**Q.1 A hollow cube of size 5 cm is taken with a thickness of 1 cm. It is made of smaller cubes of size 1 cm. If only 2 faces of the outer surface of the cube are painted, totally how many faces of the smaller cubes remain unpainted ?**

**A) 485**

**B) 486**

**C) 487**

**D) 488**

**Ans. D**

**The big cube is completely hollow.. But, the thickness of its side is 1cm & this is made up of smaller cubes.**

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**Just like the walls of a room. The room is hollow, but, its thickness is due to smaller bricks. The 4 walls of this room is painted from the outside. And, you are asked totally how many sides of all the bricks are now unpainted.**

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**First, you need to know how many bricks (small cubes) are there.**

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**Size of big cube = 5cm**

**Total volume of the big cube = 5*5*5 = 125cm^3**

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**Size of the hollow cube inside the big cube = 3cm**

**Volume of the hollow space inside the big cube = 3*3*3 = 27cm^3**

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**Therefore, volume occupied by small cubes (or volume of thickness) = 125 – 27 = 98cm^3**

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**Size of each small cube = 1cm**

**Volume of each small cube = 1*1*1 = 1cm^3**

**Total number of small cubes in wall = 98 / 1 = 98**

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**In short, 98 small cubes make up the wall of the big cube.**

**Each cube has 6 faces, so 98 cubes have = 98*6 faces = 588**

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**The four sides of the big cube have 100 painted faces.**

**Because each big side has 25 faces of the small cubes.**

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**Therefore, total unpainted faces = 588 – 100 = 488**

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**Q.2 A large cube is painted green on all six faces & then cut into a certain number of smaller but identical cubes. It was found that amongst the smaller cubes, there were 8 cubes which had no face painted at all. What is the maximum number of cubes that were cut from large original cube ?**

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**A) 46**

**B) 56**

**C) 64 **

**D) 54**

**Ans. C**

**When a cube is painted & cut into smaller n pieces along each side then the total number of smaller cubes obtained will be (n*n*n). From these, remove the outer layer of cubes, that is 2 from each face, then there will be [(n-2) * (n-2) * (n-2)] cubes left. Now. Number of cubes that do not have any face painted is (2*2*2)**

**As, (n-2) = 2**

**i.e. n=4**

**Therefore, the maximum number of cubes that were cut from original cube are 4*4*4= 64 Cube**

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**Q.3 Pointing towards a boy, Veena said, “He is the son of the only son of my grandfather”. How is that boy related to Veena?**

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**A) Brother**

**B) Nephew**

**C) Father**

**D) None of these**

**Ans. A**

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**EXPLANATION:**

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**Q.4 Pointing to Kapil, Shilpa said, “His mother’s brother is the father of my son Ashish”. How is Kapil related to Shilpa?**

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**A) Sister–in–law**

**B) Nephew**

**C) Niece**

**D) Aunt**

**Ans. B**

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**Kapil is the nephew of Shilpa. **

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**Q.5 A cube is painted green on all the sides. Then it is cut into 512 cubes of equal size. How many of smaller cubes are painted on 1 side ?**

**A) 18**

**B) 216**

**C) 12**

**D) 14**

**Ans. B**

**Cube root of 512 is 8. Si, cube is 8*8*8. Each side of the cube has 64 visible small cube faces, 4 of which are corners (3 green sides) & 24 more are edges (2 coloured sides), so there are only 36 small cubes with one green face on each of the six sides of the large cube.**

**So, 36*6= 216**

**Shortcut:**

**Learn this: Vertices are: 8**

**Edges are: 12**

**Faces are :6**

***To Find 1 face painted:**

**(N/n-2)2 *6**

**= (8/1-2)2 *6 = 36 *6**

**=216**

***Similarly if it asks you to find 2 faces painted, then use this formula,**

**(N/n-2) * 12**

***For 3 faces painted, use this formula,**

**just find how many vertices are there & these are 8 which is answer**

***For finding no. of smaller cubes: (N/n)3**

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**Q.6 Find the number of triangles in the given figure.**

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**A) 22 **

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**B) 24**

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**C) 26**

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**D) 28**

**Ans. D**

**The figure may be labelled as shown.**

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**The simplest triangles are AGH, GFO, LFO, DJK, EKP, PEL and IMN i.e. 7**

**in number.**

**The triangles having two components each are GFL, KEL, AMO, NDP, BHN,**

** CMJ, NEJ and HFM i.e. 8 in number.**

**The triangles having three components each are IOE, IFP, BIF and CEI i.e. 4 in **

**number.**

**The triangles having four components each are ANE and DMF i.e. 2 in number.**

**The triangles having five components each are FCK, BGE and ADL i.e. 3 in **

**Number.**

**The triangles having six components each are BPF, COE, DHF and AJE i.e. 4 in**

**Number**

**Total number of triangles in the figure = 7 + 8 + 4 + 2 + 3 + 4 = 28.**