Question 1 :
If r is a real number such that |r| < 1 and if a = 5(1−r), then
Question 2 :
The number of real roots of the equation 1 + 3<sup>x/2</sup> = 2<sup>x</sup>, is
Question 3 :
The solution set contained in R of the inequation 3<sup>x</sup> + 3<sup>1 − x</sup> − 4 < 0, is
Question 4 :
A function {tex} f ( x ) {/tex} is defined for all real {tex} x {/tex} and satisfied {tex} f ( x + y ) = {/tex} {tex} f ( x y ) \forall x , y . {/tex} If {tex} f ( 1 ) = - 1 , {/tex} then {tex} f ( 2006 ) {/tex} equals
Question 5 :
The function {tex} f ( x ) = \sin [ \log ( x + \sqrt {\left. x ^ { 2 } + 1 \right) } ] \text { is } {/tex}
Question 6 :
The number of solution(s) of the equation {tex} x ^ { 2 } - 2 - 2 [ x ] = 0 {/tex} ([.] denotes the greatest integer function) is (are)
Question 7 :
If A, B and C are non-empty sets, then (A - B) ∪ (B - A) equals
Question 8 :
The function {tex} f ( x ) = \left( x ^ { 2 } + 2 x + c \right) / \left( x ^ { 2 } + 4 x + 3 c \right) {/tex} has the range {tex} ( - \infty , \infty ) {/tex} for the allowed values of {tex} x \in R {/tex} if<br>
Question 9 :
In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is
Question 10 :
Let {tex} f : R \rightarrow R {/tex} be a function such that {tex} f ( 2 - x ) = f ( 2 + x ) {/tex} and {tex} f ( 4 - x ) = f ( 4 + x ) , {/tex} for all {tex} x \in R {/tex} and {tex} \int \limits_ { 0 } ^ { 2 } f ( x ) d x = 5 . {/tex} Then the value of {tex} \int \limits_ { 10 } ^ { 50 } f ( x ) d x {/tex} is<br>
Question 11 :
If$\text{\ \ }\frac{\sin x}{\sin y} = \frac{1}{2},\frac{\cos x}{\cos y} = \frac{3}{2},\ $where $x,\ y \in \left( 0,\frac{\pi}{2} \right),$ then the value of tan (x + y) is equal to
Question 12 :
The solution of the inequality log<sub>1/2</sub>sin x>log<sub>1/2</sub>cos x in (0, 2π) is
Question 13 :
If in a Δ ABC, 3a = b + c, then the value of $\cot\frac{B}{2}\cot\frac{C}{2}$ is
Question 15 :
If c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup>, 2 s = a + b + c, then 4s(s−a)(s−b)(s−c)=
Question 16 :
The value of $\frac{\sin\left( B + A \right) + \ cos(B - A)}{\sin\left( B - A \right) + \cos\left( B + A \right)}$ is equal to
Question 17 :
If $A + B + C = \frac{3\pi}{2}$, then cos 2A + cos 2B + cos 2C=
Question 18 :
If cos x = 3cos y, then $2\tan\frac{y - x}{2}$ is equal to
Question 19 :
The number of values of x ∈ [0,2 π] that satisfy cot x − cosec x = 2sin x, is
Question 20 :
If 12cot<sup>2</sup>θ − 31 cosec θ + 32 = 0, then the value of sin θ is
Question 21 :
The number of solutions of the equation tan x + sec x = 2cos x lying in the interval [0, 2π] is
Question 22 :
If θ ∈ [0, 5π] and r ∈ R such that 2sin θ = r<sup>4</sup> − 2r<sup>2</sup> + 3, then the maximum number of values of the pair (r, θ) is
Question 23 :
3 (sinx − cos x)<sup>4</sup> + 6(sinx + cos x)<sup>2</sup> + 4(sin<sup>6</sup>x + cos<sup>6</sup>x) is equal to
Question 24 :
If $a_{n + 1} = \sqrt{\frac{1}{2}\left( 1 + a_{n} \right)}$, then $\cos\left( \frac{\sqrt{1 - a_{0}^{2}}}{a_{1}a_{2}a_{3}\ldots to\ \infty} \right) =$
Question 25 :
The maximum value of$\ \sin{\left( x + \frac{\pi}{6} \right) + \cos{\left( x + \frac{\pi}{6} \right)\ }}$ in the interval $\left( 0,\frac{\pi}{2} \right)$ is attained at
