Elementary mathematics and science content with the pedagogical issues of teaching both subjects, as asked in upper-primary teacher papers.
The value of 18 ÷ 3 + 5 × 6 - 1 is:
31
65
40
35
Correct answer
35
Why this answer is correct
Concept used:
Calculation:
18 ÷ 3 + 5 × 6 - 1
⇒ 6 + 30 - 1
⇒ 35
∴ The correct answer is 35
If \(\rm \left(\frac{3}{5}\right)^5 \times\left(\frac{3}{5}\right)^{-3}=\left(\frac{3}{5}\right)^{5 x-3} \), then the value of x is :
1
0
2
\(\frac{2}{5}\)
Correct answer
1
Why this answer is correct
Concept used:
Ab × Ac = A(b + c)
Calculation:
\(\rm \left(\frac{3}{5}\right)^5 \times\left(\frac{3}{5}\right)^{-3}=\left(\frac{3}{5}\right)^{5 x-3} \)
⇒ \((\frac{3}{5})^{(5 - 3)}\) = \((\frac{3}{5})^{5x - 3}\)
⇒ \((\frac{3}{5})^2\) = \((\frac{3}{5})^{5x - 3}\)
Comparing, we get,
(5x - 3) = 2
⇒ x = 1
∴ The correct answer is 1
If 21168 = p4q3r2, where p, q and r are prime numbers, then the value of (4p + 3q - 2r) is :
6
7
2
3
Correct answer
3
Why this answer is correct
Given:
21168 = p4q3r2
Calculation:
To identify p, q, r, we have to factorize 21168
21168 = 24 × 33 × 72 comparing with p4q3r2 we get p = 2, q = 3, and r = 7
Then the value of (4p + 3q - 2r) = (4 × 2 + 3 × 3 - 2 × 7) = (8 + 9 - 14) = 3
∴ The correct answer is 3
The value of [(-20) ÷ 4] × (-2) + 6 × 3 - 1 is:
27
22
32
30
Correct answer
27
Why this answer is correct
Concept used:
Calculation:
[(-20) ÷ 4] × (-2) + 6 × 3 - 1
⇒ [- 5] × (- 2) + 18 - 1
⇒ 10 + 18 - 1
⇒ 28 - 1
⇒ 27
∴ The correct answer is 27
The fractions \(\frac{30}{25}, \frac{27}{81}, \frac{22}{25} \) and \( \frac{45}{22}\) are written in ascending order as:
\(\frac{27}{81}, \frac{30}{25}, \frac{45}{22}, \frac{22}{25} \)
\(\frac{27}{81}, \frac{45}{22}, \frac{22}{25}, \frac{30}{25} \)
\(\frac{45}{22}, \frac{30}{25}, \frac{22}{25}, \frac{27}{81} \)
\(\frac{27}{81}, \frac{22}{25}, \frac{30}{25}, \frac{45}{22}\)
Correct answer
\(\frac{27}{81}, \frac{22}{25}, \frac{30}{25}, \frac{45}{22}\)
Why this answer is correct
Calculation:
This question can be solved by simply converting the fractions in decimal forms and then comparing them.
30/25 = 1.2
27/81 = 0.333
22/25 = 0.88
45/22 = 2.045
∴ The correct ascending order is = \(\frac{27}{81}, \frac{22}{25}, \frac{30}{25}, \frac{45}{22}\)
The marked price of an article is Rs. 420. A shopkeeper gives a discount of 15% on the marked price and still makes a profit of 19%. What is the cost price of the article?
Rs. 270
Rs. 290
Rs. 300
Rs. 320
Correct answer
Rs. 300
Why this answer is correct
Calculation:
A 15% discount given on MP = 420, then SP = 420 × 17/20 = 357
Selling at the price of Rs. 357, the shopkeeper makes 19% profit, then CP = 357 × 100/119 = Rs. 300
∴ The correct answer is Rs. 300
If \(\rm x+\frac{1}{x}=4 \), then the value of \(\rm x^4+\frac{1}{x^4}\) is:
128
194
196
256
Correct answer
194
Why this answer is correct
Concept used:
Calculation:
Here, x + 1/x = 4, then x2 + 1/x2 = 42 - 2 = 14,
Again if x2 + 1/x2 = 14, then x4 + 1/x4 = 142 - 2 = 194
∴ The correct answer is 194
One of the factors of the expression x2y - xy2 + x - y is:
x2 - y2
x + y
xy + 1
xy
Correct answer
xy + 1
Why this answer is correct
Calculation:
(x2y - xy2)+ (x - y)
⇒ xy(x - y) + (x - y)
⇒ (x - y)(xy + 1)
∴ From the given options, (xy + 1) is one of the factors of the given expression
In the product (x - 2)(x - 3)(x - 5), the co-efficient of x2 is:
-30
10
1
-10
Correct answer
-10
Why this answer is correct
Calculation:
Multiply the first two factors, we get
(x - 2)(x - 3)
⇒ x2 - 3x - 2x + 6
⇒ x2 - 5x + 6
Now multiply the result above with the third factor (x - 5), we get
(x - 5)(x2 - 5x + 6)
⇒ x3 - 5x2 + 6x - 5x2 + 25x - 30
⇒ x3 - 10x2 + 31x - 30
Here, we see the co-efficient of x2 is - 10
∴ The correct answer is - 10
If x2 + y2 = 6 and x3y + xy3 + x2 + y2 = 36, then the value of xy is :
4
3
5
6
Correct answer
5
Why this answer is correct
Shortcut Trick
Factor out the common factor (x2 + y2) from the given expression: xy(x2 + y2) + (x2 + y2) = 36
Substitute the value x2 + y2 = 6 into the equation: 6(xy) + 6 = 36
⇒ 6(xy + 1) = 36
⇒ xy + 1 = 6 ⇒ xy = 5
∴ The correct answer is 5.
Alternate Method
Given:
x2 + y2 = 6 --- (1)
x3y + xy3 + x2 + y2 = 36 --- (2)
Method: Factorization by grouping
⇒ xy(x2 + y2) + (x2 + y2) = 36
⇒ (x2 + y2)(xy + 1) = 36
Substitute the value of (x2 + y2) from equation (1):
⇒ 6 × (xy + 1) = 36
⇒ xy + 1 = 36 ÷ 6
⇒ xy + 1 = 6
⇒ xy = 6 − 1
⇒ xy = 5
∴ The correct answer is 5.
Additional Information
Factorization by Grouping
This method involves rearranging the terms of a polynomial into groups that share a common factor, such as a(x + y) + b(x + y) = (a + b)(x + y).
Algebraic Identity for Squares
The square of a binomial is given by (x + y)2 = x2 + y2 + 2xy. This is useful for finding the sum of squares when the product is known.
Substitution Method
In a system of equations, a known value or expression from one equation is substituted into another to reduce the number of variables and solve the problem.
Which of the following is a Pythagorean triplet ?
12, 15, 19
8, 13, 18
5, 12, 13
14, 20, 24
Correct answer
5, 12, 13
Why this answer is correct
Calculation:
According to the given options, option 3
52 + 122 = 169 = 132 holds Pythagoras theorem
So, 5, 12, 13 are Pythagorean triplet
∴ The correct answer is option 3
If one of the angles of a triangle is 120°, then the angle between the bisectors of the other two angles is :
60°
120°
130°
150°
Correct answer
150°
Why this answer is correct
Shortcut Trick
In any triangle, the angle formed by the internal bisectors of two angles (say ∠B and ∠C) at the incenter (I) is related to the third angle (∠A) by the formula:
Angle between bisectors = 90° + (∠A ÷ 2)
Given ∠A = 120°
⇒ Required Angle = 90° + (120° ÷ 2)
⇒ Required Angle = 90° + 60° = 150°
∴ The correct answer is 150°ree;.
Alternate Method
Given:
One angle of the triangle (∠A) = 120°
The other two angles are ∠B and ∠C.
Property Used: Sum of interior angles of a triangle is 180°.
⇒ In ΔABC, ∠A + ∠B + ∠C = 180°
⇒ 120° + ∠B + ∠C = 180°
⇒ ∠B + ∠C = 180° − 120° = 60°
Let I be the point of intersection of the bisectors of ∠B and ∠C.
⇒ In ΔBIC, the sum of angles is also 180°:
⇒ ∠BIC + ∠IBC + ∠ICB = 180°
⇒ Since BI and CI are bisectors, ∠IBC = ∠B ÷ 2 and ∠ICB = ∠C ÷ 2.
⇒ ∠BIC + (∠B ÷ 2) + (∠C ÷ 2) = 180°
⇒ ∠BIC + (∠B + ∠C) ÷ 2 = 180°
⇒ ∠BIC + (60° ÷ 2) = 180°
⇒ ∠BIC + 30° = 180°
⇒ ∠BIC = 180° − 30° = 150°
∴ The correct answer is 150°ree;.
Additional Information
Incenter
The point where all internal angle bisectors of a triangle meet. It is the center of the incircle.
Angle at Excenter
The angle formed by the external bisectors of two angles of a triangle is given by the formula 90° − (∠A ÷ 2).
Angle in a Straight Line
Internal bisector and external bisector of the same angle are always perpendicular (form a 90° angle) to each other.
A regular polygon of 6 sides has :
3 lines of symmetry
6 lines of symmetry
12 lines of symmetry
9 lines of symmetry
Correct answer
6 lines of symmetry
Why this answer is correct
Concept used:
In a regular polygon, the lines of symmetry are equal to the number of sides
Calculation:
According to the given concept,
A regular polygon of 6 sides (Hexagon) has 6 lines of symmetry
∴ The correct answer is 6 lines of symmetry
In triangles ABC and DEF, ∠C = ∠F, AC = DF and BC = EF. If AB = 4x - 2 and DE = 3x - 1, then the value of x is:
1
2
3
4
Correct answer
1
Why this answer is correct
Shortcut Trick
Given AC = DF, BC = EF, and the included angle ∠C = ∠F.
By the SAS (Side-Angle-Side) congruence rule, ΔABC ≅ ΔDEF.
In congruent triangles, corresponding parts are equal (CPCT), so AB = DE.
⇒ 4x − 2 = 3x − 1
⇒ 4x − 3x = 2 − 1
⇒ x = 1
∴ The correct answer is 1.
Alternate Method
Given: In ΔABC and ΔDEF, AC = DF, BC = EF, ∠C = ∠F. Also, AB = 4x − 2 and DE = 3x − 1.
Formula Used: SAS Congruence Criterion and CPCT (Corresponding Parts of Congruent Triangles).
Applying the SAS Congruence Rule:
⇒ Side AC = Side DF (Given)
⇒ Angle ∠C = Angle ∠F (Given)
⇒ Side BC = Side EF (Given)
⇒ ∴ ΔABC ≅ ΔDEF (By SAS criterion)
Since the triangles are congruent, their corresponding sides are equal:
⇒ AB = DE
⇒ 4x − 2 = 3x − 1
⇒ 4x − 3x = 2 − 1
⇒ x = 1
∴ The correct answer is 1.
Additional Information
SAS Congruence Rule
Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
CPCT Property
Corresponding Parts of Congruent Triangles are always equal in measure (sides and angles).
Other Congruence Criteria
Triangles can also be proven congruent using SSS (Side-Side-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side).
Which of the following triplets can form a polyhedron, if the numbers represent the number of faces, vertices and edges, respectively ?
4, 4 and 5
6, 8 and 12
6, 8 and 10
5, 6 and 8
Correct answer
6, 8 and 12
Why this answer is correct
Concept used :
Euler's formula for polyhedra states that F + V - E = 2. This formula holds true for all polyhedra, regardless of their shape or size.
Calculation :
The triplets (4, 4, 5) and (5, 6, 8) do not satisfy Euler's formula, so they cannot form polyhedra.
The triplet (6, 8, 10) satisfies Euler's formula, but it is not possible to form a polyhedron with 6 faces that are all triangles.
The minimum number of edges for a polyhedron with 6 faces that are all triangles is 9, but the triplet (6, 8, 10) only has 10 edges.
The triplet (6, 8, 12) satisfies Euler's formula and it is possible to form a polyhedron with 6 faces that are all triangles.
The minimum number of edges for a polyhedron with 6 faces that are all triangles is 9, and the triplet (6, 8, 12) has 12 edges, so it is possible to form a polyhedron with these properties.
Therefore, the only triplet that can form a polyhedron is (6, 8, 12).
The heights of two right circular cylinders are in the ratio of 8 ∶ 5 and their radii of bases are in the ratio of 1 ∶ 2. The ratio of their volumes is :
2 ∶ 5
1 ∶ 4
16 ∶ 25
9 ∶ 7
Correct answer
2 ∶ 5
Why this answer is correct
Concept used:
The volume of right circular cylinder = π × r2 × h
Calculation:
Let's assume that the height of the cylinder is 8x and 5x and the radius is k and 2k
So, the ratio of the volume of the two cylinders is
π × k2 × 8x : π × (2k)2 × 5x
⇒ 8 : 4 × 5
⇒ 2 : 5
∴ The correct answer is 2 : 5
If each edge of a solid cube is increased by 100%, then the percentage increase in its surface area is :
100
200
300
400
Correct answer
300
Why this answer is correct
Concept used:
The total Surface area of a Cube = 6 × side2
Calculation:
Let's assume that each side of the cube be 10 unit
So, TSA = 6 × 102 = 600
Now, if each side increased by 100%, then the sides becomes = 10 × 200/100 = 20
Now, TSA = 6 × 202 = 2400
TSA increased (2400 - 600) = 1800
Percentage increase in TSA = 1800/600 × 100 = 300%
∴ The correct answer is 300%
If the sum of the parallel sides of a trapezium is 40 cm and its height is 10 cm, then the area of the trapezium is:
400 cm2
200 cm2
40 cm2
100 cm2
Correct answer
200 cm2
Why this answer is correct
Concept used:
Area of trapezium = 1/2 × (Sum of parallel sides) × (distance between them)
Calculation:
According to the given formula,
Area of trapezium = 1/2 × 40 × 10 = 200 cm2
∴ The correct answer is 200 cm2
Following table shows marks obtained by 40 students. What is the mode of this data ?
| Marks obtained | 42 | 36 | 30 | 45 | 50 |
|---|---|---|---|---|---|
| No. of students | 7 | 10 | 13 | 8 | 2 |
50
30
10
36
Correct answer
30
Why this answer is correct
Concept used:
The number which has. the highest frequency is called the mode.
Calculation:
According to the given data set,
Marks 30 is obtained by 13 numbers of students which is the highest,
So, the mode of this data = 30
∴ The correct answer is 30
Cards numbered as 12, 13, 14, . . ., 51, (one number on one card) are kept in a box and mixed thoroughly. One card is chosen at random from the box. What is the probability that the number on the chosen card is a composite number between 21 and 42 ?
\(\frac{1}{2} \)
\(\frac{7}{20} \)
\(\frac{3}{8} \)
\(\frac{17}{40}\)
Correct answer
\(\frac{3}{8} \)
Why this answer is correct
Concept used:
Probability of any event = favorable number of cases / Total number of cases
Calculation:
Total number of cases from 12 to 51 is 40
Favorable number of cases = composite number between 21 and 42 = (22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40) total 15 numbers
So, required probability = 15/40 = 3/8
∴ The correct answer is 3/8
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.