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CBSE(NCERT) · Grade 10 · Maths

CBSE(NCERT) GRADE 10 MATHS 2023 SET4

50 questions from this Grade 10 Maths paper. Log in as a Grade 10 student to view solutions.

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Grade 10 / Maths
Q1 mcq 1 mark
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

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Q2 mcq 1 mark
The roots of the equation $x^2 + 3x - 10 = 0$ are :
  • A. 2, –5
  • B. –2, 5
  • C. 2, 5
  • D. –2, –5

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Q3 mcq 1 mark
The next term of the A.P. : 6, 24, 54 is :
  • A. 60
  • B. 96
  • C. 72
  • D. 216

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Q4 mcq 1 mark
The distance of the point (–1, 7) from x-axis is :
  • A. –1
  • B. 7
  • C. 6
  • D. 50

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Q5 mcq 1 mark
What is the area of a semi-circle of diameter 'd' ?
  • A. $\frac{1}{16} \pi d^2$
  • B. $\frac{1}{4} \pi d^2$
  • C. $\frac{1}{8} \pi d^2$
  • D. $\frac{1}{2} \pi d^2$

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Q6 mcq 1 mark
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

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Q7 mcq 1 mark
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

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Q8 mcq 1 mark
If α, β are zeroes of the polynomial $x^2 – 1$, then value of (α + β) is :
  • A. 2
  • B. 1
  • C. –1

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Q9 mcq 1 mark
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°

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Q10 mcq 1 mark
sec θ when expressed in terms of cot θ, is equal to :
  • A. $\frac{1+\cot^2\theta}{\cot\theta}$
  • B. $\frac{1}{\sqrt{1+\cot^2\theta}}$
  • C. $\frac{\sqrt{1+\cot^2\theta}}{\cot\theta}$
  • D. $\frac{1-\cot^2\theta}{\cot\theta}$

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Q11 mcq 1 mark
Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equals to 3 is :
  • A. 1/9
  • B. 2/9
  • C. 1/6
  • D. 1/12

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Q12 mcq 1 mark
In the given figure, PABC ~ PQPR. 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

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Q14 mcq 1 mark
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°

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Q15 mcq 1 mark
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 3 cm
  • B. 2 cm
  • C. 2 2 cm
  • D. 3 cm

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Q16 mcq 1 mark
In Δ 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

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Q17 mcq 1 mark
If α, β are the zeroes of the polynomial p(x) = 4x^2 – 3x – 7, then (1/α + 1/β) is equal to :
  • A. 7/3
  • B. -7/3
  • C. 3/7
  • D. -3/7

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Q18 mcq 1 mark
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. 1/13
  • B. 9/13
  • C. 4/13
  • D. 12/13

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Q19 mcq 1 mark
Assertion (A) : The probability that a leap year has 53 Sundays is 2/7. Reason (R) : The probability that a non-leap year has 53 Sundays is 5/7.
  • A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • B. Both Assertion (A) and Reason (R) are true and Reason (R) is not 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.

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Q20 mcq 1 mark
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.
  • A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • B. Both Assertion (A) and Reason (R) are true and Reason (R) is not 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.

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Q20 short answer 2 marks
Find the sum and product of the roots of the quadratic equation $2x^2 - 9x + 4 = 0$.

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Q21 short answer 2 marks
Find the discriminant of the quadratic equation $4x^2 - 5 = 0$ and hence comment on the nature of roots of the equation.

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Q23 short answer 2 marks
Evaluate $\frac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

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Q24 short answer 2 marks
If a fair coin is tossed twice, find the probability of getting ‘atmost one head’.

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Q24 short answer 2 marks
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.

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Q25 short answer 3 marks
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.

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Q26 short answer 3 marks
Which term of the A.P. : 65, 61, 57, 53, .................. is the first negative term ?

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Q27 short answer 3 marks
Prove that $\sqrt{5}$ is an irrational number.

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Q28 short answer 3 marks
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.

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Q29 short answer 3 marks
Prove that $\frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} = \tan A$

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Q30 short answer 3 marks
Prove that $\sec A (1 - \sin A) (\sec A + \tan A) = 1$

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Q30 long answer 3 marks
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.

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Q31 long answer 3 marks
Find the value of ‘p’ for which the quadratic equation $px(x - 2) + 6 = 0$ has two equal real roots.

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Q33 long answer 5 marks
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 $\sqrt{3} = 1.73$)

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Q34 long answer 5 marks
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.

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Q34 long answer 5 marks
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.

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Q35 long answer 5 marks
D is a point on the side BC of a triangle ABC such that $\angle ADC = \angle BAC$, prove that $CA^2 = CB.CD$.

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Q35 long answer 5 marks
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.

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Q36 long answer 5 marks
If AD and PM are medians of triangles ABC and PQR, respectively where $\triangle ABC \sim \triangle PQR$, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$.

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Reading Passage

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.

Q39 short answer 1 mark
Represent the following information algebraically (in terms of x and y).

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Q40 short answer 2 marks
What is the prize amount for hockey?

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Reading Passage

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 :

Q41 short answer 2 marks
Prize amount on which game is more and by how much ?

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Q42 short answer 1 mark
What will be the total prize amount if there are 2 students each from two games ?

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Reading Passage

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 :

Q43 short answer 1 mark
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?

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Q44 short answer 2 marks
What is the area of square PQRS?

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Q45 short answer 2 marks
What is the length of diagonal PR in square PQRS?

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Q46 short answer 1 mark
If S divides CA in the ratio K:1, what is the value of K, where point A is (200, 800)?

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Reading Passage

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 :

Q47 short answer 1 mark
What is the total perimeter of the parking area?

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Q48 short answer 2 marks
What is the total area of parking and the two quadrants?

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Q49 short answer 2 marks
What is the ratio of area of playground to the area of parking area?

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Q50 short answer 1 mark
Find the cost of fencing the playground and parking area at the rate of ` 2 per unit.

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