JIPMAT 2024 Question Paper

Reason (R) states that the sum of a geometric series having n terms is given by $$S_{n} = \dfrac{a(1-r^{n})}{1-r}$$, where a is the $$1^{st}$$ term and r is the common ratio. We know that this is the sum of geometric series, thus the reason is true.

Now Assertion (A) states that the sum of n terms of the Progression $$1+\dfrac{1}{2}+\dfrac{1}{2^{2}}+\dfrac{1}{2^{3}}+.... is \dfrac{2^{(n-1)}-1}{2^{(n-1)}}$$

Here a = 1 and r = 1/2. We will substitute these values in the formula given in (R).

$$S_n=\dfrac{1(1-\left(\frac{1}{2}\right)^n)}{1-\frac{1}{2}}$$

$$S_n=\dfrac{2\left(2^n-1\right)}{2^n}$$

$$S_n=\dfrac{2^n-1}{2^{n-1}}$$

This is not equal to $$\dfrac{2^{(n-1)}-1}{2^{(n-1)}}$$. Thus, the assertion is false. 

Let the efficiency of B be 10x. A is 40% less efficient than B, thus his efficiency will be 6x. Let the efficiency of C be y. Let the total work be W units. 

B can do the same work in 20% less time than C. This means - 

$$0.8\times\dfrac{W}{y}=\dfrac{W}{10x}$$

$$y=8x$$

Thus, efficiency of C be 8x. 

A and B together can complete 80% work in 12 days.

$$0.8W=\left(10x+6x\right)\times12$$

$$0.8W=192x$$

$$W=240x$$

We have to find out the number of days B and C will take to complete 60% of the work. Let us assume that they take D days to complete this amount of work.

$$0.6W=\left(10x+8x\right)\times D$$

$$144x=18x\times D$$

$$D=8$$ days. 

Probability of A speaks truth $$=P\left(A\right)=0.75$$
Probability of A speaks lie $$=P\left(\overline{A}\right)=0.25$$
Probability of B speaks truth $$=P\left(B\right)=0.8$$
Probability of B speaks lie $$=P\left(\overline{B}\right)=0.2$$

A and B will contradict each other when A is speaking the truth and B is speaking lie or A is speaking lie and B is speaking the truth.

$$P(C)=P\left(A\right)P\left(\overline{B}\right)+P\left(B\right)P\left(\overline{A}\right)$$

$$P(C)=0.75\times0.2+0.8\times0.25$$

$$P(C)=0.15+0.2=0.35$$

Thus, the chances of A and B contradicting each other is 35%. 

Let the cost price of each article be x.

By selling two articles for Rs 800, a person gains the cost price of three articles.

Gain = SP - CP

3x = 2 * 800 - 2x

5x = 1600

x = 320

Thus, the cost price of the article is Rs 320, and the selling price of the article is Rs 800

Profit %age $$=\dfrac{SP-CP}{CP}\times100=\dfrac{800-320}{320}\times100$$

Profit %age $$=150\%$$

Let the three equal distances be denoted by D. 

Total Distance covered $$=D+D+D=3D$$

Total time travelled $$=\dfrac{D}{a}+\dfrac{D}{b}+\dfrac{D}{c}=D\left(\dfrac{ab+bc+ca}{abc}\right)$$

Average speed = Total Distance Covered / Total Time travelled

$$=\dfrac{3D}{D\left(\frac{ab+bc+ca}{abc}\right)}$$

$$=\dfrac{3abc}{\left(ab+bc+ca\right)}$$

$$\alpha$$ is given as the root of the equation $$ax^2+bx+c=0$$. Thus -

$$a\alpha^2+b\alpha+c=0$$

$$a\alpha+b=-\dfrac{c}{\alpha}\rightarrow1$$

$$\beta$$ is given as the root of the equation $$ax^2+bx+c=0$$. Thus -

$$a\beta^2+b\beta+c=0$$

$$a\beta+b=-\dfrac{c}{\beta}\rightarrow2$$

We need to find the value of $$\dfrac{1}{a\alpha+b}+\dfrac{1}{a\beta+b}$$.

$$\dfrac{1}{a\alpha+b}+\dfrac{1}{a\beta+b}=-\dfrac{\alpha}{c}-\dfrac{\beta}{c}$$

$$\dfrac{1}{a\alpha+b}+\dfrac{1}{a\beta+b}=-\dfrac{\alpha+\beta}{c}$$

$$\dfrac{1}{a\alpha+b}+\dfrac{1}{a\beta+b}=-\dfrac{-b}{ac}$$

$$\dfrac{1}{a\alpha+b}+\dfrac{1}{a\beta+b}=\dfrac{b}{ac}$$

For the triangle to be a right-angled triangle, the sum of squares of the two smaller sides is equal to the square of the largest side. 

(A) 20 cm, 22 cm, 24 cm
$$20^2+22^2=400+484=884$$
$$24^2=576$$
Not a right angled triangle

(B) 15 cm, 32 cm, 37 cm
$$15^2+32^2=225+1024=1249$$
$$37^2=1369$$
Not a right angled triangle

(C) 11 cm, 60 cm, 61 cm
$$11^2+60^2=121+3600=3721$$
$$61^2=3721$$
This is a right angled triangle

(D) 6 cm, 8 cm, 10 cm
$$6^2+8^2=36+64=100$$
$$10^2=100$$
This is a right angled triangle

$$x=1$$ will make the polynomial $$x^{3}-6x^{2}+11x-6=0$$. Thus, $$x-1$$ is a factor of the given polynomial, so we will divide the given polynomial by this factor. Thus, we will get 

$$x^{3}-6x^{2}+11x-6=(x-1)(x^2-5x+6)$$

$$x^{3}-6x^{2}+11x-6=(x-1)(x-2)(x-3)$$

Thus, the factors of the given polynomial are $$x-1$$, $$x-2$$, and $$x-3$$. 

A circle is the most efficient shape — it covers the most area for a given perimeter, or needs the least perimeter for a given area.

As a regular polygon gets more sides, it starts to resemble a circle, and its perimeter for the same area becomes smaller.

1. Equilateral Triangle (3 sides): Fewest sides, largest perimeter.

2. Square (4 sides): More sides, smaller perimeter than a triangle.

3. Regular Pentagon (5 sides): More sides, smaller perimeter than a square.

4. Circle (infinite sides): Most efficient, smallest perimeter for given area.

Therefore, the perimeters in ascending order are:

(A) Circle < (D) Regular pentagon < (B) Square < (C) Equilateral triangle.

$$(x-3)$$ is a factor of $$x^{2}-px-q$$ and $$5x^{2}-3px-15q$$, thus, at x = 3, the value of polynomial will be equal to 0. Therefore, 

$$3p+q=9$$ and $$9p+15q=45$$

Solving the above equations, we will get $$p=\dfrac{5}{2}\ and\ q=\dfrac{3}{2}$$