x)² sin²x + x²Consider the statementsStatement 1: There exists some x ∈ ℝ such that f(x) + 2x = 2(1 + x²)Statement 2: There exists some x ∈ ℝ such that 2f(x) + 1 = 2x(x + 1)

a) Both 1 and 2 are true

b) 1 is true and 2 is false

c) 1 is false and 2 is true

d) Both 1 and 2 are false