Question 1 |

An ant walks in a straight line on a plane leaving behind a trace of its
movement. The initial position of the ant is at point P facing east.

The ant first turns 72^{\circ} anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.

1. moves forward for 10 cm.

2. turns 144^{\circ} clockwise.

The pattern made by the trace left behind by the ant is

The ant first turns 72^{\circ} anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.

1. moves forward for 10 cm.

2. turns 144^{\circ} clockwise.

The pattern made by the trace left behind by the ant is

A | |

B | |

C | |

D |

Question 1 Explanation:

MTA - Marks to All

Question 2 |

Consider the following equations of straight lines:

Line L1: 2x - 3y = 5

Line L2: 3x + 2y = 8

Line L3: 4x - 6y = 5

Line L4: 6x - 9y = 6

Which one among the following is the correct statement?

Line L1: 2x - 3y = 5

Line L2: 3x + 2y = 8

Line L3: 4x - 6y = 5

Line L4: 6x - 9y = 6

Which one among the following is the correct statement?

L1 is parallel to L2 and L1 is perpendicular to L3 | |

L2 is parallel to L4 and L2 is perpendicular to L1 | |

L3 is perpendicular to L4 and L3 is parallel to L2 | |

L4 is perpendicular to L2 and L4 is parallel to L3 |

Question 3 |

In a partnership business the monthly investment by three friends for the first
six months is in the ratio 3: 4: 5. After six months, they had to increase their
monthly investments by 10%, 15% and 20%, respectively, of their initial
monthly investment. The new investment ratio was kept constant for the next
six months.

What is the ratio of their shares in the total profit (in the same order) at the end of the year such that the share is proportional to their individual total investment over the year?

What is the ratio of their shares in the total profit (in the same order) at the end of the year such that the share is proportional to their individual total investment over the year?

22 : 23 : 24 | |

22 : 33 : 50 | |

33 : 46 : 60 | |

63 : 86 : 110 |

Question 4 |

Both the numerator and the denominator of frac{3}{4} are increased by a positive integer, x, and those of frac{15}{17} are decreased by the same integer. This operation results in the same value for both the fractions.

What is the value of x?

What is the value of x?

1 | |

2 | |

3 | |

4 |

Question 5 |

x:y:z=\frac{1}{2}:\frac{1}{3}:\frac{1}{4}

What is the value of frac{x+z-y}{y}

What is the value of frac{x+z-y}{y}

0.75 | |

1.25 | |

2.25 | |

3.25 |

Question 6 |

In the square grid shown on the left, a person standing at P2 position is required
to move to P5 position.

The only movement allowed for a step involves, "two moves along one direction followed by one move in a perpendicular direction". The permissible directions for movement are shown as dotted arrows in the right.

For example, a person at a given position Y can move only to the positions marked X on the right.

Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is

The only movement allowed for a step involves, "two moves along one direction followed by one move in a perpendicular direction". The permissible directions for movement are shown as dotted arrows in the right.

For example, a person at a given position Y can move only to the positions marked X on the right.

Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is

4 | |

5 | |

6 | |

7 |

Question 6 Explanation:

Minimum number of steps required are:

P2 -> Q4 -> S3 -> T5 -> R4 -> P5

P2 -> Q4 -> S3 -> T5 -> R4 -> P5

Question 7 |

The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam.

From the data presented above, which one of the following is CORRECT?

mean \gt mode \gt median | |

mode \gt median \gt mean | |

mode \gt mean \gt median | |

median \gt mode \gt mean |

Question 7 Explanation:

Mean=\frac{3 \times 3+9\times 4+11\times 5+7\times 6+14\times 7+2\times 8+4\times 9}{3+9+11+7+14+2+4}=5.84

\begin{aligned} Median&=\frac{\left ( \frac{n}{2} \right )^{th}+\left ( \frac{n}{2} +1\right )^{th}}{2}\\ &=\frac{\left ( \frac{50}{2} \right )^{th}+\left ( \frac{50}{2} +1\right )^{th}}{2}\\ &=\frac{25^{th}+26^{th}}{2}\\ &=\frac{6+6}{2}=6 \end{aligned}

Mode = frequently occured observation (data with high frequency)=7

\begin{aligned} Median&=\frac{\left ( \frac{n}{2} \right )^{th}+\left ( \frac{n}{2} +1\right )^{th}}{2}\\ &=\frac{\left ( \frac{50}{2} \right )^{th}+\left ( \frac{50}{2} +1\right )^{th}}{2}\\ &=\frac{25^{th}+26^{th}}{2}\\ &=\frac{6+6}{2}=6 \end{aligned}

Mode = frequently occured observation (data with high frequency)=7

Question 8 |

P invested Rs. 5000 per month for 6 months of a year and Q invested Rs. x per
month for 8 months of the year in a partnership business. The profit is shared in
proportion to the total investment made in that year.

If at the end of that investment year, Q receives \frac{4}{9} of the total profit, what is the value of x (in Rs)?

If at the end of that investment year, Q receives \frac{4}{9} of the total profit, what is the value of x (in Rs)?

2500 | |

3000 | |

4687 | |

8437 |

Question 8 Explanation:

Let total profit =K

So, profit byQ=\frac{4}{9}K

profit by P=K-\frac{4}{9}K=\frac{5}{9}K

\begin{aligned} \frac{\text{Profit by P}}{\text{Profit by Q}}&=\frac{5}{4}\\ \frac{5}{4}&=\frac{C_PT_P}{C_QT_Q}\\ &=\frac{5000 \times 6 }{x \times 8}\\ x&=\frac{5000 \times 6 \times 4}{8 \times 5}\\ x&=3000 \end{aligned}

So, profit byQ=\frac{4}{9}K

profit by P=K-\frac{4}{9}K=\frac{5}{9}K

\begin{aligned} \frac{\text{Profit by P}}{\text{Profit by Q}}&=\frac{5}{4}\\ \frac{5}{4}&=\frac{C_PT_P}{C_QT_Q}\\ &=\frac{5000 \times 6 }{x \times 8}\\ x&=\frac{5000 \times 6 \times 4}{8 \times 5}\\ x&=3000 \end{aligned}

Question 9 |

Two straight lines pass through the origin (x_0,y_0)=(0,0). One of them passes
through the point (x_1,y_1)=(1,3) and the other passes through the point
(x_2,y_2)=(1,2).

What is the area enclosed between the straight lines in the interval [0,1] on the x-axis?

What is the area enclosed between the straight lines in the interval [0,1] on the x-axis?

0.5 | |

1 | |

1.5 | |

2 |

Question 9 Explanation:

Equation of first straight line passing through (0,
0) and (1,3)

\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{3-0}{1-0}(x-0) \\ y&=3x \end{aligned}

Equation of second stragith line passing through (0,0) and (1,2)

\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{2-0}{1-0}(x-0) \\ y&=2x \end{aligned}

Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5

\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{3-0}{1-0}(x-0) \\ y&=3x \end{aligned}

Equation of second stragith line passing through (0,0) and (1,2)

\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{2-0}{1-0}(x-0) \\ y&=2x \end{aligned}

Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5

Question 10 |

In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equals parts. The length of each subdivided part in cm is an integer. The minimum area of the triangle PQR possible, in cm^2, is

18 | |

24 | |

48 \sqrt{3}
| |

144 \sqrt{3} |

Question 10 Explanation:

For \left(\frac{a}{4}, \frac{a}{6}, \frac{a}{8}\right) to be integer, a must be LCM of 4, 6 and 8. So a = 24

\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}

There are 10 questions to complete.

And solution??

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