Question(s) from Search: IIT

The differential equation representing the family of curves  where c is a positive parameter, is of

a) Order 1

b) Order 2

c) Degree 3

d) Degree 4

Let a, b, c be real numbers with a² + b² + c² = 1. Show that the equation represents a straight line
 = 0

Let , then the set  is

a)  

b)  

c)  

d)  ϕ

A normal is drawn at a point  of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is  

If f(x) = 3x – 5 then  

a) is given by

b) is given by

c) does not exist because f is not one-one

d) does not exist because f is not onto

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4

Area bounded by  and

mn squares of equal size are arranged to form a rectangle of dimension m by n, where m and n are natural numbers. Two squares will be called neighbours if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in the neighbouring squares. Show that this is possible only if all the numbers used are equal.

Let A =
 
AU₁ =  , AU₂ =  and AU₃ =
 

a) 3

b) −3

c)  

d) 2

The domain of definition of  is

a)  

b)  

c)  

d)  

Let f : ℝ → ℝ be defined by f(x) = 2x + sinx for all x  ℝ. Then f is

a) One to one and onto

b) One to one but not onto

c) Onto but not one to one

d) Neither one to one nor onto

Range of    ;   x  ℝ is

a) (1, ∞)

b)

c)

d)

Let a, b, c, ε R and α, β be roots of  such that  and  then show that .

If  where
. Given F(5) = 5, then f(10) is equal to

a) 5

b) 10

c) 0

d) 15

Subjective problems
Let .  Find all real values of x for which y takes real values.

a) [− 1, 2)

b)  [3, ∞)

c) [− 1, 2) ∪ [3, ∞)

d) None of the above

Let R be the set of real numbers and f : R → R be such that for all x and y in R, . Prove that f(x) is constant.

The number of points in the interval $\left[-\sqrt{13},\sqrt{13}\right]$

in which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value is

a) 8

b) 2

c) 4

d) 0

If the integers m and n are chosen at random between 1 and 100 then the probability that a number of form  is divisible by 5, equals

a)

b)

c)

d)

Show that the integral
 =

where y = x^1/6

If $\alpha =\underset{0}{\overset{1}{\int }}{e}^{\left(9x+{3\tan }^{-1}x\right)}\left(\frac{12+{9x}^2}{1+x^2}\right)\mathrm{dx}$

Where ${\tan }^{-1}x$ takes only principal values then the value of $\left({\log }_e\left|1+\alpha \right|-\frac{3\pi }{4}\right)$ is

a) 6

b) 9

c) 8

d) 11

The intercept on X axis made by the tangent to the curve $y=\underset{0}{\overset{x}{\int }}\left|t\right|\mathrm{dt},t\in R$

which is parallel to the line y = 2x are equal to

a) ±1

b) ±2

c) ±3

d) ±4

The common tangent to the curve x² + y² = 2 and the parabola y² = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area (in square units) of the quadrilateral PQRS is

a) 3

b) 6

c) 9

d) 15

(One or more correct answers)
Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A ∪ B) = P (A) + P (B) – P (A ∩ B) then

a) P (B/A) = P (B) – P (A)

b) P (Aʹ – Bʹ) = P (Aʹ) – P (Bʹ)

c) P (A U B)ʹ = P (Aʹ) P (Bʹ)

d) P (A/B) = P (A)

For any integer n, the integral
 has the value

a) π

b) 1

c) 0

d) None of these

The area (in square units) of the region described by (x, y) : y² < 2x and y ≥ 4x – 1 is

a) $\frac{7}{32}$

b) $\frac{9}{32}$

c) $\frac{3}{2}$

d) $\frac{5}{3}$