Possible values for HCF = 60
Q. HCF of (9000, X) = 60; where X is a natural number less than 9000. Find the number of possible values of X A. 24 B. 40 C. 30 D. 36 If the HCF of two numbers is h, the two numbers can be assumed as (h × a) and (h × b), where a and b are co-prime. Here, we have (60 × 150) and X = (60 × k). How many values can k assume such that it is co-prime to 150? B. 40 Since HCF is 60, X has to be a multiple of 60, say X = 60 × k9000 = 60 × 150For HCF to be 60, k & 150 must be co-prime. Also for X to be less than 9000, k < 150.Thus, we need to find how many numbers less than 150 are co-prime to 150. This [...]
Adding mineral water to milk
Q. A milkman mixes 5 litres of mineral water with 20 litres of pure milk. The mixture is sold at a price per litre that is 20% more than the cost per litre of milk, and the milkman makes a profit of 40%. Find the ratio of the cost per litre of mineral water to that of milk. A. \(\frac{2}{7}\) B. \(\frac{3}{7}\) C. \(\frac{6}{13}\) D. \(\frac{4}{7}\) Since selling rate is 20% i.e. 1/5 more than cost of milk, assume cost of milk to be Rs. 5/lt, and cost of mineral water to be Rs. x/lt. Find total cost and total sales. And use the data that profit% is 40%. A. \(\frac{2}{7}\) Let's assume C.P. of 1 lt pure milk = 5, and C.P. of 1 lt mineral water = x ⇒ S.P. of 1 lt of mixture = 6 Total volume sold = [...]
f(5+x) = f(5-x), find sum of roots
Q.1 If f(5+x) = f(5-x) for every real x, and f(x) = 0 has four distinct real roots, then what is the sum of these roots? TITA type i.e. Type In The Answer type 1. Re-write the given relation in the form, f(x) = f(?) 2. If p is a root, can you find another root related to p? What are their sum? 20 f(5 + x) = f(5 – x) Denoting (5 + x) = y, we have x = y – 5. And (5 – x) = 5 – (y – 5) i.e. 10 – y. Replacing (5 + x) with y and (5 – x) with (10 – y), f(y) = f(10 – y), for all values of y. If p be a root of f(x) ⇒ f(p) = 0, And using the earlier relation, f(10 – p) = 0 [...]
Kitchen – spills milk in honey jar
Q. In Mrs. White's kitchen, Mr. White accidentally spilled some milk into a jar of honey. In order to compensate for the spilling, he decided to replace 100 ml of the mixture with 100 ml of pure honey such that the ratio of milk to honey in the mixture becomes 5 : 19. If the jar initially had 450 ml of pure honey, then how much milk was spilled in the jar? A. 25 ml B. 150 ml C. 50 ml D. None of the above B. 150 ml This is a tough question. We will either have to proceed with using a variable. Or if we want to avoid variable & equations, then we will have to use options. Approach 1: Using options ... [A] If 25 ml was spilled, then the volume at end will be 450+25–100+100 = 475. Dividing this in [...]
Expenditure proportional to square of number of people
In a family, there are 'n' persons. The expenditure of rice per month is directly proportional to 4 times the square of the number of persons of the family. If one of them left the family there was a decrease in consumption of 28 kg rice per month. Then initially how many persons were in the family? TITA type i.e. Type In The Answer type 4 Initial consumption = \(k \times 4{n}^{2}\) Final consumption = \(k \times 4 {(n -1)} ^{2}\) It is given that: \(k \times \left(4{n}^{2}- 4\left({n}^{2}\; - 2n + 1\right) \right) = 28\) ⇒ k × 4 × (2n – 1) = 28 ⇒ k × (2n – 1) = 7 7 is a prime number and (2n – 1) is a whole number. And 7 can be factorised as 1 × 7 or 7 × 1. If (2n – 1) [...]
Number of solutions to a equation in log
Q. The equation \(\log_{x}{3}-\log_{3}{x}=2\) has A. no real solution B. exactly one real solution C. exactly two real solutions D. infinitely many real solutions C. exactly two real solutions \(\log_{3}{x}-\log_{x}{3}=2\) \(\log_{3}{x}-\frac{1}{\log_{3}{x}}=2\) Let \(\log_{3}{x}=K\) \(K-\frac{1}{K}=2\) \({K}^{2}-2K-1=0\) \(K=\frac{-(-2)\pm\sqrt{{-2}^{2}-4(1)(-1)}}{2(1)}\) \(K=\frac{2\pm\sqrt{8}}{2}\) \(K=\frac{2\pm2\sqrt{2}}{2}\) \(K=1+\sqrt{2}\) OR \(K=1-\sqrt{2}\) Two real solutions
Logs, proportion, AP GP HP
Q. If \( \frac{\log_{4}x}{2} = \frac{\log_{6}x}{k} = \frac{\log_{9}x}{4} \), find the value of k. A. 8/3 B. 10/3 C. 3 D. \( \sqrt{3} \) Find the relation between 4, 6 & 9 and then using it, the relation between \(\log_{x}4, \; \log_{x}6, \; \log_{x}9\) and then replace them with terms related to 2, k & n. A. 8/3 Approach 1: 4, 6, 9 are in GP ⇒ \(\log_{x}4, \; \log_{x}6, \; \log_{x}9 \) are in AP ⇒ \(\log_{4}x, \; \log_{6}x, \; \log_{9}x \) are in HP ⇒ 2n, kn, 4n are in HP ⇒ 2, k, 4 are in HP ⇒ \( k = \frac{2 \times 2 \times 4}{2 + 4} \) Approach 2: If the given ratios is equal to n, then \(\log_{4}x=2n \implies \log_{x}4=\frac{1}{2n}\) Similarly, \(\log_{x}6=\frac{1}{kn}\) and \(\log_{x}9 =\frac{1}{4n}\) Next, 62 = 4 × 9 \(\implies 2\log_{x}6 = \log_{x}4 + \log_{x}9\) i.e. [...]
Finding limits to a series
There are a set of unusual questions which can stump even the better prepared ones. Since they dont really fall into any topic conveniently, one usually has no prior knowledge to start them. Here we present some such questions ... Q. $$S=\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \frac{7}{8} \times ...... \times \frac{99}{100}$$ Which of the following is true about S? A. \(S < \frac{1}{10}\) B. \(\frac{1}{10} \le S < \frac{1}{5}\) C. \(\frac{1}{5} \le S < \frac{1}{2}\) D. \(S \ge \frac{1}{2}\) Wouldn't it be nice if we had .... $$\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times ...... \times \frac{99}{100}$$ So, compare some form of S with the above. What form? S currently has 50 fractions, whereas the above convenient expression has 99 fractions. So the required form of S also needs to have 99 fractions for easy comparison .... or else [...]
Geometric Progression.
Qs. In an infinite G.P., each term is equal to three times the sum of all the terms that follow it. Also the sum of the first two terms is 15. Now find the sum when this G.P. is added to infinity ? A. 16 B. 12 C. 13 D. 15 Assume any GP, then using the relation given find the value of common ratio. Plug-in common ratio in the information given about sum of first two terms and find first term. Now we have everything what is needed to find out the sum of an infinite G.P. . Plug-in those values and calculate. A. 16 Let the terms of GP be a,ar,ar2….. with "a" being the first term and "r" being the common ratio. It is given that, Each term =3*(Sum of all the terms that follow) Let's take the first term, [...]
Mixtures
Qs. Find the weight and percentage of zinc in a zinc-copper alloy, which when alloyed with 3 kgs of pure zinc contains 90% zinc and when alloyed with 2 kgs of 90% zinc contains 84% zinc. A. 2.4 kg or 80% B. 1.4 kg or 88% C. 3.4 kg or 60% D. 7.4 kg or 18% The unknowns here are the weight of zinc and copper in the zinc-copper alloy. Assume these to be x and y kgs. Write the equations for the two cases of alloying. We will now have 2 variables and 2 equations. Solve simultaneously to get x and y. A. 2.4 kg or 80% Amount of zinc in the zinc-copper alloy (let's call it alloy A)say x kg and y kg respectively. Total weight of the alloy = (x + y) kg When an alloy is formed with alloy [...]