Board Paper of Class 10 2023 Maths (Standard) Delhi(Set 1) - Solutions

General Instructions
Read the following instructions carefully and follow them:
1. This question paper contains 38 questions. All questions are compulsory.
2. Question paper is divided into FIVE sections Section A, B, C, D and E.
3. In section A, question number 1 to 18 are multiple choice questions (MCQs) and question number 19 and 20 are Assertion - Reason based questions of 1 mark each.
4. In section B, question number 21 to 25 are very short answer (VSA) type questions of 2 marks each.
5 in section C, question number 26 to 31 are short answer (SA) type questions carrying 3 marks each.
6. In section D, question number 32 to 35 are long answer (LA) type questions carrying 5 marks each.
7. In section E, question number 36 to 38 are case based integrated units of assessment questions carrying 4 marks each. Internal choice is provided in 2 marks question in each case study.
8. There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 2 questions in Section C, 2 questions in Section D and 3 questions in Section E.
9. Draw neat figures wherever required. Take π = 22/7 wherever required if not stated.
10. Use of calculators is not allowed.

Question 1
The ratio of HCF to LCM of the least composite number and the least prime number is :
(a) 1 : 2
(b) 2 : 1
(c) 1 : 1
(d) 1 : 3 VIEW SOLUTION

Question 2
The roots of the equation x² + 3x − 10 = 0 are :
(a) 2, −5
(b) −2, 5
(c) 2, 5
(d) −2, −5 VIEW SOLUTION

Question 3
The next term of the A.P. : $\sqrt{6}, \sqrt{24}, \sqrt{54}$ is :
(a) $\sqrt{60}$
(b) $\sqrt{96}$
(c) $\sqrt{72}$
(d) $\sqrt{216}$ VIEW SOLUTION

Question 4
The distance of the point (−1, 7) from x-axis is :
(a) −1
(b) 7
(c) 6
(d) $\sqrt{50}$ VIEW SOLUTION

Question 5
What is the area of a semi-circle of diameter 'd'?
(a) $\frac{1}{16} {πd}^{2}$
(b) $\frac{1}{4} {πd}^{2}$
(c) $\frac{1}{8} {πd}^{2}$
(d) $\frac{1}{2} {πd}^{2}$ VIEW SOLUTION

Question 6
The empirical relation between the mode, median and mean of a distribution is :
(a) Mode = 3 Median − 2 Mean
(b) Mode = 3 Mean − 2 Median
(c) Mode = 2 Median − 3 Mean
(d) Mode = 2 Mean − 3 Median VIEW SOLUTION

Question 7
The pair of linear equations 2x = 5y + 6 and 15y = 6x − 18 represents two lines which are :
(a) intersecting
(b) parallel
(c) coincident
(d) either intersecting or parallel VIEW SOLUTION

Question 8
If α, β are zeroes of the polynomial x² − 1, then value of (α + β) is :
(a) 2
(b) 1
(c) −1
(d) 0 VIEW SOLUTION

Question 9
If a pole 6 m high casts a shadow $2 \sqrt{3} m$ long on the ground, then sun's elevation is :
(a) 60^∘
(b) 45^∘
(c) 30^∘
(d) 90^∘ VIEW SOLUTION

Question 10
Sec θ when expressed in terms of cot θ, is equal to :
(a) $\frac{1 + \cot^{2} θ}{\cot θ}$
(b) $\sqrt{1 + \cot^{2} θ}$
(c) $\sqrt{\frac{1 + \cot^{2} θ}{\cot θ}}$
(d) $\sqrt{\frac{1 - \cot^{2} θ}{\cot θ}}$ VIEW SOLUTION

Question 11
Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equals to 3 is:
(a) $\frac{1}{9}$
(b) $\frac{2}{9}$
(c) $\frac{1}{6}$
(d) $\frac{1}{12}$ VIEW SOLUTION

Question 12

In the given figure, ∆ABC ∼ ∆QPR. If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; then the value of x is:
(a) 3.6 cm
(b) 2.5 cm
(c) 10 cm
(d) 3.2 cm VIEW SOLUTION

Question 13
The distance of the point (−6, 8) from origin is:
(a) 6
(b) −6
(c) 8
(d) 10 VIEW SOLUTION

Question 14
In the given figure, PQ is a tangent to the circle with centre O. If ∠OPQ = x, ∠POQ = y, then x + y is:

(a) 45^∘
(b) 90^∘
(c) 60^∘
(d) 180^∘ VIEW SOLUTION

Question 15
In the given figure, TA is a tangent to the circle with centre O such that OT = 4 cm, ∠OTA = 30^∘, then length of TA is:

(a) $2 \sqrt{3} cm$
(b) 2 cm
(c) $2 \sqrt{2} cm$
(d) $\sqrt{3} cm$ VIEW SOLUTION

Question 16
In ∆ABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.

(a) 12 cm
(b) 20 cm
(c) 6 cm
(d) 14 cm VIEW SOLUTION

Question 17
If α, β are the zeroes of the polynomial p(x) = $4 x^{2} - 3 x - 7,$ then $(\frac{1}{α} + \frac{1}{β})$ is equal to:
(a) $\frac{7}{3}$
(b) $\frac{- 7}{3}$
(c) $\frac{3}{7}$
(d) $\frac{- 3}{7}$ VIEW SOLUTION

Question 18
A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not an ace is:
(a) $\frac{1}{13}$
(b) $\frac{9}{13}$
(c) $\frac{4}{13}$
(d) $\frac{12}{13}$ VIEW SOLUTION

Question 19
Assertion (A) : The probability that a leap year has 53 Sundays is $\frac{2}{7}$ .
Reason (R) : The probability that a non-leap year has 53 Sundays is $\frac{5}{7}$ .
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true. VIEW SOLUTION

Question 20
Assertion (A) : a, b, c are in A.P. if and only if 2b = a + c.
Reason (R) : The sum of first n odd natural numbers is n².
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true. VIEW SOLUTION

Question 21
Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers? VIEW SOLUTION

Question 22
If one zero of the polynomial p(x) = 6x² + 37x − (k − 2) is reciprocal of the other, then find the value of k. VIEW SOLUTION

Question 23
Find the sum and product of the roots of the quadratic equation 2x² − 9x + 4 = 0.
OR

Find the discriminant of the quadratic equation 4x² − 5 = 0 and hence comment on the nature of roots of the equation. VIEW SOLUTION

Question 24
If a fair coin is tossed twice, find the probability of getting 'atmost one head'. VIEW SOLUTION

Question 25
Evaluate $\frac{5 \cos^{2} 60 ° + 4 \sec^{2} 30 ° - \tan^{2} 45 °}{\sin^{2} 30 ° + \cos^{2} 30 °}$
OR

If A and B are acute angles such that sin (A − B) = 0 and 2 cos (A + B) − 1 = 0, then find angles A and B. VIEW SOLUTION

Question 26
Which terms of the A.P. : 65, 61, 57, 53, ...................... is the first negative term?
OR

How many terms are there in an A.P. whose first and fifth terms are −14 and 2, respectively and the last term is 62. VIEW SOLUTION

Question 27
Prove that $\sqrt{5}$ is an irrational number. VIEW SOLUTION

Question 28
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre. VIEW SOLUTION

Question 29
$Prove that \frac{sinA - 2 \sin^{3} A}{2 \cos^{3} A - cosA} = tanA$
OR

Prove that sec A(1 – sin A)(sec A + tan A) = 1. VIEW SOLUTION

Question 30
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. VIEW SOLUTION

Question 31
Find the value of 'p' for which the quadratic equation px(x – 2) + 6 = 0 has two equal real roots. VIEW SOLUTION

Question 32
A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30º and 60º, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use $\sqrt{3}$ = 1.73)
OR

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60º and the angle of depression of its foot is 30º. Determine the height of the tower. VIEW SOLUTION

Question 33
D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC, prove that CA² = CB.CD
OR

If AD and PM are medians of triangles ABC and PQR, respectively when ∆ABC ~ ∆PQR, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$ . VIEW SOLUTION

Question 34
A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its total length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model. VIEW SOLUTION

Question 35

The monthly expenditure on milk in 200 families of a Housing Society is given below:

Monthly Expenditure (in ₹)	1000-1500	1500-2000	2000-2500	2500-3000	3000-3500	3500-4000	4000-4500	4500-5000
Number of families	24	40	33	x	30	22	16	7

Find the value of x and also, find the median and mean expenditure on milk. VIEW SOLUTION

Question 36
Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹x per student and Cricket ₹y per student. School 'P' decided to award a total of ₹9,500 for the two games to 5 and 4 students respectively; while school 'Q' decided to award ₹7,370 for the two games to 4 and 3 students respectively.

Based on the above information, answer the following questions:
(i) Represent the following information algebraically (in terms of x and y).
(ii) (a) What is the prize amount for hockey?
OR

(b) Prize amount on which game is more and by how much?
(iii) What will be the total prize amount if there are 2 students each from two games? VIEW SOLUTION

Question 37
Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.

Based on the above information, answer the following questions:

(i) Taking O as origin, coordinates of P are (–200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S?

(ii) (a) What is the area of square PQRS?
OR

(b) What is the length of diagonal PR in square PQRS?

(iii) If S divides CA in the ratio K : 1, what is the value of K, where point A is (200, 800)? VIEW SOLUTION

Question 38
Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a
hill, which will have adequate space for parking.

After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.

Based on the above information, answer the following questions:

(i) What is the total perimeter of the parking area?

(ii) (a) What is the total area of parking and the two quadrants?
OR

(b) What is the ratio of area of playground to the area of parking area?

(iii) Find the cost of fencing the playground and parking area at the rate of ₹2 per unit. VIEW SOLUTION

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