← Combined Technical (NG)
Past Questions — 2024
Area of an equilateral triangle with side length a is equal to:
A)
(V3/2)a
B)
(V3/2) a2
C)
(3/4) a
D)
(V3/4)a^2
To find the area of an equilateral triangle with side length 'a', we use the standard formula for the area of a triangle: Area = (1/2) × base × height.
1. **Finding the Height (h):** In an equilateral triangle, the altitude (height) bisects the base, creating two right-angled triangles. Using the Pythagorean theorem:
h² + (a/2)² = a²
h² = a² - a²/4
h² = 3a²/4
h = (√3/2)a
2. **Calculating the Area:**
Area = (1/2) × base × height
Area = (1/2) × a × (√3/2)a
Area = (√3/4)a²
Comparing this result to the options:
- Option (a) (V3/2)a represents the height of the triangle, not the area.
- Option (b) (V3/2)a² is incorrect as the denominator should be 4.
- Option (c) (3/4)a is dimensionally incorrect (linear instead of squared) and numerically wrong.
- Option (d) (V3/4)a² matches our derived formula exactly.
The equation of x-axis is:
A)
y=0
B)
x=0
C)
y=x
D)
x=y
In a 2D Cartesian coordinate system, the x-axis is the horizontal line where every point on that line has a vertical displacement of zero. For any point (x, y) lying on the x-axis, the y-coordinate is always 0, regardless of the value of x. Therefore, the mathematical equation representing the x-axis is y = 0.
Analysis of other options:
- Option b (x=0): This is the equation of the y-axis, where all points have a horizontal displacement of zero.
- Option c (y=x): This is the equation of a diagonal line passing through the origin at a 45-degree angle.
- Option d (x=y): This is identical to option c and represents the same diagonal line.
Sides of two similar triangles are in the ratio 4:9. Areas of these triangles are in the ratio-
A)
2:3
B)
4:9
C)
81:16
D)
16:81
According to the fundamental theorem of geometry regarding similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given:
Ratio of sides = 4 : 9
Formula:
Area Ratio = (Side Ratio)²
Area Ratio = (4 / 9)²
Area Ratio = (4² / 9²)
Area Ratio = 16 / 81
Therefore, the ratio of the areas is 16:81.
Analysis of options:
- Option (a) 2:3 is the square root of the area ratio, not the area ratio itself.
- Option (b) 4:9 is the ratio of the sides, not the areas.
- Option (c) 81:16 is the inverse ratio (Area of larger triangle to smaller triangle).
- Option (d) 16:81 is the correct square of the given side ratio.
Which of the following is the equation of the line passing through the point (2, 3) and parallel to the x-axis?
A)
y=2
B)
y=3
C)
x=2
D)
x=3
Step-by-step reasoning:
1. **Understanding Parallelism to the X-axis**: A line parallel to the x-axis is a horizontal line. In a coordinate plane, every point on a horizontal line has the same y-coordinate, regardless of the x-value.
2. **General Equation**: The general equation for any line parallel to the x-axis is given by 'y = k', where 'k' is a constant representing the y-intercept.
3. **Applying the Given Point**: The line must pass through the point (2, 3). In this coordinate pair, x = 2 and y = 3.
4. **Determining the Constant**: Since the line is horizontal (parallel to the x-axis), the y-coordinate must remain constant for all points on the line. Therefore, k must be 3.
5. **Final Equation**: The equation is y = 3.
Analysis of other options:
- Option (a) y=2: This is a horizontal line, but it passes through points where y=2, not (2, 3).
- Option (c) x=2: This is a vertical line (parallel to the y-axis) passing through (2, 3).
- Option (d) x=3: This is a vertical line passing through points where x=3.
If the coordinates of a point are (4, 7), then what is the distance between the point and the x-axis?
A)
4
B)
7
C)
11
D)
None of these
To find the distance of a point (x, y) from the x-axis, we look at the absolute value of its y-coordinate.
1. Given Point: (4, 7). Here, the x-coordinate (abscissa) is 4 and the y-coordinate (ordinate) is 7.
2. The x-axis is the line where y = 0.
3. The perpendicular distance from any point (x, y) to the x-axis is given by the formula: Distance = |y|.
4. In this case, Distance = |7| = 7 units.
Analysis of options:
- Option (a) 4 is the distance of the point from the y-axis (the x-coordinate).
- Option (b) 7 is the distance of the point from the x-axis (the y-coordinate), which is correct.
- Option (c) 11 is the sum of the coordinates, which is irrelevant to the distance from an axis.
D and E are the midpoints of side AB and AC of an equilateral triangle ABC, respectively and BC = 6m. If DE || BC, then the length (in cm) of DE is:
A)
2.5
B)
3
C)
5
D)
6
1. Given: Triangle ABC is an equilateral triangle with side BC = 6m.
2. Points D and E are the midpoints of sides AB and AC, respectively.
3. According to the Midpoint Theorem in geometry, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.
4. Therefore, since D and E are midpoints of AB and AC, the segment DE is parallel to BC and its length is given by: DE = 1/2 * BC.
5. Substituting the given value: DE = 1/2 * 6m = 3m.
6. Note on units: The question asks for the length in 'cm' but provides the side in 'm' and the options as single digits. In the context of competitive exams, this is typically a typographical error in the unit label of the question (asking for cm instead of m), as the numerical value 3 matches option (b) perfectly. If converted literally to cm, the answer would be 300cm, which is not an option.
7. Thus, the numerical length is 3.
If two triangles are similar, then their corresponding sides are:
A)
Equal
B)
Proportional
C)
Unequal
D)
Parallel
In geometry, two triangles are defined as 'similar' (denoted by the symbol ~) if they have the same shape but not necessarily the same size. For two triangles to be similar, two conditions must be met: 1. Their corresponding angles must be equal. 2. Their corresponding sides must be in the same ratio, which means they are proportional. For example, if triangle ABC is similar to triangle DEF, then AB/DE = BC/EF = AC/DF.
Analysis of options:
- (a) Equal: This describes 'congruent' triangles, which are a specific subset of similar triangles where the ratio is 1:1. Not all similar triangles are equal in size.
- (b) Proportional: This is the defining characteristic of similarity for sides.
- (c) Unequal: While sides may be unequal, this is not a mathematical property or requirement; they must follow a specific ratio.
- (d) Parallel: Parallelism refers to the relationship between lines in a plane, not the relationship between corresponding sides of two different triangles.
The line y = 2x - 1 intersects the y-axis at the point:
A)
(0, -1)
B)
(1,9)
C)
(0, 1)
D)
(-1, 0)
To find where a line intersects the y-axis (the y-intercept), we must set the x-coordinate to 0, because every point on the y-axis has an x-value of 0.
Step-by-step calculation:
1. Given the equation of the line: y = 2x - 1
2. Substitute x = 0 into the equation:
y = 2(0) - 1
y = 0 - 1
y = -1
3. The point of intersection is therefore (x, y) = (0, -1).
Analysis of options:
- Option (a) (0, -1) matches our calculation.
- Option (b) (1, 9) is incorrect; if x=1, y=2(1)-1=1, not 9.
- Option (c) (0, 1) is incorrect; this would be the intercept if the equation were y = 2x + 1.
- Option (d) (-1, 0) is the x-intercept (where y=0), not the y-intercept.
If sin θ=1/2 and cos θ= √3/2, then what is the value of tan θ?
A)
1/3
B)
√3/3
C)
√3
D)
1/2
To find the value of tan θ, we use the fundamental trigonometric identity: tan θ = sin θ / cos θ.
Given:
sin θ = 1/2
cos θ = √3/2
Substituting these values into the formula:
tan θ = (1/2) / (√3/2)
To divide by a fraction, we multiply by its reciprocal:
tan θ = (1/2) * (2/√3)
the 2s cancel out, leaving:
tan θ = 1/√3
To rationalize the denominator, multiply both the numerator and the denominator by √3:
tan θ = (1 * √3) / (√3 * √3) = √3/3.
Verification of options:
- Option (a) 1/3 is incorrect.
- Option (b) √3/3 is the rationalized form of 1/√3, which is correct.
- Option (c) √3 is the value of cot θ in this case, not tan θ.
- Option (d) 1/2 is the value of sin θ, not tan θ.
The area of the square whose perimeter is 45.2 cm:
A)
250.43 cm²
B)
127.69 cm²
C)
142 cm²
D)
64.8 cm²
Step-by-step solution:
1. **Identify the given information**: The perimeter (P) of the square is 45.2 cm.
2. **Formula for the perimeter of a square**: P = 4 × side (s).
3. **Calculate the length of one side**:
s = P / 4
s = 45.2 / 4
s = 11.3 cm
4. **Formula for the area of a square**: Area (A) = side² (s²).
5. **Calculate the area**:
A = (11.3)²
A = 11.3 × 11.3 = 127.69 cm².
Comparing this result to the options:
- Option (a) 250.43 cm² is incorrect.
- Option (b) 127.69 cm² matches our calculation exactly.
- Option (c) 142 cm² is incorrect.
- Option (d) 64.8 cm² is incorrect.
Sin(90° - x) equals to _
A)
cosx
B)
cotx
C)
cosec x
D)
secx
This question is based on the trigonometric complementary angle identities. In trigonometry, the complementary angle identity for the sine function states that the sine of an angle is equal to the cosine of its complement. Mathematically, this is expressed as: sin(90° - θ) = cos(θ).
Applying this to the given problem:
1. We are given the expression sin(90° - x).
2. According to the complementary angle formula, sin(90° - x) = cos(x).
Evaluating the other options:
- Option b (cot x): cot x is equal to tan(90° - x).
- Option c (cosec x): cosec x is the reciprocal of sin x, not the complement.
- Option d (sec x): sec x is equal to cosec(90° - x).
Therefore, the correct identity is cosx.
From a point P on a level ground, the angle of elevation of the top tower is 30°. If the tower is 100 m high, find the distance of point P from the foot of the tower is:
A)
149m
B)
156m
C)
173m
D)
200m
1. Identify the given values:
- Height of the tower (Opposite side, h) = 100 m
- Angle of elevation (θ) = 30°
- Distance from point P to the foot of the tower (Adjacent side, d) = ?
2. Use the trigonometric ratio for tangent, which relates the opposite side to the adjacent side:
tan(θ) = Opposite / Adjacent
tan(30°) = 100 / d
3. Substitute the known value of tan(30°):
tan(30°) = 1/√3 (approximately 0.57735)
1/√3 = 100 / d
4. Solve for d:
d = 100 * √3
Using the value of √3 ≈ 1.732:
d = 100 * 1.732 = 173.2 m
5. Comparing the result with the given options, 173.2 m rounds to 173 m.
Option (a) 149m, (b) 156m, and (d) 200m are mathematically incorrect based on the tangent ratio for 30°.
If the area of a circle is 154 cm², then its perimeter is -
A)
11cm
B)
22cm
C)
44cm
D)
55cm
Step-by-step solution:
1. **Given**: Area of the circle (A) = 154 cm².
2. **Formula for Area**: The area of a circle is given by the formula A = πr², where 'r' is the radius.
3. **Find the Radius (r)**:
154 = (22/7) * r²
r² = (154 * 7) / 22
r² = 7 * 7
r² = 49
r = √49 = 7 cm.
4. **Formula for Perimeter (Circumference)**: The perimeter of a circle is given by the formula P = 2πr.
5. **Calculate Perimeter**:
P = 2 * (22/7) * 7
P = 2 * 22
P = 44 cm.
Comparing this result to the options:
- Option (a) 11cm is incorrect.
- Option (b) 22cm is incorrect (this is only πr).
- Option (c) 44cm matches our calculation.
- Option (d) 55cm is incorrect.
A circle has a number of tangents equal to -
A)
0
B)
1
C)
2
D)
Infinite
A tangent to a circle is defined as a straight line that touches the circle at exactly one point. By definition, every single point on the circumference of a circle can serve as a point of tangency. Since a circle consists of an infinite number of points along its boundary (circumference), and a unique tangent can be drawn at each of these points, there are an infinite number of tangents that can be drawn to a circle.
Analysis of options:
- Option (a) 0: Incorrect, as tangents clearly exist for any circle.
- Option (b) 1: Incorrect, as you can draw tangents at any point on the circle, not just one.
- Option (c) 2: Incorrect; while only two tangents can be drawn from a single point *outside* the circle, the question asks for the total number of tangents the circle *has* in total.
- Option (d) Infinite: Correct, because there are infinite points on the circumference.
Corresponding sides of two similar triangles are in the ratio of 2:3. If the area of the small triangle is 48 cm², then the area of large triangle is:
A)
230 cm²
B)
106 cm²
C)
107 cm²
D)
108 cm²
According to the properties of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
1. Given: Ratio of corresponding sides = 2:3
2. Therefore, the ratio of their areas = (2)² : (3)² = 4 : 9
3. Let the area of the small triangle be A1 and the area of the large triangle be A2.
4. We are given A1 = 48 cm².
5. Using the ratio: A1 / A2 = 4 / 9
6. Substitute the value of A1: 48 / A2 = 4 / 9
7. Solve for A2: A2 = (48 * 9) / 4
8. A2 = 12 * 9 = 108 cm².
Comparing this result to the options:
- Option a (230 cm²), b (106 cm²), and c (107 cm²) are mathematically incorrect based on the theorem of similar triangles.
The area of a rhombus whose vertices are (3, 0), (4, 5), (-1, 4) and (-2,-1) taken in order, is:
A)
12 sq.units
B)
24sq.units
C)
30sq.units
D)
32 sq.units
To find the area of a rhombus given its vertices, the most efficient method is to use the formula for the area of a rhombus: Area = (1/2) * d1 * d2, where d1 and d2 are the lengths of the diagonals.
Step 1: Identify the vertices.
Let A = (3, 0), B = (4, 5), C = (-1, 4), and D = (-2, -1).
Step 2: Calculate the lengths of the diagonals.
Diagonal 1 (AC) connects (3, 0) and (-1, 4):
d1 = √[(-1 - 3)² + (4 - 0)²]
d1 = √[(-4)² + (4)²] = √[16 + 16] = √32 = 4√2 units.
Diagonal 2 (BD) connects (4, 5) and (-2, -1):
d2 = √[(-2 - 4)² + (-1 - 5)²]
d2 = √[(-6)² + (-6)²] = √[36 + 36] = √72 = 6√2 units.
Step 3: Calculate the area.
Area = (1/2) * d1 * d2
Area = (1/2) * (4√2) * (6√2)
Area = (1/2) * 24 * (√2 * √2)
Area = (1/2) * 24 * 2
Area = 24 sq. units.
Alternatively, using the Shoelace Formula for the area of a polygon:
Area = 1/2 |(x1y2 + x2y3 + x3y4 + x4y1) - (y1x2 + y2x3 + y3x4 + y4x1)|
Area = 1/2 |(3*5 + 4*4 + (-1)*(-1) + (-2)*0) - (0*4 + 5*(-1) + 4*(-2) + (-1)*3)|
Area = 1/2 |(15 + 16 + 1 + 0) - (0 - 5 - 8 - 3)|
Area = 1/2 |(32) - (-16)|
Area = 1/2 |32 + 16| = 1/2 * 48 = 24 sq. units.
Therefore, option (b) is correct.
Which of the following is a coordinate point on the y-axis?
A)
(0, 1)
B)
(1, 0)
C)
(1, 1)
D)
(0, 0)
In a 2D Cartesian coordinate system, a point is represented as (x, y).
1. For a point to lie on the y-axis, its horizontal distance from the origin must be zero. Therefore, the x-coordinate must be 0.
2. Analyzing the options:
- Option (a) (0, 1): The x-coordinate is 0. This point lies exactly on the y-axis at a height of 1 unit. This is the correct answer.
- Option (b) (1, 0): The y-coordinate is 0. This point lies on the x-axis.
- Option (c) (1, 1): Neither coordinate is 0. This point lies in the first quadrant.
- Option (d) (0, 0): This is the origin. While it technically lies on both axes, in the context of multiple-choice questions identifying a specific axis point, a point with a non-zero y-value and zero x-value is the definitive answer for 'a point on the y-axis'. However, (0, 1) is the most distinct example of a point specifically on the y-axis.
If ΔPQR is an equilateral triangle and PQ = 6 cm, then the area of ΔPQR is:
A)
18cm²
B)
9√3 cm²
C)
36cm²
D)
27√3 cm²
Step-by-step solution:
1. **Identify the given information**: The triangle ΔPQR is an equilateral triangle, meaning all three sides are equal. The side length (s) is given as PQ = 6 cm.
2. **Recall the formula for the area of an equilateral triangle**: The area (A) is calculated using the formula: A = (√3 / 4) * s².
3. **Substitute the given value**:
A = (√3 / 4) * (6)²
A = (√3 / 4) * 36
4. **Simplify the expression**:
A = √3 * (36 / 4)
A = 9√3 cm².
Analysis of options:
- Option (a) 18cm²: Incorrect. This is likely obtained by incorrectly using (1/2 * base * height) without calculating the correct height of the triangle.
- Option (b) 9√3 cm²: Correct. Matches the calculated result.
- Option (c) 36cm²: Incorrect. This is simply the square of the side length (s²).
- Option (d) 27√3 cm²: Incorrect. This would be the result if the side length were 6√3 or if the formula was applied incorrectly.
Find the value of cos 135°.
A)
1/2
B)
√2
C)
-√2
D)
-1/√2
To find the value of cos 135°, we can use trigonometric identities.
1. **Quadrant Analysis**: The angle 135° lies in the Second Quadrant (between 90° and 180°). In the second quadrant, the cosine function is negative.
2. **Reference Angle**: The reference angle for 135° is calculated as:
Reference Angle = 180° - 135° = 45°.
3. **Calculation**:
cos(135°) = -cos(45°)
We know that cos(45°) = 1/√2 (or √2/2).
Therefore, cos(135°) = -1/√2.
**Analysis of Options**:
- Option (a) 1/2: This is the value of cos 60°.
- Option (b) √2: This is impossible as the value of cosine cannot exceed 1.
- Option (c) -√2: This is impossible as the value of cosine cannot be less than -1.
- Option (d) -1/√2: This is the correct value based on the second quadrant property and the reference angle of 45°.
If ΔPQR ~ ΔXYZ and ∠P=50°, ∠Q=60°, then ∠Y is:
A)
50°
B)
60°
C)
70°
D)
80°
Option B is correct: 60°.
What is the value of sin 0° + cos 0°?
A)
0
B)
2
C)
1
D)
∞
To solve the expression sin 0° + cos 0°, we use the standard trigonometric values for the angle 0°:
1. The value of sin 0° is 0. (In a unit circle, the sine represents the y-coordinate, which is 0 at 0 radians/degrees).
2. The value of cos 0° is 1. (In a unit circle, the cosine represents the x-coordinate, which is 1 at 0 radians/degrees).
Substituting these values into the expression:
sin 0° + cos 0° = 0 + 1 = 1.
Analysis of options:
- Option (a) 0 is incorrect because it only considers the value of sin 0°.
- Option (b) 2 is incorrect as it does not match the sum of the trigonometric values.
- Option (c) 1 is correct.
- Option (d) ∞ is incorrect as both functions are defined and finite at 0°.
Evaluate (cosec θ + cot θ)(cosec θ - cot θ).
A)
0
B)
1
C)
2
D)
3
The given expression is in the form of (a + b)(a - b), which is a standard algebraic identity known as the difference of squares: (a + b)(a - b) = a² - b².
Step 1: Apply the identity to the trigonometric expression:
(cosec θ + cot θ)(cosec θ - cot θ) = cosec² θ - cot² θ
Step 2: Use the fundamental trigonometric identity relating cosecant and cotangent:
We know that 1 + cot² θ = cosec² θ.
Step 3: Rearrange the identity to find the value of cosec² θ - cot² θ:
cosec² θ - cot² θ = 1
Therefore, the value of the expression is 1.
Analysis of options:
- Option (a) 0 is incorrect because the difference between the squares of cosec θ and cot θ is constant (1), not zero.
- Option (b) 1 is correct based on the Pythagorean identity.
- Option (c) 2 and (d) 3 are incorrect as they do not satisfy the trigonometric identity.
If r is the radius of the sphere, then the surface area of the sphere is given by;
A)
4πr
B)
2πr²
C)
πr²
D)
4πr²
The surface area of a sphere is defined as the total area of the exterior surface of the sphere. Mathematically, the formula for the surface area (A) of a sphere with radius 'r' is A = 4πr².
Analysis of options:
- Option (a) 4πr: This is not a standard formula for area (it lacks the squared dimension required for area).
- Option (b) 2πr²: This represents the curved surface area of a hemisphere (half a sphere).
- Option (c) πr²: This is the formula for the area of a circle (a 2D cross-section of the sphere).
- Option (d) 4πr²: This is the correct standard geometric formula for the total surface area of a sphere.
(sin A - cos A)² is equal to
A)
1 + 2sinAcosA
B)
1 - 2sinAcosA
C)
2sinAcosA - 1
D)
2sinAcosA + 1
To solve the expression (sin A - cos A)², we use the algebraic identity (a - b)² = a² - 2ab + b².
Step 1: Apply the identity to the trigonometric expression:
(sin A - cos A)² = (sin A)² - 2(sin A)(cos A) + (cos A)²
Step 2: Rearrange the terms:
= (sin² A + cos² A) - 2sin A cos A
Step 3: Use the fundamental trigonometric identity sin² A + cos² A = 1:
= 1 - 2sin A cos A
Comparing this result with the given options:
- Option (a) 1 + 2sinAcosA is the expansion of (sin A + cos A)².
- Option (b) 1 - 2sinAcosA matches our result.
- Option (c) 2sinAcosA - 1 is the negative of our result.
- Option (d) 2sinAcosA + 1 is incorrect.
Out of the following, which is the Pythagorean triplet?
A)
(1,5, 10)
B)
(3,4, 5)
C)
(2, 2, 2)
D)
(5,5, 2)
A Pythagorean triplet consists of three positive integers (a, b, c) such that they satisfy the Pythagorean theorem: a² + b² = c², where 'c' is the largest number (the hypotenuse).
Let's test each option:
Option (a): (1, 5, 10)
1² + 5² = 1 + 25 = 26
10² = 100
Since 26 ≠ 100, this is not a Pythagorean triplet.
Option (b): (3, 4, 5)
3² + 4² = 9 + 16 = 25
5² = 25
Since 25 = 25, this satisfies the theorem. This is the most basic and well-known Pythagorean triplet.
Option (c): (2, 2, 2)
2² + 2² = 4 + 4 = 8
2² = 4
Since 8 ≠ 4, this is not a Pythagorean triplet.
Option (d): (5, 5, 2)
(Note: The largest number must be the hypotenuse, so we check 2² + 5² = 4 + 25 = 29, which is not 5²=25. Or if we take 5 as the hypotenuse: 2² + 5² = 29 ≠ 25).
Since no combination satisfies a² + b² = c², this is not a Pythagorean triplet.
Therefore, option (b) is the correct answer.