List of practice Questions

A racing track is build around an elliptical ground whose equation is given by \( 9x^2 + 16y^2 = 144 \).The width of the track is 3 m as shown below :
Smoking increases the risk of lung problems.
A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. If a person is selected at random from these 50 people
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed.
The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
In an online jackpot, there is one first prize of Rs 3,00,000, two second prizes of Rs 2,00,000 each and three third prizes of Rs 50,000 each.
A total of 1,00,000 jackpot tickets each costing Rs 100 were sold there by raising a fund of Rs 1,00,00,000. Rohan bought one ticket.
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of Rs 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of Rs 1, ten subscribers will discontinue. Assume that the company increased the annual fee by Rs x.
Roundabouts are often made on busy roads to ease the traffic and avoid red lights.
One such round-about is made such that the equation representing its boundary is given by \( C_1 : x^2 + y^2 = 64 \). There is a circular pond with a fountain in the middle of the roundabout whose equation is given by \( C_2 : x^2 + y^2 = 4 \). Based on the given information
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box \(i\) is selected and a cap picked out is \(i/6\), where \(i = 1, 2, 3\).
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.
At a birthday party, children are being served orange juice in conical cups. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0.1 cm\(^3\)/s.
There is a triangular park \(ABC\) in the society. The park is divided into two sections as shown in the figure.
In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed.
The vertices of the triangular park that is ABC are \(A(0, 4), B(-2, 0)\) and \(C(3, 0)\).
Based on the above information, answer the following questions :
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII.
As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII.
Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination.
Based on the above information, answer the following questions.
Two vertical light poles of height \(22 \text{ m}\) and \(16 \text{ m}\) stand on the opposite sides of a \(20 \text{ m}\) wide road. Two ladders of length \(l_1\) and \(l_2\) are placed from a common point \(R\) on the road at a distance of \(x \text{ m}\) from the smaller pole.
Based on the above information,
A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have a 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that
A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. If for a tumbler, \( V \) is its volume, \( h \) the height and \( r \) the radius of the circular base, then:
There are three types of vaccines \( A_1, A_2, A_3 \), available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine \( A_1 \), 35% of the population was given Vaccine \( A_2 \) and 40% of the population was given Vaccine \( A_3 \). The survey also stated that the probabilities that Vaccines \( A_1, A_2 \) and \( A_3 \) would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, find the probability that:
A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition.
7 students out of 80 are selected for the project and are categorized into two sets such that :
Girl students belong to Set A = \( A = \{G_1, G_2, G_3, G_4\} \) and
Boy students belong to Set B = \( B = \{B_1, B_2, B_3\} \) .
A photon and an electron, each of \(10\,\text{eV}\) energy, move in free space. The ratio of linear momentum of electron \(P_e\) to that of photon \(P_{ph}\), \[ \frac{P_e}{P_{ph}} \] is :
There are 3 students from section A, 5 from section B and 4 from section C. In how many ways can they occupy 6 seats such that the leftmost seat is occupied by a student of section A and the remaining seats are occupied by 3 students of section B and 2 students of section C?
When two electric charges are placed at a distance apart in air, the electrostatic force between them is \(400\text{ N}\). If \(\frac13\) of the space between the two charges is filled with a medium of dielectric constant \(4\), then the electrostatic force between them is:
Refer the given image carefully:
Explain any two measures that can be taken by the Central Bank to control the indicated macroeconomic problem.
What is 3r in grid square 3501?
Find the differential of \( x^{\cot x} + \frac{2x^2-3}{2x^2-x+2} \) with respect to \( x \).
Find the foot of the perpendicular from the point \( (0, 2, 3) \) on the line \( \frac{-x-3}{-5} = \frac{1-y}{-2} = \frac{3z+12}{9} \) and hence find the length of the perpendicular.
Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.
Solve the differential equation \( (x - \sin y) \, dy + \tan y \, dx = 0 \).