## Lakshya Education MCQs

Question: Triangle PQR is equilateral with sides of length 5 units. O is any point in the interior of triangle PQR. Segment OL, OM, and ON are perpendicular to PQ, PR and QR respectively. Find the sum of segments OL, OM and ON.
Options:
 A. 10√32 B. 3√32 C. 5√32 D. Data insufficient
: C Conventional Method Join PO, OQ and OR There are 3 triangles whose sum of areas = area of the equilateral triangle Area of a triangle =12× base × height Thus Area of POR=12×PR×OM Area of QOR=12×QR×ON Area of POQ=12×PQ×OL Sum = Area of POR+Area of QOR+Area of POQ= Area of PQR =12×PR×OM+12×QR×ON+12×PQ×OL Area of an equilateral triangle =34×a2 , where 'a' is the length of 1 side
34×52=12×PR×(OM+OL+ON) (Since, PR = QR = PQ)
2534=12×5×(OM+OL+OM)
OM+ON+OL=532 Shortcut:- Assumption Note the most important word in the question-ANY. We can assume ANY point inside the triangle. We will consider a point infinitely close to P. Thus, the diagram will look as follows. The answer can now be obtained in a single step.
OL = 0, OM =0 and ON is the height of an equilateral triangle =532
Thus, OL+OM+ON=532

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## More Questions on This Topic :

Question 1. Two rugby teams (thirty players) have been selected to play for their school. Five of the players speak Spanish, Afrikaans and English. Nine of them speak only Spanish and English. Twenty speak Afrikaans, of which twelve also speak Spanish. Eighteen speak English. No one speaks only Spanish. How many players speak only English?___

: From the data given in the question, we can draw a Venn diagram as follows Since N = 30, the sum of the shaded region is 10. Hence 9 + x = 10 x = 1.
Question 2. Consider three numbers a, b and c Max(a,b,c)+Min(a,b,c)=13. Median (a,b,c)-Mean(a,b,c)=2. Find the median of a, b and c.
1.    11.5
2.    9
3.    9.5
4.    12.5
5.    5.12 am
: C

The numbers can be arranged in ascending order min < median < max

We have min + max =13

Mean=min+med+max3=median12+med3=2

On solving we get median =9.5.
Question 3. Two circles are placed in an equilateral triangle as shown in Fig. What is the ratio of the area of smaller circle to that of equilateral triangle?
: C  Question 4. For the given equation x9 + 5x8 - x3 + 7x+3=0 how many maximum real roots are possible?
1.    3
2.    5
3.    6
4.    7
: B

Let f(x) =x9 + 5x8 - x3 + 7x+3 = 0 We can solve this question easily using Descartes' Rule According to Descartes' rule maximum number of positive real roots = number of sign changes in f(x) = 2 Similarly, Maximum number of negative real roots = number of sign changes in f(-x) Note: To find f(-x) replace "x” by "-x” in each instance therefore f(-x) = -x9 + 5x8 +x3 -7x+3 Maximum number of negative real roots = number of sign changes in f(-x) = 3 Zero cannot be a root because constant part is also involved in equation. So maximum number of real roots = 2 + 3 = 5

DESCARTES' RULE (Points to Remember)
Maximum number of positive real roots = number of sign changes in f(x)
Maximum number of negative real roots = number of sign changes inf(-x)
If a constant term is present, do not consider 0 as a root or else,you need to consider it
Question 5. Find the inverse of
: A

Conventional method Replacing f(x) with y, the given function is y= (4-(x-7)3)1/5 To find the inverse function, we need to write an equation for x in terms of y Raising both sides to the power 5, we get y5 = 4 - (x-7)3 (x-7)3 = 4- y5 Taking cube root on both sides x-7 = (4 - y5)1/3 x = 7 + (4 - y5)1/3 We have an expression for x interms of y. We get f-1(x) if we replace y with x in the above equation Hence, f-1(x) = 7 + (4 - x5)1/3. Answer option (a) Shortcut What is the basic definition of an inverse function? An Inverse function is one which can be expressed as f(y) = x when f(x) = y Using, this very definition, we can use a common sense approach to arrive at the answer In no time at all 1) Substitute a suitable value for x (it can be any value). in this case, for our convenience we will take x=7. Then at x=7, f(x)= 41//5 2) Now use the definition of inverse function. Put x=41/5 in each of the answer options and see where you get f(x)=7. You are just exchanging "x” and "y” values! This is the concept of inverse function after all! Only option (a) gives 7 at x=41/5 All other options can be eliminated as for x=41/5, we do not get 7. Answer is option (a)
Question 6. ABCDEF is a regular hexagon, with O as the centre. There is another regular hexagon with 2 of the end points PQ inscribed in the above hexagon (again with centre O). Find the area of the unshaded region, given that FPQC lies on a straight line and FP=CQ=2 and FP = 13 FO.

1.    20√3
2.    24√3
3.    18√3
4.    28√3
The hexagon can be graphically divided as follows Hence, there are totally 54 equilateral triangles out of which 24 are unshaded. Each of the triangles have aside 2. Total unshaded area = 34 x 4 x 24 = 243