Board Paper of Class 10 2023 Maths (Basic) Delhi(Set 2) - 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
Let E be an event such that P(not E) = $\frac{1}{5},$ then P(E) is equal to:
(a) $\frac{1}{5}$
(b) $\frac{2}{5}$
(c) 0
(d) $\frac{4}{5}$ VIEW SOLUTION

Question 2
If p(x) = x² + 5x + 6, then p(−2) is :
(a) 20
(b) 0
(c) −8
(d) 8 VIEW SOLUTION

Question 3
The mode of the numbers 2, 3, 3, 4, 5, 4, 4, 5, 3, 4, 2, 6, 7 is:
(a) 2
(b) 3
(c) 4
(d) 5 VIEW SOLUTION

Question 4
How many tangents can be drawn to a circle from a point on it?
(a) One
(b) Two
(c) Infinite
(d) Zero VIEW SOLUTION

Question 5
A quadratic equation whose one root is 2 and the sum of whose roots is zero, is:
(a) $x^{2} + 4 = 0$
(b) $x^{2} - 2 = 0$
(c) $4 x^{2} - 1 = 0$
(d) $x^{2} - 4 = 0$ VIEW SOLUTION

Question 6
Which of the following is not a quadratic equation?
(a) $2 {(x - 1)}^{2} = 4 x^{2} - 2 x + 1$
(b) $2 x - x^{2} = x^{2} + 5$
(c) ${(\sqrt{2} x + \sqrt{3})}^{2} + x^{2} = 3 x^{2} - 5 x$
(d) ${(x^{2} + 2 x)}^{2} = x^{4} + 3 + 4 x^{3}$ VIEW SOLUTION

Question 7
A quadratic polynomial whose sum and product of zeroes are 2 and −1 respectively is:
(a) x² + 2x +1
(b) x² − 2x − 1
(c) x² + 2x − 1
(d) x² − 2x + 1 VIEW SOLUTION

Question 8
(HCF × LCM) for the numbers 30 and 70 is:
(a) 2100
(b) 21
(c) 210
(d) 70 VIEW SOLUTION

Question 9
The length of the arc of a circle of radius 14 cm which subtends an angle of 60^∘ at the centre of the circle is:
(a) $\frac{44}{3} cm$
(b) $\frac{88}{3} cm$
(c) $\frac{308}{3} cm$
(d) $\frac{616}{3} cm$ VIEW SOLUTION

Question 10
If the radius of a semi-circular protractor is 7 cm, then its perimeter is :
(a) 11 cm
(b) 14 cm
(c) 22 cm
(d) 36 cm VIEW SOLUTION

Question 11
The angle of elevation of the top of a 15 m high tower at a point $15 \sqrt{3} m$ away from the base of the tower is
(a) 30°
(b) 45°
(c) 60°
(d) 90° VIEW SOLUTION

Question 12
$(\frac{2}{3} \sin 0 ° - \frac{4}{5} \cos 0 °)$ is equal to:
(a) $\frac{2}{3}$
(b) $\frac{- 4}{5}$
(c) 0
(d) $\frac{- 2}{15}$ VIEW SOLUTION

Question 13
From a well-shuffled deck of 52 cards, a card is drawn at random. What is the probability of getting king of hearts?
(a) $\frac{1}{52}$
(b) $\frac{1}{26}$
(c) $\frac{1}{13}$
(d) $\frac{12}{13}$ VIEW SOLUTION

Question 14
The number $(5 - 3 \sqrt{5} + \sqrt{5})$ is :
(a) an integer
(b) a rational number
(c) an irrational number
(d) a whole number VIEW SOLUTION

Question 15
If the pair of linear equations $x - y = 1, x + k y = 5$ has a unique solution x = 2, y = 1, then the value of k is:
(a) −2
(b) −3
(c) 3
(d) 4 VIEW SOLUTION

Question 16
If ∆ABC ∼ ∆DEF and $∠ A = 47 °, ∠ E = 83 °$ then $∠ C$ is equal :
(a) 47^∘
(b) 50^∘
(c) 83^∘
(d) 130^∘ VIEW SOLUTION

Question 17
The length of the tangent from an external point A to a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is:
(a) 7 cm
(b) 5 cm
(c) $\sqrt{7}$ cm
(d) 25 cm VIEW SOLUTION

Question 18
The pair of linear equations x + 2y + 5 = 0 and −3x − 6y + 1 = 0 has :
(a) a unique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution VIEW SOLUTION

Question 19
Assertion (A): If one root of the quadratic equation 4x² − 10x + (k − 4) = 0 is reciprocal of the other, then value of k is 8.
Reason (R): Roots of the quadratic equation x² − x + 1 = 0 are real.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of 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 tangent to a circle is perpendicular to the radius through the point of contact.
Reason (R): The lengths of tangents drawn from an external point to a circle are equal.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) gives the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) does not give the correct explanation of 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
Find the discriminant of the quadratic equation 3x²−2x + $\frac{1}{3}$ = 0 and hence find the nature of its roots.
OR

Find the roots of the quadratic equation x² − x − 2 = 0. VIEW SOLUTION

Question 22
In the adjoining figure, A, B and C are points on OP, OQ and OR respectively such that AB||PQ and AC||PR. Show that BC||QR.
VIEW SOLUTION

Question 23
If $\sin α = \frac{1}{2},$ then find the value of $(3 \cos α - 4 \cos^{3} α) .$ VIEW SOLUTION

Question 24
Find the coordinates of the point which divides the join of A(−1, 7) and B(4, −3) in the ratio 2 : 3.
OR

If the points A (2, 3), B (−5, 6), C (6, 7) and D (p, 4) are the vertices of a parallelogram ABCD, find the value of p. VIEW SOLUTION

Question 25
PA and PB are tangents drawn to the circle with centre O as shown in the figure. Prove that ∠APB = 2 ∠OAB.
VIEW SOLUTION

Question 26
Find the area of the sector of a circle of radius 7 cm and of central angle 90°. Also, find the area of corresponding major sector. VIEW SOLUTION

Question 27
If α, β are zeroes of the quadratic polynomial x² − 5x + 6, form another quadratic polynomial whose zeroes are $\frac{1}{α}, \frac{1}{β} .$ VIEW SOLUTION

Question 28
A die is rolled once. Find the probability of getting:
(i) an even prime number.
(ii) a number greater than 4.
(iii) an odd number. VIEW SOLUTION

Question 29
$Prove that \frac{1 + \tan^{2} A}{1 + \cot^{2} A} = \sec^{2} A - 1$ VIEW SOLUTION

Question 30
Prove that the lengths of tangents drawn from an external point to a circle are equal.
OR

Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P.
VIEW SOLUTION

Question 31
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes $\frac{1}{2}$ if we only add 1 to the denominator. What is the fraction?
OR

For which value of 'k' will the following pair of linear equations have no solution ?
3x + y = 1
(2k – 1) x + (k – 1) y = 2k + 1 VIEW SOLUTION

Question 32
Find the sum of first 51 terms of an A.P. whose second and third terms are 14 and 18, respectively.
OR

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. VIEW SOLUTION

Question 33
The distribution below gives the weights of 30 students of a class. Find the median weight of the students:

Weight in kg 40-45 45-50 50-55 55-60 60-65 65-70 70-75

Number of Students 2 3 8 6 6 3 2

VIEW SOLUTION

Question 34
The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends. Length of the cylindrical part is 7m and radius of cylindrical part is $\frac{7}{2} m .$
Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.
VIEW SOLUTION

Question 35
The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30º than when it was 60º. Find the height of the tower.
OR

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. VIEW SOLUTION

Question 36
Observe the figures given below carefully and answer the questions:
Figure A

Figure B

Figure C

(i) Name the figure(s) wherein two figures are similar.

(ii) Name the figure(s) wherein the figures are congruent.

(iii) (a) Prove that congruent triangles are also similar but not the converse.
OR

(b) What more is least needed for two similar triangles to be congruent?
VIEW SOLUTION

Question 37
Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle. One such campaign board made by class X student of the school is shown in the figure.

Based on the above information, answer the following questions:

(i) Find the coordinates of the point of intersection of diagonals AC and BD.

(ii) Find the length of the diagonal AC.

(iii) (a) Find the area of the campaign Board ABCD.
OR

(b) Find the ratio of the length of side AB to the length of the diagonal AC.
VIEW SOLUTION

Question 38
Khushi wants to organize her birthday party. Being health conscious, she decided to serve only fruits in her birthday party. She bought 36 apples and 60 bananas and decided to distribute fruits equally among all.

Based on the above information, answer the following questions:

(i) How many guests Khushi can invite at the most?

(ii) How many apples and bananas will each guest get ?

(iii) (a) If Khushi decides to add 42 mangoes, how many guests Khushi can invite at the most ?
OR

(b) If the cost of l dozen of bananas is ₹60, the cost of 1 apple is ₹15 and cost of 1 mango is ₹20, find the total amount spent on 60 bananas, 36 apples and 42 mangoes.
VIEW SOLUTION

More Board Paper Solutions for Class 10 Maths

Board Paper Solutions for Other Subjects

Weight in kg	40-45	45-50	50-55	55-60	60-65	65-70	70-75
Number of Students	2	3	8	6	6	3	2