Instructions

For the following questions answer them individually

Question 45

Let n be the least positive integer such that 168 is a factor of $$1134^{n}$$. If m is the least positive integer such that $$1134^{n}$$ is a factor of $$168^{m}$$, then m + n equals

Question 46

If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y $$ equals

Question 48

If $$x$$ and $$y$$ are real numbers such that $$x^{2} + (x - 2y - 1)^{2} = -4y(x + y)$$, then the value $$x - 2y$$ is

Question 49

The number of integer solutions of equation $$2|x|(x^{2}+1) = 5x^{2}$$ is

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Question 50

The equation $$x^{3} + (2r + 1)x^{2} + (4r - 1)x + 2 =0$$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

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Question 51

Let $$\alpha$$ and $$\beta$$ be the two distinct roots of the equation $$2x^{2} - 6x + k = 0$$, such that ( $$\alpha + \beta$$) and $$\alpha \beta$$ are the distinct roots of the equation $$x^{2} + px + p = 0$$. Then, the value of 8(k - p) is

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Question 52

A mixture P is formed by removing a certain amount of coffee from a coffee jar andÂ replacing the same amount with cocoa powder. The same amount is again removedÂ from mixture P and replaced with same amount of cocoa powder to form a newÂ mixture Q. If the ratio of coffee and cocoa in the mixture Q is 16 : 9, then the ratio ofÂ cocoa in mixture P to that in mixture Q is

Question 53

The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimumÂ duration, in minutes, after 8.48 am when this angle increases by 50% is

Question 54

Gita sells two objects A and B at the same price such that she makes a profit of 20%Â on object A and a loss of 10% on object B. If she increases the selling price such thatÂ objects A and B are still sold at an equal price and a profit of 10% is made on object B,Â then the profit made on object A will be nearest to

Question 55

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of $$x$$ km till then, whereÂ $$x$$ is a whole number and is palindromic, i.e., $$x$$ remains unchanged when its digits are reversed. At the endÂ of the trip, the car had travelled a total of 26862 km till then, this number again being palindromic. IfÂ Brishti never drove at more than 110 km/h, then the greatest possible average speed at which she droveÂ during the trip, in km/h, was

Question 56

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls hasÂ the same marks while each of the boys has the same marks. If the marks of any girl isÂ at most double the marks of any boy, but not less than the marks of any boy, then theÂ number of possible distinct integer values of the total marks of 2 girls and 6 boys is

Question 57

The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7,Â respectively. In the first year, they get salary hikes of 20%, 25% and 20%, respectively.Â In the second year, Sita and Mita get salary hikes of 40% and 25%, respectively, andÂ the salary of Gita becomes equal to the mean salary of the three friends. The salaryÂ hike of Gita in the second year is

Question 58

Arvind travels from town A to town B, and Surbhi from town B to town A, bothÂ starting at the same time along the same route. After meeting each other, ArvindÂ takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. IfÂ Arvind travelled at a speed of 54 km/h, then the distance, in km, between town AÂ and town B is

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Question 59

Anil invests Rs. 22000 for 6 years in a certain scheme with 4% interest per annum,Â compounded half-yearly. Sunil invests in the same scheme for 5 years, and thenÂ reinvests the entire amount received at the end of 5 years for one year at 10%Â simple interest. If the amounts received by both at the end of 6 years are same, thenÂ the initial investment made by Sunil, in rupees, is

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Question 60

The amount of job that Amal, Sunil and Kamal can individually do in a day, are inÂ harmonic progression. Kamal takes twice as much time as Amal to do the sameÂ amount of job. If Amal and Sunil work for 4 days and 9 days, respectively, KamalÂ needs to work for 16 days to finish the remaining job. Then the number of daysÂ Sunil will take to finish the job working alone, is

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Question 61

Let C be the circle $$x^{2} + y^{2} + 4x - 6y - 3 = 0$$ and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to $$60^{\circ}$$. Then, the point at which L touches the line $$x$$ = 6 is

Question 62

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 :Â 4. If AC and BD intersect at the point E, then AE : CE equals

Question 63

In a right-angled triangle âˆ†ABC, the altitude AB is 5 cm, and the base BC is 12 cm. PÂ and Q are two points on BC such that the areas of $$\triangleÂ ABP, \triangleÂ ABQ$$ and $$\triangleÂ ABC$$ are inÂ arithmetic progression. If the area of âˆ†ABC is 1.5 times the area of $$\triangleÂ ABP$$, the lengthÂ of PQ, in cm, is

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Question 64

For some positive and distinct real numbers $$x, y$$ and z, if $$\frac{1}{\sqrt{y}+\sqrt{z}}$$ is the arithmetic mean of $$\frac{1}{\sqrt{x}+\sqrt{z}}$$ and $$\frac{1}{\sqrt{x}+\sqrt{y}}$$, then the relationship which will always hold true, is

Question 66

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on theÂ first day, the number of organisms on any day is equal to 3 more than twice the number on the previousÂ day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is

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SP - CP - Discount - Marked Price
Simple and Compound Interest
Work - Efficiency
Number of Factors
Average of n terms
Percentage change
Discriminant Formulas
Quadratic Roots Formulas
Properties of circle and triangle
Quadratic Equation - Given Roots.
AM GM HM Formulas
Similarlity and Congruency of triangles
Angles subtended on a circle
Linear and Quadratic inequalities
Modulus properties
Work - Time & Efficiency
Clocks - Relative Velocity
Speed - Distance - Time
G.P. - Formulas and Properties
Triangle - Area
Arithmetic Mean
Circle - Properties
AM GM HM Inequality
Replacement of solution
Propertie of tangents and chords in a circle
Properties of Factors
Arrangement, permutation and combination formulas
Properties of logarithm

Linear Equations
Inequalities
Quadratic Equations
Averages, Ratios & Proportions
Profit & Loss
Number Systems
Time, Distance & Work
Geometry
Progressions & Series
Probability Combinatorics
Venn Diagrams
Data Sufficiency
Logarithms, Surds & Indices
Functions, Graphs & Statistics
Miscellaneous

DI Basics
DI Charts
Data Interpretation
DI Data Change Over Period
Tables With Missing Values
DI Venn Diagrams
DI Special Charts
DI Maxima & Minima
Quant-Based DI
DI Connected Sets
DI Miscellaneous
LR Arrangement
LR Selections With Conditions
LR Coins & Weights
LR Truth & Lie
LR Puzzles
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LR Games & Tournaments
2D & 3-D LR
Quant-Based LR
LR Miscellaneous