Prove that secθ - tanθ secθ + tanθ = cos 2 θ 1 + sinθ 2 = 1 - 2 secθ tanθ + 2 tan 2 θ .

Given, to prove that, secθ - tanθ secθ + tanθ = cos 2 θ 1 + sinθ 2 = 1 - 2 secθ tanθ + 2 tan 2 θ Let A = secθ - tanθ secθ + tanθ , B = cos 2 θ 1 + sinθ 2 and C = 1 - 2 secθ tanθ + 2 tan 2 θ Taking A and B first: ⇒ We have to prove secθ - tanθ secθ + tanθ = cos 2 θ 1 + sinθ 2 . . . . . . . i In LHS in equation i , secθ - tanθ secθ + tanθ ⇒ 1 cos θ - sin θ cos θ 1 cos θ + sin θ cos θ ⇒ 1 - sin θ 1 + sin θ ⇒ 1 - sin θ 1 + sin θ × 1 + sin θ 1 + sin θ ⇒ 1 - sin 2 θ 1 + sin θ 2 Using, sin 2 θ + cos 2 θ = 1 ⇒ cos 2 θ 1 + sin θ 2 = R . H . S of equation i ⇒ secθ - tanθ secθ + tanθ = cos 2 θ 1 + sinθ 2 . . . . . i i Now, taking A and C : ⇒ We have to prove that secθ - tanθ secθ + tanθ = 1 - 2 secθ tanθ + 2 tan 2 θ . . . . iii Taking L.H.S of equation i i i , ⇒ secθ - tanθ secθ + tanθ ⇒ secθ - tanθ secθ + tanθ × secθ - tanθ secθ - tanθ ⇒ secθ - tanθ 2 sec 2 θ - tan 2 θ Using, sec 2 θ - tan 2 θ = 1 ⇒ secθ - tanθ 2 1 = sec 2 θ - 2 secθ tanθ + tan 2 θ ⇒ 1 + tan 2 θ - 2 secθ tanθ + tan 2 θ ⇒ 1 - 2 secθ tanθ + 2 tan 2 θ = R . H . S of equation i i i ⇒ secθ - tanθ secθ + tanθ = 1 - 2 secθ tanθ + 2 tan 2 θ . . . . . . . iv From equations i i and i v , we get that secθ - tanθ secθ + tanθ = cos 2 θ 1 + sinθ 2 = 1 - 2 secθ tanθ + 2 tan 2 θ Hence, Proved.

Find the mode and median of the following frequency distribution. x 10 11 12 13 14 15 f 1 4 7 5 9 3

Table for given data can be represented as: x i f i Cumulative Frequency C . F 10 1 1 11 4 5 12 7 12 13 5 17 14 9 26 15 3 29 Total ∑ f i = 29 n = 29 ∴ n = 29 odd ∴ Median = n + 1 2 th term = 29 + 1 2 th term = 15 th term = 13 The value having the highest frequency is known as the mode of the distribution. Here, highest frequency is 9 , against the value 14 . So, the mode is 14 . Hence, median is 13 and mode is 14 .

E M B I B E

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 7

EASY

10th ICSE

IMPORTANT

Earn 100

A lot of $20$ bulbs contain $4$ defective ones. One bulb is drawn at random from the lot. Suppose the bulb drawn is not defective and is not replaced. Now, one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

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Important Questions on Sample Question Paper

EASY

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 7

A lot of

20

bulbs contain

4

defective ones. One bulb is drawn at random from the lot. Suppose the bulb drawn is defective and it is not replaced. Now, one bulb is drawn at random from the rest. What is the probability that this bulb is defective?

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 8

Prove that

\frac{secθ - tanθ}{secθ + tanθ} = \frac{\cos^{2} θ}{{(1 + sinθ)}^{2}} = 1 - 2 secθ tanθ + 2 \tan^{2} θ

.

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 8

If

a = \frac{b + c}{2}, c = \frac{a + b}{2}

and

b

is the mean proportional between

a and c

, then prove that

\frac{1}{a} + \frac{1}{c} = \frac{2}{b}

.

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 8

Find the equations of the altitudes of the triangle whose vertices are

A (7, - 1), B (- 2, 8) and C (1, 2)

.

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 9

Draw a

△ A B C

in which

B C = 6 cm, ∠ B = 45 ° and A B = 7 cm

. Draw the circumcircle of

△ A B C

.

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 9

The sum of a number and its positive square root is

\frac{6}{25}

. The number is

\frac{1}{a}

, find the value of

a

.

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 9

Find the mode and median of the following frequency distribution.

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$3$

HARD

10th ICSE

IMPORTANT

All In One ICSE Mathematics Class 10>Chapter 22 - Sample Question Paper>SAMPLE QUESTION PAPER>Q 10

$A, B and C$ are three points on a circle. The tangent at $C$ meets $B A$ produced at $T$ . Given that $∠ A T C = 36 °$ and that $∠ A C T = 48 °$ , then calculate the angle subtended by $A B$ at the centre of the circle.