CBSE Class 10 Maths Basic 2023 Question Paper 430/5/3 with Solutions

The sum of the first 50 odd natural numbers is :

1. (A)5000
2. (B)2500
3. (C)2550
4. (D)5050

In the given figure, AD = 2 cm, DB = 3 cm, DE = 2.5 cm and DE $\parallel$ BC. The value of x is :

1. (A)6 cm
2. (B)3.75 cm
3. (C)6.25 cm
4. (D)7.5 cm

A circle is of radius 3 cm. The distance between two of its parallel tangents is :

1. (A)12 cm
2. (B)6 cm
3. (C)3 cm
4. (D)4.5 cm

The median class for the data given below is :
Class: 20 – 40, 40 – 60, 60 – 80, 80 – 100, 100 – 120
Frequency: 10, 12, 14, 13, 17

1. (A)80 – 100
2. (B)20 – 40
3. (C)40 – 60
4. (D)60 – 80

Mean and median of some data are 32 and 30 respectively. Using empirical relation, mode of the data is :

1. (A)36
2. (B)26
3. (C)30
4. (D)20

In two triangles $△PQR$ and $△ABC$, it is given that $\frac{AB}{BC}=\frac{PQ}{PR}$. For these two triangles to be similar, which of the following should be true ?

1. (A)$\angle A=\angle P$
2. (B)$\angle B=\angle Q$
3. (C)$\angle B=\angle P$
4. (D)CA = QR

If $\sin \theta =\frac{3}{4}$, then $\frac{({\sec }^2\theta -1){\cos }^2\theta }{\sin \theta }$ equals :

1. (A)$\frac{3}{5}$
2. (B)$\frac{3}{4}$
3. (C)$\frac{4}{3}$
4. (D)$\frac{9}{16}$

The $8^{th}$ term of an A.P. is 17 and its $14^{th}$ term is 29. The common difference of this A.P. is :

1. (A)3
2. (B)2
3. (C)5
4. (D)$-2$

In the given figure, O is the centre of the circle and PA is a tangent to the circle. If $\angle OAB=60^{\circ }$, then $\angle OPA$ is equal to :

1. (A)$60^{\circ }$
2. (B)$30^{\circ }$
3. (C)$15^{\circ }$
4. (D)$20^{\circ }$

One card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is a red king is :

1. (A)$\frac{1}{52}$
2. (B)$\frac{1}{26}$
3. (C)$\frac{2}{26}$
4. (D)$\frac{2}{13}$

If the lines represented by equations $3x+2my=2$ and $2x+5y+1=0$ are parallel, then the value of m is :

1. (A)$\frac{2}{5}$
2. (B)$-\frac{5}{4}$
3. (C)$\frac{3}{2}$
4. (D)$\frac{15}{4}$

$△ABC\sim △DEF$ and their perimeters are 32 cm and 24 cm respectively. If AB = 10 cm, then DE equals :

1. (A)8 cm
2. (B)7.5 cm
3. (C)15 cm
4. (D)$5\sqrt{3}$ cm

The two roots of the equation $3x^2-2\sqrt{6}x+2=0$ are :

1. (A)real and distinct
2. (B)not real
3. (C)real and equal
4. (D)rational

If $\sin \theta =\frac{a}{b}$, then $\sec \theta$ is equal to $(0\le \theta \le 90^{\circ })$ :

1. (A)$\frac{a}{\sqrt{b^2-a^2}}$
2. (B)$\frac{b}{\sqrt{b^2-a^2}}$
3. (C)$\frac{\sqrt{b^2-a^2}}{b}$
4. (D)$\frac{\sqrt{b^2-a^2}}{a}$

The distance between the points A(0, 6) and B($-$ 6, 2) is :

1. (A)6 units
2. (B)$2\sqrt{6}$ units
3. (C)$2\sqrt{13}$ units
4. (D)$13\sqrt{2}$ units

The value(s) of k for which the roots of quadratic equation $x^2+4x+k=0$ are real, is :

1. (A)$k\ge 4$
2. (B)$k\le 4$
3. (C)$k\ge -4$
4. (D)$k\le -4$

HCF of $(3^4\times 2^2\times 7^3)$ and $(3^2\times 5\times 7)$ is :

1. (A)630
2. (B)63
3. (C)729
4. (D)567

If one zero of the quadratic polynomial $k{x}^2+3x+k$ is 2, then the value of k is :

1. (A)$-\frac{6}{5}$
2. (B)$\frac{6}{5}$
3. (C)$\frac{5}{6}$
4. (D)$-\frac{5}{6}$

Assertion (A) : The surface area of largest sphere that can be inscribed in a hollow cube of side ‘a’ cm is $\pi a^2$ cm${}^2$.
Reason (R) : The surface area of a sphere of radius ‘r’ is $\frac{4}{3}\pi r^3$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Assertion (A) : When two coins are tossed together, the probability of getting no tail is $\frac{1}{4}$.
Reason (R) : The probability P(E) of an event E satisfies $0\le$ P(E) $\le 1$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

In the given figure, tangents AB and AC are drawn to a circle centred at O. If $\angle OAB=60^{\circ }$ and OB = 5 cm, find lengths OA and AC.

Find the sum of the first 20 terms of the A.P. : $-\frac{29}{3},-9,-\frac{25}{3},-\frac{23}{3},......$ .

Evaluate :
$\frac{\sin 30^{\circ }+\tan 45^{\circ }}{\sec 30^{\circ }+\cot 45^{\circ }}$

For A = $30^{\circ }$ and B = $60^{\circ }$, verify that :
$\sin (A+B)=\sin A\cos B+\cos A\sin B$.

Find LCM of 480 and 256 using prime factorization.

Show that A(1, 2), B(5, 4), C(3, 8) and D($-$1, 6) are vertices of a parallelogram ABCD.

Show that the points A(3, 0), B(6, 4) and C($-$1, 3) are vertices of a right-angled triangle.

Find mean of the following data :
Class: 0 – 15, 15 – 30, 30 – 45, 45 – 60, 60 – 75, 75 – 90
Frequency: 12, 15, 11, 20, 16, 6

Determine the ratio in which the point P(a, $-$ 2) divides the line segment joining the points A($-$ 4, 3) and B(2, $-$ 4). Also, find the value of ‘a’.

In the given figure, in $△ABC$ points D and E are mid-points of sides BC and AC respectively. If given vertices are A(4, $-$ 2), B(2, $-$ 2) and C($-$ 6, $-$ 7), then verify the result DE = $\frac{1}{2}$ AB.

ABC is an isosceles triangle with AB = AC, circumscribed about a circle. Prove that BC is bisected at E.

Prove that :
$\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{\sec \theta -1}}=2\text{ cosec }\theta$

Sabina went to a bank ATM to withdraw ₹ 2,000. She received ₹ 50 and ₹ 100 notes only. If Sabina got 25 notes in all, how many notes of ₹ 50 and ₹ 100 did she receive ?

Five years ago, Amit was thrice as old as Baljeet. Ten years hence, Amit shall be twice as old as Baljeet. What are their present ages ?

Prove that $11+3\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.

Divide 16 into two parts such that twice the square of the greater part, exceeds the square of the smaller part by 164.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream, than to return to the same point. Find the speed of the stream and total time of the journey.

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then prove that the other two sides are divided in the same ratio.

From the top of a building 50 m high, the angles of depression of the top and bottom of a tower are observed to be $30^{\circ }$ and $60^{\circ }$. Find the height of the tower and distance between the building and the tower. (Take $\sqrt{3}$ = 1.73)

In the given figure, AB is a chord of a circle of radius 7 cm and centred at O. Find the area of the shaded region if $\angle AOB=90^{\circ }$. Also, find length of minor arc AB.

AB and CD are arcs of two concentric circles of radii 3.5 cm and 10.5 cm respectively and centred at O. Find the area of the shaded region if $\angle AOB=60^{\circ }$. Also, find the length of arc CD.

Singing bowls (hemispherical in shape) are commonly used in sound healing practices. Mallet (cylindrical in shape) is used to strike the bowl in a sequence to produce sound and vibration.
One such bowl is shown here whose dimensions are :
Hemispherical bowl has outer radius 6 cm and inner radius 5 cm.
Mallet has height of 10 cm and radius 2 cm.
Based on the above, answer the following questions :
(i) What is the volume of the material used in making the mallet ? (1)
(ii) The bowl is to be polished from inside. Find the inner surface area of the bowl. (1)
(iii) (a) Find the volume of metal used to make the bowl. (2)
OR
(iii) (b) Find total surface area of the mallet. (Use $\pi =3.14$) (2)

Rainbow is an arch of colours that is visible in the sky after rain or when water droplets are present in the atmosphere. The colours of the rainbow are generally, red, orange, yellow, green, blue, indigo and violet. Each colour of the rainbow makes a parabola. We know that any quadratic polynomial $p(x)=a{x}^2+bx+c$ $(a\ne 0)$ represents a parabola on the graph paper.
Based on the above, answer the following questions :
(i) The graph of a rainbow $y=f(x)$ is shown in the figure. Write the number of zeroes of the curve. (1)
(ii) If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeroes will it have ? (1)
(iii) (a) If a rainbow is represented by the quadratic polynomial $p(x)=x^2+(a+1)x+b$, whose zeroes are 2 and $-$ 3, find the value of a and b. (2)
OR
(iii) (b) The polynomial $x^2-2x-(7p+3)$ represents a rainbow. If $-$ 4 is a zero of it, find the value of p. (2)

Some students were asked to list their favourite colour. The measure of each colour is shown by the central angle of a pie chart given below :
Study the pie chart and answer the following questions :
(i) If a student is chosen at random, then find the probability of his/her favourite colour being white ? (1)
(ii) What is the probability of his/her favourite colour being blue or green ? (1)
(iii) (a) If 15 students liked the colour yellow, how many students participated in the survey ? (2)
OR
(iii) (b) What is the probability of the favourite colour being red or blue ? (2)