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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?

    A
    B
    C
    D

    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

    A
    0
    B
    40
    C
    10
    D
    20

    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?

    A
    B
    C
    D

    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

    A
    58
    B
    67
    C
    50
    D
    61

    Question 5.

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

    A
    B
    C
    D

    Question 6.

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

    A
    infinite
    B
    0
    C
    1
    D
    2

    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

    A
    90
    B
    80
    C
    70
    D
    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

    A
    6
    B
    4
    C
    7
    D
    5

    Question 9.

    If x = (4096)7+43(4096)^{7+4√3}, then which of the following equals 64?

    A
    x7/2x4/3\frac{x^{7/2}}{x^{4/√3}}
    B
    x7x43\frac{x^{7}}{x^{4√3}}
    C
    x7/2x23\frac{x^{7/2}}{x^{2√3}}
    D
    x7x23\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

    A
    5544
    B
    4466
    C
    4864
    D
    5050

    Question 11.

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

    A
    B
    C
    D

    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

    A
    B
    C
    D

    Question 13.

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

    A
    8
    B
    10
    C
    12
    D
    9

    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

    A
    87
    B
    82
    C
    78
    D
    75

    Question 15.

    How many distinct positive integer-valued solutions exist to the equation (x27x+11)(x213x+42)(x^{2} - 7x + 11)^{(x^{2} - 13x + 42)} = 1?

    A
    6
    B
    2
    C
    4
    D
    8

    Question 16.

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

    A
    B
    C
    D

    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

    A
    264
    B
    232
    C
    243
    D
    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

    A
    2π15\frac{2π}{15}
    B
    6π25\frac{6π}{25}
    C
    3π25\frac{3π}{25}
    D
    5π18\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?

    A
    12
    B
    11
    C
    13
    D
    14

    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

    A
    56
    B
    49
    C
    46
    D
    59

    Question 21.

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

    A
    log2log_{2} (1/3)
    B
    log2log_{2} (1/5)
    C
    log2-log_{2} (1/3)
    D
    log2-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

    A
    3√π(5 + 12π\frac{12}{π} )
    B
    4√π(3 + 12π\frac{12}{π} )
    C
    5√π(3 + 12π\frac{12}{π} )
    D
    3√π(52\frac{5}{2} + 6π\frac{6}{π} )

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