Search Previous Year Questions

Given:

\([ \frac{-4}{7} \div \frac{1}{12}] \div [- \frac{4}{5} \times \big{(-2)-(- \frac{2}{7}) \big} ]\)

Concept used:

Follow the BODMAS rule according to the table given below:

Calculation:

\([ \frac{-4}{7} \div \frac{1}{12}] \div [- \frac{4}{5} \times \big{(-2)-(- \frac{2}{7}) \big} ]\)

⇒ \([ \frac{-4}{7} \times 12] \div [- \frac{4}{5} \times \big{(-2)+(\frac{2}{7}) \big} ]\)

⇒ \([ \frac{-48}{7}] \div [- \frac{4}{5} \times (\frac{-12}{7}) \big} ]\)

⇒ \([ \frac{-48}{7}] \div [ \frac{48}{35} ]\)

⇒ \([ \frac{-48}{7}] \times [ \frac{35}{48} ]\)

⇒ - 5

Given:

A bag contains 6 white and 4 red balls

Concept used:

Probability of occurrence of the event, P(E) = (Number of favorable outcome, n(E))/(Number of possible outcome, n(S))

Calculation:

Here, total number of balls in the bag = 6 + 4 = 10

Number of ways in which 2 white balls can be drawn from 6 white balls and 1 red ball can be drawn from 4 red balls = n(E) = ⁶C₂ × ⁴C₁

⇒ 60

Total number of ways in which 3 balls can be drawn out of all the all the balls = n(S) = ¹⁰C₃ 

⇒ 120

So, Probability P(E) = n(E)/n(S)

⇒ 60/120 

⇒ 1/2

∴ The required answer is \(\frac{ 1}{2 }\).

Given:

sinθ = 5/13

Concept used:

Using the concept of trigonometric ratios.

Calculation:

sinθ = 5/13

cosθ = √(1 - sin²θ)

⇒ cosθ = √(1 - 25/169)

⇒ cosθ = 12/13

tanθ = sinθ/cosθ 

⇒ tanθ = (5/13)/(12/13)

⇒ tanθ = 5/12

cotθ = 1/tanθ 

⇒ cotθ = 12/5

(cotθ - tanθ)/2cotθ

⇒ (12/5 - 5/12)/(2 × 12/5)

⇒ (119/60)/(24/5)

⇒ 119/288

∴ The value of (cotθ - tanθ)/2cotθ is 119/288.

Given:

A triangle and a parallelogram have the same base 28 cm and the same area.

The height of the parallelogram is 12 cm.

Concept used:

1. Area of a parallelogram = Base × Height
2. Area of a triangle = 1/2 × Height × Base

Calculation:

Area of the parallelogram = 28 × 12 = 336 cm²

Let the length of the altitude of the triangle be H cm.

​According to the question,

1/2 × H × 28 = 336

⇒ H = 12 × 2

⇒ H = 24

∴ The length of the altitude of the triangle is 24 cm.

Given:

Perimeter of the rectangular park is 34m.

Area = 60 m²

Formula used:

Perimeter = 2(l + b)

Area = l × b

Length of diagonal =√l² + b²

Calculation:

According to question,

l + b = 17

l × b = 60

We know,

(l + b)² = l² + b² + 2 lb

⇒ 289 = l2 + b2 + 2 × 60

⇒ l2 + b2 = 169

Length of diagonal = √l2 + b2

⇒ √169

⇒ 13

 

∴ The correct answer is 13 m.

Given:

The length of the chord of a circle is 24 cm, and the perpendicular distance between the centre and the chord is 5 cm. 

Concept used:

Chord Length Using Perpendicular Distance from the Center = \(2 × \sqrt{(r^2 − d^2)}\) (Where r = radius of the circle, d = distance between the centre and the chord)

Calculation:

Let the radius of the circle be R.

Here,

AB = 24 cm long chord

OD = Perpendicular distance between the centre and chord

OB = Radius

​According to the concept,

24 = \(2 × \sqrt{(R^2 − 5^2)}\)

⇒ 12 = \(\sqrt{(R^2 − 5^2)}\)

⇒ 144 = R² - 5²

⇒ R² = 169

⇒ R = 13

∴ The radius of the circle is 13 cm.

Given:

Quadratic equation 3x2+ 6x + 3 = 0

Concept used:

if (x - a)(x - b) = 0, then

(x - a) = 0 or (x - b) = 0

x = a or x = b

Calculation:

3x2+ 6x + 3 = 0

⇒ 3x2+ 3x + 3x + 3 = 0

⇒ 3x(x + 1) + 3(x + 1) = 0

⇒ (3x + 3)(x + 1) = 0

⇒ x = -1 or -1

∴ The roots of the equation 3x2+ 6x + 3 = 0 is -1 or -1.

Shortcut Trick

For any cubic polynomial, the product of zeroes is given by the formula: −d/a.

Comparing with the given polynomial f(x) = ax³ + 4x² − 2x − 6, we have the constant term d = −6.

Product of zeroes = −(−6) ÷ a = 6/a.

Given that the product is 2 ⇒ 6/a = 2 ⇒ a = 3.

∴ The correct answer is 3.

Alternate Method

Given:

The polynomial is f(x) = ax³ + 4x² − 2x − 6.

The product of zeroes of this polynomial is 2.

Formula Used:

For a standard cubic polynomial of the form f(x) = Ax³ + Bx² + Cx + D, the product of its zeroes (αβγ) is given by:

αβγ = −D/A = −(constant term) ÷ (coefficient of x³)

Calculations:

Comparing the given polynomial f(x) = ax³ + 4x² − 2x − 6 with the standard cubic form:

⇒ A = a

⇒ B = 4

⇒ C = −2

⇒ D = −6

Now, substitute these coefficients into the product of zeroes formula:

⇒ Product of zeroes = −(−6) ÷ a

⇒ 2 = 6/a

⇒ 2 × a = 6

⇒ a = 3

Thus, the value of the coefficient 'a' is 3.

∴ The correct answer is 3.

Additional Information

Cubic Polynomial Relations

For a cubic polynomial f(x) = ax³ + bx² + cx + d with roots α, β, and γ, the relations are: Sum of roots = −b/a, Sum of products taken two at a time = c/a, and Product of roots = −d/a.

Quadratic Polynomial Relations

For a quadratic polynomial f(x) = ax² + bx + c with roots α and β, the relations are: Sum of roots = −b/a, and Product of roots = c/a.

General Polynomial Root Product

For any nth-degree polynomial, the product of all roots is given by (−1)ⁿ × (constant term) ÷ (leading coefficient).

Concept:

• n(U), n(A), and n(B) denote the number of elements in sets U, A, and B respectively.
• A ⋃ B = A + B - (A ∩ B)
• (A ⋃ B)c = U - (A ⋃ B)

Calculation:

Given, n(U) = 20, n(A) = 12, n(B) = 9 and n(A ∩ B) = 4.

According to the formula, 

n(A ⋃ B) = n(A) + n(B) - n(A ∩ B)

⇒ 12 + 9 - 4 = 17

Again, according to the formula mentioned above, 

n(A ⋃ B)c = n(U) - n(A ⋃ B) = 20 - 17 = 3

Given:

4y – 6x + 9 = 0       ----(1)

Concept used:

Equation of perpendicular line x cos a + y sin a = P       ----(2)

Calculation:

Comparing eq. (1) and (2)

cos a = – 6 and sin a = 4 and P = – 9

Slope, m = tan a = sin a/cos a = -4/6 = -2/3

∴ The correct answer is -2/3

Alternate Method

4y – 6x + 9 = 0

⇒ y = 3x/2 + 9/4

Comparing it with y = mx + c

m = 3/2

The slope of perpendicular lines

m₁m₂ = -1      where m₁ = 3/2

⇒ m₂ = -2/3