Problem 1 of 20
Let $\alpha$ and $\beta$ be the roots of $x^2 - 5x + 3 = 0$ with $\alpha \gt \beta$. If $a_n = \alpha^n -\ \beta^n$ for $n \ge 1$ then the value of $\dfrac{3a_6 + a_8}{a_7}$ is:Problem 2 of 20
The number of triples $(x,\ y,\ z)$ such that any one of these numbers is added to the product of the other two, the result is $2$, is:Problem 3 of 20
In rectangle $ABCD$, $AB = 5$ and $BC = 3$. Points $F$ and $G$ are on the line segment $CD$ so that $DF = 1$ and $GC = 2$.Lines $AF$ and $BG$ intersect at $E$. What is the area of $AEB$?
Problem 4 of 20
In the given figure, two concentric circles are shown with centre $O$. $PQRS$ is a square inscribed in the outer circle. It also circumscribes the inner circle, touching it at points $B$, $C$, $D$, and $A$.
What is the ratio of the perimeter of the outer circle to that of the quadrilateral $ABCD$?
Problem 5 of 20
How many positive integers $N$ leave a remainder of $8$ when $2008$ is divided by $N$?Problem 6 of 20
What is the product of all the roots of the equation$\sqrt{5|x|+ 8} = \sqrt{x^2 - 16}$?
Problem 7 of 20
$LCM$ of two numbers is $5775$. Which of the following cannot be their $HCF$?Problem 8 of 20
If $a$, $b$, $c$ are distinct real numbers such that $a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a}$ evaluate $abc$.Problem 9 of 20
If the equation $(\alpha^2 - 5\alpha + 6)x^2 + (\alpha^2 -3\alpha + 2)x + (\alpha^2 - 4) = 0$ has more than two roots, then the value of $\alpha$ is:Problem 10 of 20
$Mr.\ X$ and his eight children of different ages is on a family trip. His oldest child, who is $9$ years old saw a license plate with a $4$-digit number in which each of two digits appear two times. $\unicode{0x201C}$Look daddy!$\unicode{0x201D}$ she exclaims, $\unicode{0x201C}$That number is evenly divisible by the age of each us kids!$\unicode{0x201D}$ $\unicode{0x201C}$.That's right,$\unicode{0x201D}$ replies $Mr.\ X$ $\unicode{0x201C}$and the last two digits just happen to be my age$\unicode{0x201D}$. Which of the following is not the age of $Mr.\ X$'s children?Problem 11 of 20
How many numbers lie between $11$ and $1111$ which divided by $9$ leave a remainder $6$ and when divided by $21$ leave a remainder $12$?Problem 12 of 20
Two unbiased dice are rolled. What is the probability of getting a sum which is neither $7$ or $11$?Problem 13 of 20
The solution of the equation $1 + 4 + 7 + ... +x = 925$ is:Problem 14 of 20
If $tan\theta + sec\theta = 1.5$, then the value of $sin\theta$ is:Problem 15 of 20
An observer standing at the top of a tower, finds that the angle of elevation of a red bulb on the top of a light house of height $H$ is $\alpha$. Further, he finds that the angle of depression of reflection of the bulb in the ocean is $\beta$. Therefore, the height of the tower is:Problem 16 of 20
The sum of the roots of $\dfrac{1}{x+a} + \dfrac{1}{x + b} = \dfrac{1}{c}$ is zero. The product of the roots is:
Problem 17 of 20
In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at $O$ and the measure of $\angle AOB$ is $30^\circ$. If the areas of triangles $AOB$, $BOC$, $COD$ and $AOD$ are $1$, $2$, $8$ and $4$ square units respectively, what is the product of the lengths of the diagonals $AC$ and $DB$ in $sq.\ units$?Problem 18 of 20
If $sin^2x + sin^2y + sin^2z = 0$, then which of the following is $NOT$ a possible value of $cosx + cosy + cosz$?Problem 19 of 20
Find the remainder when $x^{51}$ is divided by $x^2 - 3x + 2$Problem 20 of 20
In an equilateral triangle, three coins of radii $1$ unit each are kept so that they touch each other and also sides of the triangle. The area of triangle $ABC$ i(n sq. units) is: