Applications of Basic Identities 1 - DPP (11th Elite) PDF

Applications of basic identities 1 -DPP Q1. If 3x +2y = 12 and xy = 6, find the value of 9x2 + 4y2. Q2. If find the value of Q3. If find the value of Q4. If (3a + 4b) = 16 and ab = 4, find the value of (9a2 + 16b2). Q5. If x + y = 12 and xy = 27, find the value of x3 + y3. Q6. If (a + 3b)...

Applications of basic identities 1 -DPP Q1. If 3x +2y = 12 and xy = 6, find the value of 9x2 + 4y2. Q2. If find the value of Q3. If find the value of Q4. If (3a + 4b) = 16 and ab = 4, find the value of (9a2 + 16b2). Q5. If x + y = 12 and xy = 27, find the value of x3 + y3. Q6. If (a + 3b) = 6, show that a3 + 27b3 + 54ab = 216. Q7. Evaluate 933 - 1073. Q8. Evaluate: 303 + 203 - 503.  1 1 Q9. If a + = 5, then find a3 + a a3 Q10. Prove that 2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = [(a - b)2 + (b - c)2 + (c - a)2] Q11. If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c. Q12. If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca. Q13. If a + b =10 and ab = 21, find the value of a3 + b3. Q14. If (a + b + c) = 14 and (a2 + b2 + c2) = 74, find the value of (ab + bc + ca). Q15. Prove (sin8θ - cos8θ) = (sin2θ - cos2θ) (1 - 2 sin2θ cos2θ) Q16. If 4x2 + 9y2 + z2 – 6xy – 3yz –2xz = 0 , then prove that 2x = 3y = z Q17. Find the relation between p, q, r in 36p2 + 100q2 + 4r2 - 60pq - 20qr - 12pr = 0 Q18. Find the relation between p, q, r in p2 + 64q2 + 16r2 + 8pq - 32qr + 4pr = 0 Q19. Find positive numbers a, b and c such that 2a + b + 3c = 6 and 8a3 + b3 + 27c3 = 18abc Q20. If a ≠ 2b and a3 + 8b3 = 18ab - 27 then find the value of a + 2b A -1 B 1 C 2 D -3 Q21. If a + b + c = 15 and a2 + b2 + c2 = 83, find the value of a3 + b3 + c3 - 3abc. Q22. If x + y + z = 1, xy + yz + zx = -1 and xyz = -1, find the value of x 3 + y3 + z3. Q23. Solve 2x - 1 ≤ 0 Q24. Solve 3x - 3 ≤ 2 Q25. Solve -2x + 5 ≤ 7 Q26. Represent the following on real number line x ≤ 2 ∪ x > 5. Applications of basic identities 1 -DPP solutions Q1. If 3x +2y = 12 and xy = 6, find the value of 9x2 + 4y2. Solution: Q2. If find the value of Solution: Q3. If find the value of Solution: Q4. If (3a + 4b) = 16 and ab = 4, find the value of (9a2 + 16b2). Solution: Q5. If x + y = 12 and xy = 27, find the value of x3 + y3. Solution: Q6. If (a + 3b) = 6, show that a3 + 27b3 + 54ab = 216. Solution: Q7. Evaluate 933 - 1073. Solution: Q8. Evaluate: 303 + 203 - 503. Solution:  1 1 Q9. If a + = 5, then find a3 + a a3 Solution: 3 1 3 a + a = 5 1 1 1 ⇒ 3 a + 3 +3 a × a + = 125 a a a 1 ⇒ a3 + 3 + 3 (1) (5) = 125 a 1 1 ⇒ a3 + 3 = 125 – 15 ⇒ a3 + 3 = 110 a a Q10. Prove that 2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = [(a - b)2 + (b - c)2 + (c - a)2] Solution: Q11. If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c. Solution: Q12. If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca. Solution: Q13. If a + b =10 and ab = 21, find the value of a3 + b3. Solution: Q14. If (a + b + c) = 14 and (a2 + b2 + c2) = 74, find the value of (ab + bc + ca). Solution: Q15. Prove (sin8θ - cos8θ) = (sin2θ - cos2θ) (1 - 2 sin2θ cos2θ) Solution:  2 2 2 Q16. If 4x + 9y + z – 6xy – 3yz –2xz = 0 , then prove that 2x = 3y = z Solution: Can we write the expression 4x2 + 9y2 + z2 – 6xy – 3yz – 2xz = 0 as (2x)2 + (3y)2 + (z)2 – (2x)(3y) – (3y)(z) – (2x)(z) = 0 a2 + b2 + c2 – ab – bc – ca = 0 ⇒ a = b = c ⇒ 2x = 3y = z Q17. Find the relation between p, q, r in 36p2 + 100q2 + 4r2 - 60pq - 20qr - 12pr = 0 Solution: Q18. Find the relation between p, q, r in p2 + 64q2 + 16r2 + 8pq - 32qr + 4pr = 0 Solution: Q19. Find positive numbers a, b and c such that 2a + b + 3c = 6 and 8a3 + b3 + 27c3 = 18abc Solution: Q20. If a ≠ 2b and a3 + 8b3 = 18ab - 27 then find the value of a + 2b A -1 B 1 C 2 D -3 Solution: Q21. If a + b + c = 15 and a2 + b2 + c2 = 83, find the value of a3 + b3 + c3 - 3abc. Solution: Q22. If x + y + z = 1, xy + yz + zx = -1 and xyz = -1, find the value of x 3 + y3 + z3. Solution: x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx) ⇒ x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 + 2xy + 2yz + 2zx - 3xy - 3yz - 3zx) [Adding the subtracting 2xy + 2yz + 2zx] ⇒ x3 + y3 + z3 - 3xyz = (x + y + z) {(x + y + z)2 - 3(xy + yz + zx)} ⇒ x3 + y3 + z3 - 3(- 1) = 1 × {(1)2 - 3(-1)} [Putting the values of x + y + z, xy + yz + zx and xyz] ⇒ x3 + y 3 + z 3 + 3 = 4 ⇒ x3 + y 3 + z 3 = 4 - 3 = 1 Q23. Solve 2x - 1 ≤ 0 Solution: 1/2 is not included 2x is less than or equal to 1 2 is a positive number, so if we divide by 2, inequality will remain same 1 1 x < 2 2 Q24. Solve 3x - 3 ≤ 2 Solution: 3x ≤ 5 is not included 3 is a positive number, so if we divide by 3, inequality will remain same 5 3 Q25. Solve -2x + 5 ≤ 7 Solution: -2x ≤ 7 - 5 -2x ≤ 2 To have x on L.H.S. let's divide both sides by – 2 –1 Inequality reverses when we divide or multiply by a negative number Q26. Represent the following on real number line x ≤ 2 ∪ x > 5. Solution: x ∈ (-∞, 2) ∪ (5, ∞) –∞ 2 5 ∞ Join Vedantu JEE Telegram channel NOW! Assignments Notes Daily Update https://vdnt.in/jeepro