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cat 2020 Complete Paper Solution | Quant Slot 1

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Question 1.

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

Question 2.

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

Question 3.

Veeru invested Rs 10000 at 5% simple annual interest, and exactly after two years, Joy invested Rs 8000 at 10% simple annual interest. How many years after Veeru’s investment, will their balances, i.e., principal plus accumulated interest, be equal?

Question 4.

A train travelled at one-thirds of its usual speed, and hence reached the destination 30 minutes after the scheduled time. On its return journey, the train initially travelled at its usual speed for 5 minutes but then stopped for 4 minutes for an emergency. The percentage by which the train must now increase its usual speed so as to reach the destination at the scheduled time, is nearest to

Question 5.

If $log_{4}$ 5 = ( $log_{4}$ y) ( $log_{6}$ √5), then y equals

Question 6.

The number of real-valued solutions of the equation $2^x$ + $2^{-x}$ = 2 - $(x - 2)^2$ is

infinite

Question 7.

A straight road connects points A and B. Car 1 travels from A to B and Car 2 travels from B to A, both leaving at the same time. After meeting each other, they take 45 minutes and 20 minutes, respectively, to complete their journeys. If Car 1 travels at the speed of 60 km/hr, then the speed of Car 2, in km/hr, is

100

Question 8.

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

Question 9.

If x = $(4096)^{7+4√3}$ , then which of the following equals 64?

\frac{x^{7/2}}{x^{4/√3}}

\frac{x^{7}}{x^{4√3}}

\frac{x^{7/2}}{x^{2√3}}

\frac{x^{7}}{x^{2√3}}

Question 10.

The mean of all 4 digit even natural numbers of the form 'aabb', where a>0, is

5544

4466

4864

5050

Question 11.

The number of distinct real roots of the equation $(x + \frac{1}{x})^{2}$ - 3(x + $\frac{1}{x}$ ) + 2 = 0 equals

Question 12.

A person spent Rs 50000 to purchase a desktop computer and a laptop computer. He sold the desktop at 20% profit and the laptop at 10% loss. If overall he made a 2% profit then the purchase price, in rupees, of the desktop is

Question 13.

Among 100 students, $x_{1}$ have birthdays in January, $x_{2}$ have birthdays in February, and so on. If $x_{0}$ = max( $x_{1}$ , $x_{2}$ , ..., $x_{12}$ ), then the smallest possible value of $x_{0}$ is

Question 14.

Two persons are walking beside a railway track at respective speeds of 2 and 4 km per hour in the same direction. A train came from behind them and crossed them in 90 and 100 seconds, respectively. The time, in seconds, taken by the train to cross an electric post is nearest to

Question 15.

How many distinct positive integer-valued solutions exist to the equation $(x^{2} - 7x + 11)^{(x^{2} - 13x + 42)}$ = 1?

Question 16.

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0, and y ≤ 1 is

Question 17.

A solid right circular cone of height 27 cm is cut into 2 pieces along a plane parallel to it's base at a height of 18 cm from the base. If the difference in the volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

264

232

243

256

Question 18.

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of the circle to the area of the rhombus is

\frac{2π}{15}

\frac{6π}{25}

\frac{3π}{25}

\frac{5π}{18}

Question 19.

Leaving home at the same time, Amal reaches office at 10:15 am if he travels at 8kmph, and at 9:40 am if he travels at 15kmph. Leaving home at 9:10 am, at what speed, in kmph, must he travel so as to reach office exactly at 10:00 am?

Question 20.

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

Question 21.

If y is a negative number such that $2^{y^{2}log_{3}5}$ = $5^{log_{2}3}$ , then y equals

log_{2}

(1/3)

log_{2}

(1/5)

-log_{2}

(1/3)

-log_{2}

(1/5)

Question 22.

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches opposite two sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

3√π(5 +

\frac{12}{π}

)

4√π(3 +

\frac{12}{π}

)

5√π(3 +

\frac{12}{π}

)

3√π(

\frac{5}{2}

\frac{6}{π}

)

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cat 2020 Complete Paper Solution | Quant Slot 1

Question Palette

Question 1.

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

Question 2.

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

Question 3.

Veeru invested Rs 10000 at 5% simple annual interest, and exactly after two years, Joy invested Rs 8000 at 10% simple annual interest. How many years after Veeru’s investment, will their balances, i.e., principal plus accumulated interest, be equal?

Question 4.

Question 5.

If log4log_{4}log4​ 5 = (log4log_{4}log4​ y) (log6log_{6}log6​ √5), then y equals

Question 6.

The number of real-valued solutions of the equation 2x2^x2x + 2−x2^{-x}2−x = 2 - (x−2)2(x - 2)^2(x−2)2 is

Question 7.

Question 8.

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

Question 9.

If x = (4096)7+4√3(4096)^{7+4√3}(4096)7+4√3, then which of the following equals 64?

Question 10.

The mean of all 4 digit even natural numbers of the form 'aabb', where a>0, is

Question 11.

The number of distinct real roots of the equation (x+1x)2(x + \frac{1}{x})^{2}(x+x1​)2 - 3(x + 1x\frac{1}{x}x1​ ) + 2 = 0 equals

Question 12.

A person spent Rs 50000 to purchase a desktop computer and a laptop computer. He sold the desktop at 20% profit and the laptop at 10% loss. If overall he made a 2% profit then the purchase price, in rupees, of the desktop is

Question 13.

Among 100 students, x1x_{1}x1​ have birthdays in January, x2x_{2}x2​ have birthdays in February, and so on. If x0x_{0}x0​ = max(x1x_{1}x1​, x2x_{2}x2​, ..., x12x_{12}x12​), then the smallest possible value of x0x_{0}x0​ is

Question 14.

Two persons are walking beside a railway track at respective speeds of 2 and 4 km per hour in the same direction. A train came from behind them and crossed them in 90 and 100 seconds, respectively. The time, in seconds, taken by the train to cross an electric post is nearest to

Question 15.

How many distinct positive integer-valued solutions exist to the equation (x2−7x+11)(x2−13x+42)(x^{2} - 7x + 11)^{(x^{2} - 13x + 42)}(x2−7x+11)(x2−13x+42) = 1?

Question 16.

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0, and y ≤ 1 is

Question 17.

A solid right circular cone of height 27 cm is cut into 2 pieces along a plane parallel to it's base at a height of 18 cm from the base. If the difference in the volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

Question 18.

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of the circle to the area of the rhombus is

Question 19.

Leaving home at the same time, Amal reaches office at 10:15 am if he travels at 8kmph, and at 9:40 am if he travels at 15kmph. Leaving home at 9:10 am, at what speed, in kmph, must he travel so as to reach office exactly at 10:00 am?

Question 20.

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

Question 21.

If y is a negative number such that 2y2log352^{y^{2}log_{3}5}2y2log3​5 = 5log235^{log_{2}3}5log2​3, then y equals

Question 22.

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches opposite two sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

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If $log_{4}$ 5 = ( $log_{4}$ y) ( $log_{6}$ √5), then y equals

The number of real-valued solutions of the equation $2^x$ + $2^{-x}$ = 2 - $(x - 2)^2$ is

If x = $(4096)^{7+4√3}$ , then which of the following equals 64?

The number of distinct real roots of the equation $(x + \frac{1}{x})^{2}$ - 3(x + $\frac{1}{x}$ ) + 2 = 0 equals

Among 100 students, $x_{1}$ have birthdays in January, $x_{2}$ have birthdays in February, and so on. If $x_{0}$ = max( $x_{1}$ , $x_{2}$ , ..., $x_{12}$ ), then the smallest possible value of $x_{0}$ is

How many distinct positive integer-valued solutions exist to the equation $(x^{2} - 7x + 11)^{(x^{2} - 13x + 42)}$ = 1?

If y is a negative number such that $2^{y^{2}log_{3}5}$ = $5^{log_{2}3}$ , then y equals