PGDBA 2019

$$P^c \cap Q \subseteq R^c \cap Q$$

$$P^c \cap Q^c \subseteq R^c \cap Q^c$$

=> $$\left(P^c\cap Q\right)U\ \left(P^c\cap Q^c\right)\subseteq\left(R^c\cap Q\right)U\ \left(R^c\cap Q^c\right)$$

=> $$P^c\cap\left(Q\ U\ Q^c\right)\subseteq R^c\cap\left(Q\ U\ Q^c\right)$$

=> $$P^c\subseteq R^c$$

=>$$R\subseteq P$$

We have 4 cases 
1: Largest side = 12  then Sides of the triangle are 8,10,12
2: Largest side = 10  then Sides of the triangle are 10,8,6
3: Largest side = 8  then Sides of the triangle are 8,6,4
4: Largest side = 6  then Sides of the triangle are 6,4,2
According to the properties of the triangle,
Sum of any two sides of a triangle is greater than the third side and difference of any two sides of a triangle is lesser than the third side
All the above cases satisfy the properties of the triangle except case 4 
Hence C is the correct answer.

Let's find the point of intersection of the circle $$x^2 + y^2 = 9$$ and the parabola $$y^2 = 8x$$

$$x^2 + 8x= 9$$

On solving the equation we will get the values of x as 1,-9 

Since the angle between the tangents has to be found in the first Quadrant, value of x is 1

$$\therefore value of y = 2\sqrt{2}$$

Let's find the slope of the tangent to the circle and the parabola

On differentiating the equation of the circle ,we get 

$$2x+2y\frac{\text{d}y}{\text{d}x}=0$$

Slope of the tangent to the circle $$\ m_1= -1/ x$$

On differentiating the equation of the parabola ,we get 

$$2y\frac{\text{d}y}{\text{d}x}=8$$

Slope of the tangent to the parabola $$m_2= 4/ y$$

Angle between the tangents $$\tan \theta$$= $$\frac{m_1-m_2}{1+m_1m_2}$$

                                           =$$\frac{-1/2\sqrt{2}-4/2\sqrt{2}}{1+-1/2\sqrt{2}*4/2\sqrt{2}}$$

                                           =$$\tan^{-1} (\frac{5}{\sqrt{2}})$$

Hence C is the correct answer.

Sum of the interior angles of a polygon is given by (2n-4)90

Smallest interior angle = 120

Let the number of sides of the convex-polygon be n 

$$\frac{n}{2}\left(2(120)+\left(n-1\right)\times\ 5\right)=(2n-4)90$$

n(240+5n-5)=360n-720

$$5n^2+235n=360n-720$$

$$5n^2-125n+720=0$$

$$n^2-25n+144=0$$

n=9,16

for a polygon to be convex, each interior angle should be less than 180

The interior angle when n=16 is 120+15*5=195 So n is not equal to 16

Interior angle when n =9 is 120+8*5 =160 

No of sides of the polygon =9

B is the correct answer.

Let $$\sqrt{32\sqrt{32\sqrt{32.......}}}$$ = x

$$\sqrt{32x}$$ = x

Squaring on both sides , we get 

32x=$$x^{2}$$

$$\therefore x$$ = 32

Hence B is the correct answer.

The number of elements in set A = 10

The number of elements in set B = 9

for the mapping to be onto there will be a element in set B which has 2 pre-images from set A

Two elements can be selected in $$^{10}C_2 $$ ways and the elements in set B can be arranged in 9! ways 

$$\therefore$$ No of onto functions = $$^{10}C_2 \times 9!$$

A is the correct answer

All the words starting with the letter A =3!

All the words starting with the letter E =3!

All the words starting with the letter R =3!

All the words starting with the letter WA =2!

All the words starting with the letter WEAR =1

$$\therefore$$ position of the word 'WEAR' in this list is at number 3!*3 + 2+1

=21

Hence B is the correct answer.

Multiples of 7 are {301,308 ................................1099} n=115

Multiples of 13 are {312,323,.................................1092} n = 61

Multiples of 91 are {364,455.................................1092} n= 9

Number of integers which are divisible by either 7 or 13 but not both is 115+61-2*9

=158

Hence B is the correct answer.

Let's rewrite the equation $$24x + 7y =168$$ in intercept form

$$\frac{x}{7}+\frac{y}{24}$$ = 1

$$\therefore$$ The co-ordinates of the triangle are (7,0),(0,0),(0,24)

Diameter of the circumcircle formed by the vertices is given by the hypotenuse, which is between (7,0) and (0,24).

Distance between the co-ordinates (7,0) and (0,24) =$$\sqrt{7^{2} +24^{2}}$$ =25.

Hence C is the correct answer.

An even function is symmetric about the y axis. Hence if there are 10 non-differential points, 5 of them lie towards the left side of the y-axis and 5 towards the right. Origin cannot be a non-differential point for an even function.

Hence $$f'\left(0\right)=0$$