## Question If 9^(2x – 1) – 81^(x-1) = 1944, then x is

A) 3
B) 9/4
C) 4/9
D) 1/3

Option (B)

## Solution

From CAT 2017 – Quantitative Aptitude – Algebra – Polynomials, we can see that,
9^(2x-1) – 9^(2x-2) = 1944
It can be written as 3^(4x)/9 – 3^(4x)/81 = 1944
8(3^(4x)/81) = 1944
x =9/4
Option (B)

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## Question The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to

A) 1/3
B) 2/3
C) 5/6
D) 7/6

Option (C)

## Solution

As per CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that
log(base 0.008)5^(1/2) = -1/6
Log (base 3^1/2) 3^4 = 8
So, -1/6 + 8 – 7 = 5/6
Option (C)

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

Q2: If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

Q3: Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

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## Question If x + 1 = x^2 and x > 0, then 2x^4 is

A) 6 + 4√5
B) 3 + 5√5
C) 5 + 3√5
D) 7 + 3√5

Option (D)

## Solution

As per CAT 2017 – Quantitative Aptitude – Algebra – Quadratic Equation, we can see that
x+1=x^2
Find out the roots of x = [1+/- root(5)]/2
X2 = [3 +/- √5]/2
X4 = [7 +/-3√5]/2
2×4 = 7 +/- 3√5
As the only option is 7 + 3√5 So, we go with that.
Option (D)

## CAT 2017 Questions from Quantitative Aptitude – Algebra

Quantitative Aptitude – Algebra – Quadratic Equations – Ques: The minimum possible value of the sum of the squares of the roots of the equation x^2 + (a + 3)x – (a + 5) = 0 is
Quantitative Aptitude – Algebra – Functions
Quantitative Aptitude – Algebra – Logarithms
Quantitative Aptitude – Algebra – Maxima Minima
Quantitative Aptitude – Algebra – Inequalities
Quantitative Aptitude – Algebra – Polynomials
Quantitative Aptitude – Algebra – Simple Equations

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## Question Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

A) √a
B) 2a
C) a/2
D) a

Option (D)

## Solution

As per CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that
x=3^a and y=12^a
G = √(3^a * 12^a) = 6^a
Log (base 6) 6^a = a
Option (D)

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

Q2: If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

Q3: The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to

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## Question Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is

24

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Geometry – Triangles,
BC^2 = AB^2 + AC^2 = 625
BC = 25
Shortest Distance from A to hypotenuse = altitude on BC = AP
AP * BC = AB * AC
So, AP = 12
Time taken = (12/30) * 60 mins = 24 mins

## CAT 2017 Questions from Quantitative Aptitude – Geometry

Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The shortest distance of the point (½, 1) from the curve y = |x -1| + |x + 1| is
Quantitative Aptitude – Geometry – Mensuration
Quantitative Aptitude – Geometry – Polygons – Ques: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

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## Question A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is 9 π cm^3 . Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is

6

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Geometry – Mensuration, The height of the cylinder (h) = 3
The volume = 9π
πr2h = 9π ⇒ r = √3
The radius of the ball (R) = 2
The height of O, the centre of the ball, above the line representing the top of the cylinder is say a. (a = 1) ∴ The height of the topmost point of the ball from the base of the cylinder is h + a +R = 3 + 1 + 2 = 6

## CAT 2017 Questions from Quantitative Aptitude – Geometry

Quantitative Aptitude – Geometry – Mensuration – Q1: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length.
Quantitative Aptitude – Geometry – Mensuration – Q2: A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27.
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Triangles – Q3: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is
Quantitative Aptitude – Geometry – Polygons – Ques: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

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## Question A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to

A) 10
B) 50
C) 60
D) 20

Option (B)

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Geometry – Mensuration,
Ratio of volumes of 5 smaller cubes and original big one = 1 : 1 : 8 : 27 : 27 : 64
Ratio of sides = 1 : 1 : 2 : 3 : 3 : 4
Ratio of areas = 1 : 1 : 4 : 9 : 9 : 16
The sum of the areas of the 5 smaller cubes is 24 parts while that of the big cube is 16 parts. The sum is 50% greater.
Option (B)

## CAT 2017 Questions from Quantitative Aptitude – Geometry

Quantitative Aptitude – Geometry – Mensuration – Q1: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length.
Quantitative Aptitude – Geometry – Mensuration – Q2: A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is 9 π cm^3 .
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Triangles – Q3: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is
Quantitative Aptitude – Geometry – Polygons – Ques: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

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## Question Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC. Let BPC be an arc of a circle centered at A and lying between BC and BQC. If AB has length 6 cm then the area, in sq cm, of the region enclosed by BPC and BQC is

A) 9π – 18
B) 18
C) 9π
D) 9

Option (B)

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Geometry – Circles, Let AB = a (a = 6)
CQB is a semicircle of radius a/√2
So, area of semicircle = pi*a^2/4
So, area of region enclosed by BPC, BQC = Area of tr(ABC) = 18.
Option (B)

## CAT 2017 Questions from Quantitative Aptitude – Geometry

Quantitative Aptitude – Geometry – Circles – Ques: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Triangles – Q3: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The shortest distance of the point (½, 1) from the curve y = |x -1| + |x + 1| is
Quantitative Aptitude – Geometry – Mensuration
Quantitative Aptitude – Geometry – Polygons – Ques: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

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## Question From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is

A) 225√3
B) 500 / √3
C) 275 / √3
D) 250 / √3

Option (B)

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Geometry – Triangles,
Area of triangle = root (s(s-a)(s-b)(s-c))
S = 50
Area = 250 * root(3)
Centroid divides triangle in the ratio 2:1
Area of remaining portion of triangle: 2/3 * (Area of Triangle) = 500/root(3)
Option (B)

## CAT 2017 Questions from Quantitative Aptitude – Geometry

Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The shortest distance of the point (½, 1) from the curve y = |x -1| + |x + 1| is
Quantitative Aptitude – Geometry – Mensuration
Quantitative Aptitude – Geometry – Polygons – Ques: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

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## Question The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is

A) 4π
B) 4
C) 8
D) 2π

Option (C)

## Solution

As per the question from CAT 2017 – Quantitative Aptitude – Algebra – Functions,
Remember the formula |x| + |y| = n
Here, area bounded by the region = 2n^2
In the question, n=2
So, area = 8
Option (C)

## CAT 2017 Questions from Quantitative Aptitude – Algebra – Functions

Quantitative Aptitude – Algebra – Functions – Q1: Let f(x) = 2x-5 and g(x) = 7-2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if
Quantitative Aptitude – Algebra – Functions – Q2: If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is
Quantitative Aptitude – Algebra – Functions – Q3: Let f(x) = x^2 and g(x) = 2^x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is
Quantitative Aptitude – Algebra – Functions – Q4: If f(x) = (5x+2)/(3x-5) and g(x) = x^2 – 2x – 1, then the value of g(f(f(3))) is
Quantitative Aptitude – Algebra – Functions – Q5: If f1(x) = x^2 + 11x + n and f2(x) = x, then the largest positive integer n for which the equation f1(x) = f2(x) has two distinct real roots, is
Quantitative Aptitude – Algebra – Logarithms
Quantitative Aptitude – Algebra – Quadratic Equations
Quantitative Aptitude – Algebra – Maxima Minima
Quantitative Aptitude – Algebra – Inequalities
Quantitative Aptitude – Algebra – Polynomials
Quantitative Aptitude – Algebra – Simple Equations

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