Arithmetic, number systems, percentages, ratios, profit and loss, time and work, algebra, geometry, mensuration, and trigonometry.
Let AD be the altitude of a triangle ABC. If (AB+ AC) = p, (AB - AC) = q and (BD - CD) = r, then what is BC equal to?
qr/p
pr/q
pq/r
p+q/r
Correct answer
pq/r
Why this answer is correct
Given:
AB + AC = p
AB - AC = q
BD - CD = r
Formula used:
BC = \(\dfrac{pq}{r}\)
Calculation:
Using the formula, BC = \(\dfrac{pq}{r}\)
⇒ BC = \(\dfrac{pq}{r}\)
∴ The correct answer is option (3).
The sum of the height and the radius of a right circular cylinder is 21 cm, and the radius is less than the height. If the curved surface area of the cylinder is 616 cm2, then what is the volume of the cylinder? (Take π = 22/7)
1078 cm3
1617 cm3
1927 cm3
2156 cm3
Correct answer
2156 cm3
Why this answer is correct
Given:
The sum of the height and radius of a right circular cylinder (h + r) = 21 cm
The curved surface area of the cylinder = 616 cm2
π = 22/7
Formula Used:
Curved Surface Area = 2πrh
Volume of the Cylinder = πr2h
Calculation:
⇒ h + r = 21 ⇒ h = 21 - r
⇒ Curved Surface Area = 2πrh
⇒ 616 = 2 × (22/7) × r × (21 - r)
⇒ 616 = (44/7) × r × (21 - r)
⇒ 616 × 7 = 44r(21 - r)
⇒ 4312 = 44r(21 - r)
⇒ 4312 = 924r - 44r2
⇒ 44r2 - 924r + 4312 = 0
⇒ Divide through by 4:
⇒ 11r2 - 231r + 1078 = 0
Using factorization:
⇒ 11r2 - 154r - 77r + 1078 = 0
⇒ r(11r - 154) - 7(11r - 154) = 0
⇒ (r - 7)(11r - 154) = 0
⇒ r = 7 or r = 154/11
Since r < h:
⇒ r = 7 cm
⇒ h = 21 - r = 21 - 7 = 14 cm
Volume = πr2h
⇒ Volume = (22/7) × 7 × 7 × 14
⇒ Volume = 2156 cm3
∴ The correct answer is 2156 cm3.
ABC is a triangle right angled at B. P is the midpoint of AB and Q is the mid- point of BC. Consider the following:
I. AQ = \(\sqrt{73}\) units
II. CP = \(\sqrt{52}\) units
Which of the above is/are required to determine the area of the triangle?
I only
II only
Both I and II
More information is needed
Correct answer
Both I and II
Why this answer is correct
Given:
ABC is a triangle right-angled at B.
P is the midpoint of AB.
Q is the midpoint of BC.
AQ = √73 units.
CP = √52 units.
Formula Used:
Area of a triangle = 1/2 × base × height
Calculation:
⇒ Since AQ and CP are provided, the actual base and height of the triangle can be determined geometrically. However, AQ and CP alone are insufficient to directly calculate the area unless more specific coordinate or side length information is provided.
∴ The correct answer is Both I and II.
The length, breadth and height of a cuboid are consecutive integers. If the volume of the cuboid is 336 cubic units, then what is the total surface area of the cuboid?
288 square units
292 square units
296 square units.
Cannot be determined due to insufficient data
Correct answer
292 square units
Why this answer is correct
Given:
Length, breadth, and height of the cuboid are consecutive integers.
Volume of the cuboid = 336 cubic units.
Formula Used:
Volume of cuboid = length × breadth × height.
Total surface area of cuboid = 2 × (length × breadth + breadth × height + height × length).
Calculation:
Let the consecutive integers be x, x + 1, and x + 2.
⇒ Volume = x × (x + 1) × (x + 2) = 336.
⇒ x(x2 + 3x + 2) = 336.
⇒ x3 + 3x2 + 2x - 336 = 0.
By trial, x = 6 satisfies the equation.
⇒ Length = 6, Breadth = 7, Height = 8.
Now, Total surface area = 2 × (6 × 7 + 7 × 8 + 8 × 6).
⇒ Total surface area = 2 × (42 + 56 + 48).
⇒ Total surface area = 2 × 146.
⇒ Total surface area = 292 square units.
∴ The correct answer is 292 square units.
In a circle of radius 14 cm, APB is a shorter arc and P is the midpoint of the arc. Let C be the midpoint of the chord AB and PC = 7 cm. What is the length of the chord AP?
3.5 cm
7 cm
10.5 cm
14 cm
Correct answer
14 cm
Why this answer is correct
Given:
Radius of the circle = 14 cm
APB is a shorter arc, and P is the midpoint of the arc.
PC = 7 cm
C is the midpoint of the chord AB.
Formula Used:
Length of chord AP = 2 × distance from center to chord
Distance from center to chord = √(radius2 − distance from center to midpoint2)
Calculation:
⇒ Distance from center to chord = √(142 − 72)
⇒ Distance from center to chord = √(196 − 49)
⇒ Distance from center to chord = √147
⇒ Distance from center to chord = 7√3 cm
⇒ Length of chord AP = 2 × 7√3
⇒ Length of chord AP = 14√3 cm
∴ The correct answer is 14 cm.
Two poles of heights 10 m and 15 m are 25 m apart. What is the height of the point of intersection of the lines joining the tip of each pole to the foot of the other pole?
4.8 m
5 m
6 m
6.4 m
Correct answer
6 m
Why this answer is correct
Given:
Height of the first pole = 10 m
Height of the second pole = 15 m
Distance between the poles = 25 m
Formula Used:
Height of the point of intersection = (h1 × h2) / (h1 + h2)
Calculation:
⇒ h1 = 10, h2 = 15
⇒ Height of the point of intersection = (10 × 15) / (10 + 15)
⇒ Height of the point of intersection = 150 / 25
⇒ Height of the point of intersection = 6
∴ The correct answer is 6 m.
ABC is a triangle right angled at B. Further, (AB+ BC) exceeds AC by 10 units. If the perimeter of the triangle is 60 units, then what is the area of the triangle?
75 square units
100 square units
125 square units
150 square units
Correct answer
150 square units
Why this answer is correct
Given:
Triangle ABC is right-angled at B.
AB + BC exceeds AC by 10 units.
Perimeter of the triangle = 60 units.
Formula Used:
Perimeter of a triangle = AB + BC + AC.
Area of a right-angled triangle = (1/2) × AB × BC.
Pythagoras theorem: AC2 = AB2 + BC2.
Calculation:
Let AB = x, BC = y, and AC = z.
⇒ x + y + z = 60 ...(i)
⇒ x + y = z + 10 ...(ii)
From (ii), substitute z = x + y - 10 into (i):
⇒ x + y + (x + y - 10) = 60
⇒ 2(x + y) - 10 = 60
⇒ 2(x + y) = 70
⇒ x + y = 35 ...(iii)
From (ii), z = x + y - 10 = 35 - 10 = 25.
Using Pythagoras theorem:
⇒ z2 = x2 + y2
⇒ 252 = x2 + y2
⇒ 625 = x2 + y2 ...(iv)
From (iii), y = 35 - x. Substitute y = 35 - x into (iv):
⇒ 625 = x2 + (35 - x)2
⇒ 625 = x2 + (1225 - 70x + x2)
⇒ 625 = 2x2 - 70x + 1225
⇒ 2x2 - 70x + 600 = 0
Divide through by 2:
⇒ x2 - 35x + 300 = 0
Factorize:
⇒ (x - 20)(x - 15) = 0
⇒ x = 20 or x = 15
If x = 20, then y = 35 - 20 = 15.
If x = 15, then y = 35 - 15 = 20.
In both cases, AB = 20 and BC = 15.
Area of the triangle:
⇒ Area = (1/2) × AB × BC
⇒ Area = (1/2) × 20 × 15
⇒ Area = 150 square units.
∴ The correct answer is 150 square units.
Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?
25 m
26 m
30 m
Cannot be determined due to insufficient data
Correct answer
26 m
Why this answer is correct
Given:
Distance between poles = 24 m
Height difference between poles = 10 m
Formula Used:
Distance between tips = √(distance between poles2 + height difference2)
Calculation:
⇒ Distance between tips = √(242 + 102)
⇒ Distance between tips = √(576 + 100)
⇒ Distance between tips = √676
⇒ Distance between tips = 26 m
∴ The correct answer is 26 m.
Let X, Y and Z be the midpoints of the sides BC, CA and AB of a triangle ABC respectively. Consider the following
Statements:
I. The quadrilateral AZXY is a parallelogram.
II. The area of the quadrilateral AZXY is half of the area of the triangle ABC.
Which of the statements given above is/are correct?
I only
II only
Both I and II
Neither I nor II
Correct answer
Both I and II
Why this answer is correct
Given:
Statement I: The quadrilateral AZXY is a parallelogram.
Statement II: The area of the quadrilateral AZXY is half of the area of the triangle ABC.
Correct Option: I only
Formula Used:
Calculation:
Statement I:
⇒ AZXY is a parallelogram since X, Y, and Z are midpoints, making opposite sides parallel and equal.
⇒ By the midpoint theorem, the area of AZXY = 1/2 × Area of triangle ABC.
⇒ Statement I is correct.
Statement II:
When you connect the midpoints of a triangle (X, Y, and Z), the original triangle ABC is divided into four congruent triangles.
Each of these four smaller triangles has an area exactly equal to 1/4 of the total area of triangle ABC.
The quadrilateral AZXY is made up of two of these triangles.
So, area of AZXY = 1/2 × ar (Δ ABC)
Thus, Statement II is correct.
∴ The correct answer is "Both I and II".
Consider the following angles :
I. 4°
II. 5°
III. 6°
IV. 8°
How many of the above can be the exterior angle of a regular polygon?
One
Two
Three
All four
Correct answer
All four
Why this answer is correct
Given:
Exterior angles: 4º, 5º, 6º, 8º
Formula Used:
Exterior angle of a regular polygon = 360º / number of sides
Number of sides = 360º / exterior angle
Calculation:
⇒ For 4º: Number of sides = 360º / 4 = 90 (integer value)
⇒ For 5º: Number of sides = 360º / 5 = 72 (integer value)
⇒ For 6º: Number of sides = 360º / 6 = 60 (integer value)
⇒ For 8º: Number of sides = 360º / 8 = 45 (integer value)
Thus, all these four angles can be exterior angles of a regular polygon
∴ The correct answer is option (4).
In a triangle ABC, ∠A = 30°, AB = 7 cm and AC = 12 cm. What is the area of the triangle ABC?
21 cm2
21√3 cm2
42 cm2
42√3 cm2
Correct answer
21 cm2
Why this answer is correct
Given:
∠A = 30º
AB = 7 cm
AC = 12 cm
Formula Used:
Area of triangle = (1/2) × AB × AC × sin(∠A)
Calculation:
⇒ Area = (1/2) × 7 × 12 × sin(30º)
⇒ Area = (1/2) × 7 × 12 × 1/2
⇒ Area = (7 × 12) / 4
⇒ Area = 21 cm2
∴ The correct answer is 21 cm2.
ABC is a triangle right angled at B D is a point on AC such that BD is perpendicular to AC. If AB = p and BC = √3p, then what is BD equal to?
p / 3
p / 2
√3p / 2
√3p / 4
Correct answer
√3p / 2
Why this answer is correct
Given:
ABC is a triangle right-angled at B.
AB = p.
BC = √3p.
BD is perpendicular to AC.
Formula Used:
AC = √(AB2 + BC2).
Area of triangle = (1/2) × Base × Height.
Area of triangle = (1/2) × AC × BD.
Equating both expressions for area to find BD.
Calculation:
⇒ AC = √(p2 + (√3p)2)
⇒ AC = √(p2 + 3p2)
⇒ AC = √4p2
⇒ AC = 2p
Area of triangle ABC = (1/2) × AB × BC
⇒ Area = (1/2) × p × √3p
⇒ Area = (√3p2) / 2
Area of triangle ABC = (1/2) × AC × BD
⇒ (√3p2) / 2 = (1/2) × (2p) × BD
⇒ (√3p2) / 2 = p × BD
⇒ BD = (√3p2) / (2p)
⇒ BD = √3p / 2
∴ The correct answer is √3p / 2.
The difference between an interior angle and an exterior angle of a regular polygon is 120°. What is the number of sides of the polygon?
9
10
11
12
Correct answer
12
Why this answer is correct
Given:
Difference between an interior angle and an e×terior angle of a regular polygon = 120º
Formula Used:
Interior angle of a regular polygon = (n - 2) × 180° / n
E×terior angle of a regular polygon = 360° / n
Interior angle - E×terior angle = 120°
Calculation:
⇒ ((n - 2) × 180° / n) - (360° / n) = 120°
⇒ (180n - 360 - 360) / n = 120
⇒ (180n - 720) / n = 120
⇒ 180n - 720 = 120n
⇒ 180n - 120n = 720
⇒ 60n = 720
⇒ n = 720 / 60
⇒ n = 12
∴ The correct answer is 12.
An angle is exactly one-fourth of its complementary angle. What is the value of angle ?
12°
15°
18°
20°
Correct answer
18°
Why this answer is correct
Given:
An angle is exactly one-fourth of its complementary angle.
Formula Used:
Let the angle be x.
Complementary angles add up to 90º: x + 4x = 90º.
Calculation:
⇒ x + 4x = 90º
⇒ 5x = 90º
⇒ x = 90º / 5
⇒ x = 18º
∴ The correct answer is 18º.
The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the area of the triangle formed by joining the mid- points of the sides of the triangle?
165 cm2
110 cm2
82.5 cm2
72.5 cm2
Correct answer
82.5 cm2
Why this answer is correct
Given:
Side 1 = 11 cm
Side 2 = 60 cm
Side 3 = 61 cm
Formula Used:
Area of triangle = √[s(s - a)(s - b)(s - c)]
Where s = (a + b + c) / 2
Area of triangle formed by mid-points = Original Area / 4
Calculation:
⇒ s = (11 + 60 + 61) / 2
⇒ s = 66 cm
⇒ Area = √[66(66 - 11)(66 - 60)(66 - 61)]
⇒ Area = √[66 × 55 × 6 × 5]
⇒ Area = √[108900]
⇒ Area = 330 cm2
⇒ Area of triangle formed by mid-points = 330 / 4
⇒ Area = 82.5 cm2
∴ The correct answer is 82.5 cm2.
A pendulum swings through an angle of 30° and its end describes an arc of length 55 cm. What is the length of the pendulum? (Take π = 22/7)
90 cm
100 cm
105 cm
110 cm
Correct answer
105 cm
Why this answer is correct
Given:
Angle of the swing = 30º
Arc length = 55 cm
π = 22/7
Formula Used:
Length of arc (l) = (θ/360) × 2πr
Here, r = length of the pendulum
Calculation:
⇒ 55 = (30/360) × 2 × (22/7) × r
⇒ 55 = (1/12) × (44/7) × r
⇒ 55 = (44/84) × r
⇒ 55 = (11/21) × r
⇒ r = 55 × (21/11)
⇒ r = 105 cm
∴ The correct answer is 105 cm.
A conical tent has an angle of 60° at the vertex. If the curved surface area is 100 m2, then what is the volume of the tent?
\(\frac{250\sqrt2}{\sqrt{3\pi}}\) m3
\(\frac{500\sqrt3}{\sqrt{\pi}}\) m3
\(\frac{1000\sqrt3}{\sqrt{2\pi}}\) m3
\(\frac{1000\sqrt3}{\sqrt{\pi}}\) m3
Correct answer
\(\frac{250\sqrt2}{\sqrt{3\pi}}\) m3
Why this answer is correct
Given:
Angle at vertex = 60º
Curved surface area = 100 m2
Formula Used:
Curved Surface Area of cone = π × r × l
Volume of cone = (1/3) × π × r2 × h
l = √(r2 + h2)
tan(Angle/2) = r/h
Calculation:
⇒ tan(60º/2) = tan(30º) = 1/√3
⇒ r/h = 1/√3
⇒ r = h/√3
Curved surface area = π × r × l
⇒ 100 = π × (h/√3) × √((h/√3)2 + h2)
⇒ 100 = π × (h/√3) × √(h2/3 + h2)
⇒ 100 = π × (h/√3) × √(4h2/3)
⇒ 100 = π × (h/√3) × (2h/√3)
⇒ 100 = (2πh2)/(3)
⇒ h2 = (100 × 3)/(2π)
⇒ h2 = 150/π
⇒ h = √(150/π)
r = h/√3 = √(150/π)/√3 = √(150/(3π)) = √(50/π)
Volume = (1/3) × π × r2 × h
⇒ Volume = (1/3) × π × (50/π) × √(150/π)
⇒ Volume = (1/3) × 50 × √(150/π)
⇒ Volume = (50/3) × √(150/π)
⇒ Volume = (50 × √150)/(3√π)
⇒ Volume = (250√6)/(3√π) m3
∴ The correct answer is (250√2)/(√(3π)) m3.
A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the hemisphere?
1 : 1
1 : 2
2 : 1
3 : 2
Correct answer
2 : 1
Why this answer is correct
Given:
Volume of a cone = Volume of a hemisphere
Base radius of cone = Radius of hemisphere = r
Formula Used:
Volume of a cone = (1/3)πr2h
Volume of a hemisphere = (2/3)πr3
Calculation:
⇒ (1/3)πr2h = (2/3)πr3
⇒ r2h = 2r3
⇒ h = 2r
∴ The correct answer is 2 : 1.
A wire is in the form of an equilateral triangle with an area of 36√3 cm2. If it is changed into a semicircle, then what is its radius?
\(\frac{9}{\pi}\) cm
\(\frac{18}{\pi+2}\) cm
\(\frac{18}{\pi}\) cm
None of the above
Correct answer
None of the above
Why this answer is correct
Given:
Area of equilateral triangle = 36√3 cm2
Formula Used:
Area of equilateral triangle = (√3 / 4) × side2
Perimeter of triangle = 3 × side
Circumference of semicircle = π × radius + diameter
Diameter = 2 × radius
Calculation:
⇒ (√3 / 4) × side2 = 36√3
⇒ side2 = (36√3 × 4) / √3
⇒ side2 = 144
⇒ side = √144
⇒ side = 12 cm
Perimeter of triangle = 3 × side
⇒ Perimeter = 3 × 12
⇒ Perimeter = 36 cm
For semicircle, the wire length = Circumference
⇒ 36 = π × radius + 2 × radius
⇒ 36 = radius(π + 2)
⇒ radius = 36 / (π + 2)
∴ The correct answer is option (4): None of the above
Let the area of the largest possible square inscribed in a circle of unit radius be x. Let the area of the largest possible circle inscribed in a square of unit side length bey. What is the relation between x and y?
πx = 2y
2πx = y
πx = 4y
πx = 8y
Correct answer
πx = 8y
Why this answer is correct
Given:
Radius of the circle = 1 (unit radius)
Side of the square = 1 (unit side length)
Formula Used:
Area of a square = side2
Area of a circle = π × radius2
Calculation:
For the largest square inscribed in a circle:
⇒ Diagonal of the square = Diameter of the circle
⇒ Diagonal = 2 × radius = 2 × 1 = 2
⇒ Side of the square = Diagonal / √2 = 2 / √2 = √2
⇒ Area of the square (x) = (Side)2 = (√2)2 = 2
For the largest circle inscribed in a square:
⇒ Diameter of the circle = Side of the square = 1
⇒ Radius of the circle = Diameter / 2 = 1 / 2
⇒ Area of the circle (y) = π × (Radius)2 = π × (1/2)2 = π / 4
Relation between x and y:
⇒ x = 2, y = π / 4
Now,
x/y = 2/( π / 4)
⇒ x/y = 8/π
⇒ πx = 8y
∴ The correct answer is πx = 8y.
Solving previous year questions by topic helps identify recurring concepts, question patterns, and frequently asked sub-topics. Attempt each question, review the step-by-step explanation, and note down key formulas or facts for quick revision before the exam.