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Ten data-sufficiency items in one set (number, algebra, geometry)

This type of question, asked 2 times: CDS 1 2023, CDS 1 2024. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1 (case study, 10 parts)

    CDS 1 2023

    For the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions :
    71
    A number 277XY5 (where X, Y are digits) is divisible by 25. Question : What is the value of X ? Statement I : The given number is divisible by 9. Statement II : X>5X > 5.

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    72
    Question : What are the unique values of a, b and c if 2 is a root of the equation ax2+bx+c=0ax^2 + bx + c = 0 ? Statement I : Ratio of c to a is 1. Statement II : Ratio of b to a is (−52)\left(-\frac{5}{2}\right).

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    73
    Question : Is m>nm > n, where m, n are non-zero numbers ? Statement I : mn>1\frac{m}{n} > 1. Statement II : m>2nm > 2n.

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    74
    Question : Can a circle be drawn through the points A, B and C ? Statement I : AB = 5 cm, BC = 5 cm, CA = 6 cm. Statement II : AB = 3 cm, BC = 4 cm, CA = 7 cm.

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    75
    x and y are consecutive odd integers. Question : Can the value of (x+y)(x + y) be determined uniquely ? Statement I : (x+y)4=256(x + y)^4 = 256. Statement II : (x+y)3<16(x + y)^3 < 16.

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    76
    Question : Is p2+q2+qp^2 + q^2 + q odd, where p, q are positive integers ? Statement I : 2p+q2p + q is odd. Statement II : q−2pq - 2p is odd.

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    77
    Question : What is the area of the circle C ? Statement I : An arc of length 7 cm subtends an angle 30∘30^\circ at the centre of C. Statement II : A chord of length 10 cm subtends an angle 90∘90^\circ at the centre of C.

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    78
    Question : Is triangle △\triangle right angled ? Statement I : The length of the line segment joining the mid-points of two sides of △\triangle is half of the third side of △\triangle. Statement II : The angles of △\triangle are in the ratio 1 : 2 : 3.

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    79
    The lengths of two longer sides of the triangle △\triangle are 25 cm and 24 cm. Question : What is the length of the shortest side ? Statement I : The angles of △\triangle are in the ratio 1 : 2 : 3. Statement II : The length of the perpendicular drawn on the longest side of △\triangle from its opposite vertex is 6.72 cm.

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    80
    A chord PQ of the circle C divides it into two segments such that 3 times the area of the major segment is 4 times the area of the minor segment. Question : What is the radius of C ? Statement I : Area of the minor segment is 66 square cm. Statement II : Area of the major segment is 88 square cm.

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  2. Question 2 (case study, 10 parts)

    CDS 1 2024

    Consider the following for the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions :
    61
    Let a, b, c and d be positive integers. Question : Which one of a, b, c, d is closest to the product abcd ? Statement-I : a > b > c Statement-II : c is not the smallest.

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    62
    Let mn = k, where m and n are prime numbers and k is an even number. Question : What is the value of mn−n+1mn - n + 1 ? Statement-I : m > n Statement-II : One of the numbers is 2.

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    63
    Question : If p is a positive integer, then what is the remainder when pnp^n is divided by p + 1 ? Statement-I : n is even. Statement-II : p is even.

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    64
    Question : Is xy positive ? Statement-I : x=−0.196833x = \sqrt[3]{-0.19683} Statement-II : y=x3y = \sqrt[3]{x}

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    65
    Let a, b and c be the sides of a triangle ABC. Question : Is the triangle equilateral ? Statement-I : a2+b2+c2=(ab+bc+ca)a^2 + b^2 + c^2 = (ab + bc + ca) Statement-II : 3a2+3b2+4c2=2ab+4bc+4ca3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca

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    66
    Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q. Question : Is P > Q ? Statement-I : x : y = 2 : 1 Statement-II : The angle between the two adjacent sides of the parallelogram is 60∘60^\circ.

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    67
    A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD. Question : What is the perimeter of the quadrilateral ? Statement-I : AB + DC = 10 cm Statement-II : AD + BC = 10 cm

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    68
    Question : What is the ratio of the lengths of diagonals of a rhombus ? Statement-I : One diagonal of the rhombus is equal to its side. Statement-II : The longer diagonal of the rhombus is equal to 3\sqrt{3} times its side.

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    69
    The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units. Question : What are the values of r and R ? Statement-I : r is an integer. Statement-II : R is an integer.

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    70
    P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD. Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS ? Statement-I : Area of the quadrilateral ABCD is 100 square unit. Statement-II : Area of the quadrilateral PQRS is 50 square unit.

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