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.

MEDIUM

10th CBSE

IMPORTANT

Earn 100

The given distribution shows the number of wickets taken by bowlers in inter-school competitions:

Number of wickets $0 - 3$ $3 - 6$ $6 - 9$ $9 - 12$ $12 - 15$

Number of Bowlers $9$ $3$ $5$ $3$ $1$

Draw a less than type 'ogive'. Also, find the median from the curve.

Important Questions on Model Test Paper

MEDIUM

10th CBSE

IMPORTANT

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

The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows, he cut down the arrows thrown by Bheeshm on him. Six arrows, he used to kill the rath driver of Bheeshm. With one arrow each he knocked down respectively the rath, flag and the bow of Bheeshm. Finally, with one more than four times the square root of arrows he laid Bheeshm unconscious on an arrow bed. Find the total number of arrows Arjun had.

MEDIUM

10th CBSE

IMPORTANT

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

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?

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

F

and

B C

E

such that

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

. Diagonal

D B

intersects

E F

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

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.

Number of wickets	$0 - 3$	$3 - 6$	$6 - 9$	$9 - 12$	$12 - 15$
Number of Bowlers	$9$	$3$	$5$	$3$	$1$

The given distribution shows the number of wickets taken by bowlers in inter-school competitions: Number of wickets 0-3 3-6 6-9 9-12 12-15 Number of Bowlers 9 3 5 3 1 Draw a less than type 'ogive'. Also, find the median from the curve.

Important Questions on Model Test Paper

Learn and Prepare for any exam you want !

The given distribution shows the number of wickets taken by bowlers in inter-school competitions:

Number of wickets $0 - 3$ $3 - 6$ $6 - 9$ $9 - 12$ $12 - 15$

Number of Bowlers $9$ $3$ $5$ $3$ $1$

Draw a less than type 'ogive'. Also, find the median from the curve.