Question(s) from Search: IIT

Let and  be the roots of the equation  where the coefficients p and q may be complex numbers. Let A and B represent  in the complex plane. If  and OB = OA where O is the origin, prove that .

If a, b, c be positive and not all equal, show that the value of the determinant  is negative.

If the expression  is real then the set of all possible values of x is .  .  .  .

a) x = 2nπ or mπ + π/4

b) x = nπ or mπ + π/4

c) x = 2nπ or 2mπ + π/4

d) x = nπ or 2mπ + π/4

(Fill in the blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Let f(x) = ∫e^x (x – 1) (x − 2) dx, then f(x) decreases in the interval

a) (−∞, −2)

b) (−2, −1)

c) (1, 2)

d) (2, ∞)

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

Show that  =

 =

a)

b)

c)

d)

 =

a)

b)

c)

d)

Show that
 =

Show that  =

The value of the integral
 is

a)

b)

c) π

d) None of these

The value of  is

a) 0

b) 1

c)

d)

If y is a function of x and ln (x + y) – 2xy = 0 then the value of

y’ (0) is equal to

a) 1

b) – 1

c) 2

d) 0

If  then g(x + π) equals

a) g(x) + g(π)

b) g(x) − g(π)

c) g(x) g(π)

d)

 is equal to

a) 2

b) –2

c)

d)

Let x be the Arithmetic Mean and y, z be two Geometric Means between any two positive numbers then

The locus of a variable point whose distance from  is  times its distance from the line  is

a) Ellipse

b) Parabola

c) Hyperbola

d) None of these

If  and α, β lie between 0 and  find  

a)

b)

c)

d) 2

The product of n positive real numbers is unity. Then their sum is

a) A positive integer

b) Divisible by n

c) Equal to

d) Never less than n

a) True

b) False

Coefficient of t²⁴ in (1 + t²)¹² (1 + t¹²) (1 + t²⁴) is

a)

b)

c)

d)

If tan A  then

a) True

b) False

Find the equation of the common tangent in the first quadrant to the circle  and the ellipse   . Also find the length of the intercept of the tangent between the coordinate axis.

If k =  then the numerical value of k is ……….

a)

b)

c)

d)