Question 1

If lies in the fourth quadrant and 3 +4=0, then what is the value of 2 + 2 ?

Accepted Answer

Given 3 + 4 = 0 = - 4 3 Since lies in the fourth quadrant, > 0 and From = - 4 3 , we can consider the base as 4 and the perpendicular as 3. Hypotenuse = 4^2 + 3^2 = 5 Therefore, = - 3 5 and = 4 5 . Using double angle formulas: 2 = 2 = 2 (- 3 5 ) ( 4 5 ) = - 24 25 2 = ^2 - ^2 = ( 4 5 )^2 - (- 3 5 )^2 = 16 25 - 9 25 = 7 25 Adding both values: 2 + 2 = - 24 25 + 7 25 = - 17 25 Answer: - 17 25

Question 2

If + =0= + , then what is a value of 2 + 2 +2 ( + ) ?

Accepted Answer

Let z_1 = + i and z_2 = + i . From the given conditions: z_1 + z_2 = ( + ) + i( + ) = 0 Squaring both sides: (z_1 + z_2)^2 = 0 z_1^2 + z_2^2 + 2z_1 z_2 = 0 Using De Moivre's theorem, z_1^2 = 2 + i 2 and z_2^2 = 2 + i 2 . The product term is 2z_1 z_2 = 2( ( + ) + i ( + )). Substituting these into the squared equation: ( 2 + 2 + 2 ( + )) + i( 2 + 2 + 2 ( + )) = 0 Equating the real part to zero: 2 + 2 + 2 ( + ) = 0 Answer: 0

Question 3

If 2 4 = 2 + 2 , then which one of the following is a possible value of ( + ) ?

Accepted Answer

The given equation is 2 4 = 2 + 2 We know that + = + = 1 = 2 2 Substituting = 2 , we get 2 + 2 = 2 4 The equation becomes 2 4 = 2 4 1 4 = 1 4 4 = 4 4 = ( 2 - 4 ) The general solution is 4 = 2n ( 2 - 4 ) where n is an integer. Taking the positive sign, we get: 4 = 2n + 2 - 4 4( + ) = 2n + 2 + = n 2 + 8 For n = 0, a possible value is + = 8 Answer: 8

Question 4

If x= - and y= + , then which one of the following is correct ?

Accepted Answer

Given x = - = 1 - y = + = 1 + Consider x(y+1): x(y+1) = ( 1 - ) ( 1 + + 1 ) x(y+1) = ( 1 - ) ( 1 + + ) x(y+1) = 1 + + - - - ^2 x(y+1) = (1 - ^2 ) + - x(y+1) = ^2 + - x(y+1) = ( + 1 - ) x(y+1) = 1 + - x(y+1) = + - 1 Substituting y = + : x(y+1) = y - 1 xy + x - y + 1 = 0 Answer: x-y+xy+1=0

Question 5

If = 1 3 , then what is the value of ( 2 ) ( 3 2 ) ?

Accepted Answer

Using the trigonometric identity 2 A B = (A-B) - (A+B), we have: ( 2 ) ( 3 2 ) = 1 2 [ ( 3 2 - 2 ) - ( 3 2 + 2 ) ] = 1 2 [ - (2 )] Using the double angle formula (2 ) = 2 ^2 - 1, we get: (2 ) = 2 ( 1 3 )^2 - 1 = 2 9 - 1 = - 7 9 Substituting the values of and (2 ): ( 2 ) ( 3 2 ) = 1 2 [ 1 3 - (- 7 9 ) ] = 1 2 [ 3 9 + 7 9 ] = 1 2 [ 10 9 ] = 5 9 Answer: 5 9

Question 6

If A and B are acute angles such that 2A+2B= , then what is the maximum value of A B ?

Accepted Answer

Given 2A + 2B = , which simplifies to A + B = 2 . Since A and B are acute angles, B = 2 - A. The expression to maximize is A B. Substituting B = 2 - A, we get A ( 2 - A ) = A A. Multiplying and dividing by 2, we obtain 2 A A 2 = 2A 2 . The maximum value of 2A is 1, which occurs when 2A = 2 , or A = 4 . Therefore, the maximum value of A B is 1 2 1 = 1 2 . Answer: 1 2

Question 7

x+ 3 x is maximum when x is equal to

Accepted Answer

Given expression is f(x) = x + 3 x Multiplying and dividing by 1^2 + ( 3 )^2 = 2 f(x) = 2 ( 1 2 x + 3 2 x ) f(x) = 2 ( 3 x + 3 x ) f(x) = 2 ( x - 3 ) The maximum value of ( x - 3 ) is 1, which occurs when x - 3 = 0 x = 3 Answer: 3

Question 8

In a triangle ABC, A= B+ C then what is ( B 2 )+ ( B 2 ) equal to ?

Accepted Answer

Given A = B + C Using the half-angle and sum-to-product formulas: 2 ( A 2 ) ( A 2 ) = 2 ( B+C 2 ) ( B-C 2 ) In ABC, A + B + C = , so B+C 2 = 2 - A 2 . This yields ( B+C 2 ) = ( A 2 ). Substituting this into the equation: 2 ( A 2 ) ( A 2 ) = 2 ( A 2 ) ( B-C 2 ) Since A (0, ), ( A 2 ) 0. Dividing both sides by 2 ( A 2 ): ( A 2 ) = ( B-C 2 ) Since both A 2 and B-C 2 lie in (- 2 , 2 ), we have: A 2 = B-C 2 This implies either A = B - C or A = C - B. If A = B - C, then B = A + C. Since A + B + C = ,

Question 9

If and are complementary angles such that - = 6 and m =n , then what is ( m+n m-n ) equal to ?

Accepted Answer

Since and are complementary angles, + = 2 . We are given - = 6 . Adding the two equations yields 2 = 2 3 = 3 . Subtracting the two equations yields 2 = 3 = 6 . From the given relation m = n , we have: m n = = ( 3 ) ( 6 ) = 3 1 3 = 3 Therefore, m+n m-n = 3+1 3-1 = 2. Answer: 2

Question 10

For the following two (02) items: Let A + B = p and A + B = q. What is p^2 - q^2 p^2 + q^2 equal to?

Accepted Answer

Given p = A + B and q = A + B. Squaring both equations: p^2 = ^2 A + ^2 B + 2 A B q^2 = ^2 A + ^2 B + 2 A B Adding the two equations: p^2 + q^2 = ( ^2 A + ^2 A) + ( ^2 B + ^2 B) + 2( A B + A B) p^2 + q^2 = 1 + 1 + 2 (A - B) = 2(1 + (A - B)) Subtracting q^2 from p^2: p^2 - q^2 = ( ^2 A - ^2 A) + ( ^2 B - ^2 B) - 2( A B - A B) p^2 - q^2 = - 2A - 2B - 2 (A + B) Using the sum-to-product formula 2A + 2B = 2 (A + B) (A - B): p^2 - q^2 = -2 (A + B) (A - B) - 2 (A + B) p^2 - q^2 = -2 (A + B)(1 + (A - B)

Question 11

For the following two (02) items: Let p = cosec \, 20° and q = cosec \, 70°. What is p^2 + q^2 p^2 q^2 equal to?

Accepted Answer

Given p = cosec \, 20^ and q = cosec \, 70^ . The given expression is p^2 + q^2 p^2 q^2 . This can be rewritten as 1 q^2 + 1 p^2 . Substituting the values of p and q: 1 cosec ^2 70^ + 1 cosec ^2 20^ = ^2 70^ + ^2 20^ Since 70^ = (90^ - 20^ ) = 20^ , we get: = ^2 20^ + ^2 20^ = 1 Answer: 1

Question 12

For the following two (02) items: Let A + B = p and A + B = q. What is p q equal to?

Accepted Answer

Given p = A + B and q = A + B. Using the sum-to-product formulas: p = 2 ( A+B 2 ) ( A-B 2 ) q = 2 ( A+B 2 ) ( A-B 2 ) Dividing p by q: p q = 2 ( A+B 2 ) ( A-B 2 ) 2 ( A+B 2 ) ( A-B 2 ) p q = ( A+B 2 )

Question 13

For the following two (02) items: Let p = cosec \, 20° and q = cosec \, 70°. What is ( 3 \,p 4 - q 4 ) equal to?

Accepted Answer

Given p = cosec \, 20^ = 1 20^ and q = cosec \, 70^ = 1 70^ = 1 20^ Substituting p and q into the given expression: 3 \,p 4 - q 4 = 3 4 20^ - 1 4 20^ Taking the common denominator: = 3 20^ - 20^ 4 20^ 20^ Multiplying and dividing the numerator by 2, and rewriting the denominator: = 2 ( 3 2 20^ - 1 2 20^ ) 2 (2 20^ 20^ ) Using the identities 60^ = 3 2 , 60^ = 1 2 , and 2 = 2 : = 2 ( 60^ 20^ - 60^ 20^ ) 2 40^ Using the identity (A - B) = A B - A B: = 2 (60^ - 20^ ) 2 40^ = 2 40^ 2 40^ = 1 Answer:

Question 14

Consider the following for the three (03) items that follow: Let p = 35°, q = 25° and r = (-95°). What is (p + q + r) equal to?

Accepted Answer

Given p = 35^ , q = 25^ , and r = (-95^ ). We need to find the value of p + q + r. p + q = 35^ + 25^ Using the formula A + B = 2 ( A+B 2 ) ( A-B 2 ): p + q = 2 ( 35^ + 25^ 2 ) ( 35^ - 25^ 2 ) p + q = 2 30^ 5^ Since 30^ = 1 2 : p + q = 2 ( 1 2 ) 5^ = 5^ Now, evaluating r: r = (-95^ ) = - 95^ r = - (90^ + 5^ ) = - 5^ Therefore, p + q + r = 5^ - 5^ = 0 Answer: 0

Question 15

Consider the following for the three (03) items that follow: Let p = 35°, q = 25° and r = (-95°). What is (pq + qr + rp) equal to?

Accepted Answer

Given p = 35^ , q = 25^ , and r = (-95^ ) = - (90^ + 5^ ) = - 5^ . p + q + r = 35^ + 25^ - 5^ Using A + B = 2 ( A+B 2 ) ( A-B 2 ): p + q + r = 2 30^ 5^ - 5^ = 2 ( 1 2 ) 5^ - 5^ = 0 We know that (p + q + r)^2 = p^2 + q^2 + r^2 + 2(pq + qr + rp) = 0 pq + qr + rp = - 1 2 (p^2 + q^2 + r^2) Now, p^2 + q^2 + r^2 = ^2 35^ + ^2 25^ + ^2 5^ Using ^2 = 1 - 2 2 and ^2 = 1 + 2 2 : p^2 + q^2 + r^2 = 1 - 70^ 2 + 1 - 50^ 2 + 1 + 10^ 2 = 3 2 - 1 2 ( 70^ + 50^ - 10^ ) Using C + D = 2 ( C+D 2 ) ( C-D 2 ): 70^ + 5

Question 16

Consider the following for the three (03) items that follow: Let p = 35°, q = 25° and r = (-95°). What is (p^2 + q^2 + r^2) equal to?

Accepted Answer

Given p = 35^ , q = 25^ , and r = (-95^ ). Simplifying r: r = - (90^ + 5^ ) = - 5^ We need to find the value of p^2 + q^2 + r^2: p^2 + q^2 + r^2 = ^2 35^ + ^2 25^ + ^2 5^ Using the identities ^2 = 1 - 2 2 and ^2 = 1 + 2 2 : p^2 + q^2 + r^2 = 1 - 70^ 2 + 1 - 50^ 2 + 1 + 10^ 2 = 3 2 - 1 2 ( 70^ + 50^ - 10^ ) Using the sum-to-product formula C + D = 2 ( C+D 2 ) ( C-D 2 ): 70^ + 50^ = 2 ( 70^ + 50^ 2 ) ( 70^ - 50^ 2 ) = 2 60^ 10^ = 2 ( 1 2 ) 10^ = 10^ Substituting this back into the expression: p^2

Question 17

Consider the following for the two (02) items that follow: Let p = | - ( - 90°)|. What is the maximum value of p?

Accepted Answer

Given p = | - ( - 90^ )| Using the trigonometric identity ( - 90^ ) = - (90^ - ) = - Substituting this into the expression for p p = | - (- )| = | + | The maximum value of the expression a + b is a^2 + b^2 Here, a = 1 and b = 1 Maximum value of + = 1^2 + 1^2 = 2 Therefore, the maximum value of p is 2 Answer: 2

Question 18

Consider the following for the two (02) items that follow: Let p = | - ( - 90°)|. What is the minimum value of p?

Accepted Answer

p = | - ( - 90^ )| Using the trigonometric identity ( - 90^ ) = - (90^ - ) = - , we get: p = | - (- )| = | + | Multiplying and dividing the expression inside the modulus by 2 : p = 2 | 1 2 + 1 2 | p = 2 | ( + 45^ )| Since the absolute value of any real expression is non-negative, the minimum possible value of p is 0. This minimum is achieved when ( + 45^ ) = 0, which occurs at = -45^ . Therefore, the minimum value of p is 0.

Question 19

Consider the following for the three (03) items that follow: Let p = 2 - and q = - 2 . What is (p/q) equal to?

Accepted Answer

Given p = 2 - and q = - 2 Expressing q in terms of gives: q = 1 - 1 2 q = 2 - 2 Substituting p = 2 - into the expression for q: q = p 2 Rearranging the terms yields: p q = 2 Answer: 2

Question 20

Consider the following for the three (03) items that follow: Let p = 2 - and q = - 2 . What is (p + q) equal to?

Accepted Answer

Given p = 2 - and q = - 2 . Simplifying p: p = 2 2 - = 2 - 2 2 = (2 - ) 2 = 2 Simplifying q: q = - 2 2 = 2 - 2 2 = (2 - ) 2 = 2 = 1 2 Now, adding p and q: p + q = 2 + 1 2 Using 2 = 2 : p + q = 2 + 1 2 p + q = 2 ^2 + 2 2 2 Using the identity 2 = 1 - 2 ^2 in the numerator and 2 = 2 in the denominator: p + q = 2 ^2 + (1 - 2 ^2 ) 2 2 p + q = 1 2 2 Multiplying and dividing by 2: p + q = 2 2 2 2 = 2 4 p + q = 2 4 Answer: 2 4

Trigonometric Ratios & Identities — Previous Year Questions & Solutions