In trapezium A B C D , A B ∥ D C , D C = 2 A B . E F drawn parallel to A B cuts A D in F and B C in E such that B E E C = 3 4 . Diagonal D B intersects E F at G . Prove that 7 F E = 10 A B .

Given : A trapezium A B C D with A B ∥ D C and D C = 2 A B . Given that E F drawn parallel to A B cuts A D in F and B C in E such that B E E C = 3 4 . Diagonal D B intersects E F at G . Consider ∆ B D C and ∆ B G E : ∠ 1 = ∠ 2 (Corresponding angles) ∠ 3 = ∠ 3 (Common Angle) ∆ B D C ~ ∆ B G E (By AA Criterion Similarity) B E B C = G E D C (Corresponding sides of similar triangles are proportional) So, G E D C = 3 7 G E 2 A B = 3 7 G E A B = 6 7 In ∆ B D C , G E ∥ D C Using Basic Proportionality theorem, B E E C = B G G D ......(1) In ∆ A D B , G F ∥ A B Using Basic Proportionality theorem, A F F D = B G G D ......(2) From (1) and (2), we get B E E C = A F F D = 3 4 In ∆ D G F and ∆ D B A , ∠ 4 = ∠ 5 (Corresponding angles) ∠ 6 = ∠ 6 (Common) ∆ D G F ~ ∆ D B A (By AA Criterion Similarity) Then G F A B = D F A D (Corresponding sides of similar triangles are proportional) G F A B = 4 7 E F A B = G E A B + G F A B = 6 7 + 4 7 = 10 7 Thus, 7 F E = 10 A B Hence, proved.

If a sinθ + b cosθ = c , then prove that a cosθ - b sinθ = ± a 2 + b 2 - c 2 .

Given: a sinθ + b cosθ = c Squaring on both sides, we get; a sinθ + b cosθ 2 = c 2 Using the algebaric identity a + b 2 = a 2 + 2 a b + b 2 , we get: a 2 sin 2 θ + b 2 cos 2 θ + 2 a b sinθcosθ = c 2 Using the identity sin 2 θ + cos 2 θ = 1 , we get: a 2 1 - cos 2 θ + b 2 1 - sin 2 θ + 2 a b sinθcosθ = c 2 a 2 - a 2 cos 2 θ + b 2 - b 2 sin 2 θ + 2 a b sinθcosθ = c 2 a 2 + b 2 - c 2 = a 2 cos 2 θ + b 2 sin 2 θ - 2 a b sinθcosθ Using the identity a - b 2 = a 2 - 2 a b + b 2 , we get: a 2 + b 2 - c 2 = a cosθ - b sinθ 2 a cosθ - b sinθ = ± a 2 + b 2 - c 2 Hence, proved.

Prove that tanθ 1 - tanθ - cotθ 1 - cotθ = cosθ + sin θ cosθ - sin θ .

We need to prove : tanθ 1 - tanθ - cotθ 1 - cotθ = cosθ + sin θ cosθ - sin θ We know that, tanθ = sinθ cosθ and cotθ = cosθ sinθ . sinθ cosθ 1 - sinθ cosθ - cosθ sinθ 1 - cosθ sinθ sinθ cosθ cosθ - sinθ cosθ - cosθ sinθ sinθ - cos θ sinθ sinθ cosθ - sinθ - cosθ sin θ - cos θ cosθ + sin θ cosθ - sin θ Hence, proved.

E M B I B E

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 23

MEDIUM

10th CBSE

IMPORTANT

Earn 100

A peacock is sitting on top of a pillar, which is $9 m$ high. From a point $27 m$ away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing this, the peacock pounces on the snake. If their speeds are equal, at what distance from hole is the snake caught?

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Important Questions on Model Test Paper

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 24

In Fig. $DEFG$ is a square and $∠ B A C = 90 °$ . Prove that

(i) $∆ A G F ~ ∆ D B G$

(ii) $∆ A G F ~ ∆ E F C$

(iii) $∆ D B G ~ ∆ E F C$

(iv) $D E^{2} = B C \times E C$

Question Image

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 24

In trapezium

A B C D,

A B ∥ D C

,

D C = 2 A B

.

E F

drawn parallel to

A B

cuts

A D

in

F

and

B C

in

E

such that

\frac{B E}{E C} = \frac{3}{4}

. Diagonal

D B

intersects

E F

at

G

. Prove that

7 F E = 10 A B

.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 25

Draw a circle of radius

4 cm

. Construct a pair of tangents to it, the angle between which is

60 °

. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 26

If

a sinθ + b cosθ = c

, then prove that

a cosθ - b sinθ = \pm \sqrt{a^{2} + b^{2} - c^{2}}

.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 26

Prove that

\frac{tanθ}{1 - tanθ} - \frac{cotθ}{1 - cotθ} = \frac{cosθ + \sin θ}{cosθ - \sin θ}

.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 27

A straight highway leads to the foot of the tower. A man standing at the top of the tower observes the car at an angle of depression of

30 °

, which is approaching the foot of the tower with the uniform speed. Six seconds later, the angle of depression of the car found is to be

60 °

. Find the time taken (in seconds) by the car to reach the foot of the tower from this point.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 28

A container shaped like a right circular cylinder having diameter

12 cm

and height

15 cm

is full of ice cream. The ice cream is to be filled into cones of height

12 cm

and diameter

6 cm

, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 16 - Model Test Paper>SECTION>Q 29

In Figure; $P S R$ , $R T Q$ and $P A Q$ are three semicircles of diameter $10 cm, 3 cm$ and $7 cm$ respectively. Find the perimeter of the shaded region.(Use $π = 3.14$ )