# Board Paper of Class 10 2022 Mathematics Standard Term-1 Set-4 Solutions - Solutions

#### General Instructions:

- This question paper contains
**50**questions out of which**40**questions are to be attempted. All questions carry**equal**marks. - This question paper contains
**three**Sections:**A**,**B**, and**C.** - Section
**A**has**20**questions. Attempt any**16**questions from Q.No.**1**to**20.** - Section
**B**has**20**questions. Attempt any**16**questions from Q.No.**21**to**40.** - Section
**C**contains**two**Case Studies containing 5 Questions in each case. Attempt any**4**questions from Q.No.**41**to**45**and**4**another from Q.No.**46**to**50.** - There is only
**one**correct option for every multiple choice question (MCQ). Marks will not be awarded for answering more than one option. - There is no negative marking.

#### Question 1

## The exponent of 5 in the prime factorisation of 3750 is

- 3
- 4
- 5
- 6

Ans: (b)

#### Question 2

## The graph of a polynomial P(x) cuts the x-axis at 3 points and touches it at 2 other points. The number of zeroes of P(x) is

- 1
- 2
- 3
- 5

Ans: (d)

Sol: The number of zeroes of P(x) is 5.

#### Question 3

## The values of x and y satisfying the two equations 32x + 33y = 34, 33x + 32y = 31 respectively are:

- –1, 2
- –1, 4
- 1, –2
- –1, –4

Ans: (a)

#### Question 4

## If A(3, √3), B(0, 0) and C(3, k) are the three vertices of an equilateral triangle ABC, then the value of k is

- 2
- –3
- –√3
- –√2

Ans: (c)

#### Question 5

## In figure, DE || BC, AD = 2 cm and BD = 3 cm, then ar(ΔABC) : ar(ΔADE) is equal to

- 4 : 25
- 2 : 3
- 9 : 4
- 25 : 4

Ans: (d)

#### Question 6

## If cot $\mathbf{\theta}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{3}}}$, the value of sec^{2} $\mathbf{\theta}$ + cosec^{2} $\mathbf{\theta}$ is

- 1
- $\frac{40}{9}$
- $\frac{38}{9}$
- $5\frac{1}{3}$

Ans: (d)

#### Question 7

## The area of a quadrant of a circle where the circumference of circle is 176 m, is

- 2464 m
^{2} - 1232 m
^{2} - 616 m
^{2} - 308 m
^{2}

Ans: (c)

#### Question 8

## For an event E, P(E) + P($\overline{\mathbf{E}}$) = x, then the value of x^{3} – 3 is

- –2
- 2
- 1
- –1

Ans. (a)

#### Question 9

## What is the greatest possible speed at which a girl can walk 95 m and 171 m in an exact number of minutes?

- 17 m/min
- 19 m/min
- 23 m/min
- 13 m/min

Ans: (b)

#### Question 10

## In figure, the graph of a polynomial P(x) is shown. The number of zeroes of P(x) is

- 1
- 2
- 3
- 4

Ans: (c)

#### Question 11

## Two lines are given to be parallel. The equation of one of the lines is 3x – 2y = 5. The equation of the second line can be

- 9x + 8y = 7
- –12x – 8y = 7
- –12x + 8y = 7
- 12x + 8y = 7

Ans: (c)

#### Question 12

## Three vertices of a parallelogram ABCD are A(1, 4), B(–2, 3) and C(5, 8). The ordinate of the fourth vertex D is

- 8
- 9
- 7
- 6

Ans: (b)

#### Question 13

## In ΔABC and ΔDEF, $\mathbf{\angle}\mathbf{F}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\angle}\mathbf{C}\mathbf{,}\mathbf{}\mathbf{\angle}\mathbf{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\angle}\mathbf{E}\mathbf{}\mathbf{and}\mathbf{}\mathbf{AB}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}$ DE. Then, the two triangles are

- Congruent, but not similar.
- Similar, but not congruent.
- Neither congruent nor similar.
- Congruent as well as similar.

Ans: (b)

#### Question 14

## In ΔABC right angled at B, sin A $\mathbf{=}\mathbf{}\frac{\mathbf{7}}{\mathbf{25}}$, then the value of cos C is

- $\frac{7}{25}$
- $\frac{24}{25}$
- $\frac{7}{24}$
- $\frac{24}{7}$

Ans: (a)

#### Question 15

## The minute hand of a clock is 84 cm long. The distance covered by the tip of the minute hand from 10:10 am to 10:25 am is

- 44 cm
- 88 cm
- 132 cm
- 176 cm

Ans: (c)

#### Question 16

## The probability that the drawn card from a pack of 52 cards is neither an ace nor a spade is

- $\frac{9}{13}$
- $\frac{35}{52}$
- $\frac{10}{13}$
- $\frac{19}{26}$

Ans: (a)

#### Question 17

## Three alarm clocks ring their alarms at regular intervals of 20 min, 25 min and 30 min respectively. If they first beep together at 12 noon, at what time will they beep again for the first time?

- 4 : 00 pm
- 4 : 30 pm
- 5 : 00 pm
- 5 : 30 pm

Ans: (c)

#### Question 18

## A quadratic polynomial, the product and sum of whose zeroes are 5 and 8 respectively is

- k [x
^{2}– 8x + 5] - k [x
^{2}+ 8x + 5] - k [x
^{2}– 5x + 8] - k [x
^{2}+ 5x + 8]

Ans: (a)

#### Question 19

## Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). The values of y are

- 1, –7
- –1, 7
- 2, 7
- –2, –7

Ans: (b)

#### Question 20

Ans: (a)

#### Question 21

## The greatest number which when divides 1251, 9377 and 15628 leaves remainder 1, 2 and 3 respectively is

- 575
- 450
- 750
- 625

Ans: (d)

#### Question 22

## Which of the following cannot be the probability of an event?

- 0.01
- 3%
- $\frac{16}{17}$
- $\frac{17}{16}$

Ans: (d)

#### Question 23

## The diameter of a car wheel is 42 cm. The number of complete revolutions it will make in moving 132 km is

- 10
^{4} - 10
^{5} - 10
^{6} - 10
^{3}

Ans: (b)

#### Question 24

## If θ is an acute angle and tan θ + cot θ = 2, then the value of sin^{3} θ + cos^{3} θ is

Ans: (c)

#### Question 25

## The ratio in which the line 3x + y – 9 = 0 divides the line segment joining the points (1, 3) and (2, 7) is

- 3 : 2
- 2 : 3
- 3 : 4
- 4 : 3

Ans: (c)

#### Question 26

## If x – 1 is a factor of the polynomial p(x) $\mathbf{=}\mathbf{}{\mathbf{x}}^{\mathbf{3}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{ax}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{b}\mathbf{}\mathbf{and}\mathbf{}\mathbf{a}\mathbf{}\mathbf{+}\mathbf{}\mathbf{b}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{,}$ then

- a = 5, b = –1
- a = 9, b = –5
- a = 7, b = –3
- a = 3, b = 1

Ans: (b)

#### Question 27

## If a and b are two coprime numbers, then a^{3} and b^{3} are

- Coprime
- Not coprime
- Even
- Odd

Ans: (a)

#### Question 28

## The area of a square that can be inscribed in a circle of area $\frac{\mathbf{1408}}{\mathbf{7}}$ cm^{2} is

- 321 cm
^{2} - 642 cm
^{2} - 128 cm
^{2} - 256 cm
^{2}

Ans: (c)

#### Question 29

## If A(4, –2), B(7, – 2) and C(7, 9) are the vertices of a ΔABC, then ΔABC is

- equilateral triangle
- isosceles triangle
- right angled triangle
- isosceles right angled triangle

Ans: (c)

#### Question 30

- –7
- 7
- -3
- 3

Ans: (b)

#### Question 31

## If *n* is a natural number, then 2(5^{n} + 6^{n}) always ends with

- 1
- 4
- 3
- 2

Ans: (d)

#### Question 32

## The line segment joining the points P(–3, 2) and Q(5, 7) is divided by the y-axis in the ratio

- 3 : 1
- 3 : 4
- 3 : 2
- 3 : 5

Ans: (d)

#### Question 33

## If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p^{2} – q^{2} =

- a
^{2}– b^{2} - b
^{2}– a^{2} - a
^{2}+ b^{2} - b – a

Ans: (b)

#### Question 34

## If the perimeter of a circle is half to that of a square, then the ratio of the area of the circle to the area of the square is

- 22 : 7
- 11 : 7
- 7 : 11
- 7 : 22

Ans: (d)

#### Question 35

## A dice is rolled twice. The probability that 5 will not come up either time is

- $\frac{11}{36}$
- $\frac{1}{3}$
- $\frac{13}{36}$
- $\frac{25}{36}$

Ans: (d)

Sol: A dice is rolled twice. Then possible outcomes are :

#### Question 36

## The LCM of two numbers is 2400. Which of the following CANNOT be their HCF?

- 300
- 400
- 500
- 600

Ans: (c)

#### Question 37

## In fig., PA, QB and RC are each perpendicular to AC. If x = 8 cm and z = 6 cm, then y is equal to

- $\frac{56}{7}$ cm
- $\frac{7}{56}$ cm
- $\frac{25}{7}$ cm
- $\frac{24}{7}$ cm

Ans: (d)

#### Question 38

## In a $\mathrm{\Delta ABC},\angle \mathrm{A}=\mathrm{x}\xb0,\angle \mathrm{B}=(3\mathrm{x}\u20132)\xb0,\angle \mathrm{C}=\mathrm{y}\xb0.\mathrm{Also}\angle \mathrm{C}\u2013\angle \mathrm{B}=9\xb0$

##
- 107°
- 135°
- 155°
- 145°

Ans: (a)

#### Question 39

## If sec θ + tan θ = p, then tan θ is

- $\frac{{\mathrm{p}}^{2}+1}{2\mathrm{p}}$
- $\frac{{\mathrm{p}}^{2}-1}{2\mathrm{p}}$
- $\frac{{\mathrm{p}}^{2}-1}{{\mathrm{p}}^{2}+1}$
- $\frac{{\mathrm{p}}^{2}+1}{{\mathrm{p}}^{2}-1}$

Ans: (b)

#### Question 40

## The base BC of an equilateral ΔABC lies on the y-axis. The coordinates of C are (0, –3). If the origin is the mid-point of the base BC, what are the coordinates of A and B?

Ans: (c)

Based on the above information, answer any four of the following questions :

Given information:

Let, fixed charge for 2 days = ₹x.

and additional charge (per day) = ₹y.

#### Question 41

## The situation of amount paid by Radhika, is algebraically represented by

- x – 4y = 16
- x + 4y = 16
- x – 2y = 16
- x + 2y = 16

Ans: (d)

The amount paid by Radhika, for keeping the book for 4 days = ₹16

$\Rightarrow $ Fixed charge for two days + additional charge (per day) for next 2 days = ₹16

$\Rightarrow $ x + 2y = 16

#### Question 42

## The situation of amount paid by Amruta, is algebraically represented by

- x – 2y = 11
- x – 2y = 11
- x + 4y = 22
- x – 4y = 11

Ans: (c)

Sol: The amount paid by Amruta, for keeping the book for 6 days = ₹22

$\Rightarrow $ Fixed charge for two days + additional charge (per day) for next 4 days = ₹22.

$\Rightarrow $ x + 4y = 22

#### Question 43

## What are the fixed charges for a book?

- ₹9
- ₹10
- ₹13
- ₹15

Ans: (b)

#### Question 44

## What are the additional charges for each subsequent day for a book?

- ₹6
- ₹5
- ₹4
- ₹3

Ans: (d)

Sol: The additional charges for each subsequent day for a book = y = ₹3 (As solved in Q. 43 equation (3).)

#### Question 45

## What is the total amount paid by both, if both of them have kept the book for 2 more days?

- ₹35
- ₹52
- ₹50
- ₹58

Ans: (c)

Sol: Total amount = Amount paid by Radhika for keeping the book for 6 days + Amruta for keeping the book for 8 days

= Radhika for keeping the book for 4 days + 2 days additional

+ Amruta keeping the book for 6 days + 2 days additional

= 16 + 2y + 22 + 2y

= 38 + 4y

= 38 + 4 × 3

= 38 + 12

= ₹50

#### Question 46

## The two triangular regions AOB and COD are

- Similar by AA criterion
- Similar by AA criterion
- Similar by RHS criterion
- Not similar

Ans: (a)

#### Question 47

## The ratio of the area of the ΔAOB to the area of ΔCOD, is

- 4 : 1
- 1 : 4
- 1 : 2
- 2 : 1

Ans: (b)

#### Question 48

## If the ratio of the perimeter of ΔAOB to the perimeter of ΔCOD would have been 1 : 4, then

- AB = 2 CD
- AB = 4 CD
- CD = 2 AB
- CD = 4 AB

Ans: (d)

#### Question 49

## If in Δs AOD and BOC, $\frac{\mathbf{A}\mathbf{O}}{\mathbf{B}\mathbf{C}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{A}\mathbf{D}}{\mathbf{B}\mathbf{O}}\mathbf{}\mathbf{=}\frac{\mathbf{O}\mathbf{D}}{\mathbf{O}\mathbf{C}}\mathbf{}$, then

- $\u2206\mathrm{AOD}~\u2206\mathrm{BOC}$
- $\u2206\mathrm{AOD}~\u2206\mathrm{BCO}$
- $\u2206\mathrm{ADO}~\u2206\mathrm{BCO}$
- $\u2206\mathrm{ODA}~\u2206\mathrm{BCO}$

Ans: (b)

#### Question 50

## If the ratio of areas of two similar triangles AOB and COD is 1 : 4, then which of the following statements is true?

- The ratio of their perimeters is 3 : 4.
- The corresponding altitudes have a ratio 1 : 2.
- The medians have a ratio 1 : 4.
- The angle bisectors have a ratio 1 : 16.

Ans: (b)