Maths One-Liners PDF (Solved Past Papers MCQs) -

Mathematics

(Note: These are not part of the test syllabus. These questions appeared in yesterday’s test likr ANF ASI, FGEI Assistant, ML&C ESE, and today we will discuss and explain them so that we can understand the type and pattern of questions that are asked in examinations. This will help us prepare more effectively for upcoming tests and competitive exams.)

Q. If X²Y³ = 10 and X³Y² = 8, find X⁵Y⁵.
Answer: 80

Q. 5, 10, 15, 20 … next number?
Answer: 25

Q. 4, 11, 15, 19 … next number?
Answer: 23

Looking at the numbers 4, 11, 15, 19, the pattern is determined by the differences between consecutive terms:

From 4 to 11, the difference is +7
From 11 to 15, the difference is +4
From 15 to 19, the difference is +4

Since the difference of +4 repeats, the pattern continues by adding 4 to the last number.

19 + 4 = 23

Therefore, the next number in the series is 23.

Q. Tan45° + Cot45° = ?
Answer: 2

Q. Slope of a vertical line?
Answer: Undefined (or Infinite)

The slope of a vertical line is undefined.

This is because a vertical line goes straight up and down, so it has no horizontal change (Run = 0). Since division by zero is impossible, the slope cannot be calculated.

Q. Dot product of perpendicular lines?
Answer: Zero

Q. A set has four elements then subsets containing at least 2 elements are?
Answer: 11

Total elements ( $n$ ) = 4

Total possible subsets = $2^4 = 16$

We need subsets with at least 2 elements (which means subsets having 2, 3, or 4 elements). The easiest way is to subtract the subsets that have fewer than 2 elements:

Subsets with 0 elements (Empty set) = 1
Subsets with 1 element = 4
Total subsets to remove = 1 + 4 = 5

Subsets with at least 2 elements = $16 - 5 = 11$

Q. A dice is rolled, the probability of getting a number divisible by 2 or 3 is?
Answer: 5/6

English:

When a dice is rolled, the possible outcomes are: {1, 2, 3, 4, 5, 6} (Total = 6)

Numbers divisible by 2: 2, 4, 6
Numbers divisible by 3: 3, 6

If we combine them (without repeating 6), the favorable numbers are: 2, 3, 4, 6 (Total = 4 numbers)

\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{4}{6} = \frac{2}{3}

Therefore, the answer is 2/3.

Q. Roots of x² − 4x + 4 = 0 are?
Answer: 2, 2 (Real and Equal)

Q. If log(x) + log(2) = 1 then value of x is?
Answer: 5

Q. First 10 natural numbers added gives which answer?
Answer: 55 ((Or simply: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55$ )

Q. If U is universal set and A is a subset, then A’ (Complement of A) is defined as?
Answer: U − A

Q. Probability of getting at least one head when two coins are tossed?
Answer: 3/4

When two coins are tossed, the total possible outcomes are 4:

{HH, HT, TH, TT} (where H = Head, T = Tail)

We want the probability of getting at least one head (which means 1 or 2 heads). The favorable outcomes are: HH, HT, TH (Total = 3)

\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{3}{4}

Therefore, the answer is 3/4 (or 0.75 / 75%).

Q. Derivative of e²ˣ?
Answer: 2e²ˣ

Q. If det(A) = 0 then matrix is?
Answer: Singular Matrix

اردو: اگر کسی میٹرکس (قالب) کا مکتع صفر ہو ( $\det(A) = 0$ )، تو اسے نادر قالب (Singular Matrix) کہا جاتا ہے۔

نادر قالب کا معکوس یعنی الٹ ( $A^{-1}$ ) معلوم کرنا ممکن نہیں ہوتا کیونکہ فارمولے کے مطابق رقم کو مکتع (صفر) سے تقسیم کرنا پڑتا ہے۔

Q. If 4log(x) + log(2) = 1 then value of x is?
Answer: ⁴√5 (approximately 1.495)

Q. A complex number x + iy is imaginary if?
Answer: x = 0

Q. If x + 1/x = 3, then x² + 1/x² = ?
Answer: 7

Q. For what value of m, the roots of the equation 3x² + 2x + m = 0 are equal?
Answer: 1/3

For the roots of a quadratic equation $ax^2 + bx + c = 0$ to be equal, its discriminant ( $D$ ) must be equal to zero:

D = b^2 - 4ac = 0

In the given equation, $3x^2 + 2x + m = 0$ , the coefficients are:

$a = 3$
$b = 2$
$c = m$

Substitute these values into the discriminant formula:

D = (2)^2 - 4(3)(m) = 0

Simplify and solve for $m$ :

4 - 12m = 0

12m = 4

m = \frac{4}{12}

m = \frac{1}{3}

Answer: The roots of the equation will be equal when $m = \frac{1}{3}$ .

Q. Degree of Polynomial 2x⁴y³ + x²y² + 8x is?
Answer: 7

To find the degree of a polynomial with multiple variables, you look at the sum of the exponents of the variables in each individual term. The highest sum determines the degree of the entire polynomial.

Let’s analyze the terms of the polynomial $2x^4y^3 + x^2y^2 + 8x$ :

First term ( $2x^4y^3$ ): The exponents are $4$ (for $x$ ) and $3$ (for $y$ ).
$\text{Sum} = 4 + 3 = 7$
Second term ( $x^2y^2$ ): The exponents are $2$ (for $x$ ) and $2$ (for $y$ ).
$\text{Sum} = 2 + 2 = 4$
Third term ( $8x$ ): The exponent is $1$ (for $x$ , since $x = x^1$ ).
$\text{Sum} = 1$

The highest sum of exponents is $7$ .

Answer:

The degree of the polynomial is $7$ .

Q. log(x²y³) – 2log(x) = ?
Answer: 3log(y)

Q. If sinθ = 3/5, find cosθ in the first quadrant?
Answer: 4/5

Q. If 625 is the sum of 5 consecutive odd integers, then the largest integer is?
Answer: 129

Find the middle number (Average):
$\frac{625}{5} = 125$
List the 5 consecutive odd numbers around 125: 121, 123, 125, 127, 129

Answer: The largest integer is 129.

Q. Roots of x² − 5x + 6 = 0?
Answer: 2, 3

To find the roots of $x^2 - 5x + 6 = 0$ , you can factor the quadratic equation:

Find two numbers that multiply to $6$ and add up to $-5$ . Those numbers are $-2$ and $-3$ .
Rewrite the equation:
$(x - 2)(x - 3) = 0$
Solve for x: * $x - 2 = 0 \implies \mathbf{x = 2}$
- $x - 3 = 0 \implies \mathbf{x = 3}$

Answer: The roots are $2$ and $3$ .

Q. Which one is a polynomial?
Answer: x² − 3x + √2

Exponents must be whole numbers: Variables can only have positive powers (like $x^2, x^3$ ).
No variables in the denominator: You cannot divide by a variable (e.g. $\frac{1}{x}$ is not allowed).
No variables under a radical: You cannot have a variable inside a square root (e.g., $\sqrt{x}$ is not allowed).

Q. In SI units, Nm⁻² is the unit of?
Answer: Pressure (Pascal)

Q. The value of Gravitational Constant was first determined by?
Answer: Henry Cavendish

G \approx 6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2

Q. Universal Set U = {1, 2, 3, 4, 5} and A = {2, 4}, then A’ = ?
Answer: {1, 3, 5}

Q. The sum of first 10 natural numbers is equal to?
Answer: 55

To find the sum of the first 10 natural numbers ( $1 + 2 + 3 + \dots + 10$ ), you can use the arithmetic progression formula:

\text{Sum} = \frac{n(n + 1)}{2}

= 10(10 + 1) \ 2

= 10(11) \ 2

= 110 \ 2

= 55

Q. The volume of a sphere is?
Answer: 4/3πr³

Sphere (سفیئر) ایک ایسی تھری-ڈائمینشنل (3D) گول شکل کو کہتے ہیں جو ہر طرف سے بالکل گول ہوتی ہے۔اس کو سمجھنے کا سب سے آسان طریقہ یہ ہے کہ یہ ایک گیند (Ball) کی طرح ہوتا ہے۔

Q. If y = 2x² − 4x + 3, the vertex (کونے) ہ x-coordinate is?
Answer: 1

Identify the coefficients from your equation $y = 2x^2 - 4x + 3$ 2x² − 4x + 3:

$a = 2$
$b = -4$
$c = 3$

x = -\frac{b}{2a}

x = – (-4) \ 2(2)

= 4 \ 4

x = 1

Q. A sequence: 2, 5, 8, … The general term is?
Answer: 3n − 1

Look at the difference between consecutive terms:

$5 - 2 = 3$
$8 - 5 = 3$

Since the difference is constantly 3, this is an Arithmetic Progression (A.P.).

2. Use the Formula

The formula for the general term of an Arithmetic Progression is:

a_n = a_1 + (n - 1)d

a_n = 2 + (n – 1)3

= 2 + 3n – 3

= 3n – 1

Q. The circumference باہر کا پورا گول چکر (باؤنڈری) of a circle is 44 cm. Its radius (سرکل کے بالکل بیچ (سینٹر) سے لے کر باہر کی باؤنڈری تک کا آدھا فاصلہ ) is (π = 22/7)?
Answer: 7 cm

\text{Circumference} = 2\pi r

44 = 2 X 22/7 X r

44 = 44 / 7 X r

44 X 7 \ 44 = r

r = 7

Q. Probability of head in a coin toss?
Answer: 1/2

Q. If the perimeter of a rectangle is 164, then l + w = _____?
Answer: 82

perimeter = 2(l + w )

164 = 2(l + w )

164\2 = (l + w )

82 = (l + w )

Q. A train covers 360 km. Speed is calculated by which formula?
Answer: Speed = Distance ÷ Time

Q. Value of x in 17x = _____?
Answer: Zero

Q. If 2000 units are produced in 10 days, units produced per day are _____?
Answer: 200

\text{Units per day} = \frac{\text{Total Units}}{\text{Total Days}}

Units per day = 2000 \ 20

\text{Units per day} = \frac{2000}{10} = 200

Q. If log₃ x = 4, then x = ?
Answer: 81

Q. x²−2kx−1=0 has real roots for?
Answer: All real values of k

Q. Imaginary part of 1/2 + i(-1/5)?
Answer: -1/5 i

Q. Three coins are tossed. Probability of getting exactly 2 heads?
Answer: 3/8

Q. Sum of first n natural numbers?
Answer: n(n+1)/2

Q. log₁₀ 1000 = ?
Answer: 3

For log base 10, just count the number of zeros! Since $1000$ has 3 zeros, the answer is 3.

Q. sin²θ + cos²θ = ?
Answer: 1

Q. Distance between (0,0) and (3,4)?
Answer: 5

Q. Sum of interior angles of an 11-sided polygon?
Answer: 1260°

\text{Sum of interior angles} = (n - 2) \times 180^\circ

Sum of interior angles = 11 – 2 X180°

Sum of interior angles = 9 X180°

Sum of interior angles = 1620°

Assignment

Q. If 6 men take 9 days to complete a work, how many men can complete the work in 3 days?

Q. If 6 men can make 10 sofas in 2 days, then 8 men can make 8 sofas in ______?

Q. If 10 workers can make 10 tables in 10 days, then how many days would it take for 5 workers to make 5 tables?

Q. Two taps X and Y can fill a tank in 10 hrs. and 15 hrs. respectively. If both taps are opened together, the tank will be full in?

Q. If 3 men or 4 women can construct a wall in 43 days, then the number of days that 7 men and 5 women take to construct it is?

Q. If B = [4 2; 6 5], Find |B| = _____?

Q. If 25² = 625, then the square root of 625 is _____?

Q. Which of the following is not a square number? 100, 4, 9, 24

Q. A perfect square that lies between 40 and 50 is _____?

Q. Which of the following is a perfect square? 101, 32, 66, 81

Q. The average of the numbers 0, 0, 4, 10, 5, and 5 is _____?

Q. What is the average of first 150 natural numbers?

Q. In a class of 60 students, the ratio of boys to girls is 3:2. How many girls are there in the class?

Mathematics

Q. If X²Y³ = 10 and X³Y² = 8, find X⁵Y⁵.Answer: 80

Q. 5, 10, 15, 20 … next number?Answer: 25

Q. 4, 11, 15, 19 … next number?Answer: 23

Looking at the numbers 4, 11, 15, 19, the pattern is determined by the differences between consecutive terms:

From 4 to 11, the difference is +7

From 11 to 15, the difference is +4

From 15 to 19, the difference is +4

Since the difference of +4 repeats, the pattern continues by adding 4 to the last number.

19 + 4 = 23

Therefore, the next number in the series is 23.

Q. Tan45° + Cot45° = ?Answer: 2

Q. Slope of a vertical line?Answer: Undefined (or Infinite)

The slope of a vertical line is undefined.

This is because a vertical line goes straight up and down, so it has no horizontal change (Run = 0). Since division by zero is impossible, the slope cannot be calculated.

Q. Dot product of perpendicular lines?Answer: Zero

Q. A set has four elements then subsets containing at least 2 elements are?Answer: 11

Total elements (n) = 4

Total possible subsets = 2^4 = 16

We need subsets with at least 2 elements (which means subsets having 2, 3, or 4 elements). The easiest way is to subtract the subsets that have fewer than 2 elements:

Subsets with 0 elements (Empty set) = 1

Subsets with 1 element = 4

Total subsets to remove = 1 + 4 = 5

Subsets with at least 2 elements = 16 – 5 = 11

Q. A dice is rolled, the probability of getting a number divisible by 2 or 3 is?Answer: 5/6

When a dice is rolled, the possible outcomes are: {1, 2, 3, 4, 5, 6} (Total = 6)

Numbers divisible by 2: 2, 4, 6

Numbers divisible by 3: 3, 6

If we combine them (without repeating 6), the favorable numbers are: 2, 3, 4, 6 (Total = 4 numbers)

Therefore, the answer is 2/3.

Q. Roots of x² − 4x + 4 = 0 are?Answer: 2, 2 (Real and Equal)

Q. If log(x) + log(2) = 1 then value of x is?Answer: 5

Q. First 10 natural numbers added gives which answer?Answer: 55 ((Or simply: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55)

Q. If U is universal set and A is a subset, then A’ (Complement of A) is defined as?Answer: U − A

Q. Probability of getting at least one head when two coins are tossed?Answer: 3/4

When two coins are tossed, the total possible outcomes are 4:

{HH, HT, TH, TT} (where H = Head, T = Tail)

We want the probability of getting at least one head (which means 1 or 2 heads). The favorable outcomes are: HH, HT, TH (Total = 3)

Therefore, the answer is 3/4 (or 0.75 / 75%).

Q. Derivative of e²ˣ?Answer: 2e²ˣ

Q. If det(A) = 0 then matrix is?Answer: Singular Matrix

اردو: اگر کسی میٹرکس (قالب) کا مکتع صفر ہو (det(A)=0)، تو اسے نادر قالب (Singular Matrix) کہا جاتا ہے۔

نادر قالب کا معکوس یعنی الٹ (A−1) معلوم کرنا ممکن نہیں ہوتا کیونکہ فارمولے کے مطابق رقم کو مکتع (صفر) سے تقسیم کرنا پڑتا ہے۔

Q. If 4log(x) + log(2) = 1 then value of x is?Answer: ⁴√5 (approximately 1.495)

Q. A complex number x + iy is imaginary if?Answer: x = 0

Q. If x + 1/x = 3, then x² + 1/x² = ?Answer: 7

Q. For what value of m, the roots of the equation 3x² + 2x + m = 0 are equal?Answer: 1/3

For the roots of a quadratic equation ax2+bx+c=0 to be equal, its discriminant (D) must be equal to zero:

In the given equation, 3x2+2x+m=0, the coefficients are:

a=3

b=2

c=m

Substitute these values into the discriminant formula:

Simplify and solve for m:

Answer: The roots of the equation will be equal when m=31​.

Q. Degree of Polynomial 2x⁴y³ + x²y² + 8x is?Answer: 7

To find the degree of a polynomial with multiple variables, you look at the sum of the exponents of the variables in each individual term. The highest sum determines the degree of the entire polynomial.

Let’s analyze the terms of the polynomial 2x^4y^3 + x^2 y^2 + 8x:

First term (2x^4 y^3): The exponents are 4 (for x) and 3 (for y).

Second term (x^2y^2): The exponents are 2 (for x) and 2 (for y).

Third term (8x): The exponent is 1 (for x, since x = x^1).

The highest sum of exponents is 7.

Answer:

The degree of the polynomial is 7.

Q. log(x²y³) – 2log(x) = ?Answer: 3log(y)

Q. If sinθ = 3/5, find cosθ in the first quadrant?Answer: 4/5

Q. If 625 is the sum of 5 consecutive odd integers, then the largest integer is?Answer: 129

Find the middle number (Average):

List the 5 consecutive odd numbers around 125: 121, 123, 125, 127, 129

Answer: The largest integer is 129.

Q. Roots of x² − 5x + 6 = 0?Answer: 2, 3

To find the roots of x^2 – 5x + 6 = 0, you can factor the quadratic equation:

Find two numbers that multiply to 6 and add up to -5. Those numbers are -2 and -3.

Rewrite the equation:

Solve for x: * x – 2 = 0 Which implies x = 2

x – 3 = 0 which implies x = 3

Answer: The roots are 2 and 3.

Q. Which one is a polynomial?Answer: x² − 3x + √2

Q. In SI units, Nm⁻² is the unit of?Answer: Pressure (Pascal)

Q. The value of Gravitational Constant was first determined by?Answer: Henry Cavendish

Q. If X²Y³ = 10 and X³Y² = 8, find X⁵Y⁵.
Answer: 80

Q. 5, 10, 15, 20 … next number?
Answer: 25

Q. 4, 11, 15, 19 … next number?
Answer: 23

Q. Tan45° + Cot45° = ?
Answer: 2

Q. Slope of a vertical line?
Answer: Undefined (or Infinite)

Q. Dot product of perpendicular lines?
Answer: Zero

Q. A set has four elements then subsets containing at least 2 elements are?
Answer: 11

Total elements ( $n$ ) = 4

Total possible subsets = $2^4 = 16$

Subsets with at least 2 elements = $16 - 5 = 11$

Q. A dice is rolled, the probability of getting a number divisible by 2 or 3 is?
Answer: 5/6

Q. Roots of x² − 4x + 4 = 0 are?
Answer: 2, 2 (Real and Equal)

Q. If log(x) + log(2) = 1 then value of x is?
Answer: 5

Q. First 10 natural numbers added gives which answer?
Answer: 55 ((Or simply: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55$ )

Q. If U is universal set and A is a subset, then A’ (Complement of A) is defined as?
Answer: U − A

Q. Probability of getting at least one head when two coins are tossed?
Answer: 3/4

Q. Derivative of e²ˣ?
Answer: 2e²ˣ

Q. If det(A) = 0 then matrix is?
Answer: Singular Matrix

اردو: اگر کسی میٹرکس (قالب) کا مکتع صفر ہو ( $\det(A) = 0$ )، تو اسے نادر قالب (Singular Matrix) کہا جاتا ہے۔

نادر قالب کا معکوس یعنی الٹ ( $A^{-1}$ ) معلوم کرنا ممکن نہیں ہوتا کیونکہ فارمولے کے مطابق رقم کو مکتع (صفر) سے تقسیم کرنا پڑتا ہے۔

Q. If 4log(x) + log(2) = 1 then value of x is?
Answer: ⁴√5 (approximately 1.495)

Q. A complex number x + iy is imaginary if?
Answer: x = 0

Q. If x + 1/x = 3, then x² + 1/x² = ?
Answer: 7

Q. For what value of m, the roots of the equation 3x² + 2x + m = 0 are equal?
Answer: 1/3

For the roots of a quadratic equation $ax^2 + bx + c = 0$ to be equal, its discriminant ( $D$ ) must be equal to zero:

In the given equation, $3x^2 + 2x + m = 0$ , the coefficients are:

$a = 3$

$b = 2$

$c = m$

Simplify and solve for $m$ :

Answer: The roots of the equation will be equal when $m = \frac{1}{3}$ .

Q. Degree of Polynomial 2x⁴y³ + x²y² + 8x is?
Answer: 7

Let’s analyze the terms of the polynomial $2x^4y^3 + x^2y^2 + 8x$ :

First term ( $2x^4y^3$ ): The exponents are $4$ (for $x$ ) and $3$ (for $y$ ).

Second term ( $x^2y^2$ ): The exponents are $2$ (for $x$ ) and $2$ (for $y$ ).

Third term ( $8x$ ): The exponent is $1$ (for $x$ , since $x = x^1$ ).

The highest sum of exponents is $7$ .

The degree of the polynomial is $7$ .

Q. log(x²y³) – 2log(x) = ?
Answer: 3log(y)

Q. If sinθ = 3/5, find cosθ in the first quadrant?
Answer: 4/5

Q. If 625 is the sum of 5 consecutive odd integers, then the largest integer is?
Answer: 129

Q. Roots of x² − 5x + 6 = 0?
Answer: 2, 3

To find the roots of $x^2 - 5x + 6 = 0$ , you can factor the quadratic equation:

Find two numbers that multiply to $6$ and add up to $-5$ . Those numbers are $-2$ and $-3$ .

Solve for x: * $x - 2 = 0 \implies \mathbf{x = 2}$

$x - 3 = 0 \implies \mathbf{x = 3}$

Answer: The roots are $2$ and $3$ .

Q. Which one is a polynomial?
Answer: x² − 3x + √2

Q. In SI units, Nm⁻² is the unit of?
Answer: Pressure (Pascal)

Q. The value of Gravitational Constant was first determined by?
Answer: Henry Cavendish

Q. Universal Set U = {1, 2, 3, 4, 5} and A = {2, 4}, then A’ = ?
Answer: {1, 3, 5}

Q. The sum of first 10 natural numbers is equal to?
Answer: 55

To find the sum of the first 10 natural numbers ( $1 + 2 + 3 + \dots + 10$ ), you can use the arithmetic progression formula:

Q. The volume of a sphere is?
Answer: 4/3πr³

Q. If y = 2x² − 4x + 3, the vertex (کونے) ہ x-coordinate is?
Answer: 1

Identify the coefficients from your equation $y = 2x^2 - 4x + 3$ 2x² − 4x + 3:

$a = 2$

$b = -4$

$c = 3$

Q. A sequence: 2, 5, 8, … The general term is?
Answer: 3n − 1

$5 - 2 = 3$

$8 - 5 = 3$

Q. The circumference باہر کا پورا گول چکر (باؤنڈری) of a circle is 44 cm. Its radius (سرکل کے بالکل بیچ (سینٹر) سے لے کر باہر کی باؤنڈری تک کا آدھا فاصلہ ) is (π = 22/7)?
Answer: 7 cm

Q. Probability of head in a coin toss?
Answer: 1/2

Q. If the perimeter of a rectangle is 164, then l + w = _____?
Answer: 82

Q. A train covers 360 km. Speed is calculated by which formula?
Answer: Speed = Distance ÷ Time

Q. Value of x in 17x = _____?
Answer: Zero

Q. If 2000 units are produced in 10 days, units produced per day are _____?
Answer: 200

Q. If log₃ x = 4, then x = ?
Answer: 81

Q. x²−2kx−1=0 has real roots for?
Answer: All real values of k

Q. Imaginary part of 1/2 + i(-1/5)?
Answer: -1/5 i

Q. Three coins are tossed. Probability of getting exactly 2 heads?
Answer: 3/8

Q. Sum of first n natural numbers?
Answer: n(n+1)/2

Q. log₁₀ 1000 = ?
Answer: 3

For log base 10, just count the number of zeros! Since $1000$ has 3 zeros, the answer is 3.

Q. sin²θ + cos²θ = ?
Answer: 1

Q. Distance between (0,0) and (3,4)?
Answer: 5

Q. Sum of interior angles of an 11-sided polygon?
Answer: 1260°