Q1. Find the value of tan65 – tan20 – tan65. tan20
(a) 0
(b) 1
(c) –1
(d) 2
Q2. If ∝+ β=45; find (1 + tan∝) (1 + tan β)
(a) 0
(b) 2
(c) 9
(d) 1
Q3. If Sin θ + Sin2 θ = 1, find cos12θ + 3cos10 θ + 3 cos8 θ + cos6 θ + 1
(a) 0
(b) 86
(c) 2
(d) –30
Q4. If a sin θ + b Cos θ = m, find the value of a Cos θ – b Sin θ =?
(a) ±√((a^2 + b^2)/m^2 )
(b) ±√(a^2-b^2 )
(c) (a+b)/m
(d) ±√(a^2+b^2-m^2 )
Q5. A prism stand on a triangular base whose volume is 810 cm3, if semi perimeter of triangle is 45 cm and radius of incentre is 9 cm then find; its total surface area
(a) 620 cm2
(b) 755 cm2
(c) 585 cm2
(d) 990 cm2
Q6. Two numbers are such that the sum of twice the first number and thrice the second number is 36 and sum of thrice the first number and twice the second number is 39. Which is the smaller number?
(a) 9
(b) 5
(c) 7
(d) None of these
Q7. If the sum and difference of two numbers are 20 and 8 respectively, then the difference of their square is:
(a) 12
(b) 28
(c) 80
(d) 160
Q8. A 4-digit number is formed by repeating a 2-digit number such as 2525, 3232, etc. Any number of this form is always exactly divisible by:
(a) 7
(b) 11
(c) 13
(d) Smallest 3-digit prime number
Q9. A number divided by 68 gives the quotient 269 and remainder zero. If the same number is divided by 67, then the remainder is:
(a) 0
(b) 1
(c) 2
(d) 3
Q10. The sum of the digits of a two-digit number is 81 less than the number. What is the difference between the digits of the number?
(a) 6
(b) 3
(c) 1
(d) Cannot be determined
Solutions
S1. Ans. (b)
Sol. tan (65 – 20) = (tan65 -tan20) /(1 + tan65 + tan20)
1 = (tan65 – tan20)/(1 + tan65 .tan20)
1 + tan 65 . tan 20 = tan 65 – tan 20
∴ tan 65 – tan 20 – tan 65 . tan 20 = 1
S2. Ans.(b)
Sol. By using direct method
∝+β=45°
Let ∝=45°
And β=0°
(1 + tan 45) (1 + tan 0)
(1 + 1) (1 + 0)
= 2
S3. Ans. (c)
Sol. Sin θ = 1 – Sin2 θ
=Sin θ = Cos2 θ
=(Cos^4 θ)^3+ (Cos^2 θ)^3+3Cos^6 θ (Cos^4 θ+Cos^2 θ)
=(Cos^4 θ+Cos^2 θ)^3+1
=(Sin^2 θ+Sinθ)^3+1
1 + 1 = 2
S4. Ans.(d)
Sol. (a Sin θ + b Cos θ) = m …(i)
Let (a cos θ – b sin θ) = x …(ii)
Squaring and adding (i) and (ii)
=a^2 Sin^2 θ + b^2 Cos^2 θ+2ab Sinθ Cosθ+a^2 Cos^2 θ+b^2 Sin^2 θ-2ab Sinθ Cosθ=m^2+x^2
=a^2 (Sin^2 θ+Cos^2 θ)+b^2 (Cos^2 θ+Sin^2 θ)=m^2+x^2
=a^2+b^2=m^2+x^2
=x^2=a^2+b^2-m^2
=x=±√(a^2+b^2-m^2 )
S5. Ans.(d)
Sol. r=∆/S
∆=9×45 cm^3
Volume of Prism = Area of base × height of the prism
810 = 9 × 45 × h
h = 2 cm
T.S.A = L.S. A + 2. Area of base
L.S.A = Perimeter of base × height of the prism
L.S.A = 45 × 4
T.S.A = 45 × 4 + 2 × 9 × 45
= 990 cm2
S6. Ans.(d)
Sol. Let the numbers be x and y
2x + 3y = 36
3x + 2y = 39
4x + 6y = 72
9x+6y=117
=5x = 45
∴ x = 9
2 × 9 + 3y = 36
y = (36 – 18)/3 = 6
∴ Smaller number is 6
S7. Ans.(d)
Sol. Let x and y be the numbers,
∴ x + y = 20, x – y = 8
⇒ x = 14, y = 6
∴x^2-y^2=(14)^2-(6)^2
= (14 + 6) (14 – 6) ⇒ 20 × 8 = 160.
S8. Ans.(d)
Sol. by 101 which is the smallest 3-digit prime number.
S9. Ans.(b)
Sol. The number is 68 × 269 = 18292. 18292, when divided by 67, leaves a remainder of 1.
S10. Ans.(d)
Sol. 10x + y – (x + y) = 81
or, 10x + y – x – y = 81
or, 9x ⇒ 81 ∴ x = 9
Hence, all such numbers are as follows: 90, 91, 92, 93, … 99.
