Total number of discs = 90 (i) Total number of two-digit numbers between 1 and 90 81 P (getting a two-digit number) = 81/90 = 9/10 (ii) Perfect squares between 1 and 90 are 1,4,9, 16, 25, 36, 49, 64, and 81. Therefore, total number of perfect squares between 1 and 90 is 9. P (getting a perfect squarRead more
Total number of discs = 90
(i) Total number of two-digit numbers between 1 and 90 81
P (getting a two-digit number) = 81/90 = 9/10
(ii) Perfect squares between 1 and 90 are 1,4,9, 16, 25, 36, 49, 64, and 81.
Therefore, total number of perfect squares between 1 and 90 is 9.
P (getting a perfect square) = 9/90 = 1/10
(iii) Numbers that are between 1 and 90 and divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 55, 60, 65, 70, 75, 80, 85, and 90.
Therefore, total numbers divisible by 5 = 18
Probability of getting a number divisible by 5 = 18/90 = 1/5
(i) False, Let A = 30° and B = 60° Therefore, LHS = sin(A + B) = sin(30° + 60°) = sin 90° 1 and RHS = sin A + sin B = sin 30° + sin 60° = 1/2 + √3/2 = (1+√3)/2 ≠ 1 Hence, sin (A + B) ≠ sin A + sin B (ii) True, As we know that sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 and sin 90° = 1Read more
(i) False,
Let A = 30° and B = 60°
Therefore, LHS = sin(A + B) = sin(30° + 60°) = sin 90° 1 and
RHS = sin A + sin B = sin 30° + sin 60° = 1/2 + √3/2 = (1+√3)/2 ≠ 1
Hence, sin (A + B) ≠ sin A + sin B
(ii) True,
As we know that sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 and sin 90° = 1
Hence, for the increasing values of θ, sin θ is also increasing.
(iii) False,
As we know that cos 0° = 1, cos 30° = √3/2, cos 45° = 1√2, cos 60° = 1/2 and cos 90° = 0
Hence, for the increasing values of θ, cos θ is decreasing.
(iv) False,
∵ cos 30° = √3/2, but sin 30° = 1/2.
(v) True,
∵ tan 0° = 0, we know that cot 0° = 1/tan 0° = 1/0, which is not defined.
See the explanation video of the above solution here✌
x² - 2x - 8 = x² - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x + 2)(x - 4) The value of x² - 2x - 8 is zero if x + 2 = 0 or x - 4 = 0 x = -2 or x = 4 Therefore, the zeroes of x² - 2x - 8 are -2 and 4. Now Sum of zeroes = -2 + 4 = 2 = -(-2)/1 =-(Cofficient of x)/(Cofficient of x²) Product of zeroes = (-2)Read more
x² – 2x – 8
= x² – 4x + 2x – 8
= x(x – 4) + 2(x – 4)
= (x + 2)(x – 4)
The value of x² – 2x – 8 is zero if x + 2 = 0 or x – 4 = 0
x = -2 or x = 4
Therefore, the zeroes of x² – 2x – 8 are -2 and 4. Now
Sum of zeroes = -2 + 4 = 2 = -(-2)/1 =-(Cofficient of x)/(Cofficient of x²)
Product of zeroes = (-2) x 4 = -8 =-8/1 = (Cofficient term)/Cofficient of x²)
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.
Total number of discs = 90 (i) Total number of two-digit numbers between 1 and 90 81 P (getting a two-digit number) = 81/90 = 9/10 (ii) Perfect squares between 1 and 90 are 1,4,9, 16, 25, 36, 49, 64, and 81. Therefore, total number of perfect squares between 1 and 90 is 9. P (getting a perfect squarRead more
Total number of discs = 90
See less(i) Total number of two-digit numbers between 1 and 90 81
P (getting a two-digit number) = 81/90 = 9/10
(ii) Perfect squares between 1 and 90 are 1,4,9, 16, 25, 36, 49, 64, and 81.
Therefore, total number of perfect squares between 1 and 90 is 9.
P (getting a perfect square) = 9/90 = 1/10
(iii) Numbers that are between 1 and 90 and divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 55, 60, 65, 70, 75, 80, 85, and 90.
Therefore, total numbers divisible by 5 = 18
Probability of getting a number divisible by 5 = 18/90 = 1/5
State whether the following are true or false. Justify your answer.
(i) False, Let A = 30° and B = 60° Therefore, LHS = sin(A + B) = sin(30° + 60°) = sin 90° 1 and RHS = sin A + sin B = sin 30° + sin 60° = 1/2 + √3/2 = (1+√3)/2 ≠ 1 Hence, sin (A + B) ≠ sin A + sin B (ii) True, As we know that sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 and sin 90° = 1Read more
(i) False,
Let A = 30° and B = 60°
Therefore, LHS = sin(A + B) = sin(30° + 60°) = sin 90° 1 and
RHS = sin A + sin B = sin 30° + sin 60° = 1/2 + √3/2 = (1+√3)/2 ≠ 1
Hence, sin (A + B) ≠ sin A + sin B
(ii) True,
As we know that sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 and sin 90° = 1
Hence, for the increasing values of θ, sin θ is also increasing.
(iii) False,
As we know that cos 0° = 1, cos 30° = √3/2, cos 45° = 1√2, cos 60° = 1/2 and cos 90° = 0
Hence, for the increasing values of θ, cos θ is decreasing.
(iv) False,
∵ cos 30° = √3/2, but sin 30° = 1/2.
(v) True,
∵ tan 0° = 0, we know that cot 0° = 1/tan 0° = 1/0, which is not defined.
See the explanation video of the above solution here✌
See lessEvaluate the following trigonometry questions?
(i) (sin 18°)/(cos 72°) = (sin 18°)/(cos 72°) = cos (90° - 18°))/(cos 72°) [∵ cos (90° - θ) = sin θ] = (cos 72°)/(cos 72°) = 1 (ii) (tan 26°)/(cot 64°) = (tan 26°)/(cot 64°) = (cot (90° - 26°))/(cot 64°) [∵ cot (90° - θ) = tan θ] = (cot 64°)/(cot 64°) = 1 (iii) cot 48° – sin 42° = cot 48° – sin 42°Read more
(i) (sin 18°)/(cos 72°)
= (sin 18°)/(cos 72°) = cos (90° – 18°))/(cos 72°) [∵ cos (90° – θ) = sin θ]
= (cos 72°)/(cos 72°) = 1
(ii) (tan 26°)/(cot 64°)
= (tan 26°)/(cot 64°) = (cot (90° – 26°))/(cot 64°) [∵ cot (90° – θ) = tan θ]
= (cot 64°)/(cot 64°) = 1
(iii) cot 48° – sin 42°
= cot 48° – sin 42° = cos 48° – cos (90° – 42°) [∵ cos (90° – θ) = sin θ]
(iv) cosec 31° – sec 59°
cosec 31° – sec 59° = cosec 31° – cosec (90° – 59°) [∵ cosec (90° – θ) = sec θ]
= cosec 31° – cosec 31° = 0
See here for video explanation for this question😮👇
See lessFind the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. 6x² – 3 – 7x.
6x² - 3 - 7x = 6x² - 7x - 3 = 6x² - 9x + 2x -3 = 3x(2x - 3) + 1(2x -3) = (3x +1)(2x -3) The value of 6x² - 7x - 3 is zero if 3x + 1 = 0 or 2x -3 = 0. ⇒ x = -1/3 or x = 3/2, Therefore, the zeroes of 6x² - 7x - 3 are -1/3 and 3/2. Now, Sum of zeroes = -1/3 + 3/2 = (-2 + 9)/6 = 7/6 = -(7)/6 = -(CofficiRead more
6x² – 3 – 7x
= 6x² – 7x – 3
= 6x² – 9x + 2x -3
= 3x(2x – 3) + 1(2x -3)
= (3x +1)(2x -3)
The value of 6x² – 7x – 3 is zero if 3x + 1 = 0 or 2x -3 = 0.
⇒ x = -1/3 or x = 3/2,
Therefore, the zeroes of 6x² – 7x – 3 are -1/3 and 3/2.
Now, Sum of zeroes = -1/3 + 3/2 = (-2 + 9)/6 = 7/6 = -(7)/6 = -(Cofficient of x)/Cofficient of x²
Product of zeroes = (-1/3) x 3/2 = -1/2 = (-3)/6 = Constant term/Cofficient of x².
See here 😎
See lessFind the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. x² – 2x – 8.
x² - 2x - 8 = x² - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x + 2)(x - 4) The value of x² - 2x - 8 is zero if x + 2 = 0 or x - 4 = 0 x = -2 or x = 4 Therefore, the zeroes of x² - 2x - 8 are -2 and 4. Now Sum of zeroes = -2 + 4 = 2 = -(-2)/1 =-(Cofficient of x)/(Cofficient of x²) Product of zeroes = (-2)Read more
x² – 2x – 8
= x² – 4x + 2x – 8
= x(x – 4) + 2(x – 4)
= (x + 2)(x – 4)
The value of x² – 2x – 8 is zero if x + 2 = 0 or x – 4 = 0
x = -2 or x = 4
Therefore, the zeroes of x² – 2x – 8 are -2 and 4. Now
Sum of zeroes = -2 + 4 = 2 = -(-2)/1 =-(Cofficient of x)/(Cofficient of x²)
Product of zeroes = (-2) x 4 = -8 =-8/1 = (Cofficient term)/Cofficient of x²)
Video Explanation 👇
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