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**Class XMathematics – StandardMax. Marks: 80Duration: 3 hrs.**

**General Instructions**

__Read the following instructions very carefully and strictly follow them:__

This question paper comprises four sections – A, B, C and D. This question. Paper carries 40 questions. All questions are compulsory.

**Section A:**Question Number 1 to 20 comprises of 20 questions of one mark each.**Section B:**Question Number 21 to 26 comprises of 6 questions of two marks each.**Section C:**Question Number 27 to 34 comprises of 8 questions of three marks each.**Section D:**Question Number 35 to 40 comprises of 6 questions of four marks each.- There is no overall choice in the question Paper. However, an internal choice has been provided in 2 questions of one mark, 2 questions of two marks, 3 questions of three marks and 3 questions of four marks. You have to attempt only one of the choices in such questions.
- In addition to this, separate instructions are given with each section and question, wherever necessary.
- Use of calculators is not permitted.

**Section - A**

**Question numbers 1 to 20 carry 1 mark each. Question numbers 1 to 10 are multiple choice questions.Q.1. On dividing a polynomial p(x) by x ^{2} – 4, quotient and remainder are found to be x and 3 respectively. The polynomial p(x) is(a) 3x^{2} + x - 12(b) x^{3} - 4x + 3(c) x^{2} + 3x - 4(d) x^{2} - 4x - 3Ans. bSolution.** P(x) = (divisor)×(quotient) + Remainder

=(x

= x

(a) AB

(b) BC

(c) AC

(d) AB

**Ans. **a**Solution. **Given that ACB is an isosceles triangle right angled at C.

Therefore, AC = BC Using Pythagoras theorem in the given triangle, we have

AB^{2} = AC^{2} + BC^{2}

= AC^{2} + AC^{2}

= 2AC^{2}**Q.3. The point on the x-axis which is equidistant from (─ 4, 0) and 10, 0) is(a) (7, 0)(b) (5, 0)(c) (0, 0)(d) (3, 0)Ans. **d

Solution.

The required point (x, 0) is the mid-point of the line joining points (–4, 0) and (10, 0).

So, x = (– 4+10)/2

= 6/2

= 3

Required point = (x, 0)

= (3, 0)

**ORThe centre of a circle whose end points of a diameter are (─ 6, 3) and 6, 4) is(a) (8, ─ 1)(b) (4, 7)(c) 0, 7/2(d) 4, 7/2Ans. **c

Solution.

End points of the diameter are: (–6, 3) and (6, 4)

Coordinates of the centre = ((– 6+6)/2, (3+4)/2)

= (0, 7/2)

(a) 4

(b) ± 4

(c) ─ 4

(d) 0

Ans.

Solution.

2x

Discriminant = b

Here, b = k, a = 2, and c = 2

So, Discriminant = k

= k

A quadratic equation has equal roots if its discriminant is zero.

k

k

k = ±4

**Solution.**

Difference between consecutive terms is not same. So, this is not an A.P.**Q.6. The pair of linear equations 3x/2 + 5y/3 = 7 and 9x + 10y = 14 is 23(a) consistent(b) inconsistent(c) consistent with one solution(d) consistent with many solutions**

**Ans. **b

Solution.

9x + 10y 42 ...(1)

9x + 10y 14 ...(2)

Ratios of coefficients of x and that of y are

9/9 = 10/10 = 1/1

Ratio of constants = 42/14 = 3/1 ≠ 1/1

Ratios of coefficients of x and y are equal but they are not equal to the ratio of constants.

So, the given equations represent a pair of parallel lines and so they do not have a common solution.**Q.7. In Figure-2 PQ is tangent to the circle with centre at O, at the point B. If √AOB = 100°, then √ABP is equal to(a) 50°(b) 40°(c) 60°(d) 80°**

**Ans. **a

Solution.

So, ∠OAB = ∠OBA

= (180° –100°)/2

= 40°

Now, a radius of a circle meets a tangent at 90°.

So, ∠ABP = ∠OBP – ∠OBA = 90° –40° = 50°

(a) 3

(b) 3√3

(c) 3

(d) 3

Ans.

Solution.

Volume of sphere = 4/3 πr

12π = 4/3 πr

r

r = 3

(a)

(b) m + n

(c)

(d)

Ans.

Solution.

(a) 3cm

(b) 4cm

(c) 2cm

(d) 2√2cm

**Ans.** b

So, line joining centre O and point P bisects ∠PQR.

OP bisects ∠QPR = 90°.

In ∆ OQP, ∠Q = 90° (radius meets tangent at 90°)

∠QPO = 45° = ∠QOP

Thus, OQ = PQ = 4 cm

Q.11. The probability of an event that is sure to happen is ______.

**Ans. **1**Q.12. Simplest form of is ______.****Ans.**

**Q.13. AOBC is a rectangle whose three vertices are A(0, –3), O(0, 0) and B (4, 0). The length of its diagonal is ______.**

**Ans. **In right-angled triangle AOB,**Q.14. In the formula h, u = _____.****Ans.****Q.15. All concentric circles are ______ to each other.**

**Ans. **Similar**Answer the following question numbers 16 to 20.Q.16. Find the sum of the first 100 natural numbers.**

**Ans. **1+ 2 + 3 ....100 is an A. P.

Here first term a = 1

Common difference d = 1

Sum of n terms of an A.P. =

The sum of first 100 natural numbers

= 50 x 101

= 5050**Q.17. In Figure-4 the angle of elevation of the top of a tower from a point C on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.**

**Ans.**

tan30º = AB/30

1/√3 = AB/30

= AB = 30/√3 = 10√3

So, the height of the tower is 10√3 m.**Q.18. The LCM of two numbers is 182 and their HCF is 13. If one of the numbers is 26, Find the other.**

**Ans. **LCM HCF Pr oduct of the two numbers

182 × 13 = 26 × x

So, the other number is 91.**Q.19. Form a quadratic polynomial, the sum and product of whose zeroes are (-3) and 2 respectively.Ans. **x

= x

= x

So, the required polynomial is x

**OR****Can (x ^{2} – 1) be a remainder while dividing x^{4} – 3x^{2} + 5x – 9 by (x^{2} +3)? Justify your answer with reasons.**

Therefore, for the given question x

Ans.

= 2

**Section - B**

**Question number 21 to 26 carry 2 marks each.Q.21. In the given Figure-5, DE ||AC and DF||AE.Prove that BF/FE = BE/EC.**

**Ans. **In ABC, DE AC

So, using basic proportionality theorem, we get

BD/DA = BE/EC ...(1)

In ΔBAE, DF║AE

So, using basic proportionality theorem, we get

BD/DA = BF/FE ...(2)

From (1) and (2), we get

BE/EC = BF/FE

Ans.

Let us assume, to the contrary, that 5 + 2√7 is rational.

That is, we can find coprime a and b (b ≠ 0) such that

5 = 2√7 = a/b

∴ 2√7 = a/b - 5

Rearranging this equation, we get √7 =

Since, and are integers, we get a - 5b/2b is rational, and 2 so √7 is a rational.

But this contradicts the fact that 7 is irrational.

This contradiction has arisen because of our incorrect assumption that 5 + 2√7

is rational. So, we conclude that 5 + 2√7 is irrational.

**ORCheck whether 12 ^{n} can end with the digit 0 for any natural number n. **

**Ans.** If the number 12 , for any , were to end with the digit zero, then it would be divisible by 5.

That is, the prime factorisation of 12^{n} would contain the prime 5. This is not possible 12 = ∵ 12^{n} = (2 × 2 × 3)^{n}

So, the prime numbers in the factorisation of 12^{n} are 2 and 3.

So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of 12^{n} .

So, there is no natural number n for which 12^{n} ends with the digit zero.**Q.23. If A, B and C are interior angles of a Ð ABC, then show that Ans.** Given that A, B and C are interior angles of a triangle ABC.

∴ A + b + c = 180º

or A = 180º - B - C

Now,**Q.24. In Figure 6, a quadrilateral ABCD is drawn to circumscribe a circle.Prove thatAB + CD = BC +AD.**

We have to prove that

AB + CD + BC + AD

We know that lengths of tangents drawn from a point to a circle are equal.

Therefore, from figure, we have

DR = DS, CR = CQ, AS = AP, BP = BQ

Now,

LHS = AB + CD = (AP + BP) + (CR + DR)

= (AS BQ) (CQ + DS)

= BQ + CQ + AS + DS

= BC + AD

RHS

**ORIn Figure-7, find the perimeter of √ABC, if AP = 12 cm.**

**Ans. **From the given figure, we have AP = 12 cm

Since AQ and AB are the tangent to the circle from a common point A, hence AP = AQ = 12

Similarly, PB = BD and CD = CQ

Also, AP = AB + PB and AQ = AC + CQ

Perimeter of ABC = AB + BD + CD + AC

= AB + PB + CQ + AC

(since PB = BM and CM = CQ)

= (AB + PB) + (CQ + AC)

= AP + AQ

= 12 +12

= 24 cm

Therefore, the perimeter of triangle ABC =24 cm**Q.25. Find the mode of the following distribution: Ans.**

From the given data, we have

I = 30, f

= 34.1667

∴ Mode of the given data is 34.1667.

Q.26. 2 cubes, each of volume 125 cm^{3}, are joined end to end. Find the surface area of the resulting cuboid.

Ans. Let the side of the old cube = a

The volume of the old cube = 125 cm^{3} (Given)

The volume of the cube = a^{3}

a = 125 cm^{3}

a^{3} = 5^{3}

a = 5 cm

The dimensions of the resulting cuboid are: Length,

l = 10 cm

Breadth, b = 5

Height, h = 5 cm

Total surface area of the resulting cuboid:

= 2(lb + bh + hl)

= 2[10(5)+5(5)+5(10)]

= 2[50+25+50]

= 2[125]

= 250 cm^{2}

**Section - C**

**Q.27. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 ****when 8 is added to its denominator. Find the fraction.****Ans.** Let the numerator of the fraction be x and denominator be y.

Therefore, the fraction is x/y.

According to question,

x - 1/y = 1/3

3x - 1 = y

3x - 3 = y ....(1)

and x/ y + 8 = 1/4

4x = y + 8

4x - 8 = y ....(2)

From equations 1 and 2, we get

3x - 3 = 4x - 8

4x - 3x = 8 -3

x = 5

Putting 5 in equation (1),

3x 5 - 3 = y

y = 12

So, the required fraction = 5/12

**ORThe present age of a father is three years more than three times the age of his son. Three years hence the father's age will be 10 years more than twice the age of the son. Determine their present ages.Ans.** Let the son's present age be x.

So, father's present age = 3x + 3

3 years later:

Son's age = x + 3

Father's age = 3x + 3 + 3 = 3x + 6

But, according to the given condition,

3 years later father's age = 2 x + 3 + 10

2x + 6 + 10

2x + 16

So, we can write

3x + 6 = 2x + 16

3x - 2x = 16 - 6

x = 10

So, son's present age 10 years

and father's present age 10 x 3 + 3

= 33 years**Q.28. Use Euclid Division Lemma to show that the square of any positive integer is either of the form 3q or 3q + 1 for some integer q.Ans. **Let be a positive integer and 3.

By Euclid's Algorithm,

a = 3m + r for some integer m ≥ 0 and 0 ≤ r ≤ 3.

The possible remainders are 0, 1 and 2. Therefore,

a can be 3m or 3m + 1 or 3m + 2.

Thus,

a

= 9m

= 3x (3m

3k

where k

Hence, square of any positive integer is either of the form 3q or 3q + 1 for some integer q.

Ans.

Let the coordinate of point on y-axis be (0, y).

Therefore,

Now

or 0 = -2k + 6

or k = 3

Therefore, the required ratio is 3:1.

Also,

Therefore, the given line segment is divided by the point (0, -25/4) in the ratio 3:1.

**ORShow that the points (7, 100, (-2, 5) and (3, -4) are vertices of an isosceles right triangle. **

**Ans.** Let the given points are P(7, 10), Q(-2, 5) and R(3, 4).

Now, using distance formula we find distance between these points i.e., PQ, QR and PR.

Distance between points P(7, 10) and Q(-2, 5),

Distance between points Q(-2, 5) and R(3, 4),

Distance between points P(7, 10) and R(3, 4),

Now,

PQ^{2} + QR^{2} 106 + 106

= 212 = PR^{2}

i.e., PQ^{2} + QR^{2} = PR^{2}

Therefore, points P(5, 2), Q(6, 4) and R(7, 2) form an isosceles right triangle because sides PQ and QR are equal.**Q.30. Prove that:**

Ans.

We need to find S

Firstly, we will find the value of n.

We know that

a

So, 50 = 5 + (n - 1)3

or 50 - 5 (n - 1)3

or 45/3 + 1 = n

or n = 16

We know that sum of first n terms of an AP is given by

= 8 x 55

or s

Ans.

**Steps of Construction :**

Step 1: Draw a ∠ABC with sides AB 5 cm, BC 6 cm and ∠ABC 60º.

Step 2: Draw a ray BX making an acute angle with line BC on the opposite side of vertex A.

Step 3: Locate 4 points B_{1} , B_{2} , B_{3} , B_{4} on BX such that BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4}.

Step 4: Join the points C and B_{4}.

Step 5: Through the point B_{3}, draw a line parallel to CB_{4 }intersecting line segment BC at point C'.

Step 6: Draw a line through C' parallel to the line AC' to intersect line segment AB at A'. The required triangle is ΔA'BC'.

**ORDraw a circle of radius 3.5 cm. Take a point P outside the circle at a distance of 7 cm from the centre of the circle and construct a pair of tangents to the circle from that point. **

**Ans.**

**Steps of Construction:**

Step 1: Draw a circle of radius 3.5 cm with centre at point O. Locate a point P, at a distance of 7 cm from O, and join O and P.

Step 2: Bisect OP. Let M be the mid-point of OP.

Step 3: Draw a circle with centre at M and MO as radius. Q and R are points of intersections of this circle with the circle having centre at O.

Step 4: Join PQ and PR. PQ and PR are the required tangents.**Q.33. Read the following passage and answer the question given at the end: Diwali Fair.**

**A game in a booth at a Diwali Fair involves using a spinner first. Then, if the spinner stops on an even number, the player is allowed to pick a marble from a bag. The spinner and the marbles in the bag are represented in Figure – 8.Prizes are given when a black marble is picked. Shweta plays the game once.**

**(i) What is the probability that she will be allowed to pick a marble from the bag?(ii) Suppose she is allowed to pick a marble from the bag, what is the probability of getting a prize, when it is given that the bag contains 20 balls out of which 6 are black?**

**Ans.** Numbers on spinner 1, 2, 4, 6, 8, 10

Even numbers on spinner 2, 4, 6, 8, 10

Shweta will pick black marble, if spinner stops on even number.

Therefore,

n(Even number) = 5

n(Possible number) = 6

**(i) **P Shweta allowed to pick a marble,

= P (Even number)

= n(Even number)/n(Possible number)

= 5/6

Therefore, the probability of allowing Shweta to pick a marble is 5/6.**(ii) **Since, prizes are given, when a black marble is picked.

Number of black marbles = 6

Total number of marbles = 20

Therefore, P(getting a prize) = P(a black marble)

= n(Black marbles)/n(Total marbles)

= 6/20

= 3/10

Therefore, the probabiltiy of getting prize is 3/10.

Q.34. In figure – 9, a square OPQR is inscribed in a quadrant OAQB of a circle. If the radius of circle is 6√2 cm, find the area of the shaded region.

Ans.

Given that, OQ = 6√2 cm

OPQR is a square.

Let the side of square = a

The diagonal of square a√2

Here, OQ is a diagonal of square.

⇒ a√2 = 6√2

⇒ a = 6 cm

Area of square OPQR = 6^{2}

= 36 cm^{2}

Radius of the quadrant OAQB = Diagonal of the square OPQR

= 6√2

Area of the quadrant OAQB = 90º/360º x 22/7 x (6√2)^{2}

= 396/7 cm^{2}

Area of shaded region = Area of the quadrant OAQB

- Area of square OPQR

= 144/7

= 20.6 cm^{2}

**Section - D**

**Q.35. Obtain other zeroes of the polynomialp(x) = 2x ^{4} - x^{3} - 11x^{2} + 5x + 5if two of its zeroes are √5 and - √5.**

The two zeroes of p(x) are √5 and - √5.

Therefore, (x - √5 and (x + √5) are factors of p(x).

Also, (x - √5) (x + √5) = x

and so x

Now,

Equating (x^{2} - 5) (2x + 1) (x - 1) to zero, we get the zeroes of the given polynomial.

Hence, the zeroes of the given polynomial are : 1

√5, - √5, -1/2 and 1.**OR****What minimum must be added to 2x ^{3} - 3x^{2} + 6x + 7 so that resulting polynomial will be divisible by x^{2} - 4x +8 ?**

Here, divisor is x

Divide 2x

Remainder = 10x - 33

Therefore, we should add -(10x - 33) to make it exactly divisible by x

Thus, we should add 10x + 33 to 2x

Ans.

To prove:

Construction: Draw AL ⊥ BC and DM ⊥ EF

Proof here:

In ΔALB and ΔDME

∠ALB = ∠DME Each 90°

and √B = √E Since ΔABC ∼ ΔDEF

So, ΔALB ∼ ΔDME AA similarity criterion

Ans.

Area of first square = x^{2}

And,

Area of second square = y^{2}

According to the question,

x^{2} + y^{2} = 544 ...1

Now,

Perimeter of first square = 4x

And,

Perimeter of second square = 4y

According to the question,

4x – 4y = 32 ...(2)

From equation (2) , we get

4(x – y) = 32

or, x - y = 32/4

or, x - y = 8

or, x = 8 + y ...(3)

Subsituting this value of x in equation (1), we get

x^{2} + y^{2} = 544

or, (8 + y)^{2} + y^{2} = 544

or, 64 + y^{2} = 16y + y^{2} = 544

or, 2y^{2} + 16y + 64 = 544

or, 2y^{2} + 16y + 64 - 544 = 0

or, 2y^{2} + 16y - 480 = 0

or, 2(y^{2} + 8y - 240) = 0

or, y^{2} + 8y - 240) = 0

or, y^{2} + 20y - 12y - 240 = 0

or, y(y + 20) - 12 (y + 20) = 0

or, (y + 20) (y - 12) = 0

⇒ y + 20 = 0 or y - 12 = 0

⇒ y = -20 or y = 12

Since side of a square cannot be negative, therefore y = 12.

Substituting y = 12 in equation (3), we get

x = 8 + y = 8 + 12 = 20

Therefore,

Side of first square = x = 20 cm

And,

Side of second square = y = 12 cm

**OR**

**A motorboat whose speed is 18km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.**

**Ans. **Let the speed of the stream be x km/h.

Therefore, speed of the boat upstream = (18 – x) km/h

and the speed of the boat downstream = (18 + x) km/h.

The time taken to go upstream = distance/speed

= 24/18 - x hours.

Similarly, the time taken to go downstream = 24/18 - x hours.

According to the question,

or, 24/18 + x) - 24(18 - x) = (18 + x) (18 - x)

or, 432 + 24x - 432 + 24x = 324 - x^{2}

or, x^{2} + 48x - 324 = 0

Using the quadratic formula, we get

⇒ x = 6 or x = –54

Since x is the speed of the stream, it cannot be negative.

So, we ignore the root x = –54. Therefore, x = 6 gives the speed of the stream as 6 km/h.**Q.38. A solid toy is in the form of a hemisphere surmounted by a right circular cone of same radius. The height of the cone is 10 cm and the radius of the base is 7 cm. Determine the volume of the toy. Also find the area of the coloured sheet required to cover the toy.(Use π = 22/7 and 12.2)Ans.**

Let ABC be the hemisphere and ADC be the cone standing on the base of the hemisphere.

Height of the cone (h

Radius of the cone (r

Since the hemisphere is surmounted by the right circular cone of same radius, therefore

Radius of the hemisphere (r

So,

Volume of the toy = Volume of the cone + Volume of the hemisphere

Area of the coloured sheet required to cover the toy

= CSA of hemisphere + CSA of cone

= 2πr

Where l is the slant height of the cone

So,

Area of the coloured sheet required to cover the toy

= (308 + 268.4)cm

= 576.4 cm

**(Use √3 = 1.73)Ans.**

Let BD be a pedestal of height x m and AD be a statue of height 1.6 m. The angle of elevation of the top of pedestal from a point C is 45° and that of point statue from C is 60°.

In the triangle ABC:

AB/BC = tan60°

....(1)

In the triangle DBC:

DB/BC = tan45°

Or, x/BC = 1

Or, x = BC ...(2)

By equations 1 and 2 , we get

or, √3x = 1.6 + x

√3 - 1 x = 1.6

or,

= 2.184 m

Therefore, the height of the pedestal is 2.184m.**Q.40. For the following data, draw a 'less than' ogive and hence find the median of the distribution.Ans.**

Plot the points (10, 5), (20, 20), …, (70, 100) on a graph paper.

**ORThe distribution given below show the number of wickets taken by bowlers in one-day cricket matches. Find the mean and the median of the number of wickets taken.Ans.**

n = 45

⇒ n/2 = 45/2 = 2.5

Median class 100 - 140

l = 100, n/2 = 22.5, cf = 12, f = 16, h = 40

= 100 + 26.25

= 126.25

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