1. (tan theta. cosec theta) whole square - (sin theta. sec theta) whole square= ?
a. 0
b. 1
c. -1
d. 2
2. Area of a sector of circle of r= 28cm and theta= 45 is ______
a. 96.5
b. 98.5
c. 97.5
d. 92.5
3. If HCF(210,55) is expressible in the form 210×5-55y. Find y.
4. If Sn denote the sum of first ‘n' terms of an AP. if Sn=35n then S3n : Sn equals ______
a. 4
b. 8
c. 6
d. 10
5. Radii of the circular ends of a frustum are 6cm and 14cm. If it's slant height is 10cm, it's vertical height is _______
a. 6cm
b. 4cm
c. 8cm
d. 7cm
6. If the area of a sector of a circle bounded by an arc of length 5π equals to 20πcm^2 then radius = _______
a. 12cm
b. 8cm
c. 16 cm
d. 10 cm
7. If sec theta + tan theta=x then sec theta =________
8. If ∆ABC~∆PQR, ar(∆ABC)/ ar(∆PQR) = ______ if AB/PQ=1/3
9. The value of c for which the pair of equations cx-y=2 and 6x-y=3 will have infinitely many solutions is ______
a. 3
b. -3
c. -12
d. no value
10. The value of K for which the quadratic equation 2x^2 - Kx + K=0 has equal roots is _____
a. 0
b. 0,4
c. 8
d. 0,8
Answers
1. (tan theta. cosec theta) whole square - (sin theta. sec theta) whole square=
Given,
(tan ∅ × cosec ∅)² - (sin ∅ × sec ∅)²
= (sin ∅ / cos ∅ × 1 / sin ∅)² - (sin ∅ × 1 / cos ∅)²
= (1 / cos ∅)² - (sin ∅ / cos ∅)²
= (1 - sin² ∅) / cos² ∅
= cos² ∅ / cos² ∅
= 1
Option b is correct.
2. Area of a sector of circle of r= 28cm and theta= 45 is
Area of sector = ∅/360° × πr²
= 45°/360° × 22/7 × 28²
= 308
The options are wrong.
3. If HCF(210,55) is expressible in the form 210×5-55y. Find y.
HCF of 210 and 55 = 5
Therefore, we have,
210 × 5 - 55y = 5
1050 - 5 = 55y
1045 = 55y
y = 1045/55
y = 19
4. If Sn denote the sum of first ‘n' terms of an AP. if Sn=35n then S3n : Sn equals
Sn = n/2 [2a + (n-1) d]
S3n = 3n/2 [2a + (3n-1) d]
Now consider
S3n / Sn = 3n/2 [2a + (3n-1) d] / n/2 [2a + (n-1) d]
= 3 [2a + (3n-1) d] / [2a + (n-1) d]
= 3 [(n+1)d + (3n-1)d] / [(n+1)d + (n-1)d]
= 3 [4nd] / [2nd]
= 6
⇒ S3n : Sn = 6
Option c is correct.
5. Radii of the circular ends of a frustum are 6cm and 14cm. If it's slant height is 10cm, it's vertical height is
Formula is, s = √ [(r1 - r2)² + h²]
Given,
r1 = 14 cm
r2 = 6 cm
s = 10 cm
10 = √ [(14 - 6)² + h²]
squaring on both sides, we get,
10² = (14 - 6)² + h²
100 = 8² + h²
h² = 100 - 64 = 36
The vertical height h = 6 cm
Option a is correct.
6. If the area of a sector of a circle bounded by an arc of length 5π equals to 20πcm^2 then radius =
∅/ 360 ×2πr = 5π
⇒ ∅/ 360 = 5π/2πr
∅/360 × πr²= 20π
⇒ ∅/ 360 = 20π/πr²
∴ 5π/2πr = 20π/πr²
5/2r = 20/r²
5/2 = 20/r
r = 20 × 2/5
∴ r = 8 cm
Option b is correct.
7. If sec theta + tan theta=x then sec theta =
Given,
sec ∅ + tan ∅ = x .........(1)
we use the identity,
sec² ∅ - tan² ∅ = 1
(sec ∅ - tan ∅) (sec ∅ + tan ∅) =1
(sec ∅ - tan ∅) (x) =1
sec ∅ - tan ∅ = 1/x ..........(2)
adding (1) and (2), we get,
sec ∅ + tan ∅ + sec ∅ - tan ∅ = x + 1/x
2 sec ∅ = x²+1 / x
∴ sec ∅ = x²+1 / 2x
8. If ∆ABC~∆PQR, ar(∆ABC)/ ar(∆PQR) = ______ if AB/PQ=1/3
Given,
∆ABC~∆PQR
⇒ ar(∆ABC)/ ar(∆PQR) = (AB/PQ)²
= (1/3)²
= 1²/3²
= 1/9
∴ ar(∆ABC)/ ar(∆PQR) = 1/9
9. The value of c for which the pair of equations cx-y=2 and 6x-y=3 will have infinitely many solutions is
The condition for system of equations to have infinitely many solutions is,
a1/a2 = b1/b2 = c1/c2
c/6 = -1/-1 ≠ 2/3
Therefore the system of equations will not have infinitely many solutions
Option d is correct.
10. The value of K for which the quadratic equation 2x^2 - Kx + K=0 has equal roots is
Given,
2x^2 - Kx + K=0
a = 2, b = -K, c = K
The condition for quadratic equation to have equal roots is:
b² - 4ac = 0
b² = 4ac
(-K)² = 4 (2) (K)
K² = 8K
K = 8
Option c is correct.
9. The value of c for which the pair of equations cx-y-2 and 6x-y=3 will have infinitely many solutions is
The condition for system of equations to have infinitely many solutions is,
a1/a2 = b1/b2 = c1/c2
c/6= -1/-1 # 2/3
Therefore the system of equations will not have infinitely many solutions
Option d is correct.
10. The value of K for which the quadratic equation 2x^2 - Kx + K=0 has equal roots
is
Given,
2x^2 - Kx + K=0
a = 2, b = -K, C = K
The condition for quadratic equation to have equal roots is:
b² - 4ac = 0
b² = 4ac
(-K)² = 4 (2) (K)
K² = 8K
K = 8
Option c is correct.
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