Question(s) from Search: IIT

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 circum circle of △PRS is

a) 5

b)

c) 3

d)

Let  where 0 ≤ x ≤ 1. Determine the area bounded by y = f (x), X–axis, x = 0 and x = 1.

a)

b)

c)

d)

Which of the following function is periodic?

a) f(x) = x – [x] where [x] denotes the greatest integer less than equal to the real number x

b)

c) f(x) = x cos(x)

d) None of these

A curve C has the property that the tangent drawn at any point P on C meets the co-ordinate axes at A and B, and P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve.

a) x²y = 1

b) x = y

c) xy = 1

d) x² = y

Let –1 ≤ p ≤ 1. Show that the equation 4x³ – 3x – p = 0 has a unique root in the interval  and identify it.

a) p

b) p/3

c)

d)

Find the coordinates of all points P on the ellipse , for which the area of △PON is maximum where O denotes the origin and N the feet of perpendicular from O to the tangent at P.

Determine the equation of the curve passing through origin in the form  which satisfies the differential equation

If α, β are roots of  and γ, δ are roots of  then evaluate  in terms of p, q, r, s.

If p(x) = 51x¹⁰¹ – 2323x¹⁰⁰ – 45x + 1035, using Rolle’s theorem prove that at least one root lies between .

For what values of m does the system of equations 3x + my = m, 2x – 5y = 20 have solutions satisfying x > 0, y > 0?

a) m ε (

b) m ε (

c) m ε ( ∪ (

d) m ε (

Given

 
and f(x) is a quadratic polynomial. V is a point of maximum of f(x) and ‘A’ is the point where f(x) cuts the X–axis. ‘B’ is a point such that AB subtends a right angle at V. Find the area between chord AB and f(x).

a) 125

b) 125/2

c) 125/3

d) 125/6

The larger of cos (lnθ) and ln (cosθ) if  is

a) cos(lnθ)

b) ln(cosθ)

For any real t, ,  is a point on the hyperbola x² – y² = 1. Find the area bounded by the hyperbola and the line joining the centre to the points corresponding to t₁ and –t₁.

The integral $\underset{\pi /4}{\overset{\pi /2}{\int }}{\left(2\mathrm{cosecx}\right)}^{17}\mathrm{dx}$

is equal to

a) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{16}\mathrm{du}$

b) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u+e^{-u}\right)}^{17}\mathrm{du}$

c) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}{\left(e^u-e^{-u}\right)}^{17}\mathrm{du}$

d) $\underset{0}{\overset{\log \left(1+\sqrt{2}\right)}{\int }}2{\left(e^u+e^{-u}\right)}^{}\mathrm{du}$

X and Y are two sets and f : X → Y. If  then the true statement is

a)

b)

c) ,

d)

Let a and b are non-zero real numbers. Then the equation
(ax² + by² + c) (x² – 5xy + 6y²) = 0 represents

a) Four straight lines when c = 0 and a, b are of the same sign

b) Two straight lines and a circle when a = b and c is of sign opposite to that of a.

c) Two straight lines and a hyperbola when a and b are of the same sign

d) A circle and an ellipse when a and b are of the same sign and c is of sign opposite to that of a.

Statement 1: The value of the integral $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{\mathrm{dx}}{1+\sqrt{\tan x}}$

is equal toStatement 2: $\underset{a}{\overset{b}{\int }}f\left(x\right)\mathrm{dx}=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)\mathrm{dx}$

a) Statement 1 is correct, statement 2 is correct. Statement 2 is correct explanation of statement 1

b) Statement 1 is correct, statement 2 is correct. Statement 2 is not a correct explanation of statement 1

c) Statement 1 is correct, statement 2 is false

d) Statement 1 is incorrect, statement 2 is correct

Multiple choices
If f(x) =  where [x] stands for the greatest integer function then

a)

b)

c)

d)

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. Points E and F are given by

a)

b)

c)

d)

One or more than one correct options

If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{\left(k+1\right)}{x(x+1)}\mathrm{dx}$

then

a) $I>{\log }_{e}99$

b) $I<{\log }_{e}99$

c) $I<\frac{49}{50}$

d) $I>\frac{49}{50}$

ConsiderL₁: 2x + 3y + p – 3 = 0; L₂: 2x + 3y + p + 3 = 0 where p is a real number and C : x² + y² + 6x – 10y + 30 = 0

Statement 1 – If the line L₁ is a chord of the circle C then L₂ is not always a diameter of C.

Statement 2 - If the line L₁ is a diameter of the circle C then L₂ is not a chord of the circle.
Which of the following four statements is true?

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

One or more than one correct options

If $I_n=\underset{-\pi }{\overset{\pi }{\int }}\frac{\sin \mathrm{nx}}{\left(1+n^x\right)\mathrm{sinx}}\mathrm{dx},n=\mathrm{0,}\mathrm{1,}\mathrm{2,}...$

then

a) $I_n=I_{n+2}$

b) $\sum _{n=1}^{10}{I}_{2n+1}=10\pi$

c) $\sum _{n=1}^{10}{I}_{2n}=0$

d) $I_n=I_{n+1}$

If E and F are events with P (E) ≤ P (F) and P (E ∩ F) > 0 then

a) occurrence of E ⇒ occurrence of F

b) occurrence of F ⇒ occurrence of E

c) non-occurrence of E ⇒ non-occurrence of F

d) none of the above occurrences hold

 =  

where t² = cot²x – 1

a) True

b) False

$\frac{\left(\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \frac{1+x}{1-x}\mathrm{dx}\right)}{\left(\underset{0}{\overset{1/2}{\int }}\cos 2x\log \frac{1+x}{1-x}\right)}$

equals

a) 8

b) 2

c) 4

d) 0