CBSE Class 10 Maths Standard 2023 Question Paper 30/4/2 with Solutions

Which of the following is true for all values of $\theta$ $(0^{\circ }\le \theta \le 90^{\circ })$ ?

1. (A)${\cos }^2\theta -{\sin }^2\theta =1$
2. (B)${\mathrm{cosec}}^2\theta -{\sec }^2\theta =1$
3. (C)${\sec }^2\theta -{\tan }^2\theta =1$
4. (D)${\cot }^2\theta -{\tan }^2\theta =1$

If $k+2$, $4k-6$ and $3k-2$ are three consecutive terms of an A.P., then the value of k is :

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

In $△ABC$, PQ $\parallel$ BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.

1. (A)12 cm
2. (B)20 cm
3. (C)6 cm
4. (D)14 cm

The ratio of HCF to LCM of the least composite number and the least prime number is :

1. (A)1:2
2. (B)2:1
3. (C)1:1
4. (D)1:3

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 :

1. (A)$\frac{1}{13}$
2. (B)$\frac{9}{13}$
3. (C)$\frac{4}{13}$
4. (D)$\frac{12}{13}$

In the given figure, $△ABC\sim △QPR$. If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; then the value of x is :

1. (A)3.6 cm
2. (B)2.5 cm
3. (C)10 cm
4. (D)3.2 cm

The roots of the equation $x^2+3x-10=0$ are :

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

If a pole 6 m high casts a shadow $2\sqrt{3}$ m long on the ground, then sun’s elevation is :

1. (A)$60^{\circ }$
2. (B)$45^{\circ }$
3. (C)$30^{\circ }$
4. (D)$90^{\circ }$

The distance of the point $(-6,8)$ from origin is :

1. (A)6
2. (B)$-6$
3. (C)8
4. (D)10

What is the area of a semi-circle of diameter ‘d’ ?

1. (A)$\frac{1}{16}\pi d^2$
2. (B)$\frac{1}{4}\pi d^2$
3. (C)$\frac{1}{8}\pi d^2$
4. (D)$\frac{1}{2}\pi d^2$

For the following distribution :
Class: 0-5, 5-10, 10-15, 15-20, 20-25
Frequency: 10, 15, 12, 20, 9
The sum of lower limits of median class and modal class is :

1. (A)15
2. (B)25
3. (C)30
4. (D)35

The length of tangent drawn to a circle of radius 9 cm from a point 41 cm from the centre is :

1. (A)40 cm
2. (B)9 cm
3. (C)41 cm
4. (D)50 cm

In the given figure, O is the centre of the circle and PQ is the chord. If the tangent PR at P makes an angle of $50^{\circ }$ with PQ, then the measure of $\angle POQ$ is :

1. (A)$50^{\circ }$
2. (B)$40^{\circ }$
3. (C)$100^{\circ }$
4. (D)$130^{\circ }$

A bag contains 5 red balls and $n$ green balls. If the probability of drawing a green ball is three times that of a red ball, then the value of $n$ is :

1. (A)18
2. (B)15
3. (C)10
4. (D)20

If $\alpha$, $\beta$ are zeroes of the polynomial $x^2-1$, then value of $(\alpha +\beta )$ is :

1. (A)2
2. (B)1
3. (C)$-1$
4. (D)0

If $\alpha$, $\beta$ are the zeroes of the polynomial $p(x)=4x^2-3x-7$, then $\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)$ is equal to :

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

The pair of linear equations $2x=5y+6$ and $15y=6x-18$ represents two lines which are :

1. (A)intersecting
2. (B)parallel
3. (C)coincident
4. (D)either intersecting or parallel

The distance of the point $(-1,7)$ from x-axis is :

1. (A)$-1$
2. (B)7
3. (C)6
4. (D)$\sqrt{50}$

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 $n^2$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of 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 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 $\frac{5}{7}$.

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

Evaluate : $\frac{5}{{\cot }^{2}30^{\circ }}+\frac{1}{{\sin }^{2}60^{\circ }}-{\cot }^{2}45^{\circ }+2{\sin }^{2}90^{\circ }$

If $\theta$ is an acute angle and $\sin \theta =\cos \theta$, find the value of ${\tan }^2\theta +{\cot }^2\theta -2$.

If a fair coin is tossed twice, find the probability of getting ‘atmost one head’.

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

Find the sum and product of the roots of the quadratic equation $2x^2-9x+4=0$.

Find the discriminant of the quadratic equation $4x^2-5=0$ and hence comment on the nature of roots of the equation.

If one zero of the polynomial $p(x)=6x^2+37x-(k-2)$ is reciprocal of the other, then find the value of k.

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.

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.

Find the value of ‘p’ for which the quadratic equation $px(x-2)+6=0$ has two equal real roots.

The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its $20^{th}$ term.

Rohan repays his total loan of ₹ 1,18,000 by paying every month starting with the first instalment of ₹ 1,000. If he increases the instalment by ₹ 100 every month, what amount will be paid by him in the $30^{th}$ instalment ? What amount of loan has he paid after $30^{th}$ instalment ?

Prove that $\sqrt{3}$ is an irrational number.

Prove that $\frac{\sin A-2{\sin }^{3}A}{2{\cos }^{3}A-\cos A}=\tan A$

Prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$.

From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hallowed out. Find the total surface area of the remaining solid.

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.

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 $30^{\circ }$ and $60^{\circ }$, 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 $\sqrt{3}=1.73$)

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^{\circ }$ and the angle of depression of its foot is $30^{\circ }$. Determine the height of the tower.

In the given figure, $\angle ADC=\angle BCA$ ; prove that $△ACB\sim △ADC$. Hence find BD if AC = 8 cm and AD = 3 cm.

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

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 ? (1)
(ii) (a) What is the area of square PQRS ? (2)
OR (b) What is the length of diagonal PR in square PQRS ? (2)
(iii) If S divides CA in the ratio K:1, what is the value of K, where point A is $(200,800)$ ? (1)

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 ? (1)
(ii) (a) What is the total area of parking and the two quadrants ? (2)
OR (b) What is the ratio of area of playground to the area of parking area ? (2)
(iii) Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit. (1)

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 $y$). (1)
(ii) (a) What is the prize amount for hockey ? (2)
OR (b) Prize amount on which game is more and by how much ? (2)
(iii) What will be the total prize amount if there are 2 students each from two games ? (1)