Search Previous Year Questions

Shortcut Trick

Arrange the given data in ascending order: 10, 12, 12, 13, 14, 15, 21.

Count the total number of observations (n) = 7.

Since n is odd, the Median is the middle term: ((7 + 1) ÷ 2)^th term.

The 4^th term in the sorted list is 13.

∴ The correct answer is 13.

Alternate Method

Given: Data set = {13, 15, 12, 21, 10, 14, 12}

Formula Used: For an odd number of observations (n), Median = ((n + 1) ÷ 2)^th observation.

⇒ Step 1: Arrange data

Shortcut Trick

Use the Value Substitution method by choosing a specific acute angle, let A = 30°.

Then, sec 30° = 2/√3 and tan 30° = 1/√3.

Expression = [(2/√3) − (1/√3) + 1] / [(1/√3) + (2/√3) + 1] = (1/√3 + 1) / (3/√3 + 1) = (1 + √3) / (3 + √3) = 1/√3.

Now check options for A = 30°: (1 − sin 30°) / cos 30° = (1 − 1/2) / (√3/2) = (1/2) / (√3/2) = 1/√3.

The values match, confirming that the expression simplifies to (1 − sin A) / cos A.

∴ The correct answer is (1 − sin A) / cos A.

Alternate Method

Given: (sec A − tan A + 1) / (tan A + sec A + 1)

Formula Used: sec²A − tan²A = 1

⇒ Substitute 1 = sec²A − tan²A in the numerator:

⇒ [(sec A − tan A) + (sec²A − tan²A)] / (tan A + sec A + 1)

⇒ [(sec A − tan A) + (sec A − tan A)(sec A + tan A)] / (tan A + sec A + 1)

⇒ [ (sec A − tan A) (1 + sec A + tan A) ] / (tan A + sec A + 1)

⇒ sec A − tan A

⇒ (1 / cos A) − (sin A / cos A)

⇒ (1 − sin A) / cos A

∴ The correct answer is (1 − sin A) / cos A.

Additional Information

Pythagorean Identities

Fundamental identities include sin²A + cos²A = 1, 1 + tan²A = sec²A, and 1 + cot²A = cosec²A.

Trigonometric Ratios

Reciprocal relationships: sec A = 1/cos A, cosec A = 1/sin A, and tan A = sin A/cos A.

Simplification Rules

Expressions involving (sec A ± tan A ± 1) are often simplified using the factorable difference of squares: (sec²A − tan²A) = 1.

Given:

Let assume point P is the midpoint of segment AB.

Co-ordinates of P are ( 2, 5 ) and B are ( 0, 2 )

Concept used:

If P is the midpoint of segment AB where Co-ordinates of  A is (x1, y1)

and Co-ordinates of B is (x2, y2), then, Co-ordinates of P = {(x1 + x2)/2, (y1 + y2)/2}

Calculation:

Let the co-ordinates of  A are (x, y)

According to the given formula:

Co-ordinates of P = {(x1 + x2)/2, (y1 + y2)/2}

⇒ ( x + 0 ) / 2 = 2, ( y + 2 ) / 2 = 5

⇒  x  =  4 ,  y + 2  = 10

⇒ x = 4 , y = 10 - 2

⇒ x = 4 , y = 8

∴ The co-ordinates of A is ( 4 , 8 ).

Shortcut Trick

The given expression is in the form of the sine addition identity: sin A cos B + cos A sin B = sin(A + B).

Here, substitute A = 25° and B = 65°.

Expression = sin(25° + 65°) = sin 90°.

We know that the standard value of sin 90° = 1.

∴ The correct answer is 1.

Alternate Method

Given: sin 25° cos 65° + cos 25° sin 65°

Formula Used: Complementary angle relations [cos θ = sin(90° − θ)] and the Pythagorean identity [sin²θ + cos²θ = 1].

⇒ cos 65° = sin(90° − 65°) = sin 25°

⇒ sin 65° = cos(90° − 65°) = cos 25°

⇒ Substitute these into the expression: (sin 25° × sin 25°) + (cos 25° × cos 25°)

⇒ Expression = sin² 25° + cos² 25°

⇒ Using the Trigonometric Identity sin²θ + cos²θ = 1, the value is 1.

∴ The correct answer is 1.

Additional Information

Compound Angle Formulas

sin(A + B) = sin A cos B + cos A sin B and sin(A − B) = sin A cos B − cos A sin B.

Complementary Ratios

For any angle θ, sin(90° − θ) = cos θ and tan(90° − θ) = cot θ.

Fundamental Identity

The identity sin²θ + cos²θ = 1 holds true for all real values of θ.

Given

3, 4, 5, 7, 10, 10, 10

Concept used

Mean = Average

Deviation is the difference with the given number in the series.

Calculation

Mean = \(\frac{{3 + 4 + 5 + 7 + 10 + 10 + 10}}{7}\)

Mean = 49/7

Mean = 7

Checking the mean deviation with all the numbers given in the series.

Mean deviation 

⇒ |7 - 3|, |7 - 4|, |7 - 5|, |7 - 7|, |7 - 10|, |7 - 10|, |7 - 10|

⇒ 4, 3, 2, 0, 3, 3, 3

Mean deviation = \(\frac{{3 + 4 + 2 + 3 + 3 + 3}}{7}\)

Mean deviation = 18/7

Shortcut Trick

Use the Value Substitution Method by putting θ = 60°.

Since tan 60° = √3 and cot 60° = 1/√3, substitute these values into the expression.

Expression = ((√3)² − 1) ÷ (1 − (1/√3)²) = (3 − 1) ÷ (1 − 1/3) = 2 ÷ (2/3) = 3.

Check options: tan² 60° = (√3)² = 3, which matches our result.

∴ The correct answer is tan² θ.

Alternate Method

Given Expression: (tan² θ − 1) ÷ (1 − cot² θ)

Formula Used: cot θ = 1/tan θ

tan² θ − 1

1 − (1/tan² θ) Simplifying using reciprocal identity

⇒ (tan² θ − 1) ÷ (1 − 1/tan² θ)

⇒ (tan² θ − 1) ÷ ((tan² θ − 1) / tan² θ)

⇒ (tan² θ − 1) × (tan² θ / (tan² θ − 1))

Canceling the common term (tan² θ − 1) from the numerator and denominator:

⇒ tan² θ

∴ The correct answer is tan² θ.

Additional Information

Reciprocal Identities

The function cot θ is the reciprocal of tan θ, meaning tan θ × cot θ = 1.

Pythagorean Identities

Standard identities include sin² θ + cos² θ = 1 and 1 + tan² θ = sec² θ.

Quotient Identities

Tangent can be expressed as tan θ = sin θ / cos θ and cotangent as cot θ = cos θ / sin θ.

Shortcut Trick

Number Hierarchy: Integers ⊆ Rational numbers ⊆ Real numbers.

Real numbers = {Rational} ∪ {Irrational}.

Complex numbers = {Real} + {Imaginary}.

Since Complex numbers include Imaginary parts, they are the superset of Real numbers.

Therefore, Complex numbers are not a subset of Real numbers.

∴ Answer = (4) Complex

Alternate Method

Given:

Number sets: Rational numbers, Irrational numbers, Integers, and Complex numbers.

Formula used:

Subset Property: A ⊆ B if every element of A is also in B. Hierarchy: N ⊆ W ⊆ Z ⊆ Q ⊆ R ⊆ C.

Complex Numbers (C) Real Numbers (R) Rational (Q) Integers (Z) Irrational Imaginary

Calculations:

⇒ Real numbers consist of all numbers that can be found on a number line, including Rational and Irrational numbers.

⇒ Integers (Z) = {..., −2, −1, 0, 1, 2, ...} are a subset of Rational numbers (Q).

⇒ Rational numbers (Q) and Irrational numbers (I) are subsets of Real numbers (R).

⇒ Complex numbers (C) are defined as a + bi, where a and b are real and i = √−1.

⇒ Since C includes imaginary numbers (where b ≠ 0), not every complex number is a real number.

⇒ Thus, Real numbers are a subset of Complex numbers (R ⊆ C), but Complex numbers are not a subset of Real numbers.

∴ The correct answer is (4) Complex.

Shortcut Trick

The unit digit of the volume 10648 is 8, so the unit digit of its cube root must be 2.

Split the number 10648 as 10 | 648. The number 10 lies between 2³ (8) and 3³ (27).

The smaller base is 2, which becomes the tens digit.

Combining the tens digit (2) and unit digit (2), we get 22.

∴ The correct answer is 22 cm.

Alternate Method

Given:

Volume of the cubic die = 10648 cm³

Formula Used:

Volume of a cube = a³ (where a is the side of the cube)

Calculations:

⇒ a³ = 10648

⇒ a = ³√10648

To find the cube root by prime factorization:

⇒ 10648 = 2 × 2 × 2 × 11 × 11 × 11

⇒ a = ³√(2³ × 11³)

⇒ a = 2 × 11

⇒ a = 22 cm

∴ The correct answer is 22 cm.

Additional Information

Total Surface Area of a Cube

The total surface area is the sum of the areas of all 6 faces: TSA = 6 × a².

Diagonal of a Cube

The length of the longest rod that can be placed inside a cube: Diagonal = a√3.

Lateral Surface Area

The area of the 4 side faces (excluding top and bottom): LSA = 4 × a².

Given:

Red = 6, Green = 5, Blue = 4

Formula used:

Probability = favorable outcomes ÷ total outcomes

6 5 4 Total = 15 balls

Calculations:

⇒ Total balls = 6 + 5 + 4 = 15

⇒ Not blue = 6 + 5 = 11

⇒ Probability = 11 ÷ 15

∴ The correct answer is 11/15.

Shortcut Trick

Check divisibility using known rules or small prime factors:

567: Sum of digits is 5 + 6 + 7 = 18 (divisible by 9 and 3).

323: 323 = 17 × 19 (not prime).

437: 437 = 19 × 23 (not prime).

241: No prime factors up to √241 ≈ 15.5.

∴ The correct answer is 241.

Alternate Method

To check if a number n is prime, find √n and check divisibility by all prime numbers less than or equal to √n.

Option 1: 567

⇒ Sum of digits = 5 + 6 + 7 = 18

⇒ 18 is divisible by 9, so 567 is divisible by 9. (Composite)

Option 2: 323

⇒ √323 ≈ 17.97. Primes to check: 2, 3, 5, 7, 11, 13, 17.

⇒ 323 ÷ 17 = 19. (Composite)

Option 3: 437

⇒ √437 ≈ 20.90. Primes to check: 2, 3, 5, 7, 11, 13, 17, 19.

⇒ 437 ÷ 19 = 23. (Composite)

Option 4: 241

⇒ √241 ≈ 15.52. Primes to check: 2, 3, 5, 7, 11, 13.

⇒ 241 ÷ 2, 3, 5, 7, 11, 13 are not integers.

⇒ No factors other than 1 and itself. (Prime)

∴ The correct answer is 241.

Additional Information

Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Composite Number

A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself.

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9 respectively.

Smallest Prime Number

The number 2 is the only even prime number and the smallest prime number.