Instructions

For the following questions answer them individually

Question 101

The roots of quadratic equation $$y^2 -8y + 14$$ are $$\alpha$$ and $$\beta$$. Find the value of $$(1 + \alpha + \beta^2)(1 + \beta + \alpha^2)$$

Question 103

Ram, Ravi and Ratan can alone finish an assignment in 9 days, 12 days and 15 days respectively. They decide to complete a work by working in turns. Ram works alone on Monday, Ravi does the work alone on Tuesday, followed by Ratan working alone on Wednesday & so on. What proportion of the complete work is done by Ravi?

Question 104

Let $$S_1$$ be a square of side 4 cm. Circle $$C_1$$ circumscribes the square $$S_1$$ such that all its corners are on $$C_1$$. Another square $$S_2$$ circumscribes the circle $$C_1$$. Circle $$C_2$$ circumscribes the square $$S_2$$, and square $$S_3$$ circumscribes circle $$C_2$$, & so on. If $$A_N$$ is the area between the square $$S_N$$ and the circle $$C_N$$, where N is the natural number. then the ratio of sum of all $$A_N$$ to $$A_l$$ is

Question 105

Joseph diametrically crosses a semi-circular playground and takes 48 seconds less than if he crosses the playground along the semi-circular path. If he walks 50 metres in one minute, the diameter of playground is

Question 106

Garima had only Rs. 200. Rs. 500 and Rs. 2000 notes in her wallet. She goes to Shoppers Stop. purchases some dresses and gives half of her Rs. 2000 notes & in turn receives same number of Rs. 200 notes. She then goes to a restaurant and gives all her Rs. 500 notes and receives thirty Rs. 2000 notes, which increases the number of Rs. 2000 notes she had by. 75%.- If now she has fifty Rs. 200 notes. what were the original number of Rs. 2000 and Rs. 200 notes she had at the start?

Question 107

A metallic solid is made up of a solid cylindrical base with a solid cone on its top. The radius of the base of the cone is 5 cm. and the ratio of the height of the cylinder and the cone is 3:2. A cylindrical hole is drilled through the solid with height equal to 2/$$3^{rd}$$ of the height of solid. What should be the radius (in cm) of the hole so that the volume of the hole is 1/$$3^{rd}$$ of the volume of the metallic solid after drilling?

Question 108

Nitin installed an overhead tank on the roof of his newly constructed house. Three taps are connected to the tank: 2 taps A and B to fill the tank and one tap C to empty it. Tap A alone can fill the tank in 12 hours, while tap B alone takes one and a half times more time than tap A to fill the tank completely. Tap C alone can empty a completely filled tank in 36 hours. Yesterday, to fill the tank, Nitin first opened tap A, and then after 2 hours opened tap B also. However after 6 hours he realised that tap C was open fi-om the very beginning. He quickly closes tap C. What will be the total time required to fill the tank?

Question 109

At the foot of the mountain, the angle of elevation of the summit at the top of the mountain is 45°. After ascending 100 metres, at a slope of 30° up the mountain towards the summit. the angle of elevation of the summit is 60°. Find the height of the summit.

Question 110

Land Cruiser Prado, the latest SUV from Toyota Motors, consumes diesel at the rate of $$\frac{1}{400}\left\{ \frac{1000}{x} + x\right\}$$litres per Km. when travelling at the speed of x km/hr. The diesel costs Rs. 65 per litre and the driver is paid Rs. 50 per hour. Find the steady speed that will minimize the total cost of a 1000 km trip?

CAT Averages, Ratios & Proportions

CAT Logarithms, Surds & Indices