JIPMAT 2023 Question Paper

Let , $$\sqrt{3+\sqrt{5}}=X$$

So, $$X^2=3+\sqrt{5}$$ 
              = $$\ \dfrac{\ 5}{2}+\dfrac{\ 1}{2}+\sqrt{5}$$
              = $$\ \dfrac{\ 5}{2}+\dfrac{\ 1}{2}+2\ \left(\dfrac{\ \sqrt{5}}{\sqrt{2}\times\sqrt{2}}\right)$$

              = $$\left(\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}\right)^2$$

Therefore, X = $$\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}$$
                               

Statement A : If average of x, y, z is y. 
                          x + y + z = 3y
                          So, 2y = x + z is true statement.

Statement B : The given data set can be written in ascending order = { 15, 28, 31, 32, 43, 44, 51, 56, 72} . Therefore in the data set the   median is 43.

Statement C : Mode is known as the data point which has highest frequency (or) the number which gets repeated in the set for maximum no of times. 
                    Therefore, in the given set {3} repeats for 4 times which is maximum among all. Hence a true statement.

As $$k^{2}-1$$ is divisible by 8, we can say that this particular number $$k^{2}-1$$ is even integer, as all multiples of 8 are even integers only.

So, $$k^{2}-1$$ = Even integer
      $$k^{2}$$ = Odd integer

For "(k)(k)" to be a odd number, the only possibility is that 'K' has to be a odd integer. 

The well is dug which will resemble a shape of cylinder with a diameter of 7 m and height equal to 30 m.
Hence, the volume of earth which is obtained by digging well will be = $$\pi\ \left(r\right)^2\left(h\right)$$
                              Volume = $$\dfrac{\ 22}{7}\left(3.5\right)^2\left(30\right)\ $$

As we are using the entire earth to make a platform of certain height, it will resemble a cuboid whose volume should be ''(L)(B)(H)".
                                           Given : L = 22 m, B = 14 m .
So, $$\dfrac{\ 22}{7}\left(3.5\right)^2\left(30\right)\ $$ = (22)(14)(H)

Therefore, H = 3.75 metres

For a number to be divisible by 88, it has to be divisible by both 11 & 8 .

Divisibility rule of 8 :  The number formed by last three digits of a given number has to be divisible by 8.
                 So, "65b" has to be a multiple of 8. 
                 This is only possible at b = 6. 

Divisibility rule of 11 : A number is divisible by 11 if the difference between the sum of the digits at the odd places and the sum of the digits at the even places is either 0 or a multiple of 11.
                    So, Sum of the digits at odd places = 5 + 3 + 4 + 5 = {17}.
                          Sum of the digits at even places = a + 2 + 6 + b = {a + b + 8}.
                                             The difference is |a+b-9| .

As we deduced 'b' is 6 : |a-3| which has to zero only, because it can't be 11 as the maximum value  'a' can take is a single digit number.
                                                       Therefore, a = 3.

Hence the value of 2a + 5b = 2(3) + 5(6) = 36.

The cost of pendrive A and B are in the ratio 5:6. Let the price of pendrive A be 5x and the price of pendrive B be 6x. 

Total Cost Price = 5x + 6x = 11x

She sells them at a 10% profit each, and thus her overall profit is Rs 22. 

Total Selling Price = (5x + 6x) * 1.1 = 12.1x

Profit = 12.1x - 11x = 22

Thus, x = 20

Thus, the cost of Pendrive A = Rs 100, and the cost of Pendrive B = Rs 120. Thus, overall cost price = Rs 220.

She sells Pendrive A at 20% less price, thus its selling price will be = Rs 80

She sells Pendrive B at a 30% higher price, so its selling price will be = Rs 120 * 1.3 = Rs 156.

Overall selling price = 80 + 156 = Rs 236

Thus, Total Profit = 236 - 220 = Rs 16

Given , $$\sin(\theta)= \cos(\theta)$$ .
           $$\tan(\theta)$$ = 1 
           this implies,  $$\sin(\theta)= \cos(\theta)$$ = $$\dfrac{\ 1}{\sqrt{2}}$$

Therefore , $$\sin^{4}(\theta) + \cos^{4}(\theta)$$ = $$\dfrac{\ 1}{4}+\dfrac{\ 1}{4}$$ = $$\dfrac{\ 1}{2}$$.

Let all the 25 observations be {$$X_1,X_2,X_3,.......X_25$$}.

Mean of first 13 observations = $$\dfrac{\ X_1+X_2+X_3+.....X_{13}}{13}=\ 32$$ ------Equation 1

Mean of last 13 observations = $$\dfrac{\ X_{13}+X_{14}+X_{15}+.....X_{25}}{13}=\ 39$$.......Equation 2

Adding Equation 1 & 2 : {$$X_1+X_2+X_3+......X_{25}$$} + ($$X_{13}$$) = 13(32 + 39) = 13(71) 

Mean of all 25 observations = $$\dfrac{\ X_1+X_2+X_3+.....X_{25}}{25}=\ 36$$
this implies :   {$$X_1+X_2+X_3+......X_{25}$$} = 25(36)

Therefore, $$X_{13}=13(71)-25(36)$$ = $$923-900$$ = 23.

The number of zeroes of f(x) means the no of real roots of f(x) .
The given graph cuts X axis for three times i.e at those values of 'x', the value of f(x) would be zero.
Hence, the answer is 3.

Statement A : Right Circular cylinder with r = 7, h = 10.
                     The volume = $$\pi\ \left(r\right)^2\left(h\right)$$ = $$\dfrac{\ 22}{7}\left(7\right)^2\left(10\right)=1540$$
                             Hence, its a true statement.

Statement B :  r = 14, h = 30
                       Total Surface Area = $$2\pi\left(r\right)\left(r+h\right)$$
                                                   = $$\dfrac{\ 44}{7}\left(14\right)\left(44\right)$$
                                                   = $$88\times44$$
                                                   = 3872.
                               Hence, its a true statement.