Let $\alpha$ and $\beta$ be the roots of $x^{2}-5 x+3=0$ with $\alpha>\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$ then the value of $\frac{3 a_{6}+a_{8}}{a_{7}}$ is
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
In rectangle $\mathrm{ABCD}, \mathrm{AB}=5$ and $\mathrm{BC}=3$. Points F and G are on the line segment CD so that $\mathrm{DF}=1$ and $\mathrm{GC}=2$. Lines AF and BG intersect at E . What is the area of AEB ?
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 $\mathrm{B}, \mathrm{C}, \mathrm{D}$ and A . What is the ratio of the perimeter of the outer circle to that of quadrilateral ABCD ?
How many positive integers N give a remainder 8 when 2008 is divided by N.
What is the product of all the roots of the equation $\sqrt{5|x|+8}=\sqrt{x^{2}-16}$ ?
LCM of two numbers is 5775 . Which of the following cannot be their HCF?
If $a, b, c$ are distinct real numbers such that $a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}$ evaluate $a b c$.
If the equation $\left(\alpha^{2}-5 \alpha+6\right) x^{2}+\left(\alpha^{2}-3 \alpha+2\right) x+\left(\alpha^{2}-4\right)=0$ has more than two roots, then the value of $\alpha$ is
Mr. X with 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. "Look daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. X, "and the last two digits just happen to be my age". Which of the following is not the age of one of Mr. X's children?
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 ?
Two unbiased dice are rolled. What is the probability of getting a sum which is neither 7 nor 11 ?
The solution of the equation $1+4+7+\ldots \ldots+x=925$ is
If $\tan \theta+\sec \theta=1.5$, then value of $\sin \theta$ is
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
The sum of the roots of $\frac{1}{x+a}+\frac{1}{x+b}=\frac{1}{c}$ is zero. The product of roots is
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 triangle $\mathrm{AOB}, \mathrm{BOC}, \mathrm{COD}$ 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?
If $\sin ^{2} x+\sin ^{2} y+\sin ^{2} z=0$, then which of the following is NOT a possible value of $\cos x+\cos y+\cos z ?$
Find the remainder when $x^{51}$ is divided by $x^{2}-3 x+2$.
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 (in sq. units) is
Apples dropping from apple trees were observed by many people before Newton. But why they fall, was explained by Isaac Newton postulating the law of universal gravitation. Which of the following statements best describes the situation?
A rectangular metal plate, shown in the adjacent figure has a charge of $420 \mu \mathrm{C}$ assumed to be uniformly distributed over it. Then how much is the charge over the shaded area? No part of metal plate is cut. (Circles and the diagonal are shown for clarity only. $\pi=22 / 7$ )
In the adjacent circuit, the voltages across AD , BD and CD are $2 \mathrm{ V}, 6 \mathrm{ V}$ and 8 V respectively. If resistance $R_{A}=1 \mathrm{k} \Omega$, then the values of resistances $\mathrm{R}_{\mathrm{B}}$ and $\mathrm{R}_{\mathrm{C}}$ are ____ and ____ respectively.
A new linear scale of temperature measurement is to be designed. It is called a ' Z scale' on which the freezing and boiling points of water are 20 Z and 220 Z respectively. What will be the temperature shown on the ' Z scale' corresponding to a temperature of $20^{\circ} \mathrm{C}$ on the Celsius scale?
Consider the motion of a small spherical steel body of mass $m$, falling freely through a long column of a fluid that opposes its motion with a force proportional to its speed. Initially the body moves down fast, but after some time attains a constant velocity known as terminal velocity. If weight $m g$, opposing force ( $F_{v}$ ) and buoyant force ( $F_{b}$ ) act on the body, then the correct equation relating these forces, after the terminal velocity is reached, is:
A piece of wire $P$ and three identical cells are connected in series. An amount of heat is generated in a certain time interval in the wire due to passage of current. Now the circuit is modified by replacing P with another wire Q and $N$ identical cells, all connected in series. Q is four times longer in length than P . The wire P and Q are of same material and have the same diameter. If the heat generated in second situation is also same as before in the same time interval, then find $N$.
Some waveforms among I, II, III and IV superpose (add graphically) to produce the waveforms P, Q, R and S. Among the following, match the pairs that give the correct combinations:
At any instant of time, the total energy ( $E$ ) of a simple pendulum is equal to the sum of its kinetic energy $\left(\frac{1}{2} m v^{2}\right)$ and potential energy $\left(\frac{1}{2} k x^{2}\right)$, where, $m$ is the mass, $v$ is the velocity, $x$ is the displacement of the bob and $k$ is a constant for the pendulum. The amplitude of oscillation of the pendulum is 10 cm and its total energy is 4 mJ . Find $k$.
A rigid body of mass $m$ is suspended from point O using an inextensible string of length $L$. When it is displaced through an angle $\theta$, what is the change in the potential energy of the mass? (Refer adjacent figure.)
Refer to the adjacent figure. A variable force F is applied to a body of mass 6 kg at rest. The body moves along $x$ - axis as shown. The speed of the body at $x=5 \mathrm{ m}$ and $x=6 \mathrm{ m}$ is ____ and ____ respectively.
When a charged particle with charge $q$ and mass $m$ enters uniform magnetic field $B$ with velocity $v$ at right angles to $B$, the force on the moving particle is given by $q v B$. This force acts as the centripetal force making the charged particle go in a uniform circular motion with radius $r=\frac{m v}{B q}$. Now if a hydrogen ion and a deuterium ion enter the magnetic field with velocities in the ratio $2: 1$ respectively, then the ratio of their radii will be ____
A piece of ice is floating in water at $4^{\circ} \mathrm{C}$ in a beaker. When the ice melts completely, the water level in the beaker will
In a screw-nut assembly (shown below) the nut is held fixed in its position and the screw is allowed to rotate inside it. A convex lens $(\mathrm{L})$ of focal length 6.0 cm is fixed on the nut. An object pin $(\mathrm{P})$ is attached to the screw head. The image of the object is observed on a screen Y. When the screw head is rotated through one rotation, the linear distance moved by the screw tip is 1.0 mm . The observations are made only when the image is obtained in the same orientation on the screen. At a certain position of P , the image formed is three times magnified as that of the pin height. Through how many turns should the screw head be rotated so that the image is two times magnified?
A school is located between two cliffs. When the metal bell is struck by school attendant, first echo is heard by him after 2.4 s and second echo follows after 2.0 s for him at the same position near the bell. If the velocity of sound in air is $340 \mathrm{ ms}^{-1}$ at the temperature of the surroundings, then the distance between the cliffs is approximately ____
The triangular face of a crown glass prism ABC is isosceles. Length $\mathrm{AB}=$ length AC and the rectangular face with edge AC is silvered. A ray of light is incident normally on rectangular face with edge AB . It undergoes reflections at AC and AB internally and it emerges normally through the rectangular base with edge BC . Then angle BAC of the prism is ____
The radius of curvature of a convex mirror is ' $x$ '. The distance of an object from focus of this mirror is ' $y$ '. Then what is the distance of image from the focus?
A physics teacher and his family are travelling in a car on a highway during a severe lightning storm. Choose the correct option:
A conductor in the form of a circular loop is carrying current $I$. The direction of the current is as shown. Then which figure represents the correct direction of magnetic field lines on the surfaces of the planes XY and XZ . (Consider those surfaces of the XY and XZ planes which are seen in the figure.)
A particle experiences constant acceleration for 20 s after starting from rest. If it travels a distance $S_{1}$ in the first 10 s and distance $S_{2}$ in the next 10 s , the relation between $S_{1}$ and $S_{2}$ is:
A sound wave is produced by a vibrating metallic string stretched between its ends. Four statements are given below. Some of them are correct.
(P) Sound wave is produced inside the string.
(Q) Sound wave in the string is transverse.
(R) Wavelength of the sound wave in surrounding air is equal to the wavelength of the transverse wave on the string.
(S) Loudness of sound is proportional to the square of the amplitude of the vibrating string.
Choose the correct option.
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.