If, for a positive integer n, the quadratic equation,
x(x + 1) + (x + 1) (x + 2) + .....
+ (x + n -1 ) (x + n) = 10n
has two consecutive integral solutions, then n is equal to :

• 11

• 12

• 9

• 9

The function 

• neither injective nor surjective.

• invertible

• injective but not surjective.

• injective but not surjective.

The following statement
(p → q ) → [(~p → q) → q] is

• a fallacy

• a tautology

• equivalent to ~ p → q

• equivalent to ~ p → q

For any three positive real numbers a, b and c, (25a² + b²) + 25(c² – 3ac) = 15b(3a + c), Then

• b , c and a are in G.P

• b, c and a are in A.P

• a, b and c are in A.P

• a, b and c are in A.P

5.

hyperbola passes through the point P(√2, √3) and has foci at (± 2, 0). Then the tangent to this hyperbola at P also passes through the point

• 
• 
• 
• 

Let a, b, c ∈ R. If f(x) = ax² + bx + c is such
that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀x,y ∈ R,then  is equal to

• 225

• 330

• 165

• 165

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector.Then the maximum area (in sq. m) of the flower-bed, is

• 30

• 12.5

• 10

• 10

The value of
(²¹C₁ – ¹⁰C₁) + (²¹C₂ – ¹⁰C₂) + (²¹C₃ – ¹⁰C₃) + (²¹C₄ – ¹⁰C₄) + .... +
(²¹C₁₀ – ¹⁰C₁₀) is

• 2²⁰ – 2¹⁰

• 2²¹ – 2¹¹

• 2²¹ – 2¹⁰

• 2²¹ – 2¹⁰

• 1/4

• 1/24

• 1/16

• 1/16

If 5(tan²x – cos²x) = 2cos ²x + 9, then the value of cos⁴x is

• -7/9

• -3/5

• 1/3

• 1/3