Practice Questions
3𝑥+𝑖 𝑥 𝑥 1
Q.1 Find the value of 𝑥 so that (i) | 2+𝑖 − 3−𝑖| = 1 (ii) 𝑖 + 3𝑖 2 𝑦 = 𝑖 , 𝑖𝑥 − 𝑦 = 𝑖 3 .
1+2𝑖 −2 1+2𝑖 (2+𝑖)2
Q.2 Separate into real and imaginary part also find modulus (i) ( 1−𝑖 ) (ii) ( 1−𝑖 ) ÷ { }
1+𝑖
Q.3 By using the properties of determinants prove that
(i)
𝑥 + 𝑦 + 2𝑧 𝑧 𝑧
| 𝑥 𝑦 + 𝑧 + 2𝑥 𝑥 | = 2(𝑥 + 𝑦 + 𝑧)3
𝑦 𝑦 𝑧 + 𝑥 + 2𝑦
(ii)
1 𝑎 𝑎2
|1 𝑏 𝑏 2 | = (𝑎 – 𝑏)(𝑏 – 𝑐)(𝑐 – 𝑎)
1 𝑐 𝑐2
(iii)
1 1 1
|𝛼 𝛽 𝛾 | = (𝛼 − 𝛽)(𝛽 − 𝛾)(𝛾 − 𝛼)
𝛽𝛾 𝛾𝛼 𝛼𝛽
(iv)
𝑎+𝑥 𝑎 𝑎
| 𝑎 𝑎+𝑥 𝑎 | = 𝑥 2 (3𝑎 + 𝑥)
𝑎 𝑎 𝑎+𝑥
𝑥 2 + 𝑎2 𝑥2 𝑥
Q.4 Solve by using properties of determinants |𝑏 2 + 𝑎2 𝑏2 𝑏| = 0. {𝑥 = 𝑏, 𝑥 = 𝑐}
𝑐 2 + 𝑎2 𝑐2 𝑐
Q.5 Solve the following system of equations by appropriate method.
(i) 𝑥 + 𝑦 = 0,3𝑥 + 4𝑦 + 𝑧 = −1, 𝑥 + 𝑦 + 2𝑧 = 11
(ii) 𝑥 + 𝑦 = 𝑎, 𝑥 − 4𝑦 − 𝑧 = 𝑏, 𝑥 + 2𝑦 − 𝑧 = 𝑐
Q.6 Prove that the cube roots of −125 are −5, −5𝜔, −5𝜔2 and their sum is zero. (where 𝜔
being the complex cube roots of unity)
Q.7 Find value of 𝑘 so that the roots of the equation are equal.
(i) 𝑥 2 − 2𝑥(1 + 3𝑘) + 7(3 + 2𝑘) = 0
(ii) (𝑘 + 1)𝑥 2 + 2(𝑘 + 3)𝑥 + (2𝑘 + 3) = 0, provided 𝑘 ≠ −1
Q.8 Prove that roots of the following equations are real.
(𝑏 2 − 4𝑎𝑐)𝑦 2 + 4(𝑎 + 𝑐)𝑦 − 4 = 0, ∀ 𝑎, 𝑏, 𝑐 ∈ 𝑅 and 𝑏 2 ≠ 4𝑎𝑐
Q.9 Solve:
t+16 t 1 144 256
(i) √ + √t+16 = 2 12 { ,− }
t 7 7
1 1 1
(ii) + √1 + x = 2 2 {2, }
√1+x 2
(iii) (x + 6)(x + 1)(x + 3)(x − 2) + 56 = 0 {1, −5, −2 ± 2√2 }
Q.10 If 𝛼, 𝛽 be the roots of the equation 𝑝𝑥 2 + 𝑞𝑥 + 𝑟 = 0 𝑝 ≠ 0, then form the equation whose
1 1 1 1
roots are (i) (2𝛼 + 3𝛽) and (2𝛽 + 3𝛼) (ii) 𝛼 2 + 𝛽 2 and (𝛼2 +𝛽2) (iii) 𝛼 2 + 𝛽 2 and (𝛼2 +𝛽2)
1 1
(iv) − 𝛼3 and − 𝛽3 (v) 𝛼 2 and 𝛽 2
Q.11 Find the equation whose roots are reciprocal the roots of 𝑥 2 − 6𝑥 + 8 = 0
Q.12 Solve the following system of equations.
(i) 2𝑥 + 3𝑦 = 7, 2𝑥 2 − 3𝑦 2 = −25 {(−1,3), (−13,11)}
4 3 25 25
(ii) 4𝑥 + 3𝑡 = 25, + = 2 {𝑥 = 4, 𝑡 = 3 and 𝑥 = ,𝑡 = }
𝑥 𝑡 8 6
2 2
Q.13 If cos 𝛼 = 3 and cos 𝛽 = 7 and both 𝛼 and 𝛽 are in the first quadrant find the value of
(i) sin(2𝛼) (ii) sin(2𝛽) (iii) tan(2𝛽)
Q.14 Show that:
𝑠𝑖𝑛7θ –sin5θ
(i) = 𝑡𝑎𝑛θ
𝑐𝑜𝑠7θ +cos5θ
2
(ii) 1 + cos 2𝜃 = 1+𝑡𝑎𝑛2 𝜃
(iii) sin 3𝜃 = 3 sin 𝜃 − 4 sin3 𝜃
3𝑡𝑎𝑛𝜃−tan3 𝜃
(iv) 𝑡𝑎𝑛3𝜃 = 1−3 tan2 𝜃
𝑠𝑖𝑛2 𝑐𝑜𝑠2
(v) − = sec
𝑠𝑖𝑛 𝑐𝑜𝑠
Q.15 Find the area and largest angle when sides of triangle are 3,4 and 4.
Q.16 Write domain, range and period of:
3 𝑥 1 𝜋 3𝑥
(i) 𝑓(𝑥) = 2 sin 3𝑥 (ii) 𝑓(𝑥) = − 2 cos 2 (iii) 𝑓(𝑥) = − 6 cos (12 − )
4
Q.17 Write the amplitude and range of the curve.
Q.18 The graph represents 𝑓(𝑥) = 𝐴 sin(𝐵𝑥) then find values of A and B.
(i)
(ii)
Q.19 Verify that
1+√3 8+4√3 𝜋
(i) tan−1 (3+√3) + sec −1 √6+3√3 = 3
1 7
(ii) cos (2 sin−1 3) = 9
1 √5 1
(iii) tan (2 tan−1 5 + sec −1 + 2 tan−1 8) = 2
2
77 3 87
(iv) sin−1 (85) + sin−1 (5) = cos −1 (− 425)
Q.20 Solve
(i) cos 𝑥 + sin 𝑥 − 1 = 0
𝜋
𝐺. 𝑆 = {2 + 2𝑛𝜋} ∪ {2𝑛𝜋}, 𝑛 ∈ ℤ
(ii)
2𝑠𝑖𝑛2 𝜃 + 2√2 sin 𝜃 − 3 = 0
𝜋 3𝜋
𝐺. 𝑆 = {4 + 2𝑛𝜋} ∪ { 4 + 2𝑛𝜋}, 𝑛 ∈ ℤ
(iii)
tan2 𝜃 + tan 𝜃 = 2
𝜋
𝐺. 𝑆 = { + 𝑛𝜋} ∪ {tan−1(−2) + 𝑛𝜋}, 𝑛 ∈ ℤ
4
(iv)
sin 3 − sin = 0
𝜋 𝜋
𝐺. 𝑆 = {𝑛𝜋 + 4 } ∪ {𝑛𝜋 + 3 4 } ∪ {2𝑛𝜋} ∪ {2𝑛𝜋 + 2𝜋}, 𝑛 ∈ ℤ
(v)
sec 3𝑥 = sec 𝑥
𝜋
𝐺. 𝑆 = {𝑛𝜋} ∪ { + 𝑛𝜋} , 𝑛 ∈ ℤ
2
