Cosine Rule MCQ Quiz in मल्याळम - Objective Question with Answer for Cosine Rule - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Last updated on Mar 14, 2025

നേടുക Cosine Rule ഉത്തരങ്ങളും വിശദമായ പരിഹാരങ്ങളുമുള്ള മൾട്ടിപ്പിൾ ചോയ്സ് ചോദ്യങ്ങൾ (MCQ ക്വിസ്). ഇവ സൗജന്യമായി ഡൗൺലോഡ് ചെയ്യുക Cosine Rule MCQ ക്വിസ് പിഡിഎഫ്, ബാങ്കിംഗ്, എസ്എസ്‌സി, റെയിൽവേ, യുപിഎസ്‌സി, സ്റ്റേറ്റ് പിഎസ്‌സി തുടങ്ങിയ നിങ്ങളുടെ വരാനിരിക്കുന്ന പരീക്ഷകൾക്കായി തയ്യാറെടുക്കുക

Latest Cosine Rule MCQ Objective Questions

Top Cosine Rule MCQ Objective Questions

Cosine Rule Question 1:

. Two chords of length a unit and b unit of a circle make angles 60° and 90° at the centre of a circle respectively, then the correct relation is: 

  1. b = √2a
  2. b = 2a
  3. b = √3a
  4. b = 3/2a

Answer (Detailed Solution Below)

Option 1 : b = √2a

Cosine Rule Question 1 Detailed Solution

Given:

Two chords of length 'a' unit and 'b' unit of a circle make angles 60° and 90° at the centre of a circle respectively,

The formula used:

\(\text {cos} (θ) = \frac{a^2 + b^2 - c^2}{2ab}\)

Where, 

\(a, b\) and \(c\) are the side of the triangle and \(\theta\) is the angle opposite to the side \(c\).

The Pythagoras Theorem:

\(h^2 = p^2 + b^2\)

Where, 

\(h, p \text { and } b\) are hypotenuse, perpendicular and base respectively.

Calculation:

According to the question, the required image is:

6250465ef11bc53c1b22667b 16499348548011

In Δ AOB,

The length of the chord or the base of the triangle AOB \(=a\)

Therefore, 

\(\Rightarrow \text {cos} (60^\circ) = \frac{r^2 + r^2 - a^2}{2r^2} \\ \Rightarrow \frac{1}{2} = \frac{2r^2 - a^2}{2r^2} \\ \Rightarrow 1 = \frac{2r^2 - a^2}{r^2} \\ \Rightarrow r^2 = a^2 \\ \Rightarrow a = r \,\,\,\,\,\,......(i)\)

In Δ DOC,

The length of the required chord or the base of the triangle is \(b\).

By using the Pythagoras theorem, we get,

\(\Rightarrow r^2 + r^2 = b^2 \\ \Rightarrow 2r^2 = b^2 \\ \Rightarrow b = √{2}r \,\,\,\,\,\,......(ii)\)

From equation \((i)\) and \((ii)\), we get,

\(\Rightarrow b=√{2}a\)

 ∴ The required relation is \(b=\sqrt{2}a\).

Cosine Rule Question 2:

For any ΔABC find the value of c(a⋅ cos B - b⋅ cos A)

  1. b2 - c2
  2. c2 - b2
  3. 0
  4. a2 - b2

Answer (Detailed Solution Below)

Option 4 : a2 - b2

Cosine Rule Question 2 Detailed Solution

Concept:

Cosine Rule:

cos A = \(\rm \frac{b^{2} + c^{2} - a^{2}}{2bc}\)

cos B = \(\rm \frac{c^{2} + a^{2} - b^{2}}{2ac}\)

cos C = \(\rm \frac{a^{2} + b^{2} - c^{2}}{2ab}\)

Calculation:

Given: c(a.cos B - b.cos A)

cos A = \(\rm \frac{b^{2} + c^{2} - a^{2}}{2bc}\)

⇒ b.c.cos A = \(\rm \frac {b^2 + c^2 - a^2}{2}\)      ----(1)

cos B = \(\rm \frac{c^{2} + a^{2} - b^{2}}{2ac}\)

⇒ a.c.cos B = \(\rm \frac {c^2 + a^2 - b^2}{2}\)      ----(2)

Subtract equation (2) from (1) we get

⇒ a.c.cos B - b.c.cos A = \(\rm (\frac {c^2 + a^2 - b^2}{2}) - ( \frac {b^2 + c^2 - a^2}{2})\)

⇒ c(a.cos B - b.cos A) = \(\rm \frac {({c^2 + a^2 - b^2 - b^2 - c^2 +a^2})}{2}\)

⇒ c(a.cos B - b.cos A) = \(\rm \frac {2({a^2 - b^2})}{2}\)

⇒ c(a.cos B - b.cos A) = a2 - b2

Additional Information

b(c.cos A - a.cos C) = c2 - a2

a(b.cos C - c.cos B) = b2 - c2

Cosine Rule Question 3:

The angle between the two sides of the triangle having lengths 5 cm and 10 cm is 60°, find the third side of the triangle.

  1. 5√2
  2. 3√2
  3. 4√2
  4. 2√5

Answer (Detailed Solution Below)

Option 1 : 5√2

Cosine Rule Question 3 Detailed Solution

Concept:

Cosine rule:

\(\rm \cos θ = {a^2+ b^2 - c^2\over2ab}\)

Where a, b and c are the sides of the triangle and θ is the angle between the sides a and b.

Calculation:

Given a = 5, b = 10 and θ = 60°

So according to the cosine rule,

\(\rm \cos 60= {5^2+10^2-c^2\over2(5)(10)}\)

⇒ \(\rm {1\over2}= {25\ +\ 100\ -\ c^2\over2\ \times \ 50}\)

⇒ 50 = 125 - c 2

⇒ c2 = 125 - 50

⇒ c2 = 75

c = 5√2

Additional Information

Cosine rule:

\(\rm \cos C = {a^2+ b^2 - c^2\over2ab}\)

\(\rm \cos B = {a^2+ c^2 - b^2\over2ac}\)

\(\rm \cos A = {c^2+ b^2 - a^2\over2cb}\)

Sine rule:

\(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = \frac{{\rm{b}}}{{\sin {\rm{B}}}} = \frac{{\rm{c}}}{{\sin {\rm{C}}}}\)

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