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Consider a rectangle $ABCD$ having $5,7,6,9$ points in the interior of the line segments $AB,$ $CD,$ $BC,$ $DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta -\alpha )$ is equal to :

Which of the following Combination is not right?

A new one-way road is built from M to N. Each car now has three possible routes to travel from A to B: A-M-B, A-N-B and A-M-N-B. On the road from M to N, one car takes 7 minutes and each additional car increases the travel time per car by 1 minute. Assume that any car taking the A-M-N-B route travels the A-M portion at the same time as other cars taking the A-M-B route, and the N-B portion at the same time as other cars taking the A-N-B route.
How many cars would the police department order to take the A-M-N-B route so that it is not possible for any car to reduce its travel time by not following the order while the other cars follow the order? (Assume that the police department would never order all the cars to take the same route).___

Find the minimum value of a+b+c+d in the equation $x^5-a{x}^4+b{x}^3-c{x}^2+dx-243=0,$ if it is given that the roots are positive real numbers ?

The sum of the values of '$m$' for which the equations $3x^2+4mx+2=0$ and $2x^2+3x-2=0$ may have a common root, is

Let $y=\frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+.....}}}}$ What is the value of y?

In the figure given below, If the areas of the two regions are equal then which of the following is true?

If $X=4^n-3n-1;nϵN$ and $Y=9(n-1):nϵN,$ then precisely

Does the equation y = (x - a)(x - b) intercept the x-axis at the point (2,0)?

(1) ab = -8

(2) -2 - a = b

At a cricket match, out of the 2 lakh spectators, 90% were fans of team A, 85% of Team B, 70% of Team C, 75% of Team D and 80% of Team E. What is the minimum number of spectators who are fans of all 5 teams?

Each of the ten students was allowed to improve his/her score in exactly one paper of choice with the objective of maximizing his/her final score. Everyone scored 100 in the paper in which he or she chose to improve. After that, the topper among the ten students was:

If $S_1,S_2,S_3,....S_n$ are the sums of infinite geometric series whose first terms are 1, 2, 3,...n and whose ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{(n+1)}$ respectively, then find the value of $S_1^2+S_2^2+S_3^2+....+S_6^2$

A cone, a hemisphere and a cylinder stand on equal bases and have the same height, the height being equal to the radius of the circular base. Their total surface areas are in the ratio:

1. 2

What is the ratio of Doctors in 2007 to nurses in 2010?

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



Each of $4$ boys attends a school gathering with both of his parents. The number of ways these $12$ people can be divided into groups of three each group containing a boy, a male parent, a female parent such that



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{No boy is with both of his parents in his group is}& \text{(P)}& 1\\ \text{(B)}& \text{Exactly two boys are with both their parents is N, then N is}& \text{(Q)}& 453\\ \text{(C)}& \text{Exactly two particular boys are with both of their parents is N, then N is}& \text{(R)}& 3\\ \text{(D)}& \text{Every boy is with both his parents is N then N is}& \text{(S)}& 18\end{array}$



Which of the following is CORRECT ?

In the following question, there is a sentence which is divided into four parts. Choose the fragment that carries an error (if any). Ignore punctuation errors if any.

A growing body of evidence is supporting the "use-it-or-lose-it" hypothesis of aging well. / It says staying mentally active helps keep the brain in shape. / So far, research has focused on reading books, / crossword puzzles and learning languages.

The sum of 19 terms of the series $S_n$= 4+10 + 26 + 66 + 162 + 386 + ...... is

In which of the given years the number of candidates appeared from state D has the maximum percentage of qualified candidates?

The remainder obtained when $(1^2+2^2+3^2+4^2+5^2+\dots {100}^2)$ is divided by 7 is:

The diagram shows a sequence of squares, decreasing in size and alternating between white and grey. Each new square is drawn by connecting the midpoints of the sides of the larger square that precedes it. If this process is continued indefinitely, then the fraction of the area that will ultimately be grey is

If Batch II has the password – ‘ came along net or else key lot’ , what could be the password for Batch IV i.e, 12 noon to 1 p.m

If $9^{x-\frac{1}{2}}-2^{2x-2}=4^x-3^{2x-3},$ then x is

Study the following information carefully and answer the given questions.
Amongst five friends, each got a different percentage of marks in the examination. Poonam scored more than Ben but less than Ajay. Ajay scored 70% marks. Shreya scored less marks only than Kim. The one who scored the minimum marks scored 65% marks and the one who scored the highest, scored 87% marks.
Who amongst the following scored the second lowest marks?

Total advertisement revenue generated from online version by magazine U in all the given months together is by what percent more than the total advertisement revenue generated from printed version by the same magazine in all the given months together?

If a, b, c >0, a + b + c =6 and f(x) = $\left(\frac{6}{x}\right)$ - 1, find the minimum value of f(a).f(b).f(c)?

Slope of the line joining the points A(-3,5) and B(4,-1) is _______.