The median of the following data is 525 . Find the values of x and y , if total frequency is 100 . Class 0 - 100 100 - 200 200 - 300 300 - 400 400 - 500 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 Frequency 2 5 x 12 17 20 y 9 7 4

Cumulative frequency for the given data; Class interval Frequency f Cumulative frequency CF 0 - 100 2 2 100 - 200 5 7 200 - 300 x x + 7 300 - 400 12 x + 19 400 - 500 17 x + 36 500 - 600 20 x + 56 600 - 700 y x + y + 56 700 - 800 9 x + y + 65 800 - 900 7 x + y + 72 900 - 1000 4 x + y + 76 total Σ f = 76 + x + y = 100 Given that the median is 525 . So median class is 500 - 600 Now median = l + N 2 - C F f × h where l = 500 , h = 100 , N = 100 , C F = 36 + x , f = 20 so, 525 = 500 + 100 2 - ( 36 + x ) 20 × 100 ⇒ 525 - 500 = ( 50 - ( 36 + x ) ) × 5 ⇒ 25 = ( 14 - x ) × 5 ⇒ x = 9 Also, Σ f = 76 + x + y = 100 ⇒ 9 + y = 100 - 76 ⇒ y = 15 Hence, x = 9 and y = 15

The median of the following data is 525 . Find the values of x and y , if total frequency is 100 . Class 0 - 100 100 - 200 200 - 300 300 - 400 400 - 500 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 Frequency 2 5 x 12 17 20 y 9 7 4

Cumulative frequency for the given data; Class interval Frequency f Cumulative frequency CF 0 - 100 2 2 100 - 200 5 7 200 - 300 x x + 7 300 - 400 12 x + 19 400 - 500 17 x + 36 500 - 600 20 x + 56 600 - 700 y x + y + 56 700 - 800 9 x + y + 65 800 - 900 7 x + y + 72 900 - 1000 4 x + y + 76 total Σ f = 76 + x + y = 100 Given that the median is 525 . So median class is 500 - 600 Now median = l + N 2 - C F f × h where l = 500 , h = 100 , N = 100 , C F = 36 + x , f = 20 so, 525 = 500 + 100 2 - ( 36 + x ) 20 × 100 ⇒ 525 - 500 = ( 50 - ( 36 + x ) ) × 5 ⇒ 25 = ( 14 - x ) × 5 ⇒ x = 9 Also, Σ f = 76 + x + y = 100 ⇒ 9 + y = 100 - 76 ⇒ y = 15 Hence, x = 9 and y = 15

In the figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T . Prove that ∠ PTQ = 2 ∠ OPQ .

We know that length of the tangents drawn from the external point to the circle are equal TP = TQ Angles opposite to equal sides are equal ∠ TQP = ∠ TPQ . . . . . . i Now, OP is the radius and PT is the tangent. We know that tangent at any point of the circle is perpendicular to the radius through the point of contact. So, PT ⊥ OP ∠ OPT = 90 ° ∠ OPQ + ∠ TPQ = 90 ° ∠ TPQ = 90 ° - ∠ OPQ . . . . . . . . i i In ∆ PTQ , ∠ TPQ + ∠ TQP + ∠ PTQ = 180 ° ∠ TPQ + ∠ TPQ + ∠ PTQ = 180 ° [ Using ( i ) ] 2 ∠ TPQ + ∠ PTQ = 180 ° 2 90 ° - ∠ OPQ + ∠ PTQ = 180 ° 180 ° - 2 ∠ OPQ + ∠ PTQ = 180 ° - 2 ∠ OPQ + ∠ PTQ = 0 ∠ PTQ = 2 ∠ OPQ Hence, proved.

For what values of k , the roots of the equation x 2 + 4 x + k = 0 are real?

We have, x 2 + 4 x + k = 0 Now we know that, For a quadratic equation a x 2 + b x + c = 0 to have real roots b 2 - 4 a c ≥ 0 That is, b 2 ≥ 4 a c In the given quadratic equation a = 1 , b = 4 and c = k Then, 4 2 ≥ 4 1 k 4 ≥ k Therefore, the values of k ≤ 4 .

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 16 - Sample Paper 2>Exercise>Q 37

MEDIUM

10th Uttar Pradesh Board

IMPORTANT

Earn 100

Find a quadratic polynomial whose zeros are reciprocals of the zeros of the polynomials $f (x) = a x^{2} + b x + c, a \neq 0, c \neq 0 .$

Important Questions on Sample Paper 2

HARD

10th Uttar Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 16 - Surface Areas and Volumes>Exercise>Q 26

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is

10 c m

, and its base is of radius

3.5 c m

. Find the total surface area of the article.
Question Image

HARD

10th Uttar Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 16 - Sample Paper 2>Exercise>Q 38

The perpendicular from the square corner of a right-triangle cuts the opposite side into two parts of $2$ and $3$ $centimetres$ of length. Prove that the two small right triangles cut by the perpendicular have the same angles.

Question Image

HARD

10th Uttar Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 16 - Sample Paper 2>Exercise>Q 40

Consider a cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 8 cm and height 6 cm is completely immersed in water. Find the volume of water left in the cylinder.

MEDIUM

10th Uttar Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 16 - Sample Paper 2>Exercise>Q 41

The median of the following data is $525$ . Find the values of $x$ and $y$ , if total frequency is $100$ .

Class	$0 - 100$	$100 - 200$	$200 - 300$	$300 - 400$	$400 - 500$	$500 - 600$	$600 - 700$	$700 - 800$	$800 - 900$	$900 - 1000$
Frequency	$2$	$5$	$x$	$12$	$17$	$20$	$y$	$9$	$7$	$4$