(cos²θ –1)(1 + tan²θ) + 2tan²θ = 1

⇒ (-sin² θ) sec² θ + 2 tan² θ = 1

⇒ (-sin² θ/cos² θ) + 2 tan² θ = 1

⇒ (-tan² θ) + 2 tan² θ = 1

⇒ tan² θ = 1

⇒ tan θ = 1

And 0° ≤ θ ≤ 90°

∴ θ = 45

Shortcut Trick

(cos²θ –1)(1 + tan²θ) + 2tan²θ = 1

Put θ = 45

(cos²45 –1)(1 + tan²45) + 2tan²45 = 1

⇒ (1/2 – 1) (1 + 1) + 2 × 1 = 1

⇒ (-1/2) × 2 + 2 = 1

⇒ (-1) + 2 = 1

⇒ 1 = 1 (satisfied)

Given:

Calculations:

According to question,

(17% + 19% + 16%) of 350

= 52% of 350

= 0.52 × 350

= 182

Hence, The Required value is 182.

Given a + b = 8, Cubing both sides

⇒ (a + b)³ = a³ + b³ + 3ab (a + b) ,

Using given data

⇒ 8³ = 152 + 3ab × (8)

⇒ (512 – 152)/24 = ab

⇒ ab = 15

∴ the value of ab is 15

Given:

Average age = 24 years

Age, P : Q = 3 : 2

LCM (age of P, Q) = 48

Calculation:

Let the ages of P and Q be ‘3x’ and ‘2x’ respectively.

⇒ LCM of ages of P and Q = 3 × 2x = 48

⇒ 6x = 48

⇒ x = 8

Age of P = 3x = 24 years

Age of Q = 2x = 16 years

Let age of R be ‘R’

Now,

(24 + 16 + R)/3 = 24

⇒ 40 + R = 72

⇒ R = 32 years

∴ LCM of ages of P, Q and R = LCM of 24, 16 and 32 = 96

Given:

In Δ ABC, DE || BC

AD = 7 cm

AE = 3.5 cm

DB = 6 cm

Concept used:

Basic Proportionality Theorem - According to this rule, if a line is drawn parallel to one side of a triangle and intersects the other two sides as well, the line will always split the two sides in the same proportion, i.e., if DE || BC, then 

Calculation:

Given:

L1 || L2

Concept:

Alternate angles are equal if lines are parallel.

The Sum of the interior angles of a triangle is 180º.

The sum of interior and exterior angle is 180º.

Calculation:

From △QRS,

⇒ 20° + z + 62° = 180°

⇒ z = 180° - 62° - 20°

⇒ z = 98°.

L1 || L2

Alternative angles ∠USP = ∠SPQ (Alternative angles)

∠USP = ∠SPQ = 46°

⇒ 46° + y + z = 180°

⇒ 46° + y + 98° = 180°  (⇒ z = 98°)

⇒ y = 180° - 144° = 36° 

From △PQS

⇒ x + y + 46° = 180°

⇒ x + 36° + 46° = 180°

⇒ x = 180° - 46° - 36°

⇒ x = 98°

Therefore, the values of x, y, z are 98°, 36°, 98°.

Given:

Principal = Rs. 8000

Rate = 10%

Formula used:

SI = (P × t × r)/100

CI = P(1 + r/100)^t - P

P = Principal

t = time

r = rate

Calculation:

SI = (8000 × 12 × 10)/(100 × 5)

⇒ Rs. 1920

CI = 8000[1 + 10/100]² × [1 + 4/100] - 8000

⇒ 8000 × 11/10 × 11/10 × 26/25 - 8000

⇒ 10067.2 - 8000

⇒ 2067.2

Difference = 2067.2 - 1920 = 147.2

∴ Required difference is Rs. 147.2

Shortcut Trick

So, the difference of CI and SI = 80 + 32 + 32 + 3.2

∴ The Difference of CI and SI = 147.2.

Calculation:

Let the initial oranges with the fruit seller be x.

1st selling = 0.45x + 1

Remaining = x - (0.45x + 1) = 0.55x - 1

2nd selling = 1/5 × (0.55x - 1) = 0.11x - 0.2 + 2 = 0.11x + 1.8

Remaining after second selling = 0.55x - 1 - (0.11x + 1.8) = 0.55x - 0.11x - 1 - 1.8 = 0.44x - 2.8

3rd selling = 90% × (0.44x - 2.8)

Remaining after 3rd selling = 0.1 × (0.44x - 2.8) = 0.044x - 0.28

According to the question-

⇒ 0.044x - 0.28 = 5

⇒ 0.044x = 5.28

∴ The number of oranges was 120.

Alternate Method

At last, he sells 90% of the remaining oranges after selling the oranges to a second customer, then he has 10% of the remaining oranges.

10% of the remaining oranges after selling the oranges to the second customer = 5

So remaining oranges after selling the oranges to the second customer = 100% of the remaining oranges after selling the oranges to the second customer = 50 oranges

He gave 2 extra oranges to the second customer, so the remaining oranges = 50 + 2

He sells 20% of the remaining oranges to the second customer, so he has 80% of the remaining oranges = 52

100% of remaining oranges after selling the oranges to the first customer = (52/4) * 5 = 65 oranges

He gave 1 extra orange to the first customer, so the total oranges after selling 45% of the oranges = 65 + 1 = 66 oranges

(100% - 45% = 55%) of total oranges = 66

so

100% of oranges = (66/55) * 100 = 120 oranges

Concept used:

Sum of all the angles of a quadrilateral is 360°

Calculation:

Let's assume that, each of the three equal angles be k, then

3k + 120 = 360

⇒ 3k = (360 - 120)

⇒ k = 240/3 = 80° 

∴ The value of each equal angle is 80°

Given:

The expression 323 ÷ 17 × √841 + 12²

Concept used:

Following the BODMAS rule according to the table given below:

Calculation:

323 ÷ 17 × √841 + 12²

= 19 × 29 + 12²

= 551 + 144

= 695