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CBSE Class 10 Mathematics Question Paper 2017 (All India Set 1) with Solutions

The complete CBSE Class 10 Mathematics board paper of 2017, Q.P. code All India Set 1: all 31 questions exactly as set, with marks. 16 solutions are free below. The Class 10 Maths PYQ Book: Solved Board Papers 2016-2026 (CBSE) solves all 17 papers in full and sorts every question chapter-wise.

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CBSE Board Paper

2017 · Q.P. Code All India Set 1

Time: 3 hours
Maximum marks: 90

Section A · 1 mark each

What is the common difference of an A.P. in which $a_{21} - a_{7} = 84$ ?

If the angle between two tangents drawn from an external point P to a circle of radius $a$ and centre O, is $6 0^{\circ}$ , then find the length of OP.

If a tower 30 m high, casts a shadow $103$ m long on the ground, then what is the angle of elevation of the sun?

The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

Section B · 2 marks each

Find the value of $p$ , for which one root of the quadratic equation $p x^{2} - 14 x + 8 = 0$ is 6 times the other.

Which term of the progression $20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \dots$ is the first negative term?

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

A circle touches all the four sides of a quadrilateral ABCD. Prove that $A B + C D = B C + D A$ .

A line intersects the y-axis and x-axis at the points P and Q respectively. If $(2, - 5)$ is the mid-point of PQ, then find the coordinates of P and Q.

10.

If the distances of $P (x, y)$ from $A (5, 1)$ and $B (- 1, 5)$ are equal, then prove that $3 x = 2 y$ .

Section C · 3 marks each

11.

If $a d \neq = b c$ , then prove that the equation $(a^{2} + b^{2}) x^{2} + 2 (a c + b d) x + (c^{2} + d^{2}) = 0$ has no real roots.

12.

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

13.

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

14.

A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

15.

In what ratio does the point $(\frac{24}{11}, y)$ divide the line segment joining the points $P (2, - 2)$ and $Q (3, 7)$ ? Also find the value of $y$ .

16.

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region.

17.

In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If $∠ A O B = 6 0^{\circ}$ , find the area of the shaded region. [Use $π = \frac{22}{7}$ ]

18.

Water in a canal, 5.4 m wide and 1.8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation?

not in 2026-27 syllabus

19.

The slant height of a frustum of a cone is 4 cm and the perimeters of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

not in 2026-27 syllabus

20.

The dimensions of a solid iron cuboid are 4.4 m × 2.6 m × 1.0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe.

not in 2026-27 syllabus

Section D · 4 marks each

21.

Solve for $x$ : $\frac{1}{x + 1} + \frac{3}{5 x + 1} = \frac{5}{x + 4}$ , $x \neq = - 1, - \frac{1}{5}, - 4$

22.

Two taps running together can fill a tank in $3 \frac{1}{13}$ hours. If one tap takes 3 hours more than the other to fill the tank, then how much time will each tap take to fill the tank?

23.

If the ratio of the sum of the first $n$ terms of two A.Ps is $(7 n + 1) : (4 n + 27)$ , then find the ratio of their $9^{th}$ terms.

24.

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

25.

In the given figure, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and X′Y′ at B. Prove that $∠ A O B = 9 0^{\circ}$ .

26.

Construct a triangle ABC with side $B C = 7$ cm, $∠ B = 4 5^{\circ}$ , $∠ A = 10 5^{\circ}$ . Then construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of the $△ A B C$ .

not in 2026-27 syllabus

27.

An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are $4 5^{\circ}$ and $3 0^{\circ}$ respectively. Find the width of the river. [Use $3 = 1.732$ ]

28.

If the points $A (k + 1, 2 k)$ , $B (3 k, 2 k + 3)$ and $C (5 k - 1, 5 k)$ are collinear, then find the value of $k$ .

not in 2026-27 syllabus

29.

Two different dice are thrown together. Find the probability that the numbers obtained have (i) even sum, and (ii) even product.

30.

In the given figure, ABCD is a rectangle of dimensions 21 cm × 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.

31.

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

not in 2026-27 syllabus

Solutions: CBSE 2017 · All India Set 1

$a_{21} - a_{7} = (a + 20 d) - (a + 6 d) = 14 d = 84 \Rightarrow d = 6$ .

Solution in the PYQ book (₹99).

$tan θ = \frac{30}{10 3} = 3 \Rightarrow θ = 6 0^{\circ}$ .

Solution in the PYQ book (₹99).

Let the roots be $α$ and $6 α$ .

Sum: $7 α = \frac{14}{p} \Rightarrow α = \frac{2}{p}$ . Product: $6 α^{2} = \frac{8}{p} \Rightarrow 6 \cdot \frac{4}{p ^{2}} = \frac{8}{p} \Rightarrow p = 3$ .

(Check: $3 x^{2} - 14 x + 8 = (3 x - 2) (x - 4)$ , roots $\frac{2}{3}$ and $4 = 6 \times \frac{2}{3}$ ✓)

Solution in the PYQ book (₹99).

Let AB be a chord of a circle with centre O, and let the tangents at A and B meet at P.

$P A = P B$ (tangents from an external point), so in $△ P A B$ , $∠ P A B = ∠ P B A$ .

So the tangents make equal angles with the chord.

(Alternatively: $∠ P A B = 9 0^{\circ} - ∠ O A B$ and $∠ P B A = 9 0^{\circ} - ∠ O B A$ , and $∠ O A B = ∠ O B A$ since $O A = O B$ .)

Solution in the PYQ book (₹99).

Let $P = (0, b)$ and $Q = (a, 0)$ . Mid-point $= (\frac{a}{2}, \frac{b}{2}) = (2, - 5) \Rightarrow a = 4$ , $b = - 10$ .

$P = (0, - 10)$ , $Q = (4, 0)$ .

Q10

Solution in the PYQ book (₹99).

Q11

$D = 4 (a c + b d)^{2} - 4 (a^{2} + b^{2}) (c^{2} + d^{2}) = 4 [a^{2} c^{2} + 2 ab c d + b^{2} d^{2} - a^{2} c^{2} - a^{2} d^{2} - b^{2} c^{2} - b^{2} d^{2}]$

$= - 4 (a^{2} d^{2} - 2 ab c d + b^{2} c^{2}) = - 4 (a d - b c)^{2}$

Since $a d \neq = b c$ , $(a d - b c)^{2} > 0$ , so $D < 0$ : the equation has no real roots.

Q12

Solution in the PYQ book (₹99).

Q13

Let the tower be $h$ m and the angle at C be $θ$ ; the angle at D is $9 0^{\circ} - θ$ .

$tan θ = \frac{h}{4}$ and $tan (9 0^{\circ} - θ) = \frac{h}{16}$ .

In the right triangle of D, the side opposite $(9 0^{\circ} - θ)$ is $h$ , and $tan (9 0^{\circ} - θ) = \frac{1}{t a n θ}$ (the two acute angles of a right triangle swap opposite and adjacent sides).

$\frac{h}{16} = \frac{4}{h} \Rightarrow h^{2} = 64 \Rightarrow h = 8$ m.

Q14

Solution in the PYQ book (₹99).

Q15

Let the ratio be $k : 1$ : $\frac{3 k + 2}{k + 1} = \frac{24}{11} \Rightarrow 33 k + 22 = 24 k + 24 \Rightarrow k = \frac{2}{9}$ . Ratio $= 2 : 9$ .

$y = \frac{2 ( 7 ) + 9 ( - 2 )}{11} = - \frac{4}{11}$ .

Q16

Solution in the PYQ book (₹99).

Q17

Area of the ring $= π (4 2^{2} - 2 1^{2}) = \frac{22}{7} \times 1323 = 4158$ cm².

Unshaded part ACDB $= \frac{6 0 ^{\circ}}{36 0 ^{\circ}} \times 4158 = 693$ cm².

Shaded $= 4158 - 693 = 3465$ cm².

Q18

Solution in the PYQ book (₹99).

Q19

$2 π r_{1} = 18$ , $2 π r_{2} = 6 \Rightarrow π (r_{1} + r_{2}) = 12$ .

CSA $= π (r_{1} + r_{2}) l = 12 \times 4 = 48$ cm².

Q20

Solution in the PYQ book (₹99).

Q21

$\frac{5 x + 1 + 3 x + 3}{( x + 1 ) ( 5 x + 1 )} = \frac{5}{x + 4} \Rightarrow (8 x + 4) (x + 4) = 5 (5 x^{2} + 6 x + 1)$

$8 x^{2} + 36 x + 16 = 25 x^{2} + 30 x + 5 \Rightarrow 17 x^{2} - 6 x - 11 = 0 \Rightarrow (17 x + 11) (x - 1) = 0$

$x = 1$ or $x = - \frac{11}{17}$ .

Q22

Solution in the PYQ book (₹99).

Q23

$\frac{S _{n}}{S _{n}^{'}} = \frac{\frac{n}{2} [ 2 a + ( n - 1 ) d ]}{\frac{n}{2} [ 2 a ^{'} + ( n - 1 ) d ^{'} ]} = \frac{a + \frac{n - 1}{2} d}{a ^{'} + \frac{n - 1}{2} d ^{'}} = \frac{7 n + 1}{4 n + 27}$ .

For the 9th terms we need $\frac{n - 1}{2} = 8$ , i.e. $n = 17$ :

$\frac{a + 8 d}{a ^{'} + 8 d ^{'}} = \frac{7 ( 17 ) + 1}{4 ( 17 ) + 27} = \frac{120}{95} = \frac{24}{19}$

Q24

Solution in the PYQ book (₹99).

Q25

Let XY touch the circle at D and X′Y′ at E. Join OC, OD, OE.

$△ O D A ≅ △ O C A$ (OD = OC, AD = AC, OA common), so $∠ O A D = ∠ O A C$ , i.e. $∠ O A B = \frac{1}{2} ∠ D A B$ .

Similarly $∠ O B A = \frac{1}{2} ∠ E B A$ .

$X Y ∥ X^{'} Y^{'}$ , so $∠ D A B + ∠ E B A = 18 0^{\circ}$ (co-interior). Hence $∠ O A B + ∠ O B A = 9 0^{\circ}$ and $∠ A O B = 9 0^{\circ}$ .

Q26

Solution in the PYQ book (₹99).

Q27

Bank at $4 5^{\circ}$ : $\frac{300}{t a n 4 5 ^{\circ}} = 300$ m from the point below the plane. Bank at $3 0^{\circ}$ : $\frac{300}{t a n 3 0 ^{\circ}} = 3003 = 519.6$ m.

Width $= 300 + 519.6 = 819.6$ m.

Q28

Solution in the PYQ book (₹99).

Q29

Total outcomes = 36.

(i) The sum is even when both numbers are even ( $3 \times 3 = 9$ ) or both odd (9): 18 outcomes. $P = \frac{18}{36} = \frac{1}{2}$ .

(ii) The product is odd only when both are odd (9 outcomes), so it is even in $36 - 9 = 27$ outcomes. $P = \frac{27}{36} = \frac{3}{4}$ .

Q30

Solution in the PYQ book (₹99).

Q31

Let the rainfall be $h$ m. Water on the roof = volume of the tank:

$22 \times 20 \times h = \frac{22}{7} \times 1^{2} \times 3.5 = 11 \Rightarrow h = \frac{11}{440} = 0.025$ m $= 2.5$ cm.

Views: harvesting rain-water recharges ground water and reduces shortage in summer; every building should store and reuse its roof water.

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