CBSE Class 10 Maths Basic 2023 Question Paper 430/5/2 with Solutions

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 $\cos \theta =\frac{3}{7}$, then $\frac{\cos \theta }{1-{\sin }^2\theta }$ is equal to :

1. (A)$\frac{3}{\sqrt{40}}$
2. (B)$\frac{3}{7}$
3. (C)$\frac{7}{3}$
4. (D)$\frac{7}{\sqrt{40}}$

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

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

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

In the given figure, ABC is a triangle in which AD = 1.6 cm, BD = 4.8 cm, AE = 1.1 cm and EC = 2.2 cm. Then :

1. (A)DE $\parallel$ BC
2. (B)DE = $\frac{1}{2}$BC
3. (C)DE = BC
4. (D)DE is not parallel to BC (DE $\nparallel$ BC)

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 a circle, PQ is a chord and the tangent PT at P makes an angle of $50^{\circ }$ with PQ. The measure of $\angle POQ$ is :

1. (A)$130^{\circ }$
2. (B)$100^{\circ }$
3. (C)$90^{\circ }$
4. (D)$75^{\circ }$

Two dice are rolled together. The probability that the sum of the numbers that appeared is 9, is :

1. (A)$\frac{5}{36}$
2. (B)$\frac{1}{9}$
3. (C)$\frac{1}{12}$
4. (D)$\frac{1}{6}$

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)$-\frac{5}{6}$

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

1. (A)40
2. (B)560
3. (C)1680
4. (D)1120

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

1. (A)$6\sqrt{2}$ units
2. (B)12 units
3. (C)$2\sqrt{6}$ units
4. (D)6 units

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 sum of the first 100 even natural numbers is :

1. (A)10100
2. (B)2550
3. (C)5050
4. (D)10010

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)$k\le -4$

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.

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.

Find the sum of the first 20 terms of the A.P. : $\frac{2}{3},0,-\frac{2}{3},-\frac{4}{3},......$ .

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.

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 HCF of 660 and 704 by prime factorization.

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.

Prove that :
$\frac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }+\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }=\frac{2{\sec }^2\theta }{{\tan }^2\theta -1}$

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

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 $8+5\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

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.

A 2-digit number is four times the sum of its digits and twice the product of its digits. Find the number.

The length of the rectangle exceeds its breadth by 8 cm and the area of the rectangle is 240 cm${}^2$. Find the dimensions of the rectangle.

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.

A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of lighthouse from $60^{\circ }$ to $45^{\circ }$. Find the speed of the boat. (Use $\sqrt{3}$ = 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)

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)

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)